Research Article Optimal Control of a Make-to-Stock System ...downloads.hindawi.com/journals/ddns/2014/301309.pdfOptimal Control of a Make-to-Stock System with Outsourced Production

Research ArticleOptimal Control of a Make-to-Stock System with OutsourcedProduction and Price-Sensitive Demand

Liuxin Chen1 Gang Hao2 and Huimin Wang1

1 Business School Hohai University No 8 Fochen West Road Jiangning District Nanjing City 211100 China2Department of Management Science City University of Hong Kong Kowloon Hong Kong

Correspondence should be addressed to Liuxin Chen lindawugehotmailcom

Received 31 March 2014 Accepted 27 May 2014 Published 6 July 2014

Academic Editor Xiaolin Xu

Copyright copy 2014 Liuxin Chen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We consider a make-to-stock system with controllable demand rate (by varying product selling price) and adjustable service rate(by outsourcing production) With one outsourcing alternative and a choice of either high or low price the system decides at anypoint in time whether to produce or even outsource for additional capacity as well as which price to sell the product at We showin the paper that the optimal control policy is of dynamic threshold type all decisions are based on the product inventory positionwhich represents the state of the system there is a state dependent base stock level to decide on production and a higher level onoutsourcing and there is a state dependent threshold which divides the choice of high and low prices

1 Introduction

A case study of Mattel the worldrsquos largest toy makerwas done by Johnson [1] with a focus on its productioncapacity management In particular Johnson reported thatMattel owned a state-of-the-art die-cast facility in PenangMalaysia that was operating at full capacity to produce die-cast toy vehicles Due to surge of demand for Hot Wheelsa core line of product Mattel considered several optionsto expand production capacity including one through theVendor Operations Asia Division to outsource productionin Asia-Pacific VOA added flexibility to Mattelrsquos in-housemanufacturing capability and was one of the companyrsquos mostvaluable assets In the meantime Mattel managed demandfor the Hot Wheels through a new marketing strategy thatchanged the assortment mix of cars every two weeks

In this paper we consider a single-productmake-to-stocksystem that has the option to increase production capacity byoutsourcing to external contract manufacturers The systemscan also manage product demand through adjusting its sell-ing price For the basic setting of one outsourcing alternativeand a choice of either high or low price the system optimalcontrol problem is to decide at any point in time whether toproduce at the in-house facility or to outsource for additionalcapacity as well as which price to sell the product at We

model the production processes at the in-house and externalfacilities by exponential times of different means and thedemand process by a Poisson process with a price-dependentrate Thus mathematically the problem is optimal control ofan 1198721198721 make-to-stock queue with discretely adjustableproduction and demand rates

With the objective to maximize the total discountedprofit we show in the paper that the optimal control policyis of dynamic threshold type all decisions are based onthe product inventory position which represents the stateof the system there is a state dependent base stock level todecide on production and a higher level on outsourcing andthere is a state dependent threshold which divides the choiceof high and low prices Furthermore we show that for agiven outsourcing production capacity all three thresholdsof the optimal control policy and the associated optimalprofit are decreasing in the outsourcing costThis implies thatoutsourcing to lower cost facilitieswill lead to lower inventoryholdings and lower selling price but higher profit

There is a rich literature on the optimal control of1198721198721

make-to-stock queues most of which take demand as exoge-nous and solve for the optimal control of the production rateTypically the optimal policy is of base stock type (produceat the maximum rate when the inventory holding falls belowcertain level otherwise halt production) which was first

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 301309 12 pageshttpdxdoiorg1011552014301309

2 Discrete Dynamics in Nature and Society

proved by Gavish and Graves [2] and Sobel [3] for the singleproduct and single machine case Later works of Zheng andZipkin [4] Wein [5] Veach and Wein [6] and Bertsimasand Paschalidis [7] attempt to extend the base stock policyto the multiple product cases There are also extensions tothe case of single product but with multiple demand classeslike Ha [8ndash10] Another direction of extension has beento incorporate more detailed modeling of the productionfacility For example Kapuscinski and Tayur [11] model theproduction process by a tandem queue and Feng and Yan[12] and Feng and Xiao [13] deal with unreliable productionfacilities

Li [14] and Chen et al [15 16] are three works thatincorporate controls on both production and demand pro-cesses in a make-to-stock queue optimization problem Li[14] assumes a continuous spectrum of product selling pricesand corresponding demand rates and hence manages toderive a concave and differentiable profit function in termsof the production rate and the selling price which yieldsa qualitative characterization of the optimal policy Chenet al [15] allow discrete choices of prices and derive anefficient algorithm to compute the optimal policy as well as itsqualitative characterization Chen et al [16] consider a make-to-stock manufacturing system with batch production anddiscrete choices of price and derive the characterization ofthe optimal control policy Similar work on a make-to-orderqueue is done by Ata and Shneorson [17] Carr and Duenyas[18] study the optimal control of a mixture of make-to-stock and make-to-order queues Our work adds in anotherdimensionwith the outsourcing option to expand productioncapacity

The rest of the paper is organized as follows Section 2describes precisely the system model and defines an opti-mization problem that solves for the optimal policy Section 3characterizes the optimal threshold policy proves its globaloptimality amongst all nonanticipative control policies anddiscusses its relationship to the cost of outsourcing Section 4briefly discusses the extension to multiple price choicesSection 5 concludes the paper with a summary of the resultsand possible extensions in the future research

To streamline presentation of the paper we state in themain body of the paper all the results without proofs andcollect all the proofs in the appendix

2 Problem Formulation

Themake-to-stock system of concern in the paper has an in-house facility with a production rate 120583 and a unit productioncost 119887 The system can outsource production to an externalfacility which can produce at a rate 119886 and a per unit cost 119888 Weassume that the existing in-house facility has a lower variableproduction cost than the external facility that is 119887 lt 119888 whichholds true in the case ofMattel for example and is the reasonfor keeping the in-house facility We also assume that theproduction processes at both facilities are random and followexponential distributions

The demand process for the product is assumed to bea Poisson process with a price-dependent rate Specificallythere are two selling prices high 119901

1and low 119901

2 which

correspond to two demand rates 1205821and 120582

2 We assume that

1205821lt 1205822and

12058221199012minus 12058211199011

1205822minus 1205821

gt 119887 (1)

which indicates that the marginal profit gain from switchingprice fromhigh to low is greater than the in-house productioncost 119887 Also to ensure system stability we assume that 120583+119886 gt1205821When a demand arrives it is filled from the finished

goods inventory if possible otherwise it is added to a waitingqueue which is served in first-come-first-serve order Thefinished goods inventory carries a holding cost of ℎ+ per unitproduct per unit time and the backordering cost for demandunmet at arrival is ℎminus per waiting demand per unit timeDefine the inventory cost function ℎ(119909) = ℎ

+119909++ ℎminus119909minus

where 119909+ = max[119909 0] and 119909minus = max[minus119909 0] We will considerthe total discounted profit assuming a discount factor 120574 Toensure that it ismore profitable to produce to fill demand thanto backlog forever we assume that 119887 lt 119888 lt ℎminus120574

We specify a dynamic control policy for the system by119906 = 120583(119905) 119886(119905) 119901(119905) 119905 gt 0 where 120583(119905) = 0 or 1 representingin-house production is off or on similarly 119886(119905) = 0 or1 representing outsourced production being off or on and119901(119905) = 119901

1or 1199012representing the price charged at time 119905 A

policy 119906 is called nonanticipatory if at all 119905 gt 0 120583(119905) 119886(119905)and 119901(119905) depend only on information prior to 119905 Let U bethe collection of all nonanticipatory control policies Undera given 119906 isin U denote the total demand sold at price 119901

119894up to

time 119905 gt 0 by 119873119906119894(119905) 119894 = 1 2 the total in-house production

by 119875119906119868(119905) and the total outsourced production by 119875119906

119874(119905) Then

the product inventory level at time 119905 is given by

119883119906(119905) = 119909 + 119875

119880

119868(119905) + 119875

119906

119874(119905) minus 119873

119906

1(119905) minus 119873

119906

2(119905) (2)

where 119909 is the initial inventory at 119905 = 0Consequently the total discounted profit under policy 119906

is

119881119906(119909) = 119864[int

infin

0

119890minus120574119905

(

2

sum

119894=1

119901119894119889119873119906

119894(119905) minus 119887119889119875

119906

119868(119905)

minus119888119889119875119906

119874(119905) minus ℎ (119883

119906(119905)) 119889119905)]

(3)

A policy 119906lowast isin U is said to be optimal if it solves thefollowing optimization problem

119881119906lowast

(119909) = sup119906isinU

119881119906(119909) (4)

The optimal solution to this semi-Markov decision prob-lem can be characterized by the following Hamilton-Jacobi-Bellman (HJB) equation (cf Chapter 7 of [19])

0 = minus 120574119881 (119909) minus ℎ (119909)

+max120572=0120583

120572 [119881 (119909 + 1) minus 119881 (119909) minus 119887]

Discrete Dynamics in Nature and Society 3

+max120573=0119886

120573 [119881 (119909 + 1) minus 119881 (119909) minus 119888]

+max119895=12

120582119895[119881 (119909 minus 1) minus 119881 (119909) + 119901

119895]

(5)

Since 119888 gt 119887 we have119881(119909+1)minus119881(119909)minus119888 lt 0when119881(119909+1)minus119881(119909) minus 119887 lt 0 and thus 120573 = 0 when 120572 = 0 In essence we canenvisage an effective production process with rates 120583

0= 0

1205831= 120583 and 120583

2= 120583 + 119886 corresponding to the unit production

costs 1198870= 0 119887

1= 119887 and 119887

2= (120583119887 + 119886119888)(120583 + 119886) respectively

As a result HJB equation (5) can be simplified to

0 = minus 120574119881119906(119909) minus ℎ (119909)

+ max119894=012

120583119894[119881 (119909 + 1) minus 119881 (119909) minus 119887

119894]

+max119895=12

120582119895[119881 (119909 minus 1) minus 119881 (119909) + 119901

119895]

(6)

We are especially interested in a class of control policieswhich are parameterized by three thresholds 119877 119863 and 119878with max119877119863 lt 119878 An (119877119863 119878) policy decides on pro-duction outsourcing and pricing in the following manner(1) when the inventory is above or equal to 119878 there isno production at the in-house facility and no productionoutsourcing (2) when it is below 119878 and above or equal to119863 production is on at the in-house facility but there is nooutsourcing (3) when it is below 119863 production is on atthe in-house and outsourced to the external facility (4) theproduct sale price is set low at 119901(119905) = 119901

2when it is above

119877 and the produce selling price is set low at 1199012 otherwise

the price is high at 1199011 In Section 3 below we characterize the

best (119877119863 119878) policy and verify that it satisfies the above HJBequation (6) and thus is optimal amongst all policies inU

3 Optimality of (119877119863 119878) Policy

TheHJB equation (6) can be made more specific when givenan (119877119863 119878) policy For example for an (119877119863 119878) policy with0 lt 119877 lt 119863 lt 119878 it can be simplified to the following equations

For 119909 ge 119878

0 = minus120574119881 (119909) minus ℎ+119909 + 1205822[119881 (119909 minus 1) minus 119881 (119909) + 119901

2] (7)

for119863 le 119909 lt 119878

0 = minus 120574119881 (119909) minus ℎ+119909 + 1205822[119881 (119909 minus 1) minus 119881 (119909) + 119901

2]

+ 120583 [119881 (119909 + 1) minus 119881 (119909) minus 119887]

(8)

for 119877 lt 119909 lt 119863


2]

+ (120583 + 119886) [119881 (119909 + 1) minus 119881 (119909) minus120583119887 + 119886119888

120583 + 119886]

(9)

for 0 le 119909 le 119877


1]

+ (120583 + 119886) [119881 (119909 + 1) minus 119881 (119909) minus120583119887 + 119886119888

120583 + 119886]

(10)

and for 119909 lt 0

0 = minus 120574119881 (119909) minus ℎminus119909 + 1205821[119881 (119909 minus 1) minus 119881 (119909) + 119901

1]

+ (120583 + 119886) [119881 (119909 + 1) minus 119881 (119909) minus120583119887 + 119886119888

120583 + 119886]

(11)

Let 119881(119877119863119878)(119909) be the profit function of a given (119877119863 119878)policyThe following lemma characterizes some of its limitingbehaviors

Lemma 1 The profit function of an (119877119863 119878) policy has thefollowing limits

(1) lim119909rarrplusmninfin

[119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909)] = 0(2) lim

119909rarrplusmninfin[119881(119877119863119878)

(119909) minus 119881(119877119863minus1119878)

(119909)] = 0(3) lim


(119909) minus 119881(119877minus1119863119878)

(119909)] = 0(4) lim

119909rarr+infinD119881(119877119863119878)

(119909) = minusℎ+120574

and lim119909rarrminusinfin

D119881(119877119863119878)

(119909) = ℎminus120574

Our approach is to first find the best (119877119863 119878) policy andthen to prove its global optimality

Definition 2 An (119877119863 119878) policy is said to be better than an(1198771015840 1198631015840 1198781015840) policy if 119881(119877119863119878)(119909) ge 119881

(1198771015840

1198631015840

1198781015840

)(119909) for any initial

inventory level 119909 and at least for an 119909 the inequality holdsstrictly It is said to be the best (119877119863 119878) policy if no other(119877119863 119878) policies are better

For notational simplicity we also define a first-orderdifference operatorD

D119892 (119909) = 119892 (119909) minus 119892 (119909 minus 1) for any function119892 (119909) (12)

The following lemma provides means to compare(119877119863 119878) policies

Lemma 3 Comparing with a given (119877119863 119878) policy we havethe following results

(1) the (119877119863 119878 + 1) policy is better if and only if

D119881(119877119863119878)

(119878 + 1) gt 119887 or D119881(119877119863119878+1)

(119878 + 1) gt 119887 (13)

(2) the (119877119863 + 1 119878) policy is better if and only if

D119881(119877119863119878)

(119863 + 1) gt 119888 or D119881(119877119863+1119878)

(119863 + 1) gt 119888 (14)


D119881(119877119863119878)

(119877 + 1) gt12058221199012minus 12058211199011

1205822minus 1205821

or D119881(119877+1119863119878)

(119877 + 1) gt12058221199012minus 12058211199011

1205822minus 1205821

(15)

The first comparison implies that under either the(119877119863 119878) policy or the (119877119863 119878 + 1) policy if the marginal


profit at inventory level 119909 = 119878 for producing one more unit ofproduct is higher than the in-house production cost then the(119877119863 119878 + 1) policy is better than the (119877119863 119878) policy Similarimplications can be drawn from the other two comparisons

Lemma 3 leads to the following characterization of thebest (119877119863 119878) policy

Theorem 4 If the (119877lowast 119863lowast 119878lowast) policy is the best (119877119863 119878)

policy then

(1) at the best base stock level 119878lowast

D119881(119877lowast

119863lowast

119878lowast

)(119878lowast+ 1) le 119887 lt D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast) (16)

(2) at the best outsourcing threshold119863lowast

D119881(119877lowast

119863lowast

119878lowast

)(119863lowast+ 1) le 119888 lt D119881

(119877lowast

119863lowast

119878lowast

)(119863lowast) (17)

(3) at the best price switch threshold 119877lowast

D119881(119877lowast

119863lowast

119878lowast

)(119877lowast+ 1) le

12058221199012minus 12058211199011

1205822minus 1205821

lt D119881(119877lowast

119863lowast

119878lowast

)(119877lowast)

(18)

Furthermore we have that 119881(119877lowast

119863lowast

119878lowast

)(119909) is concave in 119909

Finally we obtain the main result of the paper based onthe characteristics of Theorem 4 as stated above


policy then its profit function 119881(119877lowast

119863lowast

119878lowast

)(119909) satisfies the HJB

equation (6) Hence the (119877lowast 119863lowast 119878lowast) policy is optimal amongstall nonanticipative policies ofU

The optimal (119877lowast 119863lowast 119878lowast) policy has some importantproperties which we list below

Theorem 6 In the best (119877lowast 119863lowast 119878lowast) policy 119878lowast ge 0 and 119863lowast gt119877lowast if (120582

21199012minus 12058211199011)(1205822minus 1205821) gt 119888 otherwise 119877lowast ge 119863lowast

Theorem 6 tells us two results as the following (i)Under the best thresholds (119877lowast 119863lowast 119878lowast) policy the maximuminventory level 119878lowast is nonnegative and it is optimal to idleboth the in-house and outsourced facilities when the stocklevel is above the maximum inventory level 119878lowast (ii) Whenproduction is on at both in-house and outsourced facilitiesand the inventory is increasingly building up if the marginalprofit gain when switching price from high to low is greaterthan the outsourced production cost 119888 it is more profitable tofirst stop outsourced production than to first stop in-houseproduction and then decrease price The converse is true ifthe marginal profit gain when switching price from high tolow is smaller than the outsourced production cost 119888

Theorem 7 As for fixed sourcing production rate 119886 supposethat the variable cost 119888 of outsourced production is negotiableLet (119877lowast(119888) 119863lowast(119888) 119878lowast(119888)) policy be the optimal threshold policyassociated with a cost 119888 Then

(1) the optimal thresholds 119877lowast(119888) 119878lowast(119888) are piecewise con-stant increasing functions of 119888 but119863lowast(119888) is a piecewiseconstant decreasing function of 119888

(2) as for the optimal profit function119881(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888)

is decreasing in 119888

Theorem 7 concludes the following results (i) A lowervariable cost from outsourced production will lead to lowerproduct selling price but higher safety stock level and moreoutsourced production And (ii) a lower variable outsourcedproduction cost will lead to higher optimal long-term dis-counted profit

4 Extension to Multiple Price Choices

In this section we briefly discuss the extension of the resultsin the previous section to multiple price choices Namely wehave now119870 ge 2 possible prices to choose from for the sellingof the product 119901

1gt 1199012gt sdot sdot sdot gt 119901

119870gt 119888 with corresponding

119870 demand arrival rates 1205821lt 1205822lt sdot sdot sdot lt 120582

119870

We further assume that the profit rates are also increas-ingly ordered that is

1205821(1199011minus 119888) lt 120582

2(1199012minus 119888) lt sdot sdot sdot lt 120582

119870(119901119870minus 119888) (19)

It can be shown that if the profit rates do not follow thisorder a dominating subset of the price levels can be chosento make the other prices unattractive for selection Readersare referred to Chen Feng and Ou [20] for the analysis ondominating prices

The 119870 price choice model can be optimized by a pol-icy which maximizes the right-hand side of the followingHamilton-Jacobi-Bellman equation

0 = minus ℎ (119909) +max120572=0120583

120572 [119881 (119909 + 1) minus 119881 (119909) minus 119888]

+max120573=0119886

120583 [119881 (119909 + 1) minus 119881 (119909) minus 119888]

+ max119894=1119870

120582119894[119881 (119909 minus 1) + 119901

119894minus 119881 (119909)] minus 120574119881 (119909)

(20)

where 119881(119909) is the profit function of the policyThe natural extension of the (119877119863 119878) policy as defined at

the end of Section 2 is a 119870-level control policy characterizedby 119870 + 1 parameters

Consider (1198772 119877

119870 119863 119878) with 119877

2lt sdot sdot sdot 119877

119870lt 119878 and

119863 lt 119878 119878 is the base stock level on and above which there isno production at the in-house facility and also no productionoutsourcing And when it is below 119878 and above or equal to119863production is on at the in-house facility but no outsourcingwhen it is below 119863 production is on at the in-house andoutsourced to the external facilityThe other119870minus1 parametersare price switch thresholds when the inventory is in the range(119877119894 119877119894+1] 119894 = 1 119870 (assuming 119877

1= minusinfin and 119877

119870+1= +infin)

the product is sold at price 119901119894

For a given (1198772 119877

119870 119863 119878) policy the profit function

can be calculated recursively as

0 = minus ℎ (119909) + 120582119870[119901119870minusD119881 (119909)] minus 120574119881 (119909) 119909 ge 119878

0 = minus ℎ (119909) + 120583 [D119881 (119909) minus 119887]

+ 120582119870[119901119870minusD119881 (119909)] minus 120574119881 (119909) 119863 le 119909 lt 119878


0 = minus ℎ (119909) + 120583 [D119881 (119909) minus 119887] + 119886 [D119881 (119909) minus 119888]

+ 120582119894[119901119894minusD119881 (119909)] minus 120574119881 (119909) 119877

119894lt 119909 le 119877

119894+1

(21)

Qualitatively the 119870 + 1-level (1198772 119877

119870 119863 119878) policies

possess the following properties similar to those for the two-price case

Proposition 8 Let 119881(1198772 119877119870119863119878)(119909) be the long run totaldiscounted profit of the 119870-level (119877

2 119877

119870 119863 119878) policy when

the initial inventory is 119909 then

(1) the (1198772 119877

119870 119863 119878 + 1) policy is better than the

(1198772 119877

119870 119863 119878) policy if and only if

D119881(1198772119877119870119863119878)

(119878 + 1) gt 119887

119900119903 D119881(1198772119877119870119863119878+1)

(119878 + 1) gt 119887

(22)

(2) the (1198772 119877

119870 119863 + 1 119878) policy is better than the

(1198772 119877


D119881(1198772119877119870119863119878)

(119863 + 1) gt 119888

119900119903 D119881(1198772119877119870119863+1119878)

(119863 + 1) gt 119888

(23)

(3) the (1198772 119877

119870 119863 119878) policy cannot be better than the

(1198772 119877

119870 119863 119878 + 1) policy if 119878 lt 0

(4) the (1198772 119877

119894+ 1 119877

119870 119863 119878) policy is better than

the (1198772 119877

119894 119877


D119881(1198772119877119894119877119870119863119878)

(119877119894+ 1) gt

120582119894119901119894minus 120582119894minus1119901119894minus1

120582119894minus 120582119894minus1

(24)

or

D119881(1198772119877119894+1119877

119870119863119878)

(119877119894+ 1) gt


120582119894minus 120582119894minus1

(25)

We also have the following characterization of the best119870-level policy

Theorem 9 The best 119870-level (119877lowast2 119877

lowast

119870 119863lowast 119878lowast) policy is

characterized by the following relations

(1) the profit function 119881(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119909) is concave in

integer value of 119909(2) the best base stock level 119878lowast satisfies 119878lowast ge 0 and

D119881(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119878lowast+ 1) le 119887 lt D119881

(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119878lowast)

(26)

(3) at the optimal threshold119863lowast

D119881(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119863lowast+ 1) le 119888 lt D119881

(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119863lowast)

(27)

(4) at the best price switch thresholds 119877lowast119894 119894 = 2 119870 minus 1

D119881(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119877lowast

119894+ 1) le


120582119894minus 120582119894minus1

lt D119881(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119877lowast

119894)

(28)

The following Theorem 10 characterizes the impact ofsouring cost 119888 on the optimal discounted profit and itsoptimal thresholds

Theorem 10 As for fixed sourcing production rate 119886 sup-pose that the variable cost 119888 of outsourced production isnegotiable Let (119877lowast

2(119888) 119877

lowast

119870(119888) 119863

lowast(119888) 119878lowast(119888)) policy be the

optimal threshold policy associated with a cost 119888 Then(1) the optimal thresholds 119877

lowast

2(119888) 119877

lowast

119870(119888) 119878lowast(119888) are

piecewise constant increasing functions of 119888 but119863lowast(119888)is a piecewise constant decreasing function of 119888

(2) the optimal profit function119881(119877lowast

2(119888)119877

lowast

119870(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) is decreasing in 119888

5 Concluding Remarks

We have analyzed the optimal control of a single-productmake-to-stock system that has the option to increase produc-tion capacity by outsourcing to external contractmanufactur-ers and the option to vary product selling prices Idealizingthe system by a simple1198721198721 make-to-stock queue modelwith discretely adjustable production and demand rates weobtain a complete characterization of the optimal thresholdcontrol policy and prove beneficial impact of low costoutsourced production to the system We wish to point outthat the results can be easily extended multiple outsourcingalternatives and multiple price choices It should also bepossible to extend to the case of multiple demand classes thatcompete for the same product More challenging extensionswill be on incorporating fixed cost of outsourcing into themodel as well as havingmultiple product classes in themodel

Appendix

In order to simplify writing we let1205751= 120583 + 119886 + 120582

1+ 120574 120575

2= 120583 + 119886 + 120582

2+ 120574 (A1)

Proof of Lemma 1 First we attempt to show thatlim119909rarr+infin

[119881(119877119863119878)

(119909) minus 119881(119877119863119878)

(119909)] = 0 To this endwe let Δ(119909) = 119881(119877119863119878)(119909)minus119881(119877119863119878)(119909) By definition functionΔ(119909) can be derived recursively from the recursion of the(119877119863 119878) policy In particular for 119909 ge 119878

(1205822+ 120574) Δ (119909) = 120582

2Δ (119909 minus 1) (A2)

which leads to

Δ (119909) =1205822+ 120574

1205822

Δ (119909 + 1) = sdot sdot sdot = (1205822+ 120574

1205822

)

119899

Δ (119909 + 119899)

Δ (119909) = (1205822

1205822+ 120574

)

119909minus119878

Δ (119878)

(A3)


where 119909 ge 119878 These equations establish that if Δ(119878) ge 0 thenΔ(119909) ge 0 and Δ(119909) le Δ(119909 + 1) for all 119909 ge 119878 and further ifΔ(119878) le 0 then Δ(119909) le 0 and Δ(119909) ge Δ(119909 + 1) for all 119909 ge 119878Moreover a limiting behavior of Δ(119909) is deduced by the factthat

lim119909rarrinfin

Δ (119909) = lim119909rarrinfin

(1205822

1205822+ 120574

)

119909minus119878

Δ (119878) = 0 (A4)

That is

lim119909rarr+infin

[119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909)] = 0 (A5)

Next we are proving a parallel result that

lim119909rarrminusinfin

[119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909)] = 0 (A6)

To this end we consider 119909 le min119877 0 and definea shift operator 119879 such that 119879119891(119909) = 119891(119909 + 1) for anyfunction 119891(119909) The inverse of 119879 is expressed formally as 119879minus1 119879minus1119891(119909) = 119891(119909 minus 1) Using 119863 we can derive the correspond-

ing characteristic equations for linear recursion (11) Thencharacteristic equation of recursion (11) is

0 = 1205821119879minus1+ (120583 + 119886) 119879 minus 120575

1 (A7)

or

0 = 1205821119910minus1+ (120583 + 119886) 119910 minus 120575

1 (A8)

which has two solutions

1199101=

120575 minus radic1205752

1minus 4 (120583 + 119886) 120582

1

2 (120583 + 119886)

1199102=

120575 + radic1205752

1minus 4 (120583 + 119886) 120582

1

2 (120583 + 119886)

(A9)

It can be checked that 0 lt 1199101lt 1 lt 119910

2

Then the homogeneous solution to (11) is in the form of

(119909) = 1198621119910119909

1+ 1198622119910119909

2 (A10)

where 1198621and 119862

2are constants to be determined at 119909 rarr

minusinfin Note that when 119909 rarr minusinfin 1199101199091

increases to infin

exponentially However we do not expect 119881(119909) to grow ordiminish exponentially when119909 rarr minusinfin and hence postulatethat 119862

1= 0 To determine a particular solution to (11) we

suppose it is1198671+1198672119909where119867

119894 119894 = 1 2 are to be determined

from the following equation

minus ℎminus119909 + (120583119887 + 119886119888) minus 120582

11199011

= [1205821119879minus1+ (120583 + 119886) 119879 minus 120575

1] (1198671+ 1198672119909)

(A11)

Comparing the coefficients of constant and linear termsbetween the two sides yields

1198672=ℎminus

120574

1198671=

1

1205742[120574 (12058211199011minus (120583119887 + 119886119888)) + ℎ

minus(120583 + 119886 minus 120582

1)]

(A12)

Hence the general solution to (11) becomes

119881(119877119878119861)

(119909) = 1198622119910119909

2+ℎminus119909

120574+12058211199011minus (120583119887 + 119886119888)

120574

+ℎminus(120583 + 119886 minus 120582

1)

1205742

(A13)

Denoting by 1198622the coefficient of 119910

2for 119881(119877119863119878)(119909) it is

deduced from (A13) that

Δ (119909) = 119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909) = (1198622minus 1198622) 119910119909

2 (A14)

Then we get


[119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909)] = 0 (A15)

We can prove (2) and (3) by analogy with this logic andtherefore omit its proof

Now we turn to prove lim119909rarr+infin

D119881(119877119863119878)

(119909) = minusℎ+120574


D119881(119877119863119878)

(119909) = ℎminus120574 Similar to the pre-

ceding methodology in the proof of lim119909rarrminusinfin

[119881(119877119863119878)

(119909) minus

119881(119877119863119878minus1)

(119909)] = 0 we can write the general solution of119881(119877119863119878)

(119909) for 119909 gt 119878 as the following

119881(119877119863119878)

(119909) = 119875(1205822

1205822+ 120574

)

119909

minusℎ+119909

120574+12058221199012

120574+1205822ℎ+

1205742 (A16)

where 119875 is a constant to be determined at 119909 rarr +infin And for119909 lt min119877119863 0 we get the profit function 119881(119877119863119878)(119909) from(A13) as follows

119881(119877119863119878)

(119909) = 1198622119910119909

2+ℎminus119909

120574+12058211199011minus (120583119887 + 119886119888)

120574

+ℎminus(120583 + 119886 minus 120582

1)

1205742

(A17)

where 1198622is a constant to be determined at 119909 rarr minusinfin and

1199102=

1205751+ radic1205752

1minus 4 (120583 + 119886) (120582

1)

2 (120583 + 119886) (A18)

Combining 1205822(1205822+ 120574) isin (0 1) and 119910

2gt 1 we get

lim119909rarr+infin

D119881(119877119863119878)

(119909) = minusℎ+

120574


D119881(119877119863119878)

(119909) =ℎminus

120574

(A19)

Proof of Lemma 3 For brevity we provide only the proof that(119877119863 119878+1) policy is better than (119877119863 119878) policy if and only ifD119881(119877119863119878)

(119878 + 1) gt 119887 orD119881(119877119863119878+1)

(119878 + 1) gt 119887 The proofs ofother two essential and sufficient conditions (2) and (3) aresimilar so including them here is not additionally illustrative

We first prove inequality D119881(119877119863119878+1)

(119878 + 1) gt 119887 impliesthat (119877119863 119878+1) policy is better than (119877119863 119878) policy For that


we focus on the case that 119877 lt 119863 lt 119878 the other cases can beproved in the same fashion

Let Δ(119909) = 119881(119877119863119878+1)

(119909) minus 119881(119877119863119878)

(119909) and make use of(7)ndash(11) to obtain

0 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1) for 119909 ge 119878 + 1

minus120598 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1) for 119909 = 119878

0 = 120583Δ (119909 + 1) minus (1205822+ 120574 + 120583)Δ (119909)

+ 1205822Δ (119909 minus 1) for 119863 le 119909 lt 119878

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909)

+ 1205822Δ (119909 minus 1) for 119877 le 119909 lt 119863

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 119877

(A20)

where 120598 = 120583[D119881(119877119863119878+1)

(119878 + 1) minus 119887] 120598 gt 0We know that 120598 gt 0 While in line with Lemma 1 we get

that


[119881(119877119863119878+1)

(119909) minus 119881(119877119863119878)

(119909)] = 0 (A21)

Hence when 119872 is a large enough positive integer and119872 gt 119878119863 |119877| we can approximately write the followingequation for 119909 = minus119872

0 = (120583 + 119886) Δ (minus119872 + 1) minus 1205751Δ (minus119872) (A22)

So we can verify that the coefficient matrix of the linearequations about variable Δ(119909) minus119872 le 119909 le 119872 is totallynonpositive Hence by solving a finite approximate negativesystem of linear equations we can corroborate that Δ(119909) ge 0

for all minus119872 le 119909 le 119872 Thus we get that Δ(119909) ge 0 for all 119909whenrarr +infin At last there exists an 119909 such that Δ(119909) gt 0 because120598 gt 0

Consequently the (119877119863 119878 + 1) policy is better than(119877119863 119878) policy To this end we reemploy all above equationsbut change equation with 119909 = 119878 as follows

minus 120583 [119863119881(119877119863119878)

(119878 + 1) minus 119887]

= 120583Δ (119878 + 1) minus (120583 + 1205822+ 120574) Δ (119878) + 120582

2Δ (119878 minus 1)

(A23)

The rest follows exactly the same as aboveWe now prove the reverse And we supposeD119881

(119877119863119878)(119878+

1) le 119887 or D119881(119877119863119878+1)

(119878 + 1) le 119887 The above proof steps willlead to Δ(119909) le 0 (rather than Δ(119909) gt 0) for any 119909 that is119881(119877119863119878+1)

(119909) le 119881(119877119863119878)

(119909)

Proof of Theorem 4 Based on Lemma 1 we find that (1) (2)and (3) are true So we only need to prove 119881(119877119863119878)(119909) isconcave For simplicity we give the proof for the case of0 le 119877

lowastlt 119863lowastlt 119878lowast The other cases can be analogously

analyzed To simplify the notation we drop the superscriptlowast in the proof Define D2119881(119909) = D119881(119909) minus D119881(119909 minus 1) and

D119881(119909) = 119881(119877119863119878)

(119909) minus 119881(119877119863119878)

(119909 minus 1) then that 119881(119877119863119878)(119909) isconcave is equivalent toD2119881(119909) lt 0 for all 119909

Based on (7)ndash(11) we obtain the following system ofequations

ℎ+= minus 120582

2D2119881 (119909) minus 120574D119881 (119909) for 119909 ge 119878 + 1

ℎ+= minus 120582

2D2119881 (119878) minus (120583 + 120574)D119881 (119878) + 120583119887 for 119909 = 119878

ℎ+= 120583D

2119881 (119909 + 1) minus 120582

2D2119881 (119909)

minus 120574D119881 (119909) for 119863 + 1 le 119909 lt 119878

ℎ+= 120583D119881 (119863 + 1) minus 120575

2D119881 (119863)

+ 1205822D119881 (119863 minus 1) + 119886119888 for 119909 = 119863

ℎ+= (120583 + 119886)D

2119881 (119909 + 1) minus 120582

2D2119881 (119909)

minus 120574D119881 (119909) for 119877 + 1 lt 119909 le 119863 minus 1

ℎ+= (120583 + 119886)D

2119881 (119877 + 2) minus 120582

1D2119881 (119877 + 1)

minus (1205822minus 1205821+ 120574)D119881 (119877 + 1) + 120582

21199012minus 12058211199011

ℎ+= (120583 + 119886)D

2119881 (119909 + 1) minus 120582

1D2119881 (119909)

minus 120574D119881 (119909) for 1 le 119909 le 119877

minusℎminus= (120583 + 119886)D

2119881 (119909 + 1) minus 120582

1D2119881 (119909)

minus 120574D119881 (119909) for 119909 le 0(A24)

Evaluating the above for 119909 ge 119878+1 at 119909 and 119909minus1 and fromtheir difference we derive for 119909 ge 119878 + 2

0 = minus (1205822+ 120574)D

2119881 (119909) + 120582

2D2119881 (119909 minus 1) (A25)

Similarly we derive

0 = minus (1205822+ 120574)D

2119881 (119878 + 1) + 120582

2D2119881 (119878)

+ 120583D119881 (119878) minus 120583119887 for 119909 = 119878 + 1(A26)

0 = minus (1205822+ 120574 + 120583)D

2119881 (119878) + 120582

2D2119881 (119878 minus 1)

minus 120583D119881 (119878) + 120583119887 for 119909 = 119878(A27)

0 = 120583D2119881 (119909 + 1) minus (120582

2+ 120574 + 120583)D

2119881 (119909)

+ 1205822D2119881 (119909 minus 1) for 119863 + 2 le 119909 lt 119878

(A28)

1205983= 120583D

2119881 (119863 + 2) minus 120575

2D2119881 (119863 + 1)

+ 1205822D2119881 (119863) for 119909 = 119863 + 1

(A29)

1205984= 120583D

2119881 (119863 + 1) minus 120575

2D2119881 (119863)

+ 1205822D2119881 (119863 minus 1) for 119909 = 119863

(A30)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

2D2119881 (119909)

+ 1205822D2119881 (119909 minus 1) for 119877 + 2 lt 119909 le 119863 minus 1

(A31)


1205985= (120583 + 119886)D

2119881 (119877 + 3) minus 120575

2D2119881 (119877 + 2)

+ 1205821D2119881 (119877 + 1)

(A32)

1205986= (120583 + 119886)D

2119881 (119877 + 2) minus 120575

2D2119881 (119877 + 1)

+ 1205821D2119881 (119877)

(A33)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

1D2119881 (119909)

+ 1205821D2119881 (119909 minus 1) for 2 le 119909 le 119877

(A34)

1205987= (120583 + 119886)D

2119881 (2) minus 120575

1D2119881 (1) + 120582

1D2119881 (0) (A35)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

1D2119881 (119909)

+ 1205821D2119881 (119909 minus 1) for 119909 le 0

(A36)

where 1205983= minus119886[D119881(119863 + 1) minus 119888] 120598

4= 119886[D119881(119863) minus 119888] 120598

5=

minus(1205822minus1205821)D119881(119877+1)+(120582

21199012minus12058211199011) 1205986= (1205822minus1205821)D119881(119877)minus

(12058221199012minus 12058211199011) and 120598

7= ℎ++ ℎminus

Based on the above equations we know that

D2119881 (119878) =

ℎ++ (120583 + 120574)D119881 (119878) minus 120583119887

minus1205822

(A37)

and the (119877119863 119878) policy being the best implies D119881(119878) gt 119887therefore we haveD2119881(119878) lt 0

Equation (A26) is equivalent to the following equationafter taking into accountD2119881(119878)

(1205822+ 120574)D

2119881 (119878 + 1) = minusℎ

+minus 120574D119881 (119878) (A38)

which gives rise toD2119881(119878 + 1) lt 0 based on the above resultD119881(119878) gt 119887

Equation (A25) shows that when 119909 ge 119878 + 2 D2119881(119909) =(12058221205822+ 120574)D2119881(119909 minus 1) Thus based on D2119881(119878 + 1) lt 0 we

haveD2119881(119909) lt 0 for 119909 ge 119878 + 1By adding the two sides of (A26) and (A27) we obtain

0 = minus (1205822+ 120574)D

2119881 (119878 + 1) minus (120574 + 120583)D

2119881 (119878)

+ 1205822D2119881 (119878 minus 1)

(A39)

Therefore we haveD2119881(119878 minus 1) lt 0To consider 119909 le 119878minus 2 we define 120598

8= minus120583D2119881(119878minus 1) and

for 119909 = 119878 minus 2

1205988= minus (120582

2+ 120574 + 120583)D

2119881 (119878 minus 2) + 120582

2D2119881 (119878 minus 3) (A40)

Of course 1205987

gt 0 and 1205988

gt 0 Besides thatit is easy to prove that 120598

119894gt 0 for 119894 = 3 4 5 6

and the limit lim119909rarrminusinfin

D119881(119877119863119878)

(119909) = ℎminus120574 implies that


D2119881(119877119863119878)(119909) = 0 It can be verified that thecoefficient matrix of the linear equation about variableD2(119877119863 119878)(119909) from minus119872 to 119878minus2 is totally nonpositive where119872 is a large enough positive integer greater than max119878 |119877|ThusD2119881(119909) le 0 for all 119909 le 119878minus2 It implies thatD2119881(119909) le 0for all 119909 and thus 119881(119877119863119878)(119909) is concave function of 119909

Proof of Theorem 5 For simplicity we assume that 119877lowast lt

119863lowastlt 119878lowast and for the other cases we can argue them in

the same fashion Based on the concavity of the optimaldiscounted profit function D119881

(119877lowast

119863lowast

119878lowast

)(119909) we know that

D119881(119877lowast

119863lowast

119878lowast

)(119909) is a decreasing function of 119909 Hence for 119909 le

119877lowast

D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast) gt

12058221199012minus 12058211199011

1205822minus 1205821

(A41)

for 119909 gt 119878lowast

D119881(119877lowast

119863lowast

119889lowast

)(119909) le D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast+ 1) le 119887 (A42)

for119863lowast lt 119909 le 119878lowast

119888 ge D119881(119877lowast

119863lowast

119878lowast

)(119863lowast+ 1)

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast) ge 119887

(A43)

and for 119877lowast lt 119909 le 119878lowast

D119881(119877lowast

119863lowast

119878lowast

)(119877lowast+ 1)

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast)

(A44)

which yields

12058221199012minus 12058211199011

1205822minus 1205821

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) gt 119887 (A45)

Combining the above five inequalities we can infer that119881(119877lowast

119863lowast

119878lowast

)(119909) satisfies theHamilton-Jacobi-Bellman equation

(5) It follows therefore that the best (119877lowast 119863lowast 119878lowast) policy isglobally optimal

Proof of Theorem 6 Given an (119877119863 119878) policy with 119878 lt 0 weassume that the (119877119863 119878) policy is optimal For simplicity weonly consider the case 119877 lt 119863 lt 119878 lt 0 and the othercase 119863 lt 119877 lt 119878 lt 0 can be analogously analyzed Whilefor (119877119863 119878) policy with 119877 lt 119863 lt 119878 lt 0 we obtain thatD1198810(119909) where D119881

0(119909) = D119881

(119877119863119878)(119909) for 119909 le 119878 satisfies

the following equations

minusℎminus= minus (120582

2+ 120583 + 120574)D119881

0(119878) + 120582

2D1198810(119878 minus 1)

+ 120583119887 for 119909 = 119878

minusℎminus= 120583D119881

0(119909 + 1) minus (120583 + 120582

2+ 120574)D119881

0(119909)

+ 1205822D1198810(119909 minus 1) for 119863 + 1 le 119909 lt 119878


0(119863 + 1) minus 120575

2D1198810(119863)

+ 1205822D1198810(119863 minus 1) + 119886119888 for 119909 = 119863


minusℎminus= (120583 + 119886)D119881

0(119909 + 1) minus 120575

2D1198810(119909)

+ 1205822D1198810(119909 minus 1) for 119877 + 1 lt 119909 lt 119863

minusℎminus= (120583 + 119886)D119881

0(119877 + 2) minus 120575

2D1198810(119877 + 1)

+ 1205821D1198810(119877) + 120582

21199012minus 12058211199011 for 119909 = 119877 + 1

minusℎminus= (120583 + 119886)D119881

0(119909 + 1) minus 120575

1D1198810(119909)

+ 1205821D1198810(119909 minus 1) for 119909 le 119877

(A46)

Let Δ(119909) = D1198811(119909 + 1) minus D119881

0(119909) where D119881

1(119909) =

D119881(119877+1119863+1119878+1)

(119909) then we obtain through a series of sub-tractions

0 = minus (1205822+ 120583 + 120574) Δ (119878) + 120582

2Δ (119878 minus 1) for 119909 = 119878

0 = 120583Δ (119909 + 1) minus (120583 + 1205822+ 120574) Δ (119909)

+ 1205822Δ (119909 minus 1) for 119863 + 1 le 119909 lt 119878

0 = 120583Δ (119863 + 1) minus 1205752Δ (119863) + 120582

2Δ (119863 minus 1) for 119909 = 119863

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909)

+ 1205822Δ (119909 minus 1) for 119877 + 1 lt 119909 lt 119863

0 = (120583 + 119886) Δ (119877 + 2) minus 1205752Δ (119877 + 1)

+ 1205821Δ (119877) for 119909 = 119877 + 1

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 119877

(A47)

We approximate the above infinite system of linearequations with a finite one involving minus119872 le 119909 le 119878 with 119872being a large enough positive integer greater thanmax119878 |119877|The last equation in the finite system corresponding to 119909 =

minus119872 is


where 1205821Δ(minus119872 minus 1) is crossed out As argued before in the

proof of Lemma 3 the coefficient matrix of above linearequations about variable Δ(119909) minus119872 le 119909 le 119878 is invertibleand hence Δ(119909) = 0 for minus119872 le 119909 le 119878 And then we getΔ(119909) = 0 where119872 rarr +infin that is

D119881(119877+1119863+1119878+1)

(119909 + 1) = D119881(119877119863119878)

(119909) (A49)

Furthermore we have the following equation

119881(119877+1119863+1119878+1)

(119878 + 1) minus 119881(119877119863119878)

(119878 + 1)

= minusD119881(119877119863119878)

(119878 + 1)

+

119878

sum

119910=minus119872+1

Δ (119910) +D119881(119877+1119863+1119878+1)

(minus119872 + 1)

+ [119881(119877+1119863+1119878+1)

(minus119872) minus 119881(119877119863119878)

(minus119872)]

(A50)

Lemma 1 implies that

lim119872rarrinfin

[119881(119877+1119863+1119878+1)

(minus119872) minus 119881(119877119863119878)

(minus119872)] = 0 (A51)

Because of

lim119872rarrinfin

D119881(119877+1119863+1119878+1)

(minus119872 + 1) =ℎminus

120574(A52)

andD119881(119877119863119878)

(119878 + 1) le 119887 we get

119881(119877+1119863+1119878+1)

(119878 + 1) minus 119881(119877119863119878)

(119878 + 1) gt 0 (A53)

that is 119881(119877+1119863+1119878+1)(119878 + 1) gt 119881(119877119863119878)

(119878 + 1) However thisrelation invalidates

119881(119877+1119863+1119878+1)

(119878 + 1) le 119881(119877119863119878)

(119878 + 1) (A54)

based on the assumption that the (119877119863 119878) policy is optimalIf (12058221199012minus 12058211199011)(1205822minus 1205821) gt 119888 then we have

D119881(119877lowast

119863lowast

119878lowast

)(119877lowast) gt

12058221199012minus 12058211199011

1205822minus 1205821

gt 119888 ge D119881(119877lowast

119863lowast

119878lowast

)(119863lowast+ 1)

(A55)

Based on the concavity of the optimal discounted profitfunction D119881

(119877lowast

119863lowast

119878lowast

)(119909) we get 119877lowast lt 119863

lowast Similarly we get119877lowastgt 119863lowast if (120582

21199012minus 12058211199011)(1205822minus 1205821) lt 119888

Proof of Theorem 7 We let

119881(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) = 119881

(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119886 119888) (A56)

where (119877lowast(119888) 119863

lowast(119888) 119878lowast(119888)) means the optimal thresholds

policy corresponding to the sourcing production cost 119888where sourcing production rate 119886 is fixed

(1) In the proof we focus on the case that 0 lt 119877lowast(119888) lt

119863lowast(119888) lt 119878

lowast(119888) the other cases can be proved in the

same fashion As for a fixed sourcing production rate 119886 letD119881(119909 119888) = D119881

(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) satisfy the recursions as

follows

ℎ+= minus (120582

2+ 120574)D119881 (119909 119888)

+ 1205822D119881 (119909 minus 1 119888) for 119909 ge 119878lowast (119888) + 1

ℎ+= minus (120582

2+ 120583 + 120574)D119881 (119878

lowast(119888) 119888)

+ 1205822D119881 (119878

lowast(119888) minus 1 119888) + 120583119887 for 119909 = 119878lowast (119888)

ℎ+= 120583D119881 (119909 + 1 119888) minus (120583 + 120582

2+ 120574)D119881 (119909 119888)

+ 1205822D119881 (119909 minus 1 119888) for 119863lowast (119888) + 1 le 119909 lt 119878lowast (119888)

ℎ+= 120583D119881 (119863

lowast(119888) + 1 119888) minus 120575

2D119881 (119863

lowast(119888) 119888)

+ 1205822D119881 (119863



ℎ+= (120583 + 119886)D119881 (119909 + 1 119888) minus 120575

2D119881 (119909 119888)

+ 1205822D119881 (119909 minus 1 119888) for 119877lowast (119888) + 1 lt 119909 lt 119863lowast (119888)

ℎ+= (120583 + 119886)D119881 (119877

lowast(119888) + 2 119888) minus 120575

2D119881 (119877

lowast(119888) + 1 119888)

+ 1205821D119881 (119877

lowast(119888) 119888) + 120582

21199012minus 12058211199011

ℎ+= (120583 + 119886)D119881 (119909 + 1 119888) minus 120575

1D119881 (119909 119888)

+ 1205821D119881 (119909 minus 1 119888) for 0 lt 119909 le 119877lowast (119888)

minusℎminus= (120583 + 119886)D119881 (119909 + 1 119888) minus 120575

1D119881 (119909 119888)

+ 1205821D119881 (119909 minus 1 119888) for 119909 le 0

(A57)

And let Δ12(119909 1198881 1198882) = D119881(119909 119888

2) minusD119881(119909 119888

1) and let 120585 =

minus119886(1198882minus 1198881) which can be simplified as Δ

1(119909) without loss of

generalityWe getΔ1(119909) satisfying a recursion formulated by

subtractingD119881(119909 1198881) from (119909 119888

2) as follows

0 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1)

for 119909 ge 119878lowast (1198881) + 1

0 = minus (1205822+ 120583 + 120574) Δ (119878

lowast(1198881)) + 120582

2Δ (119878lowast(1198881) minus 1)

0 = 120583Δ (119909 + 1) minus (120583 + 1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1)

for 119863lowast (1198881) + 1 le 119909 lt 119878

lowast(1198881)

120585 = 120583Δ (119863lowast(1198881) + 1) minus 120575

2Δ (119863lowast(1198881))

+ 1205822Δ (119863lowast(1198881) minus 1) for 119909 = 119863lowast (119888

1)

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909) + 120582

2Δ (119909 minus 1)

for 119877lowast (1198881) + 1 lt 119909 lt 119863

lowast(1198881)

0 = (120583 + 119886) Δ (119877lowast(1198881) + 2) minus 120575

2Δ (119877lowast(1198881) + 1)

+ 1205821Δ (119877lowast(1198881)) for 119909 = 119877lowast (119888

1) + 1

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 0 lt 119909 le 119877lowast (119888

1)

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 0

(A58)

In line with Lemma 1 we know thatlim119909rarrminusinfin

Δ (119909) = 0 (A59)

It can be verified that the coefficient matrix of the abovelinear equations is totally nonpositive Hence by solving afinite approximate negative systemof linear equations we cancorroborate that Δ(119909) ge 0 that is

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882)

ge D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881)

(A60)

So we get

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119878lowast(1198881) 1198882)

ge D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119878lowast(1198881) 1198881) gt 119887

(A61)

which means that the best base stock level 119878lowast(1198881) + 1 can

be better than 119878lowast(1198881) minus 1 and improved when sourcing cost

is increased from 1198881 that is 119878lowast(119888

2) ge 119878

lowast(1198881) Similarly

we get 119877lowast(1198882) ge 119877

lowast(1198881) Because of thresholds 119878lowast(119888) 119877lowast(119888)

being integers for any sourcing cost 119888 we know that theoptimal thresholds 119877lowast(119888) 119878lowast(119888) are piecewise constants andincreasing functions of 119888

On other hand we will find

D119881(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) minus 119888 (A62)

is decreasing in 119888 Let

D1198811(119909 119888) = D119881

(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) minus 119888 (A63)

which satisfies the recursions as follows

ℎ++ 120574119888 = minus (120582

2+ 120574)D119881

1(119909 119888)

+ 1205822D1198811(119909 minus 1 119888) for 119909 ge 119878lowast (119888) + 1

ℎ++ 120574119888 = minus (120582

2+ 120583 + 120574)D119881

1(119878lowast 119888)

+ 1205822D1198811(119878lowastminus 1 119888) + 120583119887 for 119909 = 119878lowast (119888)

ℎ++ 120574119888 = 120583D119881

1(119909 + 1 119888) minus (120583 + 120582

2+ 120574)D119881

1(119909 119888)

+ 1205822D1198811(119909 minus 1 119888)

for 119863lowast (119888) + 1 le 119909 lt 119878lowast (119888)

ℎ++ 120574119888 = 120583D119881

1(119863lowast(119888) + 1 119888) minus 120575

2D1198811(119863lowast(119888) 119888)

+ 1205822D1198811(119863lowast(119888) minus 1 119888) for 119909 = 119863lowast (119888)

ℎ++ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

2D1198811(119909 119888)

+ 1205822D1198811(119909 minus 1 119888)

for 119877lowast (119888) + 1 lt 119909 lt 119863lowast (119888)

ℎ++ 120574119888 = (120583 + 119886)D119881

1(119877lowast(119888) + 2 119888)

minus 1205752D1198811(119877lowast(119888) + 1 119888)

+ 1205821D1198811(119877lowast(119888) 119888) + 120582

21199012minus 12058211199011

ℎ++ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

1D1198811(119909 119886)

+ 1205821D1198811(119909 minus 1 119888) for 0 lt 119909 le 119877lowast (119888)

minusℎminus+ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

1D1198811(119909 119888)

+ 1205821D1198811(119909 minus 1 119888) for 119909 le 0

(A64)


Let

D1198811(119909 1198881) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

D1198811(119909 1198882) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

(A65)

And let

Δ1(119909 1198881 1198882) = D119881

1(119909 1198882) minusD119881

1(119909 1198881) (A66)

which can be simplified as Δ1(119909) without loss of generality

Let 120578 = 120574(1198882minus 1198881) then we get that Δ

1(119909) satisfied a recursion

formulated by subtracting D1198811(119909 1198881) from D119881

1(119909 1198882) as

follows

120578 = minus (1205822+ 120574) Δ

1(119909) + 120582

2Δ1(119909 minus 1)

for 119909 ge 119878lowast (1198881) + 1

120578 = minus (1205822+ 120583 + 120574) Δ

1(119878lowast(1198881))

+ 1205822Δ1(119878lowast(1198881) minus 1) for 119909 = 119878lowast (119888

1)

120578 = 120583Δ1(119909 + 1) minus (120583 + 120582

2+ 120574) Δ

1(119909)

+ 1205822Δ1(119909 minus 1) for 119863lowast (119888

1) + 1 le 119909 lt 119878

lowast(1198881)

120578 = 120583Δ1(119863lowast(1198881) + 1) minus 120575

2Δ1(119863lowast(1198881))


1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

2Δ1(119909)


1) + 1 lt 119909 lt 119863

lowast(1198881)

120578 = (120583 + 119886) Δ1(119877lowast(1198881) + 2) minus 120575

2Δ1(119877lowast(1198881) + 1)

+ 1205821Δ1(119877lowast(1198881)) for 119909 = 119877lowast (119888

1) + 1

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)

+ 1205821Δ1(119909 minus 1) for 0 lt 119909 le 119877lowast (119888

1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)

+ 1205821Δ1(119909 minus 1) for 119909 le 0

(A67)

In line with Lemma 1 we know that


Δ1(119909) = minus (119888

2minus 1198881) (A68)

Then the preceding equations form a system of linearequations with unknown variables Δ

1(119909) Let 119872 be a large

positive integer that satisfies 119872 gt max(119878lowast |119877lowast| |119863lowast|) Thenwe can get for 119909 = minus119872

(120574 + 1205821) (1198882minus 1198881) = (120583 + 119886) Δ

1(119909 + 1) minus 120575

1Δ1(119909) (A69)

It can be verified that the coefficient matrix of the abovelinear equation about the variableΔ

1(119909)withminus119872 le 119909 le 119872 is

totally nonpositive where119872 is large enough positive integerHence by solving a finite approximate negative system of

linear equations we get Δ1(119909) le 0 for minus119872 le 119909 le 119872 And

then we have Δ1(119909) le 0 for all 119909 when119872 rarr +infin that is

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

(A70)

Hence we get

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

1) minus 1198881le 0

(A71)

that is

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) le 1198882 (A72)

which implies that (119877lowast(1198881) 119863lowast(1198881) minus 1 119878

lowast(1198881)) can be better

than (119877lowast(1198881) 119863lowast(1198881) + 1 119878

lowast(1198881)) and might be improved if

the sourcing cost is increased from 1198881 that is 119863lowast(119888

2) le

119863lowast(1198881) Because of thresholds 119863lowast(119888) being integer for any

sourcing cost 119888 we know that the optimal threshold 119863lowast(119888)

is a piecewise constant decreasing function of 119888(2) Clearly for any 119906 isin U 119881119906(119909 119888

2) lt 119881119906(119909 1198881) Then

119881119906lowast

(119909 1198882) = sup119906isinU

119881119906(119909 1198882)

le sup119906isinU

119881119906(119909 1198881) = 119881119906lowast

(119909 1198881)

(A73)

which proves that 119881119906lowast

(119909 119888) is strictly decreasing in 119888

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the National Social ScienceMajor Foundation of China under Grant 12amp ZD214

References

[1] M E Johnson ldquoDual sourcing strategies operational hedgingand outsourcing to reducing risk in low-cost countriesrdquo inSupply Chain Excellence in Emerging Economies C Y Lee andH L Lee Eds Springer 2005

[2] B Gavish and S C Graves ldquoA one-product productioninventory problem under continuous review policyrdquoOperationsResearch vol 28 no 5 pp 1228ndash1236 1980

[3] M J Sobel ldquoThe optimality of full-service policiesrdquo OperationsResearch vol 30 no 4 pp 636ndash649 1982

[4] Y-S Zheng and P Zipkin ldquoA queueing model to analyzethe value of centralized inventory informationrdquo OperationsResearch vol 38 no 2 pp 296ndash307 1990

[5] L M Wein ldquoDynamic scheduling of a make-to-stock queuerdquoOperations Research vol 40 no 4 pp 724ndash735 1996


[6] M H Veatch and L M Wein ldquoScheduling a make-to-stockqueue index policies and hedging pointsrdquoOperations Researchvol 44 no 4 pp 634ndash647 1996

[7] D Bertsimas and I C Paschalidis ldquoProbabilistic service levelguarantees in make-to-stock manufacturing systemsrdquo Opera-tions Research vol 49 no 1 pp 119ndash133 2001

[8] A Y Ha ldquoInventory rationing in a make-to-stock productionsystemwith several demand classes and lost salesrdquoManagementScience vol 43 no 8 pp 1093ndash1103 1997

[9] A Y Ha ldquoOptimal dynamic scheduling policy for a make-to-stock production systemrdquoOperations Research vol 45 no 1 pp42ndash53 1997

[10] A Y Ha ldquoStock rationing in an MEk1 make-to-stock queuerdquoManagement Science vol 46 no 1 pp 77ndash87 2000

[11] R Kapuscinski and S R Tayur ldquoOptimal policies and simula-tion based optimization for capacitated production inventorysystemsrdquo inQuantitative Models for Supply ChianManagementS R Tayur R Ganeshan and M Magazine Eds pp 7ndash40Kluwer Academic Publishers Amsterdam The Netherlands1999

[12] Y Feng and H Yan ldquoOptimal production control in a discretemanufacturing system with unreliable machines and randomdemandsrdquo IEEE Transactions on Automatic Control vol 45 no12 pp 2280ndash2296 2000

[13] Y Feng and B Xiao ldquoOptimal threshold control in discretefailure-prone manufacturing systemsrdquo IEEE Transactions onAutomatic Control vol 47 no 7 pp 1167ndash1174 2002

[14] L Li ldquoA stochastic theory of the firmrdquo Mathematics of Opera-tions Research vol 13 no 3 pp 447ndash466 1988

[15] L Chen Y Feng and J Ou ldquoJoint management of finishedgoods inventory and demand process for a make-to-stockproduct a computational approachrdquo IEEE Transactions onAutomatic Control vol 51 no 2 pp 258ndash273 2006

[16] L Chen Y Feng and J Ou ldquoCoordinating batch productionand pricing of a make-to-stock productrdquo IEEE Transactions onAutomatic Control vol 54 no 7 pp 1674ndash1680 2009

[17] B Ata and S Shneorson ldquoDynamic control of anMM1 servicesystem with adjustable arrival and service ratesrdquo ManagementScience vol 52 no 11 pp 1778ndash1791 2006

[18] S Carr and I Duenyas ldquoOptimal admission control andsequencing in a make-to-stockmake-to-order production sys-temrdquo Operations Research vol 48 no 5 pp 709ndash720 2000

[19] S M Ross Applied Probabilistic Models with OptimizationApplications Holden-Day San Francisco Calif USA 1970

[20] F Y Chen Y Feng and J Ou ldquoManagement of inventoryreplenishment and available offerings for goods sold withoptional value-added packagesrdquo IIE Transactions vol 37 no 5pp 397ndash406 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


proved by Gavish and Graves [2] and Sobel [3] for the singleproduct and single machine case Later works of Zheng andZipkin [4] Wein [5] Veach and Wein [6] and Bertsimasand Paschalidis [7] attempt to extend the base stock policyto the multiple product cases There are also extensions tothe case of single product but with multiple demand classeslike Ha [8ndash10] Another direction of extension has beento incorporate more detailed modeling of the productionfacility For example Kapuscinski and Tayur [11] model theproduction process by a tandem queue and Feng and Yan[12] and Feng and Xiao [13] deal with unreliable productionfacilities

Li [14] and Chen et al [15 16] are three works thatincorporate controls on both production and demand pro-cesses in a make-to-stock queue optimization problem Li[14] assumes a continuous spectrum of product selling pricesand corresponding demand rates and hence manages toderive a concave and differentiable profit function in termsof the production rate and the selling price which yieldsa qualitative characterization of the optimal policy Chenet al [15] allow discrete choices of prices and derive anefficient algorithm to compute the optimal policy as well as itsqualitative characterization Chen et al [16] consider a make-to-stock manufacturing system with batch production anddiscrete choices of price and derive the characterization ofthe optimal control policy Similar work on a make-to-orderqueue is done by Ata and Shneorson [17] Carr and Duenyas[18] study the optimal control of a mixture of make-to-stock and make-to-order queues Our work adds in anotherdimensionwith the outsourcing option to expand productioncapacity

The rest of the paper is organized as follows Section 2describes precisely the system model and defines an opti-mization problem that solves for the optimal policy Section 3characterizes the optimal threshold policy proves its globaloptimality amongst all nonanticipative control policies anddiscusses its relationship to the cost of outsourcing Section 4briefly discusses the extension to multiple price choicesSection 5 concludes the paper with a summary of the resultsand possible extensions in the future research

To streamline presentation of the paper we state in themain body of the paper all the results without proofs andcollect all the proofs in the appendix

2 Problem Formulation

Themake-to-stock system of concern in the paper has an in-house facility with a production rate 120583 and a unit productioncost 119887 The system can outsource production to an externalfacility which can produce at a rate 119886 and a per unit cost 119888 Weassume that the existing in-house facility has a lower variableproduction cost than the external facility that is 119887 lt 119888 whichholds true in the case ofMattel for example and is the reasonfor keeping the in-house facility We also assume that theproduction processes at both facilities are random and followexponential distributions

The demand process for the product is assumed to bea Poisson process with a price-dependent rate Specificallythere are two selling prices high 119901

1and low 119901

2 which

correspond to two demand rates 1205821and 120582

2 We assume that

1205821lt 1205822and

12058221199012minus 12058211199011

1205822minus 1205821

gt 119887 (1)

which indicates that the marginal profit gain from switchingprice fromhigh to low is greater than the in-house productioncost 119887 Also to ensure system stability we assume that 120583+119886 gt1205821When a demand arrives it is filled from the finished

goods inventory if possible otherwise it is added to a waitingqueue which is served in first-come-first-serve order Thefinished goods inventory carries a holding cost of ℎ+ per unitproduct per unit time and the backordering cost for demandunmet at arrival is ℎminus per waiting demand per unit timeDefine the inventory cost function ℎ(119909) = ℎ

+119909++ ℎminus119909minus

where 119909+ = max[119909 0] and 119909minus = max[minus119909 0] We will considerthe total discounted profit assuming a discount factor 120574 Toensure that it ismore profitable to produce to fill demand thanto backlog forever we assume that 119887 lt 119888 lt ℎminus120574

We specify a dynamic control policy for the system by119906 = 120583(119905) 119886(119905) 119901(119905) 119905 gt 0 where 120583(119905) = 0 or 1 representingin-house production is off or on similarly 119886(119905) = 0 or1 representing outsourced production being off or on and119901(119905) = 119901

1or 1199012representing the price charged at time 119905 A

policy 119906 is called nonanticipatory if at all 119905 gt 0 120583(119905) 119886(119905)and 119901(119905) depend only on information prior to 119905 Let U bethe collection of all nonanticipatory control policies Undera given 119906 isin U denote the total demand sold at price 119901

119894up to

time 119905 gt 0 by 119873119906119894(119905) 119894 = 1 2 the total in-house production

by 119875119906119868(119905) and the total outsourced production by 119875119906

119874(119905) Then

the product inventory level at time 119905 is given by

119883119906(119905) = 119909 + 119875

119880

119868(119905) + 119875

119906

119874(119905) minus 119873

119906

1(119905) minus 119873

119906

2(119905) (2)

where 119909 is the initial inventory at 119905 = 0Consequently the total discounted profit under policy 119906

is

119881119906(119909) = 119864[int

infin

0

119890minus120574119905

(

2

sum

119894=1

119901119894119889119873119906

119894(119905) minus 119887119889119875

119906

119868(119905)

minus119888119889119875119906

119874(119905) minus ℎ (119883

119906(119905)) 119889119905)]

(3)

A policy 119906lowast isin U is said to be optimal if it solves thefollowing optimization problem

119881119906lowast

(119909) = sup119906isinU

119881119906(119909) (4)

The optimal solution to this semi-Markov decision prob-lem can be characterized by the following Hamilton-Jacobi-Bellman (HJB) equation (cf Chapter 7 of [19])

0 = minus 120574119881 (119909) minus ℎ (119909)

+max120572=0120583

120572 [119881 (119909 + 1) minus 119881 (119909) minus 119887]


+max120573=0119886

120573 [119881 (119909 + 1) minus 119881 (119909) minus 119888]

+max119895=12

120582119895[119881 (119909 minus 1) minus 119881 (119909) + 119901

119895]

(5)


0= 0

1205831= 120583 and 120583


costs 1198870= 0 119887

1= 119887 and 119887

2= (120583119887 + 119886119888)(120583 + 119886) respectively


0 = minus 120574119881119906(119909) minus ℎ (119909)

+ max119894=012

120583119894[119881 (119909 + 1) minus 119881 (119909) minus 119887

119894]

+max119895=12

120582119895[119881 (119909 minus 1) minus 119881 (119909) + 119901

119895]

(6)


2when it is above






For 119909 ge 119878


2] (7)

for119863 le 119909 lt 119878


2]

+ 120583 [119881 (119909 + 1) minus 119881 (119909) minus 119887]

(8)

for 119877 lt 119909 lt 119863


2]

+ (120583 + 119886) [119881 (119909 + 1) minus 119881 (119909) minus120583119887 + 119886119888

120583 + 119886]

(9)

for 0 le 119909 le 119877


1]

+ (120583 + 119886) [119881 (119909 + 1) minus 119881 (119909) minus120583119887 + 119886119888

120583 + 119886]

(10)

and for 119909 lt 0


1]

+ (120583 + 119886) [119881 (119909 + 1) minus 119881 (119909) minus120583119887 + 119886119888

120583 + 119886]

(11)




[119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909)] = 0(2) lim


(119909) minus 119881(119877119863minus1119878)

(119909)] = 0(3) lim


(119909) minus 119881(119877minus1119863119878)

(119909)] = 0(4) lim

119909rarr+infinD119881(119877119863119878)

(119909) = minusℎ+120574


D119881(119877119863119878)

(119909) = ℎminus120574



(1198771015840

1198631015840

1198781015840








D119881(119877119863119878)

(119878 + 1) gt 119887 or D119881(119877119863119878+1)

(119878 + 1) gt 119887 (13)


D119881(119877119863119878)

(119863 + 1) gt 119888 or D119881(119877119863+1119878)

(119863 + 1) gt 119888 (14)


D119881(119877119863119878)

(119877 + 1) gt12058221199012minus 12058211199011

1205822minus 1205821

or D119881(119877+1119863119878)

(119877 + 1) gt12058221199012minus 12058211199011

1205822minus 1205821

(15)






policy then


D119881(119877lowast

119863lowast

119878lowast

)(119878lowast+ 1) le 119887 lt D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast) (16)


D119881(119877lowast

119863lowast

119878lowast

)(119863lowast+ 1) le 119888 lt D119881

(119877lowast

119863lowast

119878lowast

)(119863lowast) (17)


D119881(119877lowast

119863lowast

119878lowast

)(119877lowast+ 1) le

12058221199012minus 12058211199011

1205822minus 1205821

lt D119881(119877lowast

119863lowast

119878lowast

)(119877lowast)

(18)


119863lowast

119878lowast

)(119909) is concave in 119909




119863lowast

119878lowast










(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888)








119870


1205821(1199011minus 119888) lt 120582


119870(119901119870minus 119888) (19)



0 = minus ℎ (119909) +max120572=0120583

120572 [119881 (119909 + 1) minus 119881 (119909) minus 119888]

+max120573=0119886

120583 [119881 (119909 + 1) minus 119881 (119909) minus 119888]

+ max119894=1119870

120582119894[119881 (119909 minus 1) + 119901

119894minus 119881 (119909)] minus 120574119881 (119909)

(20)



Consider (1198772 119877

119870 119863 119878) with 119877


119870lt 119878 and



119870+1= +infin)


For a given (1198772 119877




0 = minus ℎ (119909) + 120583 [D119881 (119909) minus 119887]




+ 120582119894[119901119894minusD119881 (119909)] minus 120574119881 (119909) 119877

119894lt 119909 le 119877

119894+1

(21)


119870 119863 119878) policies



2 119877

119870 119863 119878) policy when


(1) the (1198772 119877


(1198772 119877


D119881(1198772119877119870119863119878)

(119878 + 1) gt 119887

119900119903 D119881(1198772119877119870119863119878+1)

(119878 + 1) gt 119887

(22)

(2) the (1198772 119877


(1198772 119877


D119881(1198772119877119870119863119878)

(119863 + 1) gt 119888

119900119903 D119881(1198772119877119870119863+1119878)

(119863 + 1) gt 119888

(23)

(3) the (1198772 119877


(1198772 119877

119870 119863 119878 + 1) policy if 119878 lt 0

(4) the (1198772 119877

119894+ 1 119877


the (1198772 119877

119894 119877


D119881(1198772119877119894119877119870119863119878)

(119877119894+ 1) gt


120582119894minus 120582119894minus1

(24)

or

D119881(1198772119877119894+1119877

119870119863119878)

(119877119894+ 1) gt


120582119894minus 120582119894minus1

(25)



lowast




2119877lowast

119870119863lowast

119878lowast



D119881(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119878lowast+ 1) le 119887 lt D119881

(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119878lowast)

(26)


D119881(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119863lowast+ 1) le 119888 lt D119881

(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119863lowast)

(27)


D119881(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119877lowast

119894+ 1) le


120582119894minus 120582119894minus1

lt D119881(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119877lowast

119894)

(28)



2(119888) 119877

lowast

119870(119888) 119863



lowast

2(119888) 119877

lowast

119870(119888) 119878lowast(119888) are



2(119888)119877

lowast

119870(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) is decreasing in 119888



Appendix


1+ 120574 120575

2= 120583 + 119886 + 120582

2+ 120574 (A1)


[119881(119877119863119878)

(119909) minus 119881(119877119863119878)


(1205822+ 120574) Δ (119909) = 120582

2Δ (119909 minus 1) (A2)

which leads to

Δ (119909) =1205822+ 120574

1205822

Δ (119909 + 1) = sdot sdot sdot = (1205822+ 120574

1205822

)

119899

Δ (119909 + 119899)

Δ (119909) = (1205822

1205822+ 120574

)

119909minus119878

Δ (119878)

(A3)



lim119909rarrinfin


(1205822

1205822+ 120574

)

119909minus119878

Δ (119878) = 0 (A4)

That is

lim119909rarr+infin

[119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909)] = 0 (A5)



[119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909)] = 0 (A6)



0 = 1205821119879minus1+ (120583 + 119886) 119879 minus 120575

1 (A7)

or

0 = 1205821119910minus1+ (120583 + 119886) 119910 minus 120575

1 (A8)


1199101=


1minus 4 (120583 + 119886) 120582

1

2 (120583 + 119886)

1199102=

120575 + radic1205752

1minus 4 (120583 + 119886) 120582

1

2 (120583 + 119886)

(A9)


2


(119909) = 1198621119910119909

1+ 1198622119910119909

2 (A10)

where 1198621and 119862



increases to infin







11199011

= [1205821119879minus1+ (120583 + 119886) 119879 minus 120575

1] (1198671+ 1198672119909)

(A11)


1198672=ℎminus

120574

1198671=

1

1205742[120574 (12058211199011minus (120583119887 + 119886119888)) + ℎ

minus(120583 + 119886 minus 120582

1)]

(A12)


119881(119877119878119861)

(119909) = 1198622119910119909

2+ℎminus119909

120574+12058211199011minus (120583119887 + 119886119888)

120574

+ℎminus(120583 + 119886 minus 120582

1)

1205742

(A13)


2for 119881(119877119863119878)(119909) it is


Δ (119909) = 119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909) = (1198622minus 1198622) 119910119909

2 (A14)

Then we get


[119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909)] = 0 (A15)



D119881(119877119863119878)

(119909) = minusℎ+120574


D119881(119877119863119878)



[119881(119877119863119878)

(119909) minus

119881(119877119863119878minus1)



119881(119877119863119878)

(119909) = 119875(1205822

1205822+ 120574

)

119909

minusℎ+119909

120574+12058221199012

120574+1205822ℎ+

1205742 (A16)


119881(119877119863119878)

(119909) = 1198622119910119909

2+ℎminus119909

120574+12058211199011minus (120583119887 + 119886119888)

120574

+ℎminus(120583 + 119886 minus 120582

1)

1205742

(A17)


1199102=

1205751+ radic1205752

1minus 4 (120583 + 119886) (120582

1)

2 (120583 + 119886) (A18)


2gt 1 we get

lim119909rarr+infin

D119881(119877119863119878)

(119909) = minusℎ+

120574


D119881(119877119863119878)

(119909) =ℎminus

120574

(A19)


(119878 + 1) gt 119887 orD119881(119877119863119878+1)






Let Δ(119909) = 119881(119877119863119878+1)

(119909) minus 119881(119877119863119878)


0 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1) for 119909 ge 119878 + 1

minus120598 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1) for 119909 = 119878

0 = 120583Δ (119909 + 1) minus (1205822+ 120574 + 120583)Δ (119909)

+ 1205822Δ (119909 minus 1) for 119863 le 119909 lt 119878

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909)

+ 1205822Δ (119909 minus 1) for 119877 le 119909 lt 119863

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 119877

(A20)

where 120598 = 120583[D119881(119877119863119878+1)


that


[119881(119877119863119878+1)

(119909) minus 119881(119877119863119878)

(119909)] = 0 (A21)






minus 120583 [119863119881(119877119863119878)

(119878 + 1) minus 119887]

= 120583Δ (119878 + 1) minus (120583 + 1205822+ 120574) Δ (119878) + 120582

2Δ (119878 minus 1)

(A23)


(119877119863119878)(119878+

1) le 119887 or D119881(119877119863119878+1)


(119909) le 119881(119877119863119878)

(119909)




D119881(119909) = 119881(119877119863119878)

(119909) minus 119881(119877119863119878)



ℎ+= minus 120582

2D2119881 (119909) minus 120574D119881 (119909) for 119909 ge 119878 + 1

ℎ+= minus 120582

2D2119881 (119878) minus (120583 + 120574)D119881 (119878) + 120583119887 for 119909 = 119878

ℎ+= 120583D

2119881 (119909 + 1) minus 120582

2D2119881 (119909)

minus 120574D119881 (119909) for 119863 + 1 le 119909 lt 119878

ℎ+= 120583D119881 (119863 + 1) minus 120575

2D119881 (119863)

+ 1205822D119881 (119863 minus 1) + 119886119888 for 119909 = 119863

ℎ+= (120583 + 119886)D

2119881 (119909 + 1) minus 120582

2D2119881 (119909)


ℎ+= (120583 + 119886)D

2119881 (119877 + 2) minus 120582

1D2119881 (119877 + 1)

minus (1205822minus 1205821+ 120574)D119881 (119877 + 1) + 120582

21199012minus 12058211199011

ℎ+= (120583 + 119886)D

2119881 (119909 + 1) minus 120582

1D2119881 (119909)

minus 120574D119881 (119909) for 1 le 119909 le 119877

minusℎminus= (120583 + 119886)D

2119881 (119909 + 1) minus 120582

1D2119881 (119909)

minus 120574D119881 (119909) for 119909 le 0(A24)


0 = minus (1205822+ 120574)D

2119881 (119909) + 120582

2D2119881 (119909 minus 1) (A25)

Similarly we derive

0 = minus (1205822+ 120574)D

2119881 (119878 + 1) + 120582

2D2119881 (119878)

+ 120583D119881 (119878) minus 120583119887 for 119909 = 119878 + 1(A26)

0 = minus (1205822+ 120574 + 120583)D

2119881 (119878) + 120582

2D2119881 (119878 minus 1)

minus 120583D119881 (119878) + 120583119887 for 119909 = 119878(A27)

0 = 120583D2119881 (119909 + 1) minus (120582

2+ 120574 + 120583)D

2119881 (119909)

+ 1205822D2119881 (119909 minus 1) for 119863 + 2 le 119909 lt 119878

(A28)

1205983= 120583D

2119881 (119863 + 2) minus 120575

2D2119881 (119863 + 1)

+ 1205822D2119881 (119863) for 119909 = 119863 + 1

(A29)

1205984= 120583D

2119881 (119863 + 1) minus 120575

2D2119881 (119863)

+ 1205822D2119881 (119863 minus 1) for 119909 = 119863

(A30)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

2D2119881 (119909)


(A31)


1205985= (120583 + 119886)D

2119881 (119877 + 3) minus 120575

2D2119881 (119877 + 2)

+ 1205821D2119881 (119877 + 1)

(A32)

1205986= (120583 + 119886)D

2119881 (119877 + 2) minus 120575

2D2119881 (119877 + 1)

+ 1205821D2119881 (119877)

(A33)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

1D2119881 (119909)

+ 1205821D2119881 (119909 minus 1) for 2 le 119909 le 119877

(A34)

1205987= (120583 + 119886)D

2119881 (2) minus 120575

1D2119881 (1) + 120582

1D2119881 (0) (A35)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

1D2119881 (119909)

+ 1205821D2119881 (119909 minus 1) for 119909 le 0

(A36)


4= 119886[D119881(119863) minus 119888] 120598

5=

minus(1205822minus1205821)D119881(119877+1)+(120582


(12058221199012minus 12058211199011) and 120598

7= ℎ++ ℎminus


D2119881 (119878) =

ℎ++ (120583 + 120574)D119881 (119878) minus 120583119887

minus1205822

(A37)



(1205822+ 120574)D

2119881 (119878 + 1) = minusℎ

+minus 120574D119881 (119878) (A38)




0 = minus (1205822+ 120574)D

2119881 (119878 + 1) minus (120574 + 120583)D

2119881 (119878)

+ 1205822D2119881 (119878 minus 1)

(A39)


8= minus120583D2119881(119878minus 1) and

for 119909 = 119878 minus 2

1205988= minus (120582

2+ 120574 + 120583)D

2119881 (119878 minus 2) + 120582

2D2119881 (119878 minus 3) (A40)

Of course 1205987

gt 0 and 1205988


119894gt 0 for 119894 = 3 4 5 6


D119881(119877119863119878)







(119877lowast

119863lowast

119878lowast


D119881(119877lowast

119863lowast

119878lowast


119877lowast

D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast) gt

12058221199012minus 12058211199011

1205822minus 1205821

(A41)


D119881(119877lowast

119863lowast

119889lowast

)(119909) le D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast+ 1) le 119887 (A42)


119888 ge D119881(119877lowast

119863lowast

119878lowast

)(119863lowast+ 1)

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast) ge 119887

(A43)


D119881(119877lowast

119863lowast

119878lowast

)(119877lowast+ 1)

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast)

(A44)

which yields

12058221199012minus 12058211199011

1205822minus 1205821

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) gt 119887 (A45)


119863lowast

119878lowast




0(119909) = D119881

(119877119863119878)(119909) for 119909 le 119878 satisfies



2+ 120583 + 120574)D119881

0(119878) + 120582

2D1198810(119878 minus 1)

+ 120583119887 for 119909 = 119878


0(119909 + 1) minus (120583 + 120582

2+ 120574)D119881

0(119909)

+ 1205822D1198810(119909 minus 1) for 119863 + 1 le 119909 lt 119878


0(119863 + 1) minus 120575

2D1198810(119863)

+ 1205822D1198810(119863 minus 1) + 119886119888 for 119909 = 119863


minusℎminus= (120583 + 119886)D119881

0(119909 + 1) minus 120575

2D1198810(119909)

+ 1205822D1198810(119909 minus 1) for 119877 + 1 lt 119909 lt 119863

minusℎminus= (120583 + 119886)D119881

0(119877 + 2) minus 120575

2D1198810(119877 + 1)

+ 1205821D1198810(119877) + 120582

21199012minus 12058211199011 for 119909 = 119877 + 1

minusℎminus= (120583 + 119886)D119881

0(119909 + 1) minus 120575

1D1198810(119909)

+ 1205821D1198810(119909 minus 1) for 119909 le 119877

(A46)

Let Δ(119909) = D1198811(119909 + 1) minus D119881

0(119909) where D119881

1(119909) =

D119881(119877+1119863+1119878+1)


0 = minus (1205822+ 120583 + 120574) Δ (119878) + 120582

2Δ (119878 minus 1) for 119909 = 119878

0 = 120583Δ (119909 + 1) minus (120583 + 1205822+ 120574) Δ (119909)

+ 1205822Δ (119909 minus 1) for 119863 + 1 le 119909 lt 119878

0 = 120583Δ (119863 + 1) minus 1205752Δ (119863) + 120582

2Δ (119863 minus 1) for 119909 = 119863

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909)

+ 1205822Δ (119909 minus 1) for 119877 + 1 lt 119909 lt 119863

0 = (120583 + 119886) Δ (119877 + 2) minus 1205752Δ (119877 + 1)

+ 1205821Δ (119877) for 119909 = 119877 + 1

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 119877

(A47)


minus119872 is




D119881(119877+1119863+1119878+1)

(119909 + 1) = D119881(119877119863119878)

(119909) (A49)


119881(119877+1119863+1119878+1)

(119878 + 1) minus 119881(119877119863119878)

(119878 + 1)

= minusD119881(119877119863119878)

(119878 + 1)

+

119878

sum

119910=minus119872+1

Δ (119910) +D119881(119877+1119863+1119878+1)

(minus119872 + 1)

+ [119881(119877+1119863+1119878+1)

(minus119872) minus 119881(119877119863119878)

(minus119872)]

(A50)


lim119872rarrinfin

[119881(119877+1119863+1119878+1)

(minus119872) minus 119881(119877119863119878)

(minus119872)] = 0 (A51)

Because of

lim119872rarrinfin

D119881(119877+1119863+1119878+1)


120574(A52)

andD119881(119877119863119878)

(119878 + 1) le 119887 we get

119881(119877+1119863+1119878+1)

(119878 + 1) minus 119881(119877119863119878)

(119878 + 1) gt 0 (A53)

that is 119881(119877+1119863+1119878+1)(119878 + 1) gt 119881(119877119863119878)


119881(119877+1119863+1119878+1)

(119878 + 1) le 119881(119877119863119878)

(119878 + 1) (A54)


D119881(119877lowast

119863lowast

119878lowast

)(119877lowast) gt

12058221199012minus 12058211199011

1205822minus 1205821

gt 119888 ge D119881(119877lowast

119863lowast

119878lowast

)(119863lowast+ 1)

(A55)


(119877lowast

119863lowast

119878lowast

)(119909) we get 119877lowast lt 119863


21199012minus 12058211199011)(1205822minus 1205821) lt 119888


119881(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) = 119881

(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119886 119888) (A56)

where (119877lowast(119888) 119863




119863lowast(119888) lt 119878



(119877lowast

(119888)119863lowast

(119888)119878lowast


follows

ℎ+= minus (120582

2+ 120574)D119881 (119909 119888)


ℎ+= minus (120582

2+ 120583 + 120574)D119881 (119878

lowast(119888) 119888)

+ 1205822D119881 (119878


ℎ+= 120583D119881 (119909 + 1 119888) minus (120583 + 120582

2+ 120574)D119881 (119909 119888)


ℎ+= 120583D119881 (119863

lowast(119888) + 1 119888) minus 120575

2D119881 (119863

lowast(119888) 119888)

+ 1205822D119881 (119863



ℎ+= (120583 + 119886)D119881 (119909 + 1 119888) minus 120575

2D119881 (119909 119888)


ℎ+= (120583 + 119886)D119881 (119877

lowast(119888) + 2 119888) minus 120575

2D119881 (119877

lowast(119888) + 1 119888)

+ 1205821D119881 (119877

lowast(119888) 119888) + 120582

21199012minus 12058211199011

ℎ+= (120583 + 119886)D119881 (119909 + 1 119888) minus 120575

1D119881 (119909 119888)



1D119881 (119909 119888)

+ 1205821D119881 (119909 minus 1 119888) for 119909 le 0

(A57)

And let Δ12(119909 1198881 1198882) = D119881(119909 119888

2) minusD119881(119909 119888

1) and let 120585 =





2) as follows

0 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1)

for 119909 ge 119878lowast (1198881) + 1

0 = minus (1205822+ 120583 + 120574) Δ (119878

lowast(1198881)) + 120582

2Δ (119878lowast(1198881) minus 1)

0 = 120583Δ (119909 + 1) minus (120583 + 1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1)

for 119863lowast (1198881) + 1 le 119909 lt 119878

lowast(1198881)

120585 = 120583Δ (119863lowast(1198881) + 1) minus 120575

2Δ (119863lowast(1198881))


1)

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909) + 120582

2Δ (119909 minus 1)

for 119877lowast (1198881) + 1 lt 119909 lt 119863

lowast(1198881)

0 = (120583 + 119886) Δ (119877lowast(1198881) + 2) minus 120575

2Δ (119877lowast(1198881) + 1)


1) + 1

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)


1)

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 0

(A58)


Δ (119909) = 0 (A59)


D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882)

ge D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881)

(A60)

So we get

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119878lowast(1198881) 1198882)

ge D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119878lowast(1198881) 1198881) gt 119887

(A61)




2) ge 119878


we get 119877lowast(1198882) ge 119877




D119881(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) minus 119888 (A62)


D1198811(119909 119888) = D119881

(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) minus 119888 (A63)


ℎ++ 120574119888 = minus (120582

2+ 120574)D119881

1(119909 119888)


ℎ++ 120574119888 = minus (120582

2+ 120583 + 120574)D119881

1(119878lowast 119888)


ℎ++ 120574119888 = 120583D119881

1(119909 + 1 119888) minus (120583 + 120582

2+ 120574)D119881

1(119909 119888)

+ 1205822D1198811(119909 minus 1 119888)


ℎ++ 120574119888 = 120583D119881

1(119863lowast(119888) + 1 119888) minus 120575

2D1198811(119863lowast(119888) 119888)


ℎ++ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

2D1198811(119909 119888)

+ 1205822D1198811(119909 minus 1 119888)


ℎ++ 120574119888 = (120583 + 119886)D119881

1(119877lowast(119888) + 2 119888)

minus 1205752D1198811(119877lowast(119888) + 1 119888)

+ 1205821D1198811(119877lowast(119888) 119888) + 120582

21199012minus 12058211199011

ℎ++ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

1D1198811(119909 119886)


minusℎminus+ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

1D1198811(119909 119888)

+ 1205821D1198811(119909 minus 1 119888) for 119909 le 0

(A64)


Let

D1198811(119909 1198881) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

D1198811(119909 1198882) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

(A65)

And let

Δ1(119909 1198881 1198882) = D119881

1(119909 1198882) minusD119881

1(119909 1198881) (A66)





1(119909 1198882) as

follows

120578 = minus (1205822+ 120574) Δ

1(119909) + 120582

2Δ1(119909 minus 1)

for 119909 ge 119878lowast (1198881) + 1

120578 = minus (1205822+ 120583 + 120574) Δ

1(119878lowast(1198881))


1)

120578 = 120583Δ1(119909 + 1) minus (120583 + 120582

2+ 120574) Δ

1(119909)


1) + 1 le 119909 lt 119878

lowast(1198881)

120578 = 120583Δ1(119863lowast(1198881) + 1) minus 120575

2Δ1(119863lowast(1198881))


1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

2Δ1(119909)


1) + 1 lt 119909 lt 119863

lowast(1198881)

120578 = (120583 + 119886) Δ1(119877lowast(1198881) + 2) minus 120575

2Δ1(119877lowast(1198881) + 1)


1) + 1

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)


1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)

+ 1205821Δ1(119909 minus 1) for 119909 le 0

(A67)



Δ1(119909) = minus (119888

2minus 1198881) (A68)


1(119909) Let 119872 be a large


(120574 + 1205821) (1198882minus 1198881) = (120583 + 119886) Δ

1(119909 + 1) minus 120575

1Δ1(119909) (A69)






D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

(A70)

Hence we get

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

1) minus 1198881le 0

(A71)

that is

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) le 1198882 (A72)






2) le




2) lt 119881119906(119909 1198881) Then

119881119906lowast

(119909 1198882) = sup119906isinU

119881119906(119909 1198882)

le sup119906isinU

119881119906(119909 1198881) = 119881119906lowast

(119909 1198881)

(A73)





Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+max120573=0119886

120573 [119881 (119909 + 1) minus 119881 (119909) minus 119888]

+max119895=12

120582119895[119881 (119909 minus 1) minus 119881 (119909) + 119901

119895]

(5)


0= 0

1205831= 120583 and 120583


costs 1198870= 0 119887

1= 119887 and 119887

2= (120583119887 + 119886119888)(120583 + 119886) respectively


0 = minus 120574119881119906(119909) minus ℎ (119909)

+ max119894=012

120583119894[119881 (119909 + 1) minus 119881 (119909) minus 119887

119894]

+max119895=12

120582119895[119881 (119909 minus 1) minus 119881 (119909) + 119901

119895]

(6)


2when it is above






For 119909 ge 119878


2] (7)

for119863 le 119909 lt 119878


2]

+ 120583 [119881 (119909 + 1) minus 119881 (119909) minus 119887]

(8)

for 119877 lt 119909 lt 119863


2]

+ (120583 + 119886) [119881 (119909 + 1) minus 119881 (119909) minus120583119887 + 119886119888

120583 + 119886]

(9)

for 0 le 119909 le 119877


1]

+ (120583 + 119886) [119881 (119909 + 1) minus 119881 (119909) minus120583119887 + 119886119888

120583 + 119886]

(10)

and for 119909 lt 0


1]

+ (120583 + 119886) [119881 (119909 + 1) minus 119881 (119909) minus120583119887 + 119886119888

120583 + 119886]

(11)




[119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909)] = 0(2) lim


(119909) minus 119881(119877119863minus1119878)

(119909)] = 0(3) lim


(119909) minus 119881(119877minus1119863119878)

(119909)] = 0(4) lim

119909rarr+infinD119881(119877119863119878)

(119909) = minusℎ+120574


D119881(119877119863119878)

(119909) = ℎminus120574



(1198771015840

1198631015840

1198781015840








D119881(119877119863119878)

(119878 + 1) gt 119887 or D119881(119877119863119878+1)

(119878 + 1) gt 119887 (13)


D119881(119877119863119878)

(119863 + 1) gt 119888 or D119881(119877119863+1119878)

(119863 + 1) gt 119888 (14)


D119881(119877119863119878)

(119877 + 1) gt12058221199012minus 12058211199011

1205822minus 1205821

or D119881(119877+1119863119878)

(119877 + 1) gt12058221199012minus 12058211199011

1205822minus 1205821

(15)






policy then


D119881(119877lowast

119863lowast

119878lowast

)(119878lowast+ 1) le 119887 lt D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast) (16)


D119881(119877lowast

119863lowast

119878lowast

)(119863lowast+ 1) le 119888 lt D119881

(119877lowast

119863lowast

119878lowast

)(119863lowast) (17)


D119881(119877lowast

119863lowast

119878lowast

)(119877lowast+ 1) le

12058221199012minus 12058211199011

1205822minus 1205821

lt D119881(119877lowast

119863lowast

119878lowast

)(119877lowast)

(18)


119863lowast

119878lowast

)(119909) is concave in 119909




119863lowast

119878lowast










(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888)








119870


1205821(1199011minus 119888) lt 120582


119870(119901119870minus 119888) (19)



0 = minus ℎ (119909) +max120572=0120583

120572 [119881 (119909 + 1) minus 119881 (119909) minus 119888]

+max120573=0119886

120583 [119881 (119909 + 1) minus 119881 (119909) minus 119888]

+ max119894=1119870

120582119894[119881 (119909 minus 1) + 119901

119894minus 119881 (119909)] minus 120574119881 (119909)

(20)



Consider (1198772 119877

119870 119863 119878) with 119877


119870lt 119878 and



119870+1= +infin)


For a given (1198772 119877




0 = minus ℎ (119909) + 120583 [D119881 (119909) minus 119887]




+ 120582119894[119901119894minusD119881 (119909)] minus 120574119881 (119909) 119877

119894lt 119909 le 119877

119894+1

(21)


119870 119863 119878) policies



2 119877

119870 119863 119878) policy when


(1) the (1198772 119877


(1198772 119877


D119881(1198772119877119870119863119878)

(119878 + 1) gt 119887

119900119903 D119881(1198772119877119870119863119878+1)

(119878 + 1) gt 119887

(22)

(2) the (1198772 119877


(1198772 119877


D119881(1198772119877119870119863119878)

(119863 + 1) gt 119888

119900119903 D119881(1198772119877119870119863+1119878)

(119863 + 1) gt 119888

(23)

(3) the (1198772 119877


(1198772 119877

119870 119863 119878 + 1) policy if 119878 lt 0

(4) the (1198772 119877

119894+ 1 119877


the (1198772 119877

119894 119877


D119881(1198772119877119894119877119870119863119878)

(119877119894+ 1) gt


120582119894minus 120582119894minus1

(24)

or

D119881(1198772119877119894+1119877

119870119863119878)

(119877119894+ 1) gt


120582119894minus 120582119894minus1

(25)



lowast




2119877lowast

119870119863lowast

119878lowast



D119881(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119878lowast+ 1) le 119887 lt D119881

(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119878lowast)

(26)


D119881(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119863lowast+ 1) le 119888 lt D119881

(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119863lowast)

(27)


D119881(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119877lowast

119894+ 1) le


120582119894minus 120582119894minus1

lt D119881(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119877lowast

119894)

(28)



2(119888) 119877

lowast

119870(119888) 119863



lowast

2(119888) 119877

lowast

119870(119888) 119878lowast(119888) are



2(119888)119877

lowast

119870(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) is decreasing in 119888



Appendix


1+ 120574 120575

2= 120583 + 119886 + 120582

2+ 120574 (A1)


[119881(119877119863119878)

(119909) minus 119881(119877119863119878)


(1205822+ 120574) Δ (119909) = 120582

2Δ (119909 minus 1) (A2)

which leads to

Δ (119909) =1205822+ 120574

1205822

Δ (119909 + 1) = sdot sdot sdot = (1205822+ 120574

1205822

)

119899

Δ (119909 + 119899)

Δ (119909) = (1205822

1205822+ 120574

)

119909minus119878

Δ (119878)

(A3)



lim119909rarrinfin


(1205822

1205822+ 120574

)

119909minus119878

Δ (119878) = 0 (A4)

That is

lim119909rarr+infin

[119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909)] = 0 (A5)



[119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909)] = 0 (A6)



0 = 1205821119879minus1+ (120583 + 119886) 119879 minus 120575

1 (A7)

or

0 = 1205821119910minus1+ (120583 + 119886) 119910 minus 120575

1 (A8)


1199101=


1minus 4 (120583 + 119886) 120582

1

2 (120583 + 119886)

1199102=

120575 + radic1205752

1minus 4 (120583 + 119886) 120582

1

2 (120583 + 119886)

(A9)


2


(119909) = 1198621119910119909

1+ 1198622119910119909

2 (A10)

where 1198621and 119862



increases to infin







11199011

= [1205821119879minus1+ (120583 + 119886) 119879 minus 120575

1] (1198671+ 1198672119909)

(A11)


1198672=ℎminus

120574

1198671=

1

1205742[120574 (12058211199011minus (120583119887 + 119886119888)) + ℎ

minus(120583 + 119886 minus 120582

1)]

(A12)


119881(119877119878119861)

(119909) = 1198622119910119909

2+ℎminus119909

120574+12058211199011minus (120583119887 + 119886119888)

120574

+ℎminus(120583 + 119886 minus 120582

1)

1205742

(A13)


2for 119881(119877119863119878)(119909) it is


Δ (119909) = 119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909) = (1198622minus 1198622) 119910119909

2 (A14)

Then we get


[119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909)] = 0 (A15)



D119881(119877119863119878)

(119909) = minusℎ+120574


D119881(119877119863119878)



[119881(119877119863119878)

(119909) minus

119881(119877119863119878minus1)



119881(119877119863119878)

(119909) = 119875(1205822

1205822+ 120574

)

119909

minusℎ+119909

120574+12058221199012

120574+1205822ℎ+

1205742 (A16)


119881(119877119863119878)

(119909) = 1198622119910119909

2+ℎminus119909

120574+12058211199011minus (120583119887 + 119886119888)

120574

+ℎminus(120583 + 119886 minus 120582

1)

1205742

(A17)


1199102=

1205751+ radic1205752

1minus 4 (120583 + 119886) (120582

1)

2 (120583 + 119886) (A18)


2gt 1 we get

lim119909rarr+infin

D119881(119877119863119878)

(119909) = minusℎ+

120574


D119881(119877119863119878)

(119909) =ℎminus

120574

(A19)


(119878 + 1) gt 119887 orD119881(119877119863119878+1)






Let Δ(119909) = 119881(119877119863119878+1)

(119909) minus 119881(119877119863119878)


0 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1) for 119909 ge 119878 + 1

minus120598 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1) for 119909 = 119878

0 = 120583Δ (119909 + 1) minus (1205822+ 120574 + 120583)Δ (119909)

+ 1205822Δ (119909 minus 1) for 119863 le 119909 lt 119878

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909)

+ 1205822Δ (119909 minus 1) for 119877 le 119909 lt 119863

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 119877

(A20)

where 120598 = 120583[D119881(119877119863119878+1)


that


[119881(119877119863119878+1)

(119909) minus 119881(119877119863119878)

(119909)] = 0 (A21)






minus 120583 [119863119881(119877119863119878)

(119878 + 1) minus 119887]

= 120583Δ (119878 + 1) minus (120583 + 1205822+ 120574) Δ (119878) + 120582

2Δ (119878 minus 1)

(A23)


(119877119863119878)(119878+

1) le 119887 or D119881(119877119863119878+1)


(119909) le 119881(119877119863119878)

(119909)




D119881(119909) = 119881(119877119863119878)

(119909) minus 119881(119877119863119878)



ℎ+= minus 120582

2D2119881 (119909) minus 120574D119881 (119909) for 119909 ge 119878 + 1

ℎ+= minus 120582

2D2119881 (119878) minus (120583 + 120574)D119881 (119878) + 120583119887 for 119909 = 119878

ℎ+= 120583D

2119881 (119909 + 1) minus 120582

2D2119881 (119909)

minus 120574D119881 (119909) for 119863 + 1 le 119909 lt 119878

ℎ+= 120583D119881 (119863 + 1) minus 120575

2D119881 (119863)

+ 1205822D119881 (119863 minus 1) + 119886119888 for 119909 = 119863

ℎ+= (120583 + 119886)D

2119881 (119909 + 1) minus 120582

2D2119881 (119909)


ℎ+= (120583 + 119886)D

2119881 (119877 + 2) minus 120582

1D2119881 (119877 + 1)

minus (1205822minus 1205821+ 120574)D119881 (119877 + 1) + 120582

21199012minus 12058211199011

ℎ+= (120583 + 119886)D

2119881 (119909 + 1) minus 120582

1D2119881 (119909)

minus 120574D119881 (119909) for 1 le 119909 le 119877

minusℎminus= (120583 + 119886)D

2119881 (119909 + 1) minus 120582

1D2119881 (119909)

minus 120574D119881 (119909) for 119909 le 0(A24)


0 = minus (1205822+ 120574)D

2119881 (119909) + 120582

2D2119881 (119909 minus 1) (A25)

Similarly we derive

0 = minus (1205822+ 120574)D

2119881 (119878 + 1) + 120582

2D2119881 (119878)

+ 120583D119881 (119878) minus 120583119887 for 119909 = 119878 + 1(A26)

0 = minus (1205822+ 120574 + 120583)D

2119881 (119878) + 120582

2D2119881 (119878 minus 1)

minus 120583D119881 (119878) + 120583119887 for 119909 = 119878(A27)

0 = 120583D2119881 (119909 + 1) minus (120582

2+ 120574 + 120583)D

2119881 (119909)

+ 1205822D2119881 (119909 minus 1) for 119863 + 2 le 119909 lt 119878

(A28)

1205983= 120583D

2119881 (119863 + 2) minus 120575

2D2119881 (119863 + 1)

+ 1205822D2119881 (119863) for 119909 = 119863 + 1

(A29)

1205984= 120583D

2119881 (119863 + 1) minus 120575

2D2119881 (119863)

+ 1205822D2119881 (119863 minus 1) for 119909 = 119863

(A30)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

2D2119881 (119909)


(A31)


1205985= (120583 + 119886)D

2119881 (119877 + 3) minus 120575

2D2119881 (119877 + 2)

+ 1205821D2119881 (119877 + 1)

(A32)

1205986= (120583 + 119886)D

2119881 (119877 + 2) minus 120575

2D2119881 (119877 + 1)

+ 1205821D2119881 (119877)

(A33)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

1D2119881 (119909)

+ 1205821D2119881 (119909 minus 1) for 2 le 119909 le 119877

(A34)

1205987= (120583 + 119886)D

2119881 (2) minus 120575

1D2119881 (1) + 120582

1D2119881 (0) (A35)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

1D2119881 (119909)

+ 1205821D2119881 (119909 minus 1) for 119909 le 0

(A36)


4= 119886[D119881(119863) minus 119888] 120598

5=

minus(1205822minus1205821)D119881(119877+1)+(120582


(12058221199012minus 12058211199011) and 120598

7= ℎ++ ℎminus


D2119881 (119878) =

ℎ++ (120583 + 120574)D119881 (119878) minus 120583119887

minus1205822

(A37)



(1205822+ 120574)D

2119881 (119878 + 1) = minusℎ

+minus 120574D119881 (119878) (A38)




0 = minus (1205822+ 120574)D

2119881 (119878 + 1) minus (120574 + 120583)D

2119881 (119878)

+ 1205822D2119881 (119878 minus 1)

(A39)


8= minus120583D2119881(119878minus 1) and

for 119909 = 119878 minus 2

1205988= minus (120582

2+ 120574 + 120583)D

2119881 (119878 minus 2) + 120582

2D2119881 (119878 minus 3) (A40)

Of course 1205987

gt 0 and 1205988


119894gt 0 for 119894 = 3 4 5 6


D119881(119877119863119878)







(119877lowast

119863lowast

119878lowast


D119881(119877lowast

119863lowast

119878lowast


119877lowast

D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast) gt

12058221199012minus 12058211199011

1205822minus 1205821

(A41)


D119881(119877lowast

119863lowast

119889lowast

)(119909) le D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast+ 1) le 119887 (A42)


119888 ge D119881(119877lowast

119863lowast

119878lowast

)(119863lowast+ 1)

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast) ge 119887

(A43)


D119881(119877lowast

119863lowast

119878lowast

)(119877lowast+ 1)

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast)

(A44)

which yields

12058221199012minus 12058211199011

1205822minus 1205821

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) gt 119887 (A45)


119863lowast

119878lowast




0(119909) = D119881

(119877119863119878)(119909) for 119909 le 119878 satisfies



2+ 120583 + 120574)D119881

0(119878) + 120582

2D1198810(119878 minus 1)

+ 120583119887 for 119909 = 119878


0(119909 + 1) minus (120583 + 120582

2+ 120574)D119881

0(119909)

+ 1205822D1198810(119909 minus 1) for 119863 + 1 le 119909 lt 119878


0(119863 + 1) minus 120575

2D1198810(119863)

+ 1205822D1198810(119863 minus 1) + 119886119888 for 119909 = 119863


minusℎminus= (120583 + 119886)D119881

0(119909 + 1) minus 120575

2D1198810(119909)

+ 1205822D1198810(119909 minus 1) for 119877 + 1 lt 119909 lt 119863

minusℎminus= (120583 + 119886)D119881

0(119877 + 2) minus 120575

2D1198810(119877 + 1)

+ 1205821D1198810(119877) + 120582

21199012minus 12058211199011 for 119909 = 119877 + 1

minusℎminus= (120583 + 119886)D119881

0(119909 + 1) minus 120575

1D1198810(119909)

+ 1205821D1198810(119909 minus 1) for 119909 le 119877

(A46)

Let Δ(119909) = D1198811(119909 + 1) minus D119881

0(119909) where D119881

1(119909) =

D119881(119877+1119863+1119878+1)


0 = minus (1205822+ 120583 + 120574) Δ (119878) + 120582

2Δ (119878 minus 1) for 119909 = 119878

0 = 120583Δ (119909 + 1) minus (120583 + 1205822+ 120574) Δ (119909)

+ 1205822Δ (119909 minus 1) for 119863 + 1 le 119909 lt 119878

0 = 120583Δ (119863 + 1) minus 1205752Δ (119863) + 120582

2Δ (119863 minus 1) for 119909 = 119863

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909)

+ 1205822Δ (119909 minus 1) for 119877 + 1 lt 119909 lt 119863

0 = (120583 + 119886) Δ (119877 + 2) minus 1205752Δ (119877 + 1)

+ 1205821Δ (119877) for 119909 = 119877 + 1

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 119877

(A47)


minus119872 is




D119881(119877+1119863+1119878+1)

(119909 + 1) = D119881(119877119863119878)

(119909) (A49)


119881(119877+1119863+1119878+1)

(119878 + 1) minus 119881(119877119863119878)

(119878 + 1)

= minusD119881(119877119863119878)

(119878 + 1)

+

119878

sum

119910=minus119872+1

Δ (119910) +D119881(119877+1119863+1119878+1)

(minus119872 + 1)

+ [119881(119877+1119863+1119878+1)

(minus119872) minus 119881(119877119863119878)

(minus119872)]

(A50)


lim119872rarrinfin

[119881(119877+1119863+1119878+1)

(minus119872) minus 119881(119877119863119878)

(minus119872)] = 0 (A51)

Because of

lim119872rarrinfin

D119881(119877+1119863+1119878+1)


120574(A52)

andD119881(119877119863119878)

(119878 + 1) le 119887 we get

119881(119877+1119863+1119878+1)

(119878 + 1) minus 119881(119877119863119878)

(119878 + 1) gt 0 (A53)

that is 119881(119877+1119863+1119878+1)(119878 + 1) gt 119881(119877119863119878)


119881(119877+1119863+1119878+1)

(119878 + 1) le 119881(119877119863119878)

(119878 + 1) (A54)


D119881(119877lowast

119863lowast

119878lowast

)(119877lowast) gt

12058221199012minus 12058211199011

1205822minus 1205821

gt 119888 ge D119881(119877lowast

119863lowast

119878lowast

)(119863lowast+ 1)

(A55)


(119877lowast

119863lowast

119878lowast

)(119909) we get 119877lowast lt 119863


21199012minus 12058211199011)(1205822minus 1205821) lt 119888


119881(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) = 119881

(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119886 119888) (A56)

where (119877lowast(119888) 119863




119863lowast(119888) lt 119878



(119877lowast

(119888)119863lowast

(119888)119878lowast


follows

ℎ+= minus (120582

2+ 120574)D119881 (119909 119888)


ℎ+= minus (120582

2+ 120583 + 120574)D119881 (119878

lowast(119888) 119888)

+ 1205822D119881 (119878


ℎ+= 120583D119881 (119909 + 1 119888) minus (120583 + 120582

2+ 120574)D119881 (119909 119888)


ℎ+= 120583D119881 (119863

lowast(119888) + 1 119888) minus 120575

2D119881 (119863

lowast(119888) 119888)

+ 1205822D119881 (119863



ℎ+= (120583 + 119886)D119881 (119909 + 1 119888) minus 120575

2D119881 (119909 119888)


ℎ+= (120583 + 119886)D119881 (119877

lowast(119888) + 2 119888) minus 120575

2D119881 (119877

lowast(119888) + 1 119888)

+ 1205821D119881 (119877

lowast(119888) 119888) + 120582

21199012minus 12058211199011

ℎ+= (120583 + 119886)D119881 (119909 + 1 119888) minus 120575

1D119881 (119909 119888)



1D119881 (119909 119888)

+ 1205821D119881 (119909 minus 1 119888) for 119909 le 0

(A57)

And let Δ12(119909 1198881 1198882) = D119881(119909 119888

2) minusD119881(119909 119888

1) and let 120585 =





2) as follows

0 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1)

for 119909 ge 119878lowast (1198881) + 1

0 = minus (1205822+ 120583 + 120574) Δ (119878

lowast(1198881)) + 120582

2Δ (119878lowast(1198881) minus 1)

0 = 120583Δ (119909 + 1) minus (120583 + 1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1)

for 119863lowast (1198881) + 1 le 119909 lt 119878

lowast(1198881)

120585 = 120583Δ (119863lowast(1198881) + 1) minus 120575

2Δ (119863lowast(1198881))


1)

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909) + 120582

2Δ (119909 minus 1)

for 119877lowast (1198881) + 1 lt 119909 lt 119863

lowast(1198881)

0 = (120583 + 119886) Δ (119877lowast(1198881) + 2) minus 120575

2Δ (119877lowast(1198881) + 1)


1) + 1

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)


1)

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 0

(A58)


Δ (119909) = 0 (A59)


D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882)

ge D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881)

(A60)

So we get

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119878lowast(1198881) 1198882)

ge D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119878lowast(1198881) 1198881) gt 119887

(A61)




2) ge 119878


we get 119877lowast(1198882) ge 119877




D119881(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) minus 119888 (A62)


D1198811(119909 119888) = D119881

(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) minus 119888 (A63)


ℎ++ 120574119888 = minus (120582

2+ 120574)D119881

1(119909 119888)


ℎ++ 120574119888 = minus (120582

2+ 120583 + 120574)D119881

1(119878lowast 119888)


ℎ++ 120574119888 = 120583D119881

1(119909 + 1 119888) minus (120583 + 120582

2+ 120574)D119881

1(119909 119888)

+ 1205822D1198811(119909 minus 1 119888)


ℎ++ 120574119888 = 120583D119881

1(119863lowast(119888) + 1 119888) minus 120575

2D1198811(119863lowast(119888) 119888)


ℎ++ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

2D1198811(119909 119888)

+ 1205822D1198811(119909 minus 1 119888)


ℎ++ 120574119888 = (120583 + 119886)D119881

1(119877lowast(119888) + 2 119888)

minus 1205752D1198811(119877lowast(119888) + 1 119888)

+ 1205821D1198811(119877lowast(119888) 119888) + 120582

21199012minus 12058211199011

ℎ++ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

1D1198811(119909 119886)


minusℎminus+ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

1D1198811(119909 119888)

+ 1205821D1198811(119909 minus 1 119888) for 119909 le 0

(A64)


Let

D1198811(119909 1198881) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

D1198811(119909 1198882) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

(A65)

And let

Δ1(119909 1198881 1198882) = D119881

1(119909 1198882) minusD119881

1(119909 1198881) (A66)





1(119909 1198882) as

follows

120578 = minus (1205822+ 120574) Δ

1(119909) + 120582

2Δ1(119909 minus 1)

for 119909 ge 119878lowast (1198881) + 1

120578 = minus (1205822+ 120583 + 120574) Δ

1(119878lowast(1198881))


1)

120578 = 120583Δ1(119909 + 1) minus (120583 + 120582

2+ 120574) Δ

1(119909)


1) + 1 le 119909 lt 119878

lowast(1198881)

120578 = 120583Δ1(119863lowast(1198881) + 1) minus 120575

2Δ1(119863lowast(1198881))


1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

2Δ1(119909)


1) + 1 lt 119909 lt 119863

lowast(1198881)

120578 = (120583 + 119886) Δ1(119877lowast(1198881) + 2) minus 120575

2Δ1(119877lowast(1198881) + 1)


1) + 1

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)


1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)

+ 1205821Δ1(119909 minus 1) for 119909 le 0

(A67)



Δ1(119909) = minus (119888

2minus 1198881) (A68)


1(119909) Let 119872 be a large


(120574 + 1205821) (1198882minus 1198881) = (120583 + 119886) Δ

1(119909 + 1) minus 120575

1Δ1(119909) (A69)






D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

(A70)

Hence we get

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

1) minus 1198881le 0

(A71)

that is

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) le 1198882 (A72)






2) le




2) lt 119881119906(119909 1198881) Then

119881119906lowast

(119909 1198882) = sup119906isinU

119881119906(119909 1198882)

le sup119906isinU

119881119906(119909 1198881) = 119881119906lowast

(119909 1198881)

(A73)





Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













policy then


D119881(119877lowast

119863lowast

119878lowast

)(119878lowast+ 1) le 119887 lt D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast) (16)


D119881(119877lowast

119863lowast

119878lowast

)(119863lowast+ 1) le 119888 lt D119881

(119877lowast

119863lowast

119878lowast

)(119863lowast) (17)


D119881(119877lowast

119863lowast

119878lowast

)(119877lowast+ 1) le

12058221199012minus 12058211199011

1205822minus 1205821

lt D119881(119877lowast

119863lowast

119878lowast

)(119877lowast)

(18)


119863lowast

119878lowast

)(119909) is concave in 119909




119863lowast

119878lowast










(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888)








119870


1205821(1199011minus 119888) lt 120582


119870(119901119870minus 119888) (19)



0 = minus ℎ (119909) +max120572=0120583

120572 [119881 (119909 + 1) minus 119881 (119909) minus 119888]

+max120573=0119886

120583 [119881 (119909 + 1) minus 119881 (119909) minus 119888]

+ max119894=1119870

120582119894[119881 (119909 minus 1) + 119901

119894minus 119881 (119909)] minus 120574119881 (119909)

(20)



Consider (1198772 119877

119870 119863 119878) with 119877


119870lt 119878 and



119870+1= +infin)


For a given (1198772 119877




0 = minus ℎ (119909) + 120583 [D119881 (119909) minus 119887]




+ 120582119894[119901119894minusD119881 (119909)] minus 120574119881 (119909) 119877

119894lt 119909 le 119877

119894+1

(21)


119870 119863 119878) policies



2 119877

119870 119863 119878) policy when


(1) the (1198772 119877


(1198772 119877


D119881(1198772119877119870119863119878)

(119878 + 1) gt 119887

119900119903 D119881(1198772119877119870119863119878+1)

(119878 + 1) gt 119887

(22)

(2) the (1198772 119877


(1198772 119877


D119881(1198772119877119870119863119878)

(119863 + 1) gt 119888

119900119903 D119881(1198772119877119870119863+1119878)

(119863 + 1) gt 119888

(23)

(3) the (1198772 119877


(1198772 119877

119870 119863 119878 + 1) policy if 119878 lt 0

(4) the (1198772 119877

119894+ 1 119877


the (1198772 119877

119894 119877


D119881(1198772119877119894119877119870119863119878)

(119877119894+ 1) gt


120582119894minus 120582119894minus1

(24)

or

D119881(1198772119877119894+1119877

119870119863119878)

(119877119894+ 1) gt


120582119894minus 120582119894minus1

(25)



lowast




2119877lowast

119870119863lowast

119878lowast



D119881(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119878lowast+ 1) le 119887 lt D119881

(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119878lowast)

(26)


D119881(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119863lowast+ 1) le 119888 lt D119881

(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119863lowast)

(27)


D119881(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119877lowast

119894+ 1) le


120582119894minus 120582119894minus1

lt D119881(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119877lowast

119894)

(28)



2(119888) 119877

lowast

119870(119888) 119863



lowast

2(119888) 119877

lowast

119870(119888) 119878lowast(119888) are



2(119888)119877

lowast

119870(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) is decreasing in 119888



Appendix


1+ 120574 120575

2= 120583 + 119886 + 120582

2+ 120574 (A1)


[119881(119877119863119878)

(119909) minus 119881(119877119863119878)


(1205822+ 120574) Δ (119909) = 120582

2Δ (119909 minus 1) (A2)

which leads to

Δ (119909) =1205822+ 120574

1205822

Δ (119909 + 1) = sdot sdot sdot = (1205822+ 120574

1205822

)

119899

Δ (119909 + 119899)

Δ (119909) = (1205822

1205822+ 120574

)

119909minus119878

Δ (119878)

(A3)



lim119909rarrinfin


(1205822

1205822+ 120574

)

119909minus119878

Δ (119878) = 0 (A4)

That is

lim119909rarr+infin

[119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909)] = 0 (A5)



[119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909)] = 0 (A6)



0 = 1205821119879minus1+ (120583 + 119886) 119879 minus 120575

1 (A7)

or

0 = 1205821119910minus1+ (120583 + 119886) 119910 minus 120575

1 (A8)


1199101=


1minus 4 (120583 + 119886) 120582

1

2 (120583 + 119886)

1199102=

120575 + radic1205752

1minus 4 (120583 + 119886) 120582

1

2 (120583 + 119886)

(A9)


2


(119909) = 1198621119910119909

1+ 1198622119910119909

2 (A10)

where 1198621and 119862



increases to infin







11199011

= [1205821119879minus1+ (120583 + 119886) 119879 minus 120575

1] (1198671+ 1198672119909)

(A11)


1198672=ℎminus

120574

1198671=

1

1205742[120574 (12058211199011minus (120583119887 + 119886119888)) + ℎ

minus(120583 + 119886 minus 120582

1)]

(A12)


119881(119877119878119861)

(119909) = 1198622119910119909

2+ℎminus119909

120574+12058211199011minus (120583119887 + 119886119888)

120574

+ℎminus(120583 + 119886 minus 120582

1)

1205742

(A13)


2for 119881(119877119863119878)(119909) it is


Δ (119909) = 119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909) = (1198622minus 1198622) 119910119909

2 (A14)

Then we get


[119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909)] = 0 (A15)



D119881(119877119863119878)

(119909) = minusℎ+120574


D119881(119877119863119878)



[119881(119877119863119878)

(119909) minus

119881(119877119863119878minus1)



119881(119877119863119878)

(119909) = 119875(1205822

1205822+ 120574

)

119909

minusℎ+119909

120574+12058221199012

120574+1205822ℎ+

1205742 (A16)


119881(119877119863119878)

(119909) = 1198622119910119909

2+ℎminus119909

120574+12058211199011minus (120583119887 + 119886119888)

120574

+ℎminus(120583 + 119886 minus 120582

1)

1205742

(A17)


1199102=

1205751+ radic1205752

1minus 4 (120583 + 119886) (120582

1)

2 (120583 + 119886) (A18)


2gt 1 we get

lim119909rarr+infin

D119881(119877119863119878)

(119909) = minusℎ+

120574


D119881(119877119863119878)

(119909) =ℎminus

120574

(A19)


(119878 + 1) gt 119887 orD119881(119877119863119878+1)






Let Δ(119909) = 119881(119877119863119878+1)

(119909) minus 119881(119877119863119878)


0 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1) for 119909 ge 119878 + 1

minus120598 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1) for 119909 = 119878

0 = 120583Δ (119909 + 1) minus (1205822+ 120574 + 120583)Δ (119909)

+ 1205822Δ (119909 minus 1) for 119863 le 119909 lt 119878

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909)

+ 1205822Δ (119909 minus 1) for 119877 le 119909 lt 119863

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 119877

(A20)

where 120598 = 120583[D119881(119877119863119878+1)


that


[119881(119877119863119878+1)

(119909) minus 119881(119877119863119878)

(119909)] = 0 (A21)






minus 120583 [119863119881(119877119863119878)

(119878 + 1) minus 119887]

= 120583Δ (119878 + 1) minus (120583 + 1205822+ 120574) Δ (119878) + 120582

2Δ (119878 minus 1)

(A23)


(119877119863119878)(119878+

1) le 119887 or D119881(119877119863119878+1)


(119909) le 119881(119877119863119878)

(119909)




D119881(119909) = 119881(119877119863119878)

(119909) minus 119881(119877119863119878)



ℎ+= minus 120582

2D2119881 (119909) minus 120574D119881 (119909) for 119909 ge 119878 + 1

ℎ+= minus 120582

2D2119881 (119878) minus (120583 + 120574)D119881 (119878) + 120583119887 for 119909 = 119878

ℎ+= 120583D

2119881 (119909 + 1) minus 120582

2D2119881 (119909)

minus 120574D119881 (119909) for 119863 + 1 le 119909 lt 119878

ℎ+= 120583D119881 (119863 + 1) minus 120575

2D119881 (119863)

+ 1205822D119881 (119863 minus 1) + 119886119888 for 119909 = 119863

ℎ+= (120583 + 119886)D

2119881 (119909 + 1) minus 120582

2D2119881 (119909)


ℎ+= (120583 + 119886)D

2119881 (119877 + 2) minus 120582

1D2119881 (119877 + 1)

minus (1205822minus 1205821+ 120574)D119881 (119877 + 1) + 120582

21199012minus 12058211199011

ℎ+= (120583 + 119886)D

2119881 (119909 + 1) minus 120582

1D2119881 (119909)

minus 120574D119881 (119909) for 1 le 119909 le 119877

minusℎminus= (120583 + 119886)D

2119881 (119909 + 1) minus 120582

1D2119881 (119909)

minus 120574D119881 (119909) for 119909 le 0(A24)


0 = minus (1205822+ 120574)D

2119881 (119909) + 120582

2D2119881 (119909 minus 1) (A25)

Similarly we derive

0 = minus (1205822+ 120574)D

2119881 (119878 + 1) + 120582

2D2119881 (119878)

+ 120583D119881 (119878) minus 120583119887 for 119909 = 119878 + 1(A26)

0 = minus (1205822+ 120574 + 120583)D

2119881 (119878) + 120582

2D2119881 (119878 minus 1)

minus 120583D119881 (119878) + 120583119887 for 119909 = 119878(A27)

0 = 120583D2119881 (119909 + 1) minus (120582

2+ 120574 + 120583)D

2119881 (119909)

+ 1205822D2119881 (119909 minus 1) for 119863 + 2 le 119909 lt 119878

(A28)

1205983= 120583D

2119881 (119863 + 2) minus 120575

2D2119881 (119863 + 1)

+ 1205822D2119881 (119863) for 119909 = 119863 + 1

(A29)

1205984= 120583D

2119881 (119863 + 1) minus 120575

2D2119881 (119863)

+ 1205822D2119881 (119863 minus 1) for 119909 = 119863

(A30)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

2D2119881 (119909)


(A31)


1205985= (120583 + 119886)D

2119881 (119877 + 3) minus 120575

2D2119881 (119877 + 2)

+ 1205821D2119881 (119877 + 1)

(A32)

1205986= (120583 + 119886)D

2119881 (119877 + 2) minus 120575

2D2119881 (119877 + 1)

+ 1205821D2119881 (119877)

(A33)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

1D2119881 (119909)

+ 1205821D2119881 (119909 minus 1) for 2 le 119909 le 119877

(A34)

1205987= (120583 + 119886)D

2119881 (2) minus 120575

1D2119881 (1) + 120582

1D2119881 (0) (A35)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

1D2119881 (119909)

+ 1205821D2119881 (119909 minus 1) for 119909 le 0

(A36)


4= 119886[D119881(119863) minus 119888] 120598

5=

minus(1205822minus1205821)D119881(119877+1)+(120582


(12058221199012minus 12058211199011) and 120598

7= ℎ++ ℎminus


D2119881 (119878) =

ℎ++ (120583 + 120574)D119881 (119878) minus 120583119887

minus1205822

(A37)



(1205822+ 120574)D

2119881 (119878 + 1) = minusℎ

+minus 120574D119881 (119878) (A38)




0 = minus (1205822+ 120574)D

2119881 (119878 + 1) minus (120574 + 120583)D

2119881 (119878)

+ 1205822D2119881 (119878 minus 1)

(A39)


8= minus120583D2119881(119878minus 1) and

for 119909 = 119878 minus 2

1205988= minus (120582

2+ 120574 + 120583)D

2119881 (119878 minus 2) + 120582

2D2119881 (119878 minus 3) (A40)

Of course 1205987

gt 0 and 1205988


119894gt 0 for 119894 = 3 4 5 6


D119881(119877119863119878)







(119877lowast

119863lowast

119878lowast


D119881(119877lowast

119863lowast

119878lowast


119877lowast

D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast) gt

12058221199012minus 12058211199011

1205822minus 1205821

(A41)


D119881(119877lowast

119863lowast

119889lowast

)(119909) le D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast+ 1) le 119887 (A42)


119888 ge D119881(119877lowast

119863lowast

119878lowast

)(119863lowast+ 1)

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast) ge 119887

(A43)


D119881(119877lowast

119863lowast

119878lowast

)(119877lowast+ 1)

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast)

(A44)

which yields

12058221199012minus 12058211199011

1205822minus 1205821

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) gt 119887 (A45)


119863lowast

119878lowast




0(119909) = D119881

(119877119863119878)(119909) for 119909 le 119878 satisfies



2+ 120583 + 120574)D119881

0(119878) + 120582

2D1198810(119878 minus 1)

+ 120583119887 for 119909 = 119878


0(119909 + 1) minus (120583 + 120582

2+ 120574)D119881

0(119909)

+ 1205822D1198810(119909 minus 1) for 119863 + 1 le 119909 lt 119878


0(119863 + 1) minus 120575

2D1198810(119863)

+ 1205822D1198810(119863 minus 1) + 119886119888 for 119909 = 119863


minusℎminus= (120583 + 119886)D119881

0(119909 + 1) minus 120575

2D1198810(119909)

+ 1205822D1198810(119909 minus 1) for 119877 + 1 lt 119909 lt 119863

minusℎminus= (120583 + 119886)D119881

0(119877 + 2) minus 120575

2D1198810(119877 + 1)

+ 1205821D1198810(119877) + 120582

21199012minus 12058211199011 for 119909 = 119877 + 1

minusℎminus= (120583 + 119886)D119881

0(119909 + 1) minus 120575

1D1198810(119909)

+ 1205821D1198810(119909 minus 1) for 119909 le 119877

(A46)

Let Δ(119909) = D1198811(119909 + 1) minus D119881

0(119909) where D119881

1(119909) =

D119881(119877+1119863+1119878+1)


0 = minus (1205822+ 120583 + 120574) Δ (119878) + 120582

2Δ (119878 minus 1) for 119909 = 119878

0 = 120583Δ (119909 + 1) minus (120583 + 1205822+ 120574) Δ (119909)

+ 1205822Δ (119909 minus 1) for 119863 + 1 le 119909 lt 119878

0 = 120583Δ (119863 + 1) minus 1205752Δ (119863) + 120582

2Δ (119863 minus 1) for 119909 = 119863

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909)

+ 1205822Δ (119909 minus 1) for 119877 + 1 lt 119909 lt 119863

0 = (120583 + 119886) Δ (119877 + 2) minus 1205752Δ (119877 + 1)

+ 1205821Δ (119877) for 119909 = 119877 + 1

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 119877

(A47)


minus119872 is




D119881(119877+1119863+1119878+1)

(119909 + 1) = D119881(119877119863119878)

(119909) (A49)


119881(119877+1119863+1119878+1)

(119878 + 1) minus 119881(119877119863119878)

(119878 + 1)

= minusD119881(119877119863119878)

(119878 + 1)

+

119878

sum

119910=minus119872+1

Δ (119910) +D119881(119877+1119863+1119878+1)

(minus119872 + 1)

+ [119881(119877+1119863+1119878+1)

(minus119872) minus 119881(119877119863119878)

(minus119872)]

(A50)


lim119872rarrinfin

[119881(119877+1119863+1119878+1)

(minus119872) minus 119881(119877119863119878)

(minus119872)] = 0 (A51)

Because of

lim119872rarrinfin

D119881(119877+1119863+1119878+1)


120574(A52)

andD119881(119877119863119878)

(119878 + 1) le 119887 we get

119881(119877+1119863+1119878+1)

(119878 + 1) minus 119881(119877119863119878)

(119878 + 1) gt 0 (A53)

that is 119881(119877+1119863+1119878+1)(119878 + 1) gt 119881(119877119863119878)


119881(119877+1119863+1119878+1)

(119878 + 1) le 119881(119877119863119878)

(119878 + 1) (A54)


D119881(119877lowast

119863lowast

119878lowast

)(119877lowast) gt

12058221199012minus 12058211199011

1205822minus 1205821

gt 119888 ge D119881(119877lowast

119863lowast

119878lowast

)(119863lowast+ 1)

(A55)


(119877lowast

119863lowast

119878lowast

)(119909) we get 119877lowast lt 119863


21199012minus 12058211199011)(1205822minus 1205821) lt 119888


119881(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) = 119881

(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119886 119888) (A56)

where (119877lowast(119888) 119863




119863lowast(119888) lt 119878



(119877lowast

(119888)119863lowast

(119888)119878lowast


follows

ℎ+= minus (120582

2+ 120574)D119881 (119909 119888)


ℎ+= minus (120582

2+ 120583 + 120574)D119881 (119878

lowast(119888) 119888)

+ 1205822D119881 (119878


ℎ+= 120583D119881 (119909 + 1 119888) minus (120583 + 120582

2+ 120574)D119881 (119909 119888)


ℎ+= 120583D119881 (119863

lowast(119888) + 1 119888) minus 120575

2D119881 (119863

lowast(119888) 119888)

+ 1205822D119881 (119863



ℎ+= (120583 + 119886)D119881 (119909 + 1 119888) minus 120575

2D119881 (119909 119888)


ℎ+= (120583 + 119886)D119881 (119877

lowast(119888) + 2 119888) minus 120575

2D119881 (119877

lowast(119888) + 1 119888)

+ 1205821D119881 (119877

lowast(119888) 119888) + 120582

21199012minus 12058211199011

ℎ+= (120583 + 119886)D119881 (119909 + 1 119888) minus 120575

1D119881 (119909 119888)



1D119881 (119909 119888)

+ 1205821D119881 (119909 minus 1 119888) for 119909 le 0

(A57)

And let Δ12(119909 1198881 1198882) = D119881(119909 119888

2) minusD119881(119909 119888

1) and let 120585 =





2) as follows

0 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1)

for 119909 ge 119878lowast (1198881) + 1

0 = minus (1205822+ 120583 + 120574) Δ (119878

lowast(1198881)) + 120582

2Δ (119878lowast(1198881) minus 1)

0 = 120583Δ (119909 + 1) minus (120583 + 1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1)

for 119863lowast (1198881) + 1 le 119909 lt 119878

lowast(1198881)

120585 = 120583Δ (119863lowast(1198881) + 1) minus 120575

2Δ (119863lowast(1198881))


1)

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909) + 120582

2Δ (119909 minus 1)

for 119877lowast (1198881) + 1 lt 119909 lt 119863

lowast(1198881)

0 = (120583 + 119886) Δ (119877lowast(1198881) + 2) minus 120575

2Δ (119877lowast(1198881) + 1)


1) + 1

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)


1)

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 0

(A58)


Δ (119909) = 0 (A59)


D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882)

ge D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881)

(A60)

So we get

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119878lowast(1198881) 1198882)

ge D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119878lowast(1198881) 1198881) gt 119887

(A61)




2) ge 119878


we get 119877lowast(1198882) ge 119877




D119881(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) minus 119888 (A62)


D1198811(119909 119888) = D119881

(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) minus 119888 (A63)


ℎ++ 120574119888 = minus (120582

2+ 120574)D119881

1(119909 119888)


ℎ++ 120574119888 = minus (120582

2+ 120583 + 120574)D119881

1(119878lowast 119888)


ℎ++ 120574119888 = 120583D119881

1(119909 + 1 119888) minus (120583 + 120582

2+ 120574)D119881

1(119909 119888)

+ 1205822D1198811(119909 minus 1 119888)


ℎ++ 120574119888 = 120583D119881

1(119863lowast(119888) + 1 119888) minus 120575

2D1198811(119863lowast(119888) 119888)


ℎ++ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

2D1198811(119909 119888)

+ 1205822D1198811(119909 minus 1 119888)


ℎ++ 120574119888 = (120583 + 119886)D119881

1(119877lowast(119888) + 2 119888)

minus 1205752D1198811(119877lowast(119888) + 1 119888)

+ 1205821D1198811(119877lowast(119888) 119888) + 120582

21199012minus 12058211199011

ℎ++ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

1D1198811(119909 119886)


minusℎminus+ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

1D1198811(119909 119888)

+ 1205821D1198811(119909 minus 1 119888) for 119909 le 0

(A64)


Let

D1198811(119909 1198881) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

D1198811(119909 1198882) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

(A65)

And let

Δ1(119909 1198881 1198882) = D119881

1(119909 1198882) minusD119881

1(119909 1198881) (A66)





1(119909 1198882) as

follows

120578 = minus (1205822+ 120574) Δ

1(119909) + 120582

2Δ1(119909 minus 1)

for 119909 ge 119878lowast (1198881) + 1

120578 = minus (1205822+ 120583 + 120574) Δ

1(119878lowast(1198881))


1)

120578 = 120583Δ1(119909 + 1) minus (120583 + 120582

2+ 120574) Δ

1(119909)


1) + 1 le 119909 lt 119878

lowast(1198881)

120578 = 120583Δ1(119863lowast(1198881) + 1) minus 120575

2Δ1(119863lowast(1198881))


1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

2Δ1(119909)


1) + 1 lt 119909 lt 119863

lowast(1198881)

120578 = (120583 + 119886) Δ1(119877lowast(1198881) + 2) minus 120575

2Δ1(119877lowast(1198881) + 1)


1) + 1

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)


1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)

+ 1205821Δ1(119909 minus 1) for 119909 le 0

(A67)



Δ1(119909) = minus (119888

2minus 1198881) (A68)


1(119909) Let 119872 be a large


(120574 + 1205821) (1198882minus 1198881) = (120583 + 119886) Δ

1(119909 + 1) minus 120575

1Δ1(119909) (A69)






D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

(A70)

Hence we get

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

1) minus 1198881le 0

(A71)

that is

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) le 1198882 (A72)






2) le




2) lt 119881119906(119909 1198881) Then

119881119906lowast

(119909 1198882) = sup119906isinU

119881119906(119909 1198882)

le sup119906isinU

119881119906(119909 1198881) = 119881119906lowast

(119909 1198881)

(A73)





Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











+ 120582119894[119901119894minusD119881 (119909)] minus 120574119881 (119909) 119877

119894lt 119909 le 119877

119894+1

(21)


119870 119863 119878) policies



2 119877

119870 119863 119878) policy when


(1) the (1198772 119877


(1198772 119877


D119881(1198772119877119870119863119878)

(119878 + 1) gt 119887

119900119903 D119881(1198772119877119870119863119878+1)

(119878 + 1) gt 119887

(22)

(2) the (1198772 119877


(1198772 119877


D119881(1198772119877119870119863119878)

(119863 + 1) gt 119888

119900119903 D119881(1198772119877119870119863+1119878)

(119863 + 1) gt 119888

(23)

(3) the (1198772 119877


(1198772 119877

119870 119863 119878 + 1) policy if 119878 lt 0

(4) the (1198772 119877

119894+ 1 119877


the (1198772 119877

119894 119877


D119881(1198772119877119894119877119870119863119878)

(119877119894+ 1) gt


120582119894minus 120582119894minus1

(24)

or

D119881(1198772119877119894+1119877

119870119863119878)

(119877119894+ 1) gt


120582119894minus 120582119894minus1

(25)



lowast




2119877lowast

119870119863lowast

119878lowast



D119881(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119878lowast+ 1) le 119887 lt D119881

(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119878lowast)

(26)


D119881(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119863lowast+ 1) le 119888 lt D119881

(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119863lowast)

(27)


D119881(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119877lowast

119894+ 1) le


120582119894minus 120582119894minus1

lt D119881(119877lowast

2119877lowast

119870119863lowast

119878lowast

)(119877lowast

119894)

(28)



2(119888) 119877

lowast

119870(119888) 119863



lowast

2(119888) 119877

lowast

119870(119888) 119878lowast(119888) are



2(119888)119877

lowast

119870(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) is decreasing in 119888



Appendix


1+ 120574 120575

2= 120583 + 119886 + 120582

2+ 120574 (A1)


[119881(119877119863119878)

(119909) minus 119881(119877119863119878)


(1205822+ 120574) Δ (119909) = 120582

2Δ (119909 minus 1) (A2)

which leads to

Δ (119909) =1205822+ 120574

1205822

Δ (119909 + 1) = sdot sdot sdot = (1205822+ 120574

1205822

)

119899

Δ (119909 + 119899)

Δ (119909) = (1205822

1205822+ 120574

)

119909minus119878

Δ (119878)

(A3)



lim119909rarrinfin


(1205822

1205822+ 120574

)

119909minus119878

Δ (119878) = 0 (A4)

That is

lim119909rarr+infin

[119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909)] = 0 (A5)



[119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909)] = 0 (A6)



0 = 1205821119879minus1+ (120583 + 119886) 119879 minus 120575

1 (A7)

or

0 = 1205821119910minus1+ (120583 + 119886) 119910 minus 120575

1 (A8)


1199101=


1minus 4 (120583 + 119886) 120582

1

2 (120583 + 119886)

1199102=

120575 + radic1205752

1minus 4 (120583 + 119886) 120582

1

2 (120583 + 119886)

(A9)


2


(119909) = 1198621119910119909

1+ 1198622119910119909

2 (A10)

where 1198621and 119862



increases to infin







11199011

= [1205821119879minus1+ (120583 + 119886) 119879 minus 120575

1] (1198671+ 1198672119909)

(A11)


1198672=ℎminus

120574

1198671=

1

1205742[120574 (12058211199011minus (120583119887 + 119886119888)) + ℎ

minus(120583 + 119886 minus 120582

1)]

(A12)


119881(119877119878119861)

(119909) = 1198622119910119909

2+ℎminus119909

120574+12058211199011minus (120583119887 + 119886119888)

120574

+ℎminus(120583 + 119886 minus 120582

1)

1205742

(A13)


2for 119881(119877119863119878)(119909) it is


Δ (119909) = 119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909) = (1198622minus 1198622) 119910119909

2 (A14)

Then we get


[119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909)] = 0 (A15)



D119881(119877119863119878)

(119909) = minusℎ+120574


D119881(119877119863119878)



[119881(119877119863119878)

(119909) minus

119881(119877119863119878minus1)



119881(119877119863119878)

(119909) = 119875(1205822

1205822+ 120574

)

119909

minusℎ+119909

120574+12058221199012

120574+1205822ℎ+

1205742 (A16)


119881(119877119863119878)

(119909) = 1198622119910119909

2+ℎminus119909

120574+12058211199011minus (120583119887 + 119886119888)

120574

+ℎminus(120583 + 119886 minus 120582

1)

1205742

(A17)


1199102=

1205751+ radic1205752

1minus 4 (120583 + 119886) (120582

1)

2 (120583 + 119886) (A18)


2gt 1 we get

lim119909rarr+infin

D119881(119877119863119878)

(119909) = minusℎ+

120574


D119881(119877119863119878)

(119909) =ℎminus

120574

(A19)


(119878 + 1) gt 119887 orD119881(119877119863119878+1)






Let Δ(119909) = 119881(119877119863119878+1)

(119909) minus 119881(119877119863119878)


0 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1) for 119909 ge 119878 + 1

minus120598 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1) for 119909 = 119878

0 = 120583Δ (119909 + 1) minus (1205822+ 120574 + 120583)Δ (119909)

+ 1205822Δ (119909 minus 1) for 119863 le 119909 lt 119878

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909)

+ 1205822Δ (119909 minus 1) for 119877 le 119909 lt 119863

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 119877

(A20)

where 120598 = 120583[D119881(119877119863119878+1)


that


[119881(119877119863119878+1)

(119909) minus 119881(119877119863119878)

(119909)] = 0 (A21)






minus 120583 [119863119881(119877119863119878)

(119878 + 1) minus 119887]

= 120583Δ (119878 + 1) minus (120583 + 1205822+ 120574) Δ (119878) + 120582

2Δ (119878 minus 1)

(A23)


(119877119863119878)(119878+

1) le 119887 or D119881(119877119863119878+1)


(119909) le 119881(119877119863119878)

(119909)




D119881(119909) = 119881(119877119863119878)

(119909) minus 119881(119877119863119878)



ℎ+= minus 120582

2D2119881 (119909) minus 120574D119881 (119909) for 119909 ge 119878 + 1

ℎ+= minus 120582

2D2119881 (119878) minus (120583 + 120574)D119881 (119878) + 120583119887 for 119909 = 119878

ℎ+= 120583D

2119881 (119909 + 1) minus 120582

2D2119881 (119909)

minus 120574D119881 (119909) for 119863 + 1 le 119909 lt 119878

ℎ+= 120583D119881 (119863 + 1) minus 120575

2D119881 (119863)

+ 1205822D119881 (119863 minus 1) + 119886119888 for 119909 = 119863

ℎ+= (120583 + 119886)D

2119881 (119909 + 1) minus 120582

2D2119881 (119909)


ℎ+= (120583 + 119886)D

2119881 (119877 + 2) minus 120582

1D2119881 (119877 + 1)

minus (1205822minus 1205821+ 120574)D119881 (119877 + 1) + 120582

21199012minus 12058211199011

ℎ+= (120583 + 119886)D

2119881 (119909 + 1) minus 120582

1D2119881 (119909)

minus 120574D119881 (119909) for 1 le 119909 le 119877

minusℎminus= (120583 + 119886)D

2119881 (119909 + 1) minus 120582

1D2119881 (119909)

minus 120574D119881 (119909) for 119909 le 0(A24)


0 = minus (1205822+ 120574)D

2119881 (119909) + 120582

2D2119881 (119909 minus 1) (A25)

Similarly we derive

0 = minus (1205822+ 120574)D

2119881 (119878 + 1) + 120582

2D2119881 (119878)

+ 120583D119881 (119878) minus 120583119887 for 119909 = 119878 + 1(A26)

0 = minus (1205822+ 120574 + 120583)D

2119881 (119878) + 120582

2D2119881 (119878 minus 1)

minus 120583D119881 (119878) + 120583119887 for 119909 = 119878(A27)

0 = 120583D2119881 (119909 + 1) minus (120582

2+ 120574 + 120583)D

2119881 (119909)

+ 1205822D2119881 (119909 minus 1) for 119863 + 2 le 119909 lt 119878

(A28)

1205983= 120583D

2119881 (119863 + 2) minus 120575

2D2119881 (119863 + 1)

+ 1205822D2119881 (119863) for 119909 = 119863 + 1

(A29)

1205984= 120583D

2119881 (119863 + 1) minus 120575

2D2119881 (119863)

+ 1205822D2119881 (119863 minus 1) for 119909 = 119863

(A30)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

2D2119881 (119909)


(A31)


1205985= (120583 + 119886)D

2119881 (119877 + 3) minus 120575

2D2119881 (119877 + 2)

+ 1205821D2119881 (119877 + 1)

(A32)

1205986= (120583 + 119886)D

2119881 (119877 + 2) minus 120575

2D2119881 (119877 + 1)

+ 1205821D2119881 (119877)

(A33)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

1D2119881 (119909)

+ 1205821D2119881 (119909 minus 1) for 2 le 119909 le 119877

(A34)

1205987= (120583 + 119886)D

2119881 (2) minus 120575

1D2119881 (1) + 120582

1D2119881 (0) (A35)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

1D2119881 (119909)

+ 1205821D2119881 (119909 minus 1) for 119909 le 0

(A36)


4= 119886[D119881(119863) minus 119888] 120598

5=

minus(1205822minus1205821)D119881(119877+1)+(120582


(12058221199012minus 12058211199011) and 120598

7= ℎ++ ℎminus


D2119881 (119878) =

ℎ++ (120583 + 120574)D119881 (119878) minus 120583119887

minus1205822

(A37)



(1205822+ 120574)D

2119881 (119878 + 1) = minusℎ

+minus 120574D119881 (119878) (A38)




0 = minus (1205822+ 120574)D

2119881 (119878 + 1) minus (120574 + 120583)D

2119881 (119878)

+ 1205822D2119881 (119878 minus 1)

(A39)


8= minus120583D2119881(119878minus 1) and

for 119909 = 119878 minus 2

1205988= minus (120582

2+ 120574 + 120583)D

2119881 (119878 minus 2) + 120582

2D2119881 (119878 minus 3) (A40)

Of course 1205987

gt 0 and 1205988


119894gt 0 for 119894 = 3 4 5 6


D119881(119877119863119878)







(119877lowast

119863lowast

119878lowast


D119881(119877lowast

119863lowast

119878lowast


119877lowast

D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast) gt

12058221199012minus 12058211199011

1205822minus 1205821

(A41)


D119881(119877lowast

119863lowast

119889lowast

)(119909) le D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast+ 1) le 119887 (A42)


119888 ge D119881(119877lowast

119863lowast

119878lowast

)(119863lowast+ 1)

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast) ge 119887

(A43)


D119881(119877lowast

119863lowast

119878lowast

)(119877lowast+ 1)

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast)

(A44)

which yields

12058221199012minus 12058211199011

1205822minus 1205821

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) gt 119887 (A45)


119863lowast

119878lowast




0(119909) = D119881

(119877119863119878)(119909) for 119909 le 119878 satisfies



2+ 120583 + 120574)D119881

0(119878) + 120582

2D1198810(119878 minus 1)

+ 120583119887 for 119909 = 119878


0(119909 + 1) minus (120583 + 120582

2+ 120574)D119881

0(119909)

+ 1205822D1198810(119909 minus 1) for 119863 + 1 le 119909 lt 119878


0(119863 + 1) minus 120575

2D1198810(119863)

+ 1205822D1198810(119863 minus 1) + 119886119888 for 119909 = 119863


minusℎminus= (120583 + 119886)D119881

0(119909 + 1) minus 120575

2D1198810(119909)

+ 1205822D1198810(119909 minus 1) for 119877 + 1 lt 119909 lt 119863

minusℎminus= (120583 + 119886)D119881

0(119877 + 2) minus 120575

2D1198810(119877 + 1)

+ 1205821D1198810(119877) + 120582

21199012minus 12058211199011 for 119909 = 119877 + 1

minusℎminus= (120583 + 119886)D119881

0(119909 + 1) minus 120575

1D1198810(119909)

+ 1205821D1198810(119909 minus 1) for 119909 le 119877

(A46)

Let Δ(119909) = D1198811(119909 + 1) minus D119881

0(119909) where D119881

1(119909) =

D119881(119877+1119863+1119878+1)


0 = minus (1205822+ 120583 + 120574) Δ (119878) + 120582

2Δ (119878 minus 1) for 119909 = 119878

0 = 120583Δ (119909 + 1) minus (120583 + 1205822+ 120574) Δ (119909)

+ 1205822Δ (119909 minus 1) for 119863 + 1 le 119909 lt 119878

0 = 120583Δ (119863 + 1) minus 1205752Δ (119863) + 120582

2Δ (119863 minus 1) for 119909 = 119863

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909)

+ 1205822Δ (119909 minus 1) for 119877 + 1 lt 119909 lt 119863

0 = (120583 + 119886) Δ (119877 + 2) minus 1205752Δ (119877 + 1)

+ 1205821Δ (119877) for 119909 = 119877 + 1

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 119877

(A47)


minus119872 is




D119881(119877+1119863+1119878+1)

(119909 + 1) = D119881(119877119863119878)

(119909) (A49)


119881(119877+1119863+1119878+1)

(119878 + 1) minus 119881(119877119863119878)

(119878 + 1)

= minusD119881(119877119863119878)

(119878 + 1)

+

119878

sum

119910=minus119872+1

Δ (119910) +D119881(119877+1119863+1119878+1)

(minus119872 + 1)

+ [119881(119877+1119863+1119878+1)

(minus119872) minus 119881(119877119863119878)

(minus119872)]

(A50)


lim119872rarrinfin

[119881(119877+1119863+1119878+1)

(minus119872) minus 119881(119877119863119878)

(minus119872)] = 0 (A51)

Because of

lim119872rarrinfin

D119881(119877+1119863+1119878+1)


120574(A52)

andD119881(119877119863119878)

(119878 + 1) le 119887 we get

119881(119877+1119863+1119878+1)

(119878 + 1) minus 119881(119877119863119878)

(119878 + 1) gt 0 (A53)

that is 119881(119877+1119863+1119878+1)(119878 + 1) gt 119881(119877119863119878)


119881(119877+1119863+1119878+1)

(119878 + 1) le 119881(119877119863119878)

(119878 + 1) (A54)


D119881(119877lowast

119863lowast

119878lowast

)(119877lowast) gt

12058221199012minus 12058211199011

1205822minus 1205821

gt 119888 ge D119881(119877lowast

119863lowast

119878lowast

)(119863lowast+ 1)

(A55)


(119877lowast

119863lowast

119878lowast

)(119909) we get 119877lowast lt 119863


21199012minus 12058211199011)(1205822minus 1205821) lt 119888


119881(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) = 119881

(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119886 119888) (A56)

where (119877lowast(119888) 119863




119863lowast(119888) lt 119878



(119877lowast

(119888)119863lowast

(119888)119878lowast


follows

ℎ+= minus (120582

2+ 120574)D119881 (119909 119888)


ℎ+= minus (120582

2+ 120583 + 120574)D119881 (119878

lowast(119888) 119888)

+ 1205822D119881 (119878


ℎ+= 120583D119881 (119909 + 1 119888) minus (120583 + 120582

2+ 120574)D119881 (119909 119888)


ℎ+= 120583D119881 (119863

lowast(119888) + 1 119888) minus 120575

2D119881 (119863

lowast(119888) 119888)

+ 1205822D119881 (119863



ℎ+= (120583 + 119886)D119881 (119909 + 1 119888) minus 120575

2D119881 (119909 119888)


ℎ+= (120583 + 119886)D119881 (119877

lowast(119888) + 2 119888) minus 120575

2D119881 (119877

lowast(119888) + 1 119888)

+ 1205821D119881 (119877

lowast(119888) 119888) + 120582

21199012minus 12058211199011

ℎ+= (120583 + 119886)D119881 (119909 + 1 119888) minus 120575

1D119881 (119909 119888)



1D119881 (119909 119888)

+ 1205821D119881 (119909 minus 1 119888) for 119909 le 0

(A57)

And let Δ12(119909 1198881 1198882) = D119881(119909 119888

2) minusD119881(119909 119888

1) and let 120585 =





2) as follows

0 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1)

for 119909 ge 119878lowast (1198881) + 1

0 = minus (1205822+ 120583 + 120574) Δ (119878

lowast(1198881)) + 120582

2Δ (119878lowast(1198881) minus 1)

0 = 120583Δ (119909 + 1) minus (120583 + 1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1)

for 119863lowast (1198881) + 1 le 119909 lt 119878

lowast(1198881)

120585 = 120583Δ (119863lowast(1198881) + 1) minus 120575

2Δ (119863lowast(1198881))


1)

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909) + 120582

2Δ (119909 minus 1)

for 119877lowast (1198881) + 1 lt 119909 lt 119863

lowast(1198881)

0 = (120583 + 119886) Δ (119877lowast(1198881) + 2) minus 120575

2Δ (119877lowast(1198881) + 1)


1) + 1

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)


1)

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 0

(A58)


Δ (119909) = 0 (A59)


D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882)

ge D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881)

(A60)

So we get

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119878lowast(1198881) 1198882)

ge D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119878lowast(1198881) 1198881) gt 119887

(A61)




2) ge 119878


we get 119877lowast(1198882) ge 119877




D119881(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) minus 119888 (A62)


D1198811(119909 119888) = D119881

(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) minus 119888 (A63)


ℎ++ 120574119888 = minus (120582

2+ 120574)D119881

1(119909 119888)


ℎ++ 120574119888 = minus (120582

2+ 120583 + 120574)D119881

1(119878lowast 119888)


ℎ++ 120574119888 = 120583D119881

1(119909 + 1 119888) minus (120583 + 120582

2+ 120574)D119881

1(119909 119888)

+ 1205822D1198811(119909 minus 1 119888)


ℎ++ 120574119888 = 120583D119881

1(119863lowast(119888) + 1 119888) minus 120575

2D1198811(119863lowast(119888) 119888)


ℎ++ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

2D1198811(119909 119888)

+ 1205822D1198811(119909 minus 1 119888)


ℎ++ 120574119888 = (120583 + 119886)D119881

1(119877lowast(119888) + 2 119888)

minus 1205752D1198811(119877lowast(119888) + 1 119888)

+ 1205821D1198811(119877lowast(119888) 119888) + 120582

21199012minus 12058211199011

ℎ++ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

1D1198811(119909 119886)


minusℎminus+ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

1D1198811(119909 119888)

+ 1205821D1198811(119909 minus 1 119888) for 119909 le 0

(A64)


Let

D1198811(119909 1198881) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

D1198811(119909 1198882) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

(A65)

And let

Δ1(119909 1198881 1198882) = D119881

1(119909 1198882) minusD119881

1(119909 1198881) (A66)





1(119909 1198882) as

follows

120578 = minus (1205822+ 120574) Δ

1(119909) + 120582

2Δ1(119909 minus 1)

for 119909 ge 119878lowast (1198881) + 1

120578 = minus (1205822+ 120583 + 120574) Δ

1(119878lowast(1198881))


1)

120578 = 120583Δ1(119909 + 1) minus (120583 + 120582

2+ 120574) Δ

1(119909)


1) + 1 le 119909 lt 119878

lowast(1198881)

120578 = 120583Δ1(119863lowast(1198881) + 1) minus 120575

2Δ1(119863lowast(1198881))


1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

2Δ1(119909)


1) + 1 lt 119909 lt 119863

lowast(1198881)

120578 = (120583 + 119886) Δ1(119877lowast(1198881) + 2) minus 120575

2Δ1(119877lowast(1198881) + 1)


1) + 1

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)


1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)

+ 1205821Δ1(119909 minus 1) for 119909 le 0

(A67)



Δ1(119909) = minus (119888

2minus 1198881) (A68)


1(119909) Let 119872 be a large


(120574 + 1205821) (1198882minus 1198881) = (120583 + 119886) Δ

1(119909 + 1) minus 120575

1Δ1(119909) (A69)






D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

(A70)

Hence we get

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

1) minus 1198881le 0

(A71)

that is

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) le 1198882 (A72)






2) le




2) lt 119881119906(119909 1198881) Then

119881119906lowast

(119909 1198882) = sup119906isinU

119881119906(119909 1198882)

le sup119906isinU

119881119906(119909 1198881) = 119881119906lowast

(119909 1198881)

(A73)





Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











lim119909rarrinfin


(1205822

1205822+ 120574

)

119909minus119878

Δ (119878) = 0 (A4)

That is

lim119909rarr+infin

[119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909)] = 0 (A5)



[119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909)] = 0 (A6)



0 = 1205821119879minus1+ (120583 + 119886) 119879 minus 120575

1 (A7)

or

0 = 1205821119910minus1+ (120583 + 119886) 119910 minus 120575

1 (A8)


1199101=


1minus 4 (120583 + 119886) 120582

1

2 (120583 + 119886)

1199102=

120575 + radic1205752

1minus 4 (120583 + 119886) 120582

1

2 (120583 + 119886)

(A9)


2


(119909) = 1198621119910119909

1+ 1198622119910119909

2 (A10)

where 1198621and 119862



increases to infin







11199011

= [1205821119879minus1+ (120583 + 119886) 119879 minus 120575

1] (1198671+ 1198672119909)

(A11)


1198672=ℎminus

120574

1198671=

1

1205742[120574 (12058211199011minus (120583119887 + 119886119888)) + ℎ

minus(120583 + 119886 minus 120582

1)]

(A12)


119881(119877119878119861)

(119909) = 1198622119910119909

2+ℎminus119909

120574+12058211199011minus (120583119887 + 119886119888)

120574

+ℎminus(120583 + 119886 minus 120582

1)

1205742

(A13)


2for 119881(119877119863119878)(119909) it is


Δ (119909) = 119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909) = (1198622minus 1198622) 119910119909

2 (A14)

Then we get


[119881(119877119863119878)

(119909) minus 119881(119877119863119878minus1)

(119909)] = 0 (A15)



D119881(119877119863119878)

(119909) = minusℎ+120574


D119881(119877119863119878)



[119881(119877119863119878)

(119909) minus

119881(119877119863119878minus1)



119881(119877119863119878)

(119909) = 119875(1205822

1205822+ 120574

)

119909

minusℎ+119909

120574+12058221199012

120574+1205822ℎ+

1205742 (A16)


119881(119877119863119878)

(119909) = 1198622119910119909

2+ℎminus119909

120574+12058211199011minus (120583119887 + 119886119888)

120574

+ℎminus(120583 + 119886 minus 120582

1)

1205742

(A17)


1199102=

1205751+ radic1205752

1minus 4 (120583 + 119886) (120582

1)

2 (120583 + 119886) (A18)


2gt 1 we get

lim119909rarr+infin

D119881(119877119863119878)

(119909) = minusℎ+

120574


D119881(119877119863119878)

(119909) =ℎminus

120574

(A19)


(119878 + 1) gt 119887 orD119881(119877119863119878+1)






Let Δ(119909) = 119881(119877119863119878+1)

(119909) minus 119881(119877119863119878)


0 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1) for 119909 ge 119878 + 1

minus120598 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1) for 119909 = 119878

0 = 120583Δ (119909 + 1) minus (1205822+ 120574 + 120583)Δ (119909)

+ 1205822Δ (119909 minus 1) for 119863 le 119909 lt 119878

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909)

+ 1205822Δ (119909 minus 1) for 119877 le 119909 lt 119863

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 119877

(A20)

where 120598 = 120583[D119881(119877119863119878+1)


that


[119881(119877119863119878+1)

(119909) minus 119881(119877119863119878)

(119909)] = 0 (A21)






minus 120583 [119863119881(119877119863119878)

(119878 + 1) minus 119887]

= 120583Δ (119878 + 1) minus (120583 + 1205822+ 120574) Δ (119878) + 120582

2Δ (119878 minus 1)

(A23)


(119877119863119878)(119878+

1) le 119887 or D119881(119877119863119878+1)


(119909) le 119881(119877119863119878)

(119909)




D119881(119909) = 119881(119877119863119878)

(119909) minus 119881(119877119863119878)



ℎ+= minus 120582

2D2119881 (119909) minus 120574D119881 (119909) for 119909 ge 119878 + 1

ℎ+= minus 120582

2D2119881 (119878) minus (120583 + 120574)D119881 (119878) + 120583119887 for 119909 = 119878

ℎ+= 120583D

2119881 (119909 + 1) minus 120582

2D2119881 (119909)

minus 120574D119881 (119909) for 119863 + 1 le 119909 lt 119878

ℎ+= 120583D119881 (119863 + 1) minus 120575

2D119881 (119863)

+ 1205822D119881 (119863 minus 1) + 119886119888 for 119909 = 119863

ℎ+= (120583 + 119886)D

2119881 (119909 + 1) minus 120582

2D2119881 (119909)


ℎ+= (120583 + 119886)D

2119881 (119877 + 2) minus 120582

1D2119881 (119877 + 1)

minus (1205822minus 1205821+ 120574)D119881 (119877 + 1) + 120582

21199012minus 12058211199011

ℎ+= (120583 + 119886)D

2119881 (119909 + 1) minus 120582

1D2119881 (119909)

minus 120574D119881 (119909) for 1 le 119909 le 119877

minusℎminus= (120583 + 119886)D

2119881 (119909 + 1) minus 120582

1D2119881 (119909)

minus 120574D119881 (119909) for 119909 le 0(A24)


0 = minus (1205822+ 120574)D

2119881 (119909) + 120582

2D2119881 (119909 minus 1) (A25)

Similarly we derive

0 = minus (1205822+ 120574)D

2119881 (119878 + 1) + 120582

2D2119881 (119878)

+ 120583D119881 (119878) minus 120583119887 for 119909 = 119878 + 1(A26)

0 = minus (1205822+ 120574 + 120583)D

2119881 (119878) + 120582

2D2119881 (119878 minus 1)

minus 120583D119881 (119878) + 120583119887 for 119909 = 119878(A27)

0 = 120583D2119881 (119909 + 1) minus (120582

2+ 120574 + 120583)D

2119881 (119909)

+ 1205822D2119881 (119909 minus 1) for 119863 + 2 le 119909 lt 119878

(A28)

1205983= 120583D

2119881 (119863 + 2) minus 120575

2D2119881 (119863 + 1)

+ 1205822D2119881 (119863) for 119909 = 119863 + 1

(A29)

1205984= 120583D

2119881 (119863 + 1) minus 120575

2D2119881 (119863)

+ 1205822D2119881 (119863 minus 1) for 119909 = 119863

(A30)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

2D2119881 (119909)


(A31)


1205985= (120583 + 119886)D

2119881 (119877 + 3) minus 120575

2D2119881 (119877 + 2)

+ 1205821D2119881 (119877 + 1)

(A32)

1205986= (120583 + 119886)D

2119881 (119877 + 2) minus 120575

2D2119881 (119877 + 1)

+ 1205821D2119881 (119877)

(A33)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

1D2119881 (119909)

+ 1205821D2119881 (119909 minus 1) for 2 le 119909 le 119877

(A34)

1205987= (120583 + 119886)D

2119881 (2) minus 120575

1D2119881 (1) + 120582

1D2119881 (0) (A35)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

1D2119881 (119909)

+ 1205821D2119881 (119909 minus 1) for 119909 le 0

(A36)


4= 119886[D119881(119863) minus 119888] 120598

5=

minus(1205822minus1205821)D119881(119877+1)+(120582


(12058221199012minus 12058211199011) and 120598

7= ℎ++ ℎminus


D2119881 (119878) =

ℎ++ (120583 + 120574)D119881 (119878) minus 120583119887

minus1205822

(A37)



(1205822+ 120574)D

2119881 (119878 + 1) = minusℎ

+minus 120574D119881 (119878) (A38)




0 = minus (1205822+ 120574)D

2119881 (119878 + 1) minus (120574 + 120583)D

2119881 (119878)

+ 1205822D2119881 (119878 minus 1)

(A39)


8= minus120583D2119881(119878minus 1) and

for 119909 = 119878 minus 2

1205988= minus (120582

2+ 120574 + 120583)D

2119881 (119878 minus 2) + 120582

2D2119881 (119878 minus 3) (A40)

Of course 1205987

gt 0 and 1205988


119894gt 0 for 119894 = 3 4 5 6


D119881(119877119863119878)







(119877lowast

119863lowast

119878lowast


D119881(119877lowast

119863lowast

119878lowast


119877lowast

D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast) gt

12058221199012minus 12058211199011

1205822minus 1205821

(A41)


D119881(119877lowast

119863lowast

119889lowast

)(119909) le D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast+ 1) le 119887 (A42)


119888 ge D119881(119877lowast

119863lowast

119878lowast

)(119863lowast+ 1)

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast) ge 119887

(A43)


D119881(119877lowast

119863lowast

119878lowast

)(119877lowast+ 1)

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast)

(A44)

which yields

12058221199012minus 12058211199011

1205822minus 1205821

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) gt 119887 (A45)


119863lowast

119878lowast




0(119909) = D119881

(119877119863119878)(119909) for 119909 le 119878 satisfies



2+ 120583 + 120574)D119881

0(119878) + 120582

2D1198810(119878 minus 1)

+ 120583119887 for 119909 = 119878


0(119909 + 1) minus (120583 + 120582

2+ 120574)D119881

0(119909)

+ 1205822D1198810(119909 minus 1) for 119863 + 1 le 119909 lt 119878


0(119863 + 1) minus 120575

2D1198810(119863)

+ 1205822D1198810(119863 minus 1) + 119886119888 for 119909 = 119863


minusℎminus= (120583 + 119886)D119881

0(119909 + 1) minus 120575

2D1198810(119909)

+ 1205822D1198810(119909 minus 1) for 119877 + 1 lt 119909 lt 119863

minusℎminus= (120583 + 119886)D119881

0(119877 + 2) minus 120575

2D1198810(119877 + 1)

+ 1205821D1198810(119877) + 120582

21199012minus 12058211199011 for 119909 = 119877 + 1

minusℎminus= (120583 + 119886)D119881

0(119909 + 1) minus 120575

1D1198810(119909)

+ 1205821D1198810(119909 minus 1) for 119909 le 119877

(A46)

Let Δ(119909) = D1198811(119909 + 1) minus D119881

0(119909) where D119881

1(119909) =

D119881(119877+1119863+1119878+1)


0 = minus (1205822+ 120583 + 120574) Δ (119878) + 120582

2Δ (119878 minus 1) for 119909 = 119878

0 = 120583Δ (119909 + 1) minus (120583 + 1205822+ 120574) Δ (119909)

+ 1205822Δ (119909 minus 1) for 119863 + 1 le 119909 lt 119878

0 = 120583Δ (119863 + 1) minus 1205752Δ (119863) + 120582

2Δ (119863 minus 1) for 119909 = 119863

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909)

+ 1205822Δ (119909 minus 1) for 119877 + 1 lt 119909 lt 119863

0 = (120583 + 119886) Δ (119877 + 2) minus 1205752Δ (119877 + 1)

+ 1205821Δ (119877) for 119909 = 119877 + 1

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 119877

(A47)


minus119872 is




D119881(119877+1119863+1119878+1)

(119909 + 1) = D119881(119877119863119878)

(119909) (A49)


119881(119877+1119863+1119878+1)

(119878 + 1) minus 119881(119877119863119878)

(119878 + 1)

= minusD119881(119877119863119878)

(119878 + 1)

+

119878

sum

119910=minus119872+1

Δ (119910) +D119881(119877+1119863+1119878+1)

(minus119872 + 1)

+ [119881(119877+1119863+1119878+1)

(minus119872) minus 119881(119877119863119878)

(minus119872)]

(A50)


lim119872rarrinfin

[119881(119877+1119863+1119878+1)

(minus119872) minus 119881(119877119863119878)

(minus119872)] = 0 (A51)

Because of

lim119872rarrinfin

D119881(119877+1119863+1119878+1)


120574(A52)

andD119881(119877119863119878)

(119878 + 1) le 119887 we get

119881(119877+1119863+1119878+1)

(119878 + 1) minus 119881(119877119863119878)

(119878 + 1) gt 0 (A53)

that is 119881(119877+1119863+1119878+1)(119878 + 1) gt 119881(119877119863119878)


119881(119877+1119863+1119878+1)

(119878 + 1) le 119881(119877119863119878)

(119878 + 1) (A54)


D119881(119877lowast

119863lowast

119878lowast

)(119877lowast) gt

12058221199012minus 12058211199011

1205822minus 1205821

gt 119888 ge D119881(119877lowast

119863lowast

119878lowast

)(119863lowast+ 1)

(A55)


(119877lowast

119863lowast

119878lowast

)(119909) we get 119877lowast lt 119863


21199012minus 12058211199011)(1205822minus 1205821) lt 119888


119881(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) = 119881

(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119886 119888) (A56)

where (119877lowast(119888) 119863




119863lowast(119888) lt 119878



(119877lowast

(119888)119863lowast

(119888)119878lowast


follows

ℎ+= minus (120582

2+ 120574)D119881 (119909 119888)


ℎ+= minus (120582

2+ 120583 + 120574)D119881 (119878

lowast(119888) 119888)

+ 1205822D119881 (119878


ℎ+= 120583D119881 (119909 + 1 119888) minus (120583 + 120582

2+ 120574)D119881 (119909 119888)


ℎ+= 120583D119881 (119863

lowast(119888) + 1 119888) minus 120575

2D119881 (119863

lowast(119888) 119888)

+ 1205822D119881 (119863



ℎ+= (120583 + 119886)D119881 (119909 + 1 119888) minus 120575

2D119881 (119909 119888)


ℎ+= (120583 + 119886)D119881 (119877

lowast(119888) + 2 119888) minus 120575

2D119881 (119877

lowast(119888) + 1 119888)

+ 1205821D119881 (119877

lowast(119888) 119888) + 120582

21199012minus 12058211199011

ℎ+= (120583 + 119886)D119881 (119909 + 1 119888) minus 120575

1D119881 (119909 119888)



1D119881 (119909 119888)

+ 1205821D119881 (119909 minus 1 119888) for 119909 le 0

(A57)

And let Δ12(119909 1198881 1198882) = D119881(119909 119888

2) minusD119881(119909 119888

1) and let 120585 =





2) as follows

0 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1)

for 119909 ge 119878lowast (1198881) + 1

0 = minus (1205822+ 120583 + 120574) Δ (119878

lowast(1198881)) + 120582

2Δ (119878lowast(1198881) minus 1)

0 = 120583Δ (119909 + 1) minus (120583 + 1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1)

for 119863lowast (1198881) + 1 le 119909 lt 119878

lowast(1198881)

120585 = 120583Δ (119863lowast(1198881) + 1) minus 120575

2Δ (119863lowast(1198881))


1)

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909) + 120582

2Δ (119909 minus 1)

for 119877lowast (1198881) + 1 lt 119909 lt 119863

lowast(1198881)

0 = (120583 + 119886) Δ (119877lowast(1198881) + 2) minus 120575

2Δ (119877lowast(1198881) + 1)


1) + 1

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)


1)

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 0

(A58)


Δ (119909) = 0 (A59)


D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882)

ge D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881)

(A60)

So we get

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119878lowast(1198881) 1198882)

ge D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119878lowast(1198881) 1198881) gt 119887

(A61)




2) ge 119878


we get 119877lowast(1198882) ge 119877




D119881(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) minus 119888 (A62)


D1198811(119909 119888) = D119881

(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) minus 119888 (A63)


ℎ++ 120574119888 = minus (120582

2+ 120574)D119881

1(119909 119888)


ℎ++ 120574119888 = minus (120582

2+ 120583 + 120574)D119881

1(119878lowast 119888)


ℎ++ 120574119888 = 120583D119881

1(119909 + 1 119888) minus (120583 + 120582

2+ 120574)D119881

1(119909 119888)

+ 1205822D1198811(119909 minus 1 119888)


ℎ++ 120574119888 = 120583D119881

1(119863lowast(119888) + 1 119888) minus 120575

2D1198811(119863lowast(119888) 119888)


ℎ++ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

2D1198811(119909 119888)

+ 1205822D1198811(119909 minus 1 119888)


ℎ++ 120574119888 = (120583 + 119886)D119881

1(119877lowast(119888) + 2 119888)

minus 1205752D1198811(119877lowast(119888) + 1 119888)

+ 1205821D1198811(119877lowast(119888) 119888) + 120582

21199012minus 12058211199011

ℎ++ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

1D1198811(119909 119886)


minusℎminus+ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

1D1198811(119909 119888)

+ 1205821D1198811(119909 minus 1 119888) for 119909 le 0

(A64)


Let

D1198811(119909 1198881) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

D1198811(119909 1198882) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

(A65)

And let

Δ1(119909 1198881 1198882) = D119881

1(119909 1198882) minusD119881

1(119909 1198881) (A66)





1(119909 1198882) as

follows

120578 = minus (1205822+ 120574) Δ

1(119909) + 120582

2Δ1(119909 minus 1)

for 119909 ge 119878lowast (1198881) + 1

120578 = minus (1205822+ 120583 + 120574) Δ

1(119878lowast(1198881))


1)

120578 = 120583Δ1(119909 + 1) minus (120583 + 120582

2+ 120574) Δ

1(119909)


1) + 1 le 119909 lt 119878

lowast(1198881)

120578 = 120583Δ1(119863lowast(1198881) + 1) minus 120575

2Δ1(119863lowast(1198881))


1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

2Δ1(119909)


1) + 1 lt 119909 lt 119863

lowast(1198881)

120578 = (120583 + 119886) Δ1(119877lowast(1198881) + 2) minus 120575

2Δ1(119877lowast(1198881) + 1)


1) + 1

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)


1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)

+ 1205821Δ1(119909 minus 1) for 119909 le 0

(A67)



Δ1(119909) = minus (119888

2minus 1198881) (A68)


1(119909) Let 119872 be a large


(120574 + 1205821) (1198882minus 1198881) = (120583 + 119886) Δ

1(119909 + 1) minus 120575

1Δ1(119909) (A69)






D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

(A70)

Hence we get

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

1) minus 1198881le 0

(A71)

that is

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) le 1198882 (A72)






2) le




2) lt 119881119906(119909 1198881) Then

119881119906lowast

(119909 1198882) = sup119906isinU

119881119906(119909 1198882)

le sup119906isinU

119881119906(119909 1198881) = 119881119906lowast

(119909 1198881)

(A73)





Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Let Δ(119909) = 119881(119877119863119878+1)

(119909) minus 119881(119877119863119878)


0 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1) for 119909 ge 119878 + 1

minus120598 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1) for 119909 = 119878

0 = 120583Δ (119909 + 1) minus (1205822+ 120574 + 120583)Δ (119909)

+ 1205822Δ (119909 minus 1) for 119863 le 119909 lt 119878

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909)

+ 1205822Δ (119909 minus 1) for 119877 le 119909 lt 119863

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 119877

(A20)

where 120598 = 120583[D119881(119877119863119878+1)


that


[119881(119877119863119878+1)

(119909) minus 119881(119877119863119878)

(119909)] = 0 (A21)






minus 120583 [119863119881(119877119863119878)

(119878 + 1) minus 119887]

= 120583Δ (119878 + 1) minus (120583 + 1205822+ 120574) Δ (119878) + 120582

2Δ (119878 minus 1)

(A23)


(119877119863119878)(119878+

1) le 119887 or D119881(119877119863119878+1)


(119909) le 119881(119877119863119878)

(119909)




D119881(119909) = 119881(119877119863119878)

(119909) minus 119881(119877119863119878)



ℎ+= minus 120582

2D2119881 (119909) minus 120574D119881 (119909) for 119909 ge 119878 + 1

ℎ+= minus 120582

2D2119881 (119878) minus (120583 + 120574)D119881 (119878) + 120583119887 for 119909 = 119878

ℎ+= 120583D

2119881 (119909 + 1) minus 120582

2D2119881 (119909)

minus 120574D119881 (119909) for 119863 + 1 le 119909 lt 119878

ℎ+= 120583D119881 (119863 + 1) minus 120575

2D119881 (119863)

+ 1205822D119881 (119863 minus 1) + 119886119888 for 119909 = 119863

ℎ+= (120583 + 119886)D

2119881 (119909 + 1) minus 120582

2D2119881 (119909)


ℎ+= (120583 + 119886)D

2119881 (119877 + 2) minus 120582

1D2119881 (119877 + 1)

minus (1205822minus 1205821+ 120574)D119881 (119877 + 1) + 120582

21199012minus 12058211199011

ℎ+= (120583 + 119886)D

2119881 (119909 + 1) minus 120582

1D2119881 (119909)

minus 120574D119881 (119909) for 1 le 119909 le 119877

minusℎminus= (120583 + 119886)D

2119881 (119909 + 1) minus 120582

1D2119881 (119909)

minus 120574D119881 (119909) for 119909 le 0(A24)


0 = minus (1205822+ 120574)D

2119881 (119909) + 120582

2D2119881 (119909 minus 1) (A25)

Similarly we derive

0 = minus (1205822+ 120574)D

2119881 (119878 + 1) + 120582

2D2119881 (119878)

+ 120583D119881 (119878) minus 120583119887 for 119909 = 119878 + 1(A26)

0 = minus (1205822+ 120574 + 120583)D

2119881 (119878) + 120582

2D2119881 (119878 minus 1)

minus 120583D119881 (119878) + 120583119887 for 119909 = 119878(A27)

0 = 120583D2119881 (119909 + 1) minus (120582

2+ 120574 + 120583)D

2119881 (119909)

+ 1205822D2119881 (119909 minus 1) for 119863 + 2 le 119909 lt 119878

(A28)

1205983= 120583D

2119881 (119863 + 2) minus 120575

2D2119881 (119863 + 1)

+ 1205822D2119881 (119863) for 119909 = 119863 + 1

(A29)

1205984= 120583D

2119881 (119863 + 1) minus 120575

2D2119881 (119863)

+ 1205822D2119881 (119863 minus 1) for 119909 = 119863

(A30)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

2D2119881 (119909)


(A31)


1205985= (120583 + 119886)D

2119881 (119877 + 3) minus 120575

2D2119881 (119877 + 2)

+ 1205821D2119881 (119877 + 1)

(A32)

1205986= (120583 + 119886)D

2119881 (119877 + 2) minus 120575

2D2119881 (119877 + 1)

+ 1205821D2119881 (119877)

(A33)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

1D2119881 (119909)

+ 1205821D2119881 (119909 minus 1) for 2 le 119909 le 119877

(A34)

1205987= (120583 + 119886)D

2119881 (2) minus 120575

1D2119881 (1) + 120582

1D2119881 (0) (A35)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

1D2119881 (119909)

+ 1205821D2119881 (119909 minus 1) for 119909 le 0

(A36)


4= 119886[D119881(119863) minus 119888] 120598

5=

minus(1205822minus1205821)D119881(119877+1)+(120582


(12058221199012minus 12058211199011) and 120598

7= ℎ++ ℎminus


D2119881 (119878) =

ℎ++ (120583 + 120574)D119881 (119878) minus 120583119887

minus1205822

(A37)



(1205822+ 120574)D

2119881 (119878 + 1) = minusℎ

+minus 120574D119881 (119878) (A38)




0 = minus (1205822+ 120574)D

2119881 (119878 + 1) minus (120574 + 120583)D

2119881 (119878)

+ 1205822D2119881 (119878 minus 1)

(A39)


8= minus120583D2119881(119878minus 1) and

for 119909 = 119878 minus 2

1205988= minus (120582

2+ 120574 + 120583)D

2119881 (119878 minus 2) + 120582

2D2119881 (119878 minus 3) (A40)

Of course 1205987

gt 0 and 1205988


119894gt 0 for 119894 = 3 4 5 6


D119881(119877119863119878)







(119877lowast

119863lowast

119878lowast


D119881(119877lowast

119863lowast

119878lowast


119877lowast

D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast) gt

12058221199012minus 12058211199011

1205822minus 1205821

(A41)


D119881(119877lowast

119863lowast

119889lowast

)(119909) le D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast+ 1) le 119887 (A42)


119888 ge D119881(119877lowast

119863lowast

119878lowast

)(119863lowast+ 1)

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast) ge 119887

(A43)


D119881(119877lowast

119863lowast

119878lowast

)(119877lowast+ 1)

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast)

(A44)

which yields

12058221199012minus 12058211199011

1205822minus 1205821

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) gt 119887 (A45)


119863lowast

119878lowast




0(119909) = D119881

(119877119863119878)(119909) for 119909 le 119878 satisfies



2+ 120583 + 120574)D119881

0(119878) + 120582

2D1198810(119878 minus 1)

+ 120583119887 for 119909 = 119878


0(119909 + 1) minus (120583 + 120582

2+ 120574)D119881

0(119909)

+ 1205822D1198810(119909 minus 1) for 119863 + 1 le 119909 lt 119878


0(119863 + 1) minus 120575

2D1198810(119863)

+ 1205822D1198810(119863 minus 1) + 119886119888 for 119909 = 119863


minusℎminus= (120583 + 119886)D119881

0(119909 + 1) minus 120575

2D1198810(119909)

+ 1205822D1198810(119909 minus 1) for 119877 + 1 lt 119909 lt 119863

minusℎminus= (120583 + 119886)D119881

0(119877 + 2) minus 120575

2D1198810(119877 + 1)

+ 1205821D1198810(119877) + 120582

21199012minus 12058211199011 for 119909 = 119877 + 1

minusℎminus= (120583 + 119886)D119881

0(119909 + 1) minus 120575

1D1198810(119909)

+ 1205821D1198810(119909 minus 1) for 119909 le 119877

(A46)

Let Δ(119909) = D1198811(119909 + 1) minus D119881

0(119909) where D119881

1(119909) =

D119881(119877+1119863+1119878+1)


0 = minus (1205822+ 120583 + 120574) Δ (119878) + 120582

2Δ (119878 minus 1) for 119909 = 119878

0 = 120583Δ (119909 + 1) minus (120583 + 1205822+ 120574) Δ (119909)

+ 1205822Δ (119909 minus 1) for 119863 + 1 le 119909 lt 119878

0 = 120583Δ (119863 + 1) minus 1205752Δ (119863) + 120582

2Δ (119863 minus 1) for 119909 = 119863

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909)

+ 1205822Δ (119909 minus 1) for 119877 + 1 lt 119909 lt 119863

0 = (120583 + 119886) Δ (119877 + 2) minus 1205752Δ (119877 + 1)

+ 1205821Δ (119877) for 119909 = 119877 + 1

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 119877

(A47)


minus119872 is




D119881(119877+1119863+1119878+1)

(119909 + 1) = D119881(119877119863119878)

(119909) (A49)


119881(119877+1119863+1119878+1)

(119878 + 1) minus 119881(119877119863119878)

(119878 + 1)

= minusD119881(119877119863119878)

(119878 + 1)

+

119878

sum

119910=minus119872+1

Δ (119910) +D119881(119877+1119863+1119878+1)

(minus119872 + 1)

+ [119881(119877+1119863+1119878+1)

(minus119872) minus 119881(119877119863119878)

(minus119872)]

(A50)


lim119872rarrinfin

[119881(119877+1119863+1119878+1)

(minus119872) minus 119881(119877119863119878)

(minus119872)] = 0 (A51)

Because of

lim119872rarrinfin

D119881(119877+1119863+1119878+1)


120574(A52)

andD119881(119877119863119878)

(119878 + 1) le 119887 we get

119881(119877+1119863+1119878+1)

(119878 + 1) minus 119881(119877119863119878)

(119878 + 1) gt 0 (A53)

that is 119881(119877+1119863+1119878+1)(119878 + 1) gt 119881(119877119863119878)


119881(119877+1119863+1119878+1)

(119878 + 1) le 119881(119877119863119878)

(119878 + 1) (A54)


D119881(119877lowast

119863lowast

119878lowast

)(119877lowast) gt

12058221199012minus 12058211199011

1205822minus 1205821

gt 119888 ge D119881(119877lowast

119863lowast

119878lowast

)(119863lowast+ 1)

(A55)


(119877lowast

119863lowast

119878lowast

)(119909) we get 119877lowast lt 119863


21199012minus 12058211199011)(1205822minus 1205821) lt 119888


119881(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) = 119881

(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119886 119888) (A56)

where (119877lowast(119888) 119863




119863lowast(119888) lt 119878



(119877lowast

(119888)119863lowast

(119888)119878lowast


follows

ℎ+= minus (120582

2+ 120574)D119881 (119909 119888)


ℎ+= minus (120582

2+ 120583 + 120574)D119881 (119878

lowast(119888) 119888)

+ 1205822D119881 (119878


ℎ+= 120583D119881 (119909 + 1 119888) minus (120583 + 120582

2+ 120574)D119881 (119909 119888)


ℎ+= 120583D119881 (119863

lowast(119888) + 1 119888) minus 120575

2D119881 (119863

lowast(119888) 119888)

+ 1205822D119881 (119863



ℎ+= (120583 + 119886)D119881 (119909 + 1 119888) minus 120575

2D119881 (119909 119888)


ℎ+= (120583 + 119886)D119881 (119877

lowast(119888) + 2 119888) minus 120575

2D119881 (119877

lowast(119888) + 1 119888)

+ 1205821D119881 (119877

lowast(119888) 119888) + 120582

21199012minus 12058211199011

ℎ+= (120583 + 119886)D119881 (119909 + 1 119888) minus 120575

1D119881 (119909 119888)



1D119881 (119909 119888)

+ 1205821D119881 (119909 minus 1 119888) for 119909 le 0

(A57)

And let Δ12(119909 1198881 1198882) = D119881(119909 119888

2) minusD119881(119909 119888

1) and let 120585 =





2) as follows

0 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1)

for 119909 ge 119878lowast (1198881) + 1

0 = minus (1205822+ 120583 + 120574) Δ (119878

lowast(1198881)) + 120582

2Δ (119878lowast(1198881) minus 1)

0 = 120583Δ (119909 + 1) minus (120583 + 1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1)

for 119863lowast (1198881) + 1 le 119909 lt 119878

lowast(1198881)

120585 = 120583Δ (119863lowast(1198881) + 1) minus 120575

2Δ (119863lowast(1198881))


1)

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909) + 120582

2Δ (119909 minus 1)

for 119877lowast (1198881) + 1 lt 119909 lt 119863

lowast(1198881)

0 = (120583 + 119886) Δ (119877lowast(1198881) + 2) minus 120575

2Δ (119877lowast(1198881) + 1)


1) + 1

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)


1)

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 0

(A58)


Δ (119909) = 0 (A59)


D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882)

ge D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881)

(A60)

So we get

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119878lowast(1198881) 1198882)

ge D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119878lowast(1198881) 1198881) gt 119887

(A61)




2) ge 119878


we get 119877lowast(1198882) ge 119877




D119881(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) minus 119888 (A62)


D1198811(119909 119888) = D119881

(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) minus 119888 (A63)


ℎ++ 120574119888 = minus (120582

2+ 120574)D119881

1(119909 119888)


ℎ++ 120574119888 = minus (120582

2+ 120583 + 120574)D119881

1(119878lowast 119888)


ℎ++ 120574119888 = 120583D119881

1(119909 + 1 119888) minus (120583 + 120582

2+ 120574)D119881

1(119909 119888)

+ 1205822D1198811(119909 minus 1 119888)


ℎ++ 120574119888 = 120583D119881

1(119863lowast(119888) + 1 119888) minus 120575

2D1198811(119863lowast(119888) 119888)


ℎ++ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

2D1198811(119909 119888)

+ 1205822D1198811(119909 minus 1 119888)


ℎ++ 120574119888 = (120583 + 119886)D119881

1(119877lowast(119888) + 2 119888)

minus 1205752D1198811(119877lowast(119888) + 1 119888)

+ 1205821D1198811(119877lowast(119888) 119888) + 120582

21199012minus 12058211199011

ℎ++ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

1D1198811(119909 119886)


minusℎminus+ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

1D1198811(119909 119888)

+ 1205821D1198811(119909 minus 1 119888) for 119909 le 0

(A64)


Let

D1198811(119909 1198881) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

D1198811(119909 1198882) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

(A65)

And let

Δ1(119909 1198881 1198882) = D119881

1(119909 1198882) minusD119881

1(119909 1198881) (A66)





1(119909 1198882) as

follows

120578 = minus (1205822+ 120574) Δ

1(119909) + 120582

2Δ1(119909 minus 1)

for 119909 ge 119878lowast (1198881) + 1

120578 = minus (1205822+ 120583 + 120574) Δ

1(119878lowast(1198881))


1)

120578 = 120583Δ1(119909 + 1) minus (120583 + 120582

2+ 120574) Δ

1(119909)


1) + 1 le 119909 lt 119878

lowast(1198881)

120578 = 120583Δ1(119863lowast(1198881) + 1) minus 120575

2Δ1(119863lowast(1198881))


1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

2Δ1(119909)


1) + 1 lt 119909 lt 119863

lowast(1198881)

120578 = (120583 + 119886) Δ1(119877lowast(1198881) + 2) minus 120575

2Δ1(119877lowast(1198881) + 1)


1) + 1

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)


1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)

+ 1205821Δ1(119909 minus 1) for 119909 le 0

(A67)



Δ1(119909) = minus (119888

2minus 1198881) (A68)


1(119909) Let 119872 be a large


(120574 + 1205821) (1198882minus 1198881) = (120583 + 119886) Δ

1(119909 + 1) minus 120575

1Δ1(119909) (A69)






D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

(A70)

Hence we get

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

1) minus 1198881le 0

(A71)

that is

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) le 1198882 (A72)






2) le




2) lt 119881119906(119909 1198881) Then

119881119906lowast

(119909 1198882) = sup119906isinU

119881119906(119909 1198882)

le sup119906isinU

119881119906(119909 1198881) = 119881119906lowast

(119909 1198881)

(A73)





Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1205985= (120583 + 119886)D

2119881 (119877 + 3) minus 120575

2D2119881 (119877 + 2)

+ 1205821D2119881 (119877 + 1)

(A32)

1205986= (120583 + 119886)D

2119881 (119877 + 2) minus 120575

2D2119881 (119877 + 1)

+ 1205821D2119881 (119877)

(A33)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

1D2119881 (119909)

+ 1205821D2119881 (119909 minus 1) for 2 le 119909 le 119877

(A34)

1205987= (120583 + 119886)D

2119881 (2) minus 120575

1D2119881 (1) + 120582

1D2119881 (0) (A35)

0 = (120583 + 119886)D2119881 (119909 + 1) minus 120575

1D2119881 (119909)

+ 1205821D2119881 (119909 minus 1) for 119909 le 0

(A36)


4= 119886[D119881(119863) minus 119888] 120598

5=

minus(1205822minus1205821)D119881(119877+1)+(120582


(12058221199012minus 12058211199011) and 120598

7= ℎ++ ℎminus


D2119881 (119878) =

ℎ++ (120583 + 120574)D119881 (119878) minus 120583119887

minus1205822

(A37)



(1205822+ 120574)D

2119881 (119878 + 1) = minusℎ

+minus 120574D119881 (119878) (A38)




0 = minus (1205822+ 120574)D

2119881 (119878 + 1) minus (120574 + 120583)D

2119881 (119878)

+ 1205822D2119881 (119878 minus 1)

(A39)


8= minus120583D2119881(119878minus 1) and

for 119909 = 119878 minus 2

1205988= minus (120582

2+ 120574 + 120583)D

2119881 (119878 minus 2) + 120582

2D2119881 (119878 minus 3) (A40)

Of course 1205987

gt 0 and 1205988


119894gt 0 for 119894 = 3 4 5 6


D119881(119877119863119878)







(119877lowast

119863lowast

119878lowast


D119881(119877lowast

119863lowast

119878lowast


119877lowast

D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast) gt

12058221199012minus 12058211199011

1205822minus 1205821

(A41)


D119881(119877lowast

119863lowast

119889lowast

)(119909) le D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast+ 1) le 119887 (A42)


119888 ge D119881(119877lowast

119863lowast

119878lowast

)(119863lowast+ 1)

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast) ge 119887

(A43)


D119881(119877lowast

119863lowast

119878lowast

)(119877lowast+ 1)

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) ge D119881

(119877lowast

119863lowast

119878lowast

)(119878lowast)

(A44)

which yields

12058221199012minus 12058211199011

1205822minus 1205821

ge D119881(119877lowast

119863lowast

119878lowast

)(119909) gt 119887 (A45)


119863lowast

119878lowast




0(119909) = D119881

(119877119863119878)(119909) for 119909 le 119878 satisfies



2+ 120583 + 120574)D119881

0(119878) + 120582

2D1198810(119878 minus 1)

+ 120583119887 for 119909 = 119878


0(119909 + 1) minus (120583 + 120582

2+ 120574)D119881

0(119909)

+ 1205822D1198810(119909 minus 1) for 119863 + 1 le 119909 lt 119878


0(119863 + 1) minus 120575

2D1198810(119863)

+ 1205822D1198810(119863 minus 1) + 119886119888 for 119909 = 119863


minusℎminus= (120583 + 119886)D119881

0(119909 + 1) minus 120575

2D1198810(119909)

+ 1205822D1198810(119909 minus 1) for 119877 + 1 lt 119909 lt 119863

minusℎminus= (120583 + 119886)D119881

0(119877 + 2) minus 120575

2D1198810(119877 + 1)

+ 1205821D1198810(119877) + 120582

21199012minus 12058211199011 for 119909 = 119877 + 1

minusℎminus= (120583 + 119886)D119881

0(119909 + 1) minus 120575

1D1198810(119909)

+ 1205821D1198810(119909 minus 1) for 119909 le 119877

(A46)

Let Δ(119909) = D1198811(119909 + 1) minus D119881

0(119909) where D119881

1(119909) =

D119881(119877+1119863+1119878+1)


0 = minus (1205822+ 120583 + 120574) Δ (119878) + 120582

2Δ (119878 minus 1) for 119909 = 119878

0 = 120583Δ (119909 + 1) minus (120583 + 1205822+ 120574) Δ (119909)

+ 1205822Δ (119909 minus 1) for 119863 + 1 le 119909 lt 119878

0 = 120583Δ (119863 + 1) minus 1205752Δ (119863) + 120582

2Δ (119863 minus 1) for 119909 = 119863

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909)

+ 1205822Δ (119909 minus 1) for 119877 + 1 lt 119909 lt 119863

0 = (120583 + 119886) Δ (119877 + 2) minus 1205752Δ (119877 + 1)

+ 1205821Δ (119877) for 119909 = 119877 + 1

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 119877

(A47)


minus119872 is




D119881(119877+1119863+1119878+1)

(119909 + 1) = D119881(119877119863119878)

(119909) (A49)


119881(119877+1119863+1119878+1)

(119878 + 1) minus 119881(119877119863119878)

(119878 + 1)

= minusD119881(119877119863119878)

(119878 + 1)

+

119878

sum

119910=minus119872+1

Δ (119910) +D119881(119877+1119863+1119878+1)

(minus119872 + 1)

+ [119881(119877+1119863+1119878+1)

(minus119872) minus 119881(119877119863119878)

(minus119872)]

(A50)


lim119872rarrinfin

[119881(119877+1119863+1119878+1)

(minus119872) minus 119881(119877119863119878)

(minus119872)] = 0 (A51)

Because of

lim119872rarrinfin

D119881(119877+1119863+1119878+1)


120574(A52)

andD119881(119877119863119878)

(119878 + 1) le 119887 we get

119881(119877+1119863+1119878+1)

(119878 + 1) minus 119881(119877119863119878)

(119878 + 1) gt 0 (A53)

that is 119881(119877+1119863+1119878+1)(119878 + 1) gt 119881(119877119863119878)


119881(119877+1119863+1119878+1)

(119878 + 1) le 119881(119877119863119878)

(119878 + 1) (A54)


D119881(119877lowast

119863lowast

119878lowast

)(119877lowast) gt

12058221199012minus 12058211199011

1205822minus 1205821

gt 119888 ge D119881(119877lowast

119863lowast

119878lowast

)(119863lowast+ 1)

(A55)


(119877lowast

119863lowast

119878lowast

)(119909) we get 119877lowast lt 119863


21199012minus 12058211199011)(1205822minus 1205821) lt 119888


119881(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) = 119881

(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119886 119888) (A56)

where (119877lowast(119888) 119863




119863lowast(119888) lt 119878



(119877lowast

(119888)119863lowast

(119888)119878lowast


follows

ℎ+= minus (120582

2+ 120574)D119881 (119909 119888)


ℎ+= minus (120582

2+ 120583 + 120574)D119881 (119878

lowast(119888) 119888)

+ 1205822D119881 (119878


ℎ+= 120583D119881 (119909 + 1 119888) minus (120583 + 120582

2+ 120574)D119881 (119909 119888)


ℎ+= 120583D119881 (119863

lowast(119888) + 1 119888) minus 120575

2D119881 (119863

lowast(119888) 119888)

+ 1205822D119881 (119863



ℎ+= (120583 + 119886)D119881 (119909 + 1 119888) minus 120575

2D119881 (119909 119888)


ℎ+= (120583 + 119886)D119881 (119877

lowast(119888) + 2 119888) minus 120575

2D119881 (119877

lowast(119888) + 1 119888)

+ 1205821D119881 (119877

lowast(119888) 119888) + 120582

21199012minus 12058211199011

ℎ+= (120583 + 119886)D119881 (119909 + 1 119888) minus 120575

1D119881 (119909 119888)



1D119881 (119909 119888)

+ 1205821D119881 (119909 minus 1 119888) for 119909 le 0

(A57)

And let Δ12(119909 1198881 1198882) = D119881(119909 119888

2) minusD119881(119909 119888

1) and let 120585 =





2) as follows

0 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1)

for 119909 ge 119878lowast (1198881) + 1

0 = minus (1205822+ 120583 + 120574) Δ (119878

lowast(1198881)) + 120582

2Δ (119878lowast(1198881) minus 1)

0 = 120583Δ (119909 + 1) minus (120583 + 1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1)

for 119863lowast (1198881) + 1 le 119909 lt 119878

lowast(1198881)

120585 = 120583Δ (119863lowast(1198881) + 1) minus 120575

2Δ (119863lowast(1198881))


1)

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909) + 120582

2Δ (119909 minus 1)

for 119877lowast (1198881) + 1 lt 119909 lt 119863

lowast(1198881)

0 = (120583 + 119886) Δ (119877lowast(1198881) + 2) minus 120575

2Δ (119877lowast(1198881) + 1)


1) + 1

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)


1)

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 0

(A58)


Δ (119909) = 0 (A59)


D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882)

ge D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881)

(A60)

So we get

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119878lowast(1198881) 1198882)

ge D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119878lowast(1198881) 1198881) gt 119887

(A61)




2) ge 119878


we get 119877lowast(1198882) ge 119877




D119881(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) minus 119888 (A62)


D1198811(119909 119888) = D119881

(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) minus 119888 (A63)


ℎ++ 120574119888 = minus (120582

2+ 120574)D119881

1(119909 119888)


ℎ++ 120574119888 = minus (120582

2+ 120583 + 120574)D119881

1(119878lowast 119888)


ℎ++ 120574119888 = 120583D119881

1(119909 + 1 119888) minus (120583 + 120582

2+ 120574)D119881

1(119909 119888)

+ 1205822D1198811(119909 minus 1 119888)


ℎ++ 120574119888 = 120583D119881

1(119863lowast(119888) + 1 119888) minus 120575

2D1198811(119863lowast(119888) 119888)


ℎ++ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

2D1198811(119909 119888)

+ 1205822D1198811(119909 minus 1 119888)


ℎ++ 120574119888 = (120583 + 119886)D119881

1(119877lowast(119888) + 2 119888)

minus 1205752D1198811(119877lowast(119888) + 1 119888)

+ 1205821D1198811(119877lowast(119888) 119888) + 120582

21199012minus 12058211199011

ℎ++ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

1D1198811(119909 119886)


minusℎminus+ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

1D1198811(119909 119888)

+ 1205821D1198811(119909 minus 1 119888) for 119909 le 0

(A64)


Let

D1198811(119909 1198881) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

D1198811(119909 1198882) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

(A65)

And let

Δ1(119909 1198881 1198882) = D119881

1(119909 1198882) minusD119881

1(119909 1198881) (A66)





1(119909 1198882) as

follows

120578 = minus (1205822+ 120574) Δ

1(119909) + 120582

2Δ1(119909 minus 1)

for 119909 ge 119878lowast (1198881) + 1

120578 = minus (1205822+ 120583 + 120574) Δ

1(119878lowast(1198881))


1)

120578 = 120583Δ1(119909 + 1) minus (120583 + 120582

2+ 120574) Δ

1(119909)


1) + 1 le 119909 lt 119878

lowast(1198881)

120578 = 120583Δ1(119863lowast(1198881) + 1) minus 120575

2Δ1(119863lowast(1198881))


1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

2Δ1(119909)


1) + 1 lt 119909 lt 119863

lowast(1198881)

120578 = (120583 + 119886) Δ1(119877lowast(1198881) + 2) minus 120575

2Δ1(119877lowast(1198881) + 1)


1) + 1

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)


1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)

+ 1205821Δ1(119909 minus 1) for 119909 le 0

(A67)



Δ1(119909) = minus (119888

2minus 1198881) (A68)


1(119909) Let 119872 be a large


(120574 + 1205821) (1198882minus 1198881) = (120583 + 119886) Δ

1(119909 + 1) minus 120575

1Δ1(119909) (A69)






D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

(A70)

Hence we get

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

1) minus 1198881le 0

(A71)

that is

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) le 1198882 (A72)






2) le




2) lt 119881119906(119909 1198881) Then

119881119906lowast

(119909 1198882) = sup119906isinU

119881119906(119909 1198882)

le sup119906isinU

119881119906(119909 1198881) = 119881119906lowast

(119909 1198881)

(A73)





Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minusℎminus= (120583 + 119886)D119881

0(119909 + 1) minus 120575

2D1198810(119909)

+ 1205822D1198810(119909 minus 1) for 119877 + 1 lt 119909 lt 119863

minusℎminus= (120583 + 119886)D119881

0(119877 + 2) minus 120575

2D1198810(119877 + 1)

+ 1205821D1198810(119877) + 120582

21199012minus 12058211199011 for 119909 = 119877 + 1

minusℎminus= (120583 + 119886)D119881

0(119909 + 1) minus 120575

1D1198810(119909)

+ 1205821D1198810(119909 minus 1) for 119909 le 119877

(A46)

Let Δ(119909) = D1198811(119909 + 1) minus D119881

0(119909) where D119881

1(119909) =

D119881(119877+1119863+1119878+1)


0 = minus (1205822+ 120583 + 120574) Δ (119878) + 120582

2Δ (119878 minus 1) for 119909 = 119878

0 = 120583Δ (119909 + 1) minus (120583 + 1205822+ 120574) Δ (119909)

+ 1205822Δ (119909 minus 1) for 119863 + 1 le 119909 lt 119878

0 = 120583Δ (119863 + 1) minus 1205752Δ (119863) + 120582

2Δ (119863 minus 1) for 119909 = 119863

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909)

+ 1205822Δ (119909 minus 1) for 119877 + 1 lt 119909 lt 119863

0 = (120583 + 119886) Δ (119877 + 2) minus 1205752Δ (119877 + 1)

+ 1205821Δ (119877) for 119909 = 119877 + 1

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 119877

(A47)


minus119872 is




D119881(119877+1119863+1119878+1)

(119909 + 1) = D119881(119877119863119878)

(119909) (A49)


119881(119877+1119863+1119878+1)

(119878 + 1) minus 119881(119877119863119878)

(119878 + 1)

= minusD119881(119877119863119878)

(119878 + 1)

+

119878

sum

119910=minus119872+1

Δ (119910) +D119881(119877+1119863+1119878+1)

(minus119872 + 1)

+ [119881(119877+1119863+1119878+1)

(minus119872) minus 119881(119877119863119878)

(minus119872)]

(A50)


lim119872rarrinfin

[119881(119877+1119863+1119878+1)

(minus119872) minus 119881(119877119863119878)

(minus119872)] = 0 (A51)

Because of

lim119872rarrinfin

D119881(119877+1119863+1119878+1)


120574(A52)

andD119881(119877119863119878)

(119878 + 1) le 119887 we get

119881(119877+1119863+1119878+1)

(119878 + 1) minus 119881(119877119863119878)

(119878 + 1) gt 0 (A53)

that is 119881(119877+1119863+1119878+1)(119878 + 1) gt 119881(119877119863119878)


119881(119877+1119863+1119878+1)

(119878 + 1) le 119881(119877119863119878)

(119878 + 1) (A54)


D119881(119877lowast

119863lowast

119878lowast

)(119877lowast) gt

12058221199012minus 12058211199011

1205822minus 1205821

gt 119888 ge D119881(119877lowast

119863lowast

119878lowast

)(119863lowast+ 1)

(A55)


(119877lowast

119863lowast

119878lowast

)(119909) we get 119877lowast lt 119863


21199012minus 12058211199011)(1205822minus 1205821) lt 119888


119881(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) = 119881

(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119886 119888) (A56)

where (119877lowast(119888) 119863




119863lowast(119888) lt 119878



(119877lowast

(119888)119863lowast

(119888)119878lowast


follows

ℎ+= minus (120582

2+ 120574)D119881 (119909 119888)


ℎ+= minus (120582

2+ 120583 + 120574)D119881 (119878

lowast(119888) 119888)

+ 1205822D119881 (119878


ℎ+= 120583D119881 (119909 + 1 119888) minus (120583 + 120582

2+ 120574)D119881 (119909 119888)


ℎ+= 120583D119881 (119863

lowast(119888) + 1 119888) minus 120575

2D119881 (119863

lowast(119888) 119888)

+ 1205822D119881 (119863



ℎ+= (120583 + 119886)D119881 (119909 + 1 119888) minus 120575

2D119881 (119909 119888)


ℎ+= (120583 + 119886)D119881 (119877

lowast(119888) + 2 119888) minus 120575

2D119881 (119877

lowast(119888) + 1 119888)

+ 1205821D119881 (119877

lowast(119888) 119888) + 120582

21199012minus 12058211199011

ℎ+= (120583 + 119886)D119881 (119909 + 1 119888) minus 120575

1D119881 (119909 119888)



1D119881 (119909 119888)

+ 1205821D119881 (119909 minus 1 119888) for 119909 le 0

(A57)

And let Δ12(119909 1198881 1198882) = D119881(119909 119888

2) minusD119881(119909 119888

1) and let 120585 =





2) as follows

0 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1)

for 119909 ge 119878lowast (1198881) + 1

0 = minus (1205822+ 120583 + 120574) Δ (119878

lowast(1198881)) + 120582

2Δ (119878lowast(1198881) minus 1)

0 = 120583Δ (119909 + 1) minus (120583 + 1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1)

for 119863lowast (1198881) + 1 le 119909 lt 119878

lowast(1198881)

120585 = 120583Δ (119863lowast(1198881) + 1) minus 120575

2Δ (119863lowast(1198881))


1)

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909) + 120582

2Δ (119909 minus 1)

for 119877lowast (1198881) + 1 lt 119909 lt 119863

lowast(1198881)

0 = (120583 + 119886) Δ (119877lowast(1198881) + 2) minus 120575

2Δ (119877lowast(1198881) + 1)


1) + 1

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)


1)

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 0

(A58)


Δ (119909) = 0 (A59)


D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882)

ge D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881)

(A60)

So we get

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119878lowast(1198881) 1198882)

ge D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119878lowast(1198881) 1198881) gt 119887

(A61)




2) ge 119878


we get 119877lowast(1198882) ge 119877




D119881(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) minus 119888 (A62)


D1198811(119909 119888) = D119881

(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) minus 119888 (A63)


ℎ++ 120574119888 = minus (120582

2+ 120574)D119881

1(119909 119888)


ℎ++ 120574119888 = minus (120582

2+ 120583 + 120574)D119881

1(119878lowast 119888)


ℎ++ 120574119888 = 120583D119881

1(119909 + 1 119888) minus (120583 + 120582

2+ 120574)D119881

1(119909 119888)

+ 1205822D1198811(119909 minus 1 119888)


ℎ++ 120574119888 = 120583D119881

1(119863lowast(119888) + 1 119888) minus 120575

2D1198811(119863lowast(119888) 119888)


ℎ++ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

2D1198811(119909 119888)

+ 1205822D1198811(119909 minus 1 119888)


ℎ++ 120574119888 = (120583 + 119886)D119881

1(119877lowast(119888) + 2 119888)

minus 1205752D1198811(119877lowast(119888) + 1 119888)

+ 1205821D1198811(119877lowast(119888) 119888) + 120582

21199012minus 12058211199011

ℎ++ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

1D1198811(119909 119886)


minusℎminus+ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

1D1198811(119909 119888)

+ 1205821D1198811(119909 minus 1 119888) for 119909 le 0

(A64)


Let

D1198811(119909 1198881) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

D1198811(119909 1198882) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

(A65)

And let

Δ1(119909 1198881 1198882) = D119881

1(119909 1198882) minusD119881

1(119909 1198881) (A66)





1(119909 1198882) as

follows

120578 = minus (1205822+ 120574) Δ

1(119909) + 120582

2Δ1(119909 minus 1)

for 119909 ge 119878lowast (1198881) + 1

120578 = minus (1205822+ 120583 + 120574) Δ

1(119878lowast(1198881))


1)

120578 = 120583Δ1(119909 + 1) minus (120583 + 120582

2+ 120574) Δ

1(119909)


1) + 1 le 119909 lt 119878

lowast(1198881)

120578 = 120583Δ1(119863lowast(1198881) + 1) minus 120575

2Δ1(119863lowast(1198881))


1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

2Δ1(119909)


1) + 1 lt 119909 lt 119863

lowast(1198881)

120578 = (120583 + 119886) Δ1(119877lowast(1198881) + 2) minus 120575

2Δ1(119877lowast(1198881) + 1)


1) + 1

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)


1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)

+ 1205821Δ1(119909 minus 1) for 119909 le 0

(A67)



Δ1(119909) = minus (119888

2minus 1198881) (A68)


1(119909) Let 119872 be a large


(120574 + 1205821) (1198882minus 1198881) = (120583 + 119886) Δ

1(119909 + 1) minus 120575

1Δ1(119909) (A69)






D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

(A70)

Hence we get

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

1) minus 1198881le 0

(A71)

that is

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) le 1198882 (A72)






2) le




2) lt 119881119906(119909 1198881) Then

119881119906lowast

(119909 1198882) = sup119906isinU

119881119906(119909 1198882)

le sup119906isinU

119881119906(119909 1198881) = 119881119906lowast

(119909 1198881)

(A73)





Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ℎ+= (120583 + 119886)D119881 (119909 + 1 119888) minus 120575

2D119881 (119909 119888)


ℎ+= (120583 + 119886)D119881 (119877

lowast(119888) + 2 119888) minus 120575

2D119881 (119877

lowast(119888) + 1 119888)

+ 1205821D119881 (119877

lowast(119888) 119888) + 120582

21199012minus 12058211199011

ℎ+= (120583 + 119886)D119881 (119909 + 1 119888) minus 120575

1D119881 (119909 119888)



1D119881 (119909 119888)

+ 1205821D119881 (119909 minus 1 119888) for 119909 le 0

(A57)

And let Δ12(119909 1198881 1198882) = D119881(119909 119888

2) minusD119881(119909 119888

1) and let 120585 =





2) as follows

0 = minus (1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1)

for 119909 ge 119878lowast (1198881) + 1

0 = minus (1205822+ 120583 + 120574) Δ (119878

lowast(1198881)) + 120582

2Δ (119878lowast(1198881) minus 1)

0 = 120583Δ (119909 + 1) minus (120583 + 1205822+ 120574) Δ (119909) + 120582

2Δ (119909 minus 1)

for 119863lowast (1198881) + 1 le 119909 lt 119878

lowast(1198881)

120585 = 120583Δ (119863lowast(1198881) + 1) minus 120575

2Δ (119863lowast(1198881))


1)

0 = (120583 + 119886) Δ (119909 + 1) minus 1205752Δ (119909) + 120582

2Δ (119909 minus 1)

for 119877lowast (1198881) + 1 lt 119909 lt 119863

lowast(1198881)

0 = (120583 + 119886) Δ (119877lowast(1198881) + 2) minus 120575

2Δ (119877lowast(1198881) + 1)


1) + 1

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)


1)

0 = (120583 + 119886) Δ (119909 + 1) minus 1205751Δ (119909)

+ 1205821Δ (119909 minus 1) for 119909 le 0

(A58)


Δ (119909) = 0 (A59)


D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882)

ge D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881)

(A60)

So we get

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119878lowast(1198881) 1198882)

ge D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119878lowast(1198881) 1198881) gt 119887

(A61)




2) ge 119878


we get 119877lowast(1198882) ge 119877




D119881(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) minus 119888 (A62)


D1198811(119909 119888) = D119881

(119877lowast

(119888)119863lowast

(119888)119878lowast

(119888))(119909 119888) minus 119888 (A63)


ℎ++ 120574119888 = minus (120582

2+ 120574)D119881

1(119909 119888)


ℎ++ 120574119888 = minus (120582

2+ 120583 + 120574)D119881

1(119878lowast 119888)


ℎ++ 120574119888 = 120583D119881

1(119909 + 1 119888) minus (120583 + 120582

2+ 120574)D119881

1(119909 119888)

+ 1205822D1198811(119909 minus 1 119888)


ℎ++ 120574119888 = 120583D119881

1(119863lowast(119888) + 1 119888) minus 120575

2D1198811(119863lowast(119888) 119888)


ℎ++ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

2D1198811(119909 119888)

+ 1205822D1198811(119909 minus 1 119888)


ℎ++ 120574119888 = (120583 + 119886)D119881

1(119877lowast(119888) + 2 119888)

minus 1205752D1198811(119877lowast(119888) + 1 119888)

+ 1205821D1198811(119877lowast(119888) 119888) + 120582

21199012minus 12058211199011

ℎ++ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

1D1198811(119909 119886)


minusℎminus+ 120574119888 = (120583 + 119886)D119881

1(119909 + 1 119888) minus 120575

1D1198811(119909 119888)

+ 1205821D1198811(119909 minus 1 119888) for 119909 le 0

(A64)


Let

D1198811(119909 1198881) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

D1198811(119909 1198882) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

(A65)

And let

Δ1(119909 1198881 1198882) = D119881

1(119909 1198882) minusD119881

1(119909 1198881) (A66)





1(119909 1198882) as

follows

120578 = minus (1205822+ 120574) Δ

1(119909) + 120582

2Δ1(119909 minus 1)

for 119909 ge 119878lowast (1198881) + 1

120578 = minus (1205822+ 120583 + 120574) Δ

1(119878lowast(1198881))


1)

120578 = 120583Δ1(119909 + 1) minus (120583 + 120582

2+ 120574) Δ

1(119909)


1) + 1 le 119909 lt 119878

lowast(1198881)

120578 = 120583Δ1(119863lowast(1198881) + 1) minus 120575

2Δ1(119863lowast(1198881))


1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

2Δ1(119909)


1) + 1 lt 119909 lt 119863

lowast(1198881)

120578 = (120583 + 119886) Δ1(119877lowast(1198881) + 2) minus 120575

2Δ1(119877lowast(1198881) + 1)


1) + 1

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)


1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)

+ 1205821Δ1(119909 minus 1) for 119909 le 0

(A67)



Δ1(119909) = minus (119888

2minus 1198881) (A68)


1(119909) Let 119872 be a large


(120574 + 1205821) (1198882minus 1198881) = (120583 + 119886) Δ

1(119909 + 1) minus 120575

1Δ1(119909) (A69)






D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

(A70)

Hence we get

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

1) minus 1198881le 0

(A71)

that is

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) le 1198882 (A72)






2) le




2) lt 119881119906(119909 1198881) Then

119881119906lowast

(119909 1198882) = sup119906isinU

119881119906(119909 1198882)

le sup119906isinU

119881119906(119909 1198881) = 119881119906lowast

(119909 1198881)

(A73)





Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Let

D1198811(119909 1198881) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

D1198811(119909 1198882) = D119881

(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

(A65)

And let

Δ1(119909 1198881 1198882) = D119881

1(119909 1198882) minusD119881

1(119909 1198881) (A66)





1(119909 1198882) as

follows

120578 = minus (1205822+ 120574) Δ

1(119909) + 120582

2Δ1(119909 minus 1)

for 119909 ge 119878lowast (1198881) + 1

120578 = minus (1205822+ 120583 + 120574) Δ

1(119878lowast(1198881))


1)

120578 = 120583Δ1(119909 + 1) minus (120583 + 120582

2+ 120574) Δ

1(119909)


1) + 1 le 119909 lt 119878

lowast(1198881)

120578 = 120583Δ1(119863lowast(1198881) + 1) minus 120575

2Δ1(119863lowast(1198881))


1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

2Δ1(119909)


1) + 1 lt 119909 lt 119863

lowast(1198881)

120578 = (120583 + 119886) Δ1(119877lowast(1198881) + 2) minus 120575

2Δ1(119877lowast(1198881) + 1)


1) + 1

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)


1)

120578 = (120583 + 119886) Δ1(119909 + 1) minus 120575

1Δ1(119909)

+ 1205821Δ1(119909 minus 1) for 119909 le 0

(A67)



Δ1(119909) = minus (119888

2minus 1198881) (A68)


1(119909) Let 119872 be a large


(120574 + 1205821) (1198882minus 1198881) = (120583 + 119886) Δ

1(119909 + 1) minus 120575

1Δ1(119909) (A69)






D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198882) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119909 1198881) minus 1198881

(A70)

Hence we get

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) minus 1198882

le D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

1) minus 1198881le 0

(A71)

that is

D119881(119877lowast

(1198881)119863lowast

(1198881)119878lowast

(1198881))(119863lowast(1198881) + 1 119888

2) le 1198882 (A72)






2) le




2) lt 119881119906(119909 1198881) Then

119881119906lowast

(119909 1198882) = sup119906isinU

119881119906(119909 1198882)

le sup119906isinU

119881119906(119909 1198881) = 119881119906lowast

(119909 1198881)

(A73)





Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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