Research Article Maximum Principles for Discrete and

Research ArticleMaximum Principles for Discrete and SemidiscreteReaction-Diffusion Equation

Petr Stehliacutek and Jonaacuteš Volek

Department of Mathematics and NTIS Faculty of Applied Sciences University of West Bohemia Univerzitni 830614 Pilsen Czech Republic

Correspondence should be addressed to Petr Stehlık pstehlikkmazcucz

Received 24 March 2015 Accepted 29 July 2015

Academic Editor Cengiz Cinar

Copyright copy 2015 P Stehlık and J Volek This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We study reaction-diffusion equations with a general reaction function 119891 on one-dimensional lattices with continuous or discretetime 1199061015840

119909(or Δ

119905119906119909) = 119896(119906

119909minus1minus 2119906

119909+ 119906

119909+1) + 119891(119906

119909) 119909 isin Z We prove weak and strong maximum and minimum principles for

corresponding initial-boundary value problems Whereas the maximum principles in the semidiscrete case (continuous time)exhibit similar features to those of fully continuous reaction-diffusion model in the discrete case the weak maximum principleholds for a smaller class of functions and the strong maximum principle is valid in a weaker sense We describe in detail how thevalidity of maximum principles depends on the nonlinearity and the time step We illustrate our results on the Nagumo equationwith the bistable nonlinearity

1 Introduction

Reaction-diffusion equation 120597119905119906 = 119896120597

119909119909119906 + 119891(119906) (sometimes

called FKPP equation which abbreviates Fisher KolmogorovPetrovsky Piskounov) serves as a nonlinearmodel to describea class of (biological chemical economic and so forth)phenomena in which two factors are combined Firstly thediffusion process causes the concentration of a substance(animals wealth and so forth) to spread in space Secondly alocal reaction leads to dynamics based on the concentrationvalues

For the sake of applications and correctness of numericalprocedures it makes sense to consider partially or fullydiscretized reaction-diffusion equation In certain situations(eg spatially structured environment) it is natural to studyreaction-diffusion equations with discretized space variableand continuous time (we refer to it as a semidiscrete problemand use 119906

119909(119905) = 119906(119909 119905))

1199061015840119909(119905) = 119896 (119906

119909minus1 (119905) minus 2119906119909(119905) + 119906

119909+1 (119905)) +119891 (119906119909(119905))

119909 isin Z 119905 isin [0 +infin) (1)

or for example if nonoverlapping populations are consid-ered with both time and space variables being discrete (adiscrete problem 119906

119909119905= 119906(119909 119905))

119906119909119905+ℎ

minus 119906119909119905

ℎ= 119896 (119906

119909minus1119905 minus 2119906119909119905

+119906119909+1119905) +119891 (119906

119909119905)

119909 isin Z 119905 isin 0 ℎ 2ℎ (2)

Examples of such phenomena are chemical reactions relatedto crystal formation see Cahn [1] or myelinated nerveaxons see Bell and Cosner [2] and Keener [3] Existence andnonexistence of travelling waves in those models have beenrecently studied in Chow [4] Chow et al [5] and Zinner [6]mostly with the cubic (or bistable double-well) nonlinearitiesof the form 119891(119906) = 120582119906(119906minus119886)(1minus119906) with 120582 gt 0 and 119886 isin (0 1)(this special case of FKPP equation is being referred to asNagumo equation) In contrast various reaction functionshave been proposed inmodels without spatial interaction forexample Xu et al [7]

Motivated by these facts we allow for a general form ofthe reaction function 119891 in this paper (ie we do not restrictourselves to cubic nonlinearities)We prove a priori estimates

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 791304 13 pageshttpdxdoiorg1011552015791304

2 Discrete Dynamics in Nature and Society

for discrete reaction-diffusion equation (2) and then useEulermethod to show their validity for semidiscrete reaction-diffusion equation (1) Whereas the maximum principles inthe semidiscrete case exhibit similar features to those ofcontinuous reaction-diffusion model (ie they hold undersimilar assumptions) in the discrete case the weakmaximumprinciple holds for a smaller class of functions and the strongmaximum principle is only valid in a weaker sense involvingthe domain of dependence Finally we use the maximumprinciples to get the global existence of solutions of theinitial-boundary problem for the semidiscrete case (1) Allour results are illustrated in detail in Nagumo equationswith a symmetric bistable nonlinearity that is we considerproblems (1)-(2) with 119891(119906) = 120582119906(1 minus 1199062)

Our motivation is twofold First maximum principlescould be used to obtain comparison principles (Protter andWeinberger [8]) which in turn could serve as a valuable toolin the study of traveling waves for example Bell and Cosner[2] Moreover similarly as in the case of (non)existence oftraveling wave solutions for Nagumo equations it has beenshown that discrete and semidiscrete structures influence thevalidity of maximum principles in a significant way Even thesimplest one-dimensional linear problems require additionalassumptions on the step size seeMawhin et al [9] and Stehlıkand Thompson [10] In the case of partial difference andsemidiscrete equations the strong influence of the underlyingstructure on maximum principles has been described in thelinear case for transport equation in Stehlık and Volek [11]and for diffusion-type equations in Slavık and Stehlık [12]and Friesl et al [13] (interestingly the proofs of maximumprinciples in this case are based on product integration seeSlavık [14]) Finally simple maximum principles for nonlin-ear transport equations on semidiscrete domains have beenpresented in Volek [15]

In the classical case maximum principles for diffusion(and parabolic) equations go back to Picone [16] and Levi[17] Strong maximum principles were later established byNirenberg [18] and a survey of various versions and appli-cations could be found in a classical monograph Protter andWeinberger [8]

This paper is segmented in the following way In Sec-tion 2 we briefly summarize results for the classical reaction-diffusion equation Next we prove weak and strong maxi-mum principles for the discrete case (2) (Sections 3 and 4)In the case of the initial-boundary value problem for thesemidiscrete equation (1) we provide local existence results(Section 5) and maximum principles (Section 6) which weconsequently apply to get global existence of solutions inSection 7 Our results are then applied to the Nagumoequation with a symmetric bistable nonlinearity that isproblems (1)-(2) with 119891(119906) = 120582119906(1 minus 1199062) in Section 8

2 Reaction-Diffusion PartialDifferential Equation

In order tomotivate and compare our results for the reaction-diffusionequationsondiscrete-spacedomainswith the classical

reaction-diffusion equation we briefly summarize few basicresults for the following initial-boundary problem

120597119905119906 (119909 119905) = 119896120597

119909119909119906 (119909 119905) + 119891 (119909 119905 119906 (119909 119905))

119909 isin (119886 119887) 119905 isin R+ 119896 gt 0

119906 (119909 0) = 120593 (119909) 119909 isin [119886 119887]

119906 (119886 119905) = 120585119886(119905) 119905 isin R

+

0

119906 (119887 119905) = 120585119887(119905) 119905 isin R

+

0

(3)

where 119891 (119886 119887) times R+ times R rarr R is a reaction function and120593 [119886 119887] rarr R 120585

119886 120585

119887 R+

0 rarr R are initial-boundaryconditions satisfying 120593(119886) = 120585

119886(0) and 120593(119887) = 120585

119887(0)

The following existence and uniqueness result for (3) canbe found for example in [19 page 298]

Theorem 1 Let 119879 gt 0 be arbitrary and let 119891 be uniformlyHolder continuous in 119909 and 119905 and Lipschitz in 119906 for (119909 119905) isin(119886 119887) times (0 119879] Then for all Holder continuous initial-boundaryconditions 120593 120585

119886 120585

119887problem (3) has a unique bounded solution

which is defined on [119886 119887] times [0 119879]

We define the following two numbers for the brevity

119872119879

= max119909isin[119886119887]119905isin[0119879]

120593 (119909) 120585119886(119905) 120585

119887(119905)

119898119879

= min119909isin[119886119887]119905isin[0119879]

120593 (119909) 120585119886(119905) 120585

119887(119905)

(4)

For the linear diffusion equation (ie (3) with119891(119909 119905 119906) equiv0) the maximum principle is proved for example in [8Chapter 31] For the nonlinear problem (3) (ie 119891(119909 119905 119906) =0) the following weak maximum principle holds (see [19Theorem 1])

Theorem 2 Let 119879 gt 0 be arbitrary and let 119891 be uniformlyHolder continuous in 119909 and 119905 and Lipschitz in 119906 for (119909 119905) isin(119886 119887) times (0 119879] and assume that

119891 (119909 119905119872119879) le 0 le 119891 (119909 119905 119898

119879)

forall (119909 119905) isin (119886 119887) times (0 119879] (5)

Let 119906 be a continuous solution of (3) with Holder continuousinitial-boundary conditions 120593 120585

119886 120585

119887 Then

119898119879

le 119906 (119909 119905) le 119872119879

(6)

holds for all (119909 119905) isin [119886 119887] times [0 119879]

Moreover the strong maximum principle also holds (see[19 Theorem 2])

Theorem 3 Let the assumptions ofTheorem 2 be satisfied andlet 119906 be a solution of (3) on [119886 119887] times [0 119879] If 119906(1199090 1199050) = 119872

119879(or

119906(1199090 1199050) = 119898119879) for some (1199090 1199050) isin (119886 119887) times (0 119879] then

119906 (119909 119905) = 119872119879

(119900119903 119906 (119909 119905) =119898119879)

forall (119909 119905) isin [119886 119887] times [0 1199050]

(7)

Discrete Dynamics in Nature and Society 3

3 Discrete Reaction-Diffusion EquationWeak Maximum Principles

Let us consider the initial-boundary value problem forthe discrete reaction-diffusion equation (which could beobtained eg by Euler discretization of (3))

119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119896Δ2

119909119909119906 (119909 minus 1 119905)

+ 119891 (119909 119905 119906 (119909 119905))

119909 isin (119886 119887)Z 119905 isin ℎN0 ℎ gt 0 119896 gt 0

119906 (119909 0) = 120593 (119909) 119909 isin (119886 119887)Z

119906 (119886 119905) = 120585119886(119905) 119905 isin ℎN0

119906 (119887 119905) = 120585119887(119905) 119905 isin ℎN0

(8)

where 119891 (119886 119887)Z times ℎN0 times R rarr R is a reaction function120593 (119886 119887)Z rarr R 120585

119886 120585

119887 ℎN0 rarr R are initial-boundary

conditions ℎN0 = ℎ119899 119899 isin N0 (119886 119887)Z = (119886 119887) cap Z andΔ2119909119909

119906(119909minus1 119905) = 119906(119909minus1 119905)minus2119906(119909 119905)+119906(119909+1 119905) (for brevitywe assume the space discretization step ℎ

119909= 1 but all our

results are easily extendable to an arbitrary step ℎ119909

gt 0 if weuse the diffusion constant = 119896ℎ2

119909instead of 119896 we discuss

this in detail in a specific example at the end of Section 8)Straightforwardly problem (8) has a unique solution

which is defined in [119886 119887]Z times ℎN0 since 119906(119909 119905 + ℎ) is uniquelygiven by

119906 (119909 119905 + ℎ)

=

119906 (119909 119905) + ℎ (119896Δ2119909119909

119906 (119909 minus 1 119905) + 119891 (119909 119905 119906 (119909 119905))) 119909 isin (119886 119887)Z

120585119886(119905 + ℎ) 119909 = 119886

120585119887(119905 + ℎ) 119909 = 119887

(9)

For 119879 isin ℎN0 we define the following two numbers

119872119879

= max119909isin(119886119887)Z 119905isin[0119879]ℎN0

120593 (119909) 120585119886(119905) 120585

119887(119905) (10)

119898119879

= min119909isin(119886119887)Z 119905isin[0119879]ℎN0

120593 (119909) 120585119886(119905) 120585

119887(119905) (11)

For brevity of the following assertions we formulate theassumption in the reaction function 119891

(D) Let 119879 isin ℎN0and let 119891 satisfy

2ℎ119896 minus 1ℎ

(119906 minus119898119879) le 119891 (119909 119905 119906) le


(119906 minus119872119879) (12)

for all 119909 isin (119886 119887)Z 119905 isin [0 119879]ℎN0

and 119906 isin [119898119879119872

119879]

Remark 4 The inequalities (12) imply that for all fixed 119909and 119905 the graph of function 119891(119909 119905 sdot) does not intersect theforbidden area depicted in Figure 1

Remark 5 Let us notice that for ℎ rarr 0+ the slope (2ℎ119896 minus1)ℎ goes to minusinfin that is the forbidden area from Remark 4is smaller in the sense of inclusion and it is easier to satisfyassumption (119863) if we decrease the time discretization step ℎWe illustrate this fact in Figure 1

MT

mT

h 997888rarr 0+ f

h 997888rarr 0+

u

f(x t middot)

Figure 1The forbidden area for the function119891(119909 119905 sdot) in assumption(119863) The change of this area if ℎ rarr 0+ The slope of the dashed lineis given by 2119896 minus 1ℎ see assumption (119863)

Proposition 6 Assume that 119898119879

lt 119872119879 If ℎ gt 12119896 then (119863)

does not hold for any function 119891

Note that the inequality ℎ le 12119896 is the necessarycondition for the validity of maximum principles even in thelinear case see for example [13 Theorem 24]

Proof If ℎ gt 12119896 (ie 2ℎ119896 minus 1 gt 0) then from (12) thereshould be

0 lt2ℎ119896 minus 1

ℎ(119906 minus119898

119879) le 119891 (119909 119905 119906)

le2ℎ119896 minus 1

ℎ(119906 minus119872

119879) lt 0 for 119906 isin (119898

119879119872

119879)

(13)

a contradiction

Remark 7 Notice that if 119898119879

= 119872119879then (119863) implies that

119891(119909 119905 119898119879) = 119891(119909 119905119872

119879) = 0 for all 119909 isin (119886 119887)Z and

119905 isin [0 119879]ℎN0

This situation corresponds to the case of theconstant initial-boundary conditions 120593(119909) equiv 119872

119879and 120585

119886(119905) equiv

120585119887(119905) equiv 119872

119879 From 119891(119909 119905119872

119879) = 0 and from (9) there is

119906 (119909 119905) = 120593 (119909) = 119872119879

for 119905 isin [0 119879]ℎN0

(14)

Now we state an auxiliary lemma which is crucial in theproof of the maximum principle

Lemma 8 Let 119879 isin ℎN0 let function 119891 satisfy (119863) and let 119906be the unique solution of (8) Then for all 119909 isin [119886 119887]Z and forall 119905 isin [0 119879)

ℎN0

119898119879

le 119906 (119909 119905) le 119872119879

implies that 119898119879

le 119906 (119909 119905 + ℎ) le 119872119879

(15)

Proof For the sake of brevity we only show that 119906(119909 119905 + ℎ) le119872

119879The inequality119898

119879le 119906(119909 119905+ℎ) can be proved in the same

wayLet 119905 isin ℎN0 119905 lt 119879 be arbitrary Then 119906(119886 119905 + ℎ) = 120585

119886(119905 +

ℎ) le 119872119879and 119906(119887 119905 + ℎ) = 120585

119887(119905 + ℎ) le 119872

119879trivially from


the definition of 119872119879(10) (recall that 119905 lt 119879 ie 119905 + ℎ le 119879) If

119909 isin (119886 119887)Z then we can estimate

119906 (119909 119905 + ℎ) = 119906 (119909 119905) + ℎ (119896119906 (119909 minus 1 119905) minus 2119896119906 (119909 119905)

+ 119896119906 (119909 + 1 119905) + 119891 (119909 119905 119906 (119909 119905))) le 2ℎ119896119872119879+ (1

minus 2ℎ119896) 119906 (119909 119905) + ℎ119891 (119909 119905 119906 (119909 119905))

(16)

Thanks to the assumptions (12) and 119898119879

le 119906(119909 119905) le 119872119879we

get

ℎ119891 (119909 119905 119906 (119909 119905)) le (2ℎ119896 minus 1) (119906 (119909 119905) minus119872119879) (17)

Therefore

119906 (119909 119905 + ℎ) le 2ℎ119896119872119879+ (1minus 2ℎ119896) 119906 (119909 119905)

+ ℎ119891 (119909 119905 119906 (119909 119905))

le 2ℎ119896119872119879+ (1minus 2ℎ119896) 119906 (119909 119905)

+ (2ℎ119896 minus 1) (119906 (119909 119905) minus119872119879) = 119872

119879

(18)

The weak maximum principle follows immediately

Theorem 9 Let 119879 isin ℎN0 be arbitrary let function 119891 satisfy(119863) and let 119906 be the unique solution of (8) Then

119898119879

le 119906 (119909 119905) le 119872119879

(19)

holds for all 119909 isin [119886 119887]Z and 119905 isin [0 119879]ℎN0

Proof From (10) to (11) we get 119898119879

le 119906(119909 0) le 119872119879for all

119909 isin [119886 119887]Z Immediately Lemma 8 yields that (19) holds forall 119909 isin [119886 119887]Z and 119905 isin [0 119879]

ℎN0

Remark 10 If the reaction function 119891 does not satisfy theinequalities (12) we can find a counterexample that themaximum principle does not hold in general For examplelet us consider (8) with 119886 = minus1 119887 = 1 119905 isin N0 and 120593(119909) equiv 0120585119886(119905) equiv 120585

119887(119905) equiv 0 Let us assume that for example the latter

inequality in (12) does not hold that is

119891 (0 119905 0) gt2ℎ119896 minus 1

ℎ(0minus119872

119879) = 0 (20)

for some 119905 isin N0 Assuming without loss of generality that 119905 =0 then the maximum principle is straightforwardly violatedsince

119906 (0 1) = 119896119906 (minus1 0) + (1minus 2119896) 119906 (0 0) + 119896119906 (1 0)

+ 119891 (0 0 0) = 119891 (0 0 0) gt 0 = 119872119879

(21)

In certain cases the function 119891 could fail to satisfy (119863)but could still provide a priori bounds for solutions of (8) ifthe following inequalities hold

(1198631015840) Let 119879 isin ℎN0and let there exist 119878 ge 119872

119879and 119877 le 119898

119879

such that2ℎ119896 minus 1

ℎ(119906 minus119877) le 119891 (119909 119905 119906) le


(119906 minus 119904) (22)

S

R

MT

mT

f

u

f(x t middot)

Figure 2 The example of the function 119891 that does not satisfy(119863) but satisfies (1198631015840) for some constants 119877 119878 Such a functionconsequently provides a priori bounds for solutions of (8) in thesense of Theorem 11

for all 119909 isin (119886 119887)Z and 119905 isin ℎN0such that 0 le 119905 le 119879 and

119906 isin [119877 119878]

In that case we obtain a general version of the weakmaximum principle (for the illustration of (1198631015840) see Figure 2)


119877 le 119906 (119909 119905) le 119878 (23)

holds for all 119909 isin (119886 119887)Z and 119905 isin ℎN0 such that 0 le 119905 le 119879

Proof For 119905 = 0 we have

119877 le 119898119879

le 119906 (119909 0) le 119872119879

le 119878 forall119909 isin (119886 119887)Z (24)

Nowwe can proceed analogously as in the proofs of Lemma 8andTheorem 9where we use (1198631015840) instead of (119863)We omit thedetails

Example 12 The set of nonlinear reaction functions 119891 thatcould be considered inTheorem 9 or 11 includes for example(for the detailed analysis with 119891(119909 119905 119906) = 120582119906(1 minus 1199062) seeSection 8)

(i) 119891(119909 119905 119906) = minus|119906|119901minus1119906 with 119901 gt 1(ii) the logistic function 119891(119909 119905 119906) = 119906(1 minus 119906)(iii) the bistable nonlinearity 119891(119909 119905 119906) = 120582119906(119906minus119886)(1minus119906)

119886 isin (0 1)(iv) 119891(119909 119905 119906) = 120582119906(1 minus 119906119901) where 119901 isin N(v) 119891(119909 119905 119906) = minus|119909| arctan(1199052119906)

We state the following two claims that are direct corollar-ies of Theorem 9

Corollary 13 Assume that 120585119886 120585

119887are bounded Let 119891 satisfy

(119863) for all 119879 gt 0 Then the unique solution 119906 of (8) is bounded

Corollary 14 Assume that 120593 120585119886 120585

119887are nonnegative Let 119891

satisfy (119863) for all 119879 gt 0 Then the unique solution 119906 of (8)is nonnegative


4 Discrete Reaction-Diffusion EquationStrong Maximum Principle

As in the case of classical reaction-diffusion equation (3)(Theorem 3) we naturally turn our attention to strong max-imum principles Straightforwardly the strong maximumprinciple does not hold in the discrete case in the sense ofTheorem 3

Example 15 Let us consider problem (8) with 119909 isin [minus2 2]Z119905 isin N0 119891(119909 119905 119906) equiv 0 and 119896 = 12 (note that ℎ = 12119896) andlet

120593 (119909) equiv 119872 gt 0 119909 isin minus1 0 1

120585minus2 (119905) equiv 1205852 (119905) equiv 0 119905 isin N0

(25)

Then from (9) we get

119906 (0 1) = 119906 (0 0) + 119896119906 (minus1 0) minus 2119896119906 (0 0) + 119896119906 (1 0)

+ 119891 (0 0 0) = 119872(26)

Analogously we can deduce that

119906 (minus2 1) = 119906 (2 1) = 0

119906 (minus1 1) = 119906 (1 1) =119872

2

(27)

Consequently the strong maximum principle does not hold

Nonetheless given the fact that the values of 119906(119909 119905)are given by (9) we can easily construct the domain ofdependence of (1199090 1199050)

D (1199090 1199050) = (119909 119905) isin [119886 119887]Z times ℎN0 119905 le 1199050 119909 = 1199090

plusmn 119895 119895 = 0 1 1199050 minus 119905

ℎ

(28)

and the domain of influence of (1199090 1199050)

I (1199090 1199050) = (119909 119905) isin [119886 119887]Z times ℎN0 119905 ge 1199050 119909 = 1199090

plusmn 119895 119895 = 0 1 119905 minus 1199050

ℎ

(29)

Considering the following

(11986310158401015840) Let 119879 isin ℎN0and let 119891 satisfy for all 119909 isin (119886 119887)Z 119905 isin

[0 119879]ℎN0

(a) 119891(119909 119905 119906) lt ((2ℎ119896 minus 1)ℎ)(119906 minus 119872119879) when 119906 isin

[119898119879119872

119879)

(b) 119891(119909 119905 119906) gt ((2ℎ119896 minus 1)ℎ)(119906 minus 119898119879) when 119906 isin

(119898119879119872

119879]

(c) 119891(119909 119905119872119879) le 0 and 119891(119909 119905 119898

119879) ge 0

the weaker version of the strong maximum principle followsimmediately

Theorem 16 Assume that the function 119891 satisfies (11986310158401015840) for all119879 isin ℎN0 Let 119906 be the unique solution of (8) and (1199090 1199050) isin[119886 119887]Z times ℎN0

(1) If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) then 119906(119909 119905) =

119872119879(or 119906(119909 119905) = 119898

119879) onD(1199090 1199050)

(2) If 119906(1199090 1199050) lt 119872119879(or 119906(1199090 1199050) gt 119898

119879) then 119906(119909 119905) lt

119872119879(or 119906(119909 119905) gt 119898

119879) onI(1199090 1199050)

Proof Let us only focus on the former statement of thetheorem the latter could be proved in very similar way Weshow that if the function 119891 satisfies (11986310158401015840) and 119906(1199090 1199050) = 119872

119879

for some 1199090 isin (119886 119887)Z 1199050 isin ℎN0 0 lt 1199050 le 119879 then119906(1199090 minus 1 1199050 minus ℎ) = 119906(1199090 1199050 minus ℎ) = 119906(1199090 + 1 1199050 minus ℎ) = 119872

119879

The rest follows by inductionAssume by contradiction first that 119906(1199090 minus 1 1199050 minus ℎ) lt 119872

119879

(the case 119906(1199090 + 1 1199050 minus ℎ) lt 119872119879follows easily) Using this

assumption (9) andTheorem 9 we can estimate

119906 (1199090 1199050) = ℎ119896 (119906 (1199090 minus 1 1199050 minus ℎ) + 119906 (1199090 + 1 1199050 minus ℎ))

+ (1minus 2ℎ119896) 119906 (1199090 1199050 minus ℎ)

+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ))

lt 2ℎ119896119872119879+ (1minus 2ℎ119896) 119906 (1199090 1199050 minus ℎ)

+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ)) +119872119879

minus119872119879

= (1minus 2ℎ119896) (119906 (1199090 1199050 minus ℎ) minus119872119879)

+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ)) +119872119879

(30)

Thus (11986310158401015840) yields

(1minus 2ℎ119896) (119906 (1199090 1199050 minus ℎ) minus119872119879)

+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ)) le 0(31)

Consequently there has to be119906 (1199090 1199050) lt 119872

119879 (32)

a contradictionIf 119906(1199090 1199050 minus ℎ) lt 119872

119879 then by the similar procedure as

above we obtain119906 (1199090 1199050) le (1minus 2ℎ119896) (119906 (1199090 1199050 minus ℎ) minus119872

119879)

+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ)) +119872119879

(33)

Since 119906(1199090 1199050 minus ℎ) isin [119898119879119872

119879) in this case (11986310158401015840) implies that


+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ)) lt 0(34)

Hence119906 (1199090 1199050) lt 119872

119879 (35)

a contradiction

In the case of nonconstant time discretization ℎ = ℎ(119905)wecan follow similar techniques and consider (119863) (eventually(1198631015840) or (11986310158401015840)) with ℎmax (or ℎsup for 119879 rarr infin) see Remark 5


5 Semidiscrete Reaction-Diffusion EquationLocal Existence

In this section we study the local existence of the followinginitial-boundary value problem on semidiscrete domains

119906119905(119909 119905) = 119896Δ2

119909119909119906 (119909 minus 1 119905) + 119891 (119909 119905 119906 (119909 119905))

119909 isin (119886 119887)Z 119905 isin R+

0 119896 gt 0

119906 (119909 0) = 120593 (119909) 119909 isin (119886 119887)Z

119906 (119886 119905) = 120585119886(119905) 119905 isin R

+

0

119906 (119887 119905) = 120585119887(119905) 119905 isin R

+

0

(36)

where 119906119905= 120597

119905119906 denotes the time derivative 119891 (119886 119887)Z timesR+

0 timesR rarr R is a reaction function 120593 (119886 119887)Z rarr R 120585

119886 120585

119887isin

1198621(R+

0 ) are initial-boundary conditions and R+

0 = [0 +infin)Given the fact that (36) can be interpreted as a vector

ODE we can rewrite it as

u1015840 (119905) = g (119905 u (119905)) 119905 isin R+

0

u (0) = u(37)

where u R+

0 rarr R119873 g R+

0 timesR119873 rarr R119873 is continuous andu isin R119873

Naturally we use the well-known result of Picard andLindelof to get the local existence for the initial value problem(37) (see [20 Theorem 813])

Theorem 17 Assume that g is continuous on the rectangle

119876 = (119905 u) isinR+

0 timesR119873 0le 119905 le 120572 uminus u le 120573 (38)

and satisfies the Lipschitz condition on 119876 that is there exists119871 gt 0 such that for all (1199051 u1) (1199052 u2) isin 119876

1003817100381710038171003817g (1199051 u1) minus g (1199052 u2)1003817100381710038171003817 le 119871

1003817100381710038171003817u1 minus u21003817100381710038171003817 (39)

holds Then there exists 120578 gt 0 such that (37) has a uniquesolution u defined on [0 120578]

We apply Theorem 17 to get the local existence for thesemidiscrete reaction-diffusion equation (36) We use thefollowing two assumptions

(119862cont) Let 119891(119909 119905 119906) be continuous in (119905 119906) isin R+

0times R for all

119909 isin (119886 119887)Z(119862lip) Let 119891(119909 119905 119906) be locally Lipschitz with respect to 119906 on

(119886 119887)Z times R+

0times R that is for all 119909

0isin (119886 119887)Z 1199050 isin R+

0

and 1199060isin R there exist 120572 120573 gt 0 and

119876 (1199090 1199050 119906

0) = (119909

0 119905 119906) isin (119886 119887)Z timesR

+

0

timesR 1003816100381610038161003816119905 minus 119905

0

1003816100381610038161003816 le 1205721003816100381610038161003816119906 minus 119906

0

1003816100381610038161003816 le 120573(40)

and 119871 gt 0 such that for all (1199090 1199051 119906

1) (119909

0 1199052 119906

2) isin 119876

there is1003816100381610038161003816119891 (1199090 1199051 1199061) minus119891 (1199090 1199052 1199062)

1003816100381610038161003816 le 11987110038161003816100381610038161199061 minus1199062

1003816100381610038161003816 (41)

Theorem 18 Let 119891 satisfy (119862119888119900119899119905

) and (119862119897119894119901) Then there exists

120578 gt 0 such that (36) has a unique solution defined on [119886 119887]Z times[0 120578]

Proof Since the space variable 119909 is from a finite set (119886 119887)Zproblem (36) corresponds to the following vector ODE

(((

(

119906(119886 + 1 119905)119906 (119886 + 2 119905)

119906 (119887 minus 2 119905)119906 (119887 minus 1 119905)

)))

)

1015840

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟u1015840(119905)

= 119896((

(

minus2 1 0 sdot sdot sdot 01 minus2 1 sdot sdot sdot 0

d

0 sdot sdot sdot minus2 10 sdot sdot sdot 1 minus2

))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟A

(((

(

119906(119886 + 1 119905)119906 (119886 + 2 119905)

119906 (119887 minus 2 119905)119906 (119887 minus 1 119905)

)))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟u(119905)

+ 119896(((

(

120585119886(119905)

0

0120585119887(119905)

)))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120585(119905)

+(((

(

119891(119886 + 1 119905 119906 (119886 + 1 119905))119891 (119886 + 2 119905 119906 (119886 + 2 119905))

119891 (119887 minus 2 119905 119906 (119887 minus 2 119905))119891 (119887 minus 1 119905 119906 (119887 minus 1 119905))

)))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟f(119905u(119905))

(42)

coupled with the initial condition

(

119906(119886 + 1 0)119906 (119886 + 2 0)

119906 (119887 minus 1 0)

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟u(0)

= (

120593(119886 + 1)120593 (119886 + 2)

120593 (119887 minus 1)

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120593

(43)

Thus problem (36) can be rewritten in the vector form asfollows

u1015840 (119905) = 119896Au (119905) + 119896120585 (119905) + f (119905 u (119905)) 119905 isin R+

0

u (0) = 120593(44)


Assumptions (119862cont) and (119862lip) yield that the nonlinearfunction f is continuous and satisfies Lipschitz conditionwithrespect to u on some rectangle 119876 Since the term Au is linearand therefore Lipschitz with respect to u and 120585 isin 1198621(R+

0 ) theassumptions of Theorem 17 are satisfied Consequently thereexists 120578 gt 0 such that (44) has a unique solution on [0 120578]

Remark 19 If we assume only (119862cont) then we can apply thePeano theorem [20 Theorem 827] instead of Theorem 17 toget the local existence of solutions of (36) which need not beunique

6 Semidiscrete Reaction-Diffusion EquationMaximum Principles

Having the local existence and uniqueness we focus on themaximum principles for (36) In the following analysis weapproximate the solution of (36) by the solutions of thediscrete problem (8) which arises from (36) by the explicit(Euler) discretization of the time variable

First we define the Euler polygon (see [21 I7])

Definition 20 Let ℎ gt 0 be a discretization step Consider theinitial value problem (37) on the interval [0 119879]where119879 = 119899ℎ119899 isin N Define the subdivision of interval [0 119879] as the set ofpoints 119905

119894= 119894ℎ 119894 = 0 1 119899 and for 119894 = 0 1 119899 minus 1 define

y119894+1 = y

119894+ ℎg (119905

119894 y

119894)

y0 = u (0) = u(45)

Then the continuous function y(ℎ)

[0 119879] rarr R119873 defined by

y(ℎ)

(119905) = y119894+ (119905 minus 119905

119894) g (119905

119894 y

119894) 119905

119894le 119905 le 119905

119894+1 (46)

is called Euler polygon

The following statement sums up the convergence ofEuler method (see [21 I7 Theorem 73 and I9 page 54])

Theorem 21 Let 119879 gt 0 and let g be continuous satisfyingLipschitz condition on

119876 = (119905 u) isinR+

0 timesR119873 0le 119905 le119879 uminus u le 120573 (47)

and let g be bounded by a constant 119860 gt 0 on 119876 If 119879 le 120573119860then the following hold

(a) for ℎ rarr 0+ the Euler polygons y(ℎ)

(119905) convergeuniformly to a continuous function 120599(119905) on [0 119879]

(b) 120599 isin 1198621(0 119879) and it is the unique solution of (37) on[0 119879]

We define the bounds of initial-boundary conditionssimilarly as in the discrete problem

119872119879

= max119909isin(119886119887)Z 119905isin[0119879]

120593 (119909) 120585119886(119905) 120585

119887(119905)

119898119879

= min119909isin(119886119887)Z 119905isin[0119879]

120593 (119909) 120585119886(119905) 120585

119887(119905)

(48)

Before we state the weak maximum principle we describethe connection between the discretization of (36) and theassumption (119863)

Lemma 22 Let 119879 isin R+

0 and let 119891 satisfy (119862119888119900119899119905

) (119862119897119894119901) and

(119862119904119894119892119899

) 119891(119909 119905119872119879) le 0 le 119891(119909 119905 119898

119879) for all 119909 isin (119886 119887)Z 119905 isin

[0 119879]

Then there exists 119867 gt 0 such that for all ℎ isin (0 119867)


(119906 minus119898119879) le 119891 (119909 119905 119906) le


(119906 minus119872119879) (49)

holds for all 119909 isin (119886 119887)Z 119905 isin [0 119879] and 119906 isin [119898119879119872

119879]

Proof Weprove the latter inequality in (49) by contradictionThe former inequality can be proved in the same way Let usassume that for all 119867 gt 0 there exist ℎ isin (0 119867) 119909

ℎisin (119886 119887)Z

119905ℎisin [0 119879] and 119906

ℎisin [119898

119879119872

119879] such that

119891 (119909ℎ 119905ℎ 119906

ℎ) gt


(119906ℎminus119872

119879) (50)

Therefore there exist sequences ℎ119898infin119898=1 and (119909

119898 119905119898

119906119898)infin119898=1 (we denote 119909

119898= 119909

ℎ119898 119905

119898= 119905

ℎ119898 119906

119898= 119906

ℎ119898) such

thatℎ119898

997888rarr 0+

119891 (119909119898 119905119898 119906

119898) gt

2ℎ119898119896 minus 1ℎ119898

(119906119898

minus119872119879)

(51)

First we observe that if 119906119898

= 119872119879for some 119898 isin N we get

a contradiction with (119862sign) Thus we can assume that 119906119898

lt119872

119879for all 119898 isin NNow we have to distinguish between two cases

(i) If there does not exist any subsequence 119906119898119897

infin119897=1 sub

119906119898infin119898=1 such that 119906

119898119897rarr 119872

119879then the right-hand

side of inequality in (51) goes to infinity Hencefrom (51) 119891(119909

119898 119905119898 119906

119898) also goes to infinity This

yields a contradiction with (119862cont) which impliesboundedness of the function 119891 on (119886 119887)Z times [0 119879] times[119898

119879119872

119879]

(ii) Let there exist a subsequence 119906119898119897

infin119897=1 sub 119906

119898infin119898=1 such

that 119906119898119897

rarr 119872119879 We show that we get a contradiction

with (119862lip) in this case Since the interval (119886 119887)Zis bounded there exists a convergent subsequence119909

119898119897infin119897=1 sub 119909

119898infin119898=1 such that 119909

119898119897rarr 119909 Analogically

since [0 119879] is bounded there also exists a convergentsubsequence 119905

119898119897infin119897=1 sub 119905

119898infin119898=1 such that 119905

119898119897rarr

Let 120572 120573 gt 0 and 119871 gt 0 be arbitrary Then we can find isin N sufficiently large such that

1 minus 2ℎ119898

119896

ℎ119898

ge 119871

119872119879minus119906

119898

le 120573

119909119898

= 119909

10038161003816100381610038161003816119905119898 minus 10038161003816100381610038161003816 lt 120572

(52)


If we put 119909 = 119909119898

= 119905119898

= 119906119898

and = 119872119879

and 119876(119909 ) is the rectangle from assumption (119862lip)with given 120572 gt 0 and 120573 gt 0 then (119909 ) (119909 ) isin119876(119909 ) Now from (51) (52) and (119862sign) we canestimate

119871 | minus | le1 minus 2ℎ

119898

119896

ℎ119898

(119872119879minus ) lt 119891 (119909 )

le 119891 (119909 ) minus 119891 (119909 119872119879)

= 119891 (119909 ) minus 119891 (119909 )

le1003816100381610038161003816119891 (119909 ) minus 119891 (119909 )

1003816100381610038161003816

(53)

a contradiction with (119862lip)

Remark 23 The assumption (119862sign) defines the forbiddenarea for a reaction function 119891(119909 119905 sdot) in the same way as(119863) However this area is reduced to a pair of half-linesLet us notice that it is the limit case of forbidden areas forthe discrete case if ℎ rarr 0+ (see Remark 5 and Figure 1)Moreover it is equivalent to the assumption for classicalPDEs see (5)

Theorem24 Let119879 gt 0 be arbitrary let119891 satisfy (119862cont) (119862lip)and (119862

119904119894119892119899) and let 119906 be a solution of (36) defined on [119886 119887]Z times

[0 119879] Then

119898119879

le 119906 (119909 119905) le 119872119879

(54)

holds for all 119909 isin (119886 119887)Z 119905 isin [0 119879]

Proof We prove that for all 119909 isin (119886 119887)Z 119905 isin [0 119879] thereis 119906(119909 119905) le 119872

119879 The first inequality in (54) can be proved

similarly Let us assume by contradiction that there exist 119909119888isin

(119886 119887)Z and 119905119888isin (0 119879] such that

119906 (119909119888 119905119888) gt 119872T (55)

From the continuity of the solution 119906 there exist 1199090 isin (119886 119887)Zand 1199050 isin [0 119905

119888) such that

(a) 119906(119909 119905) le 119872119879for all 119909 isin (119886 119887)Z and 119905 isin [0 1199050]

(b) 119906(1199090 1199050) = 119872119879

(c) there exists 120575 gt 0 such that

119906 (1199090 119905) gt 119872119879

on (1199050 1199050 + 120575) (56)

Let us analyze the new initial-boundary value problem(36) with the initial condition 119906(119909 1199050) at time 1199050 Let usunderstand this problem as the initial value problem for thevector ODE (44) with the initial condition at time 1199050

From (119862cont) (119862lip) we know that f(119905 u) is continuousand Lipschitz on some rectangle 119876 From (119862cont) we also getthat f is bounded by some constant 119860 gt 0 on 119876 ThereforeTheorem 21 implies that for sufficiently small discretizationsteps ℎ gt 0 and for sufficiently small interval [1199050 1199050 + 120576]

the Euler polygons y(ℎ)

(119905) converge uniformly to the uniquesolution u(119905) on [1199050 1199050 + 120576]

Notice that the node points of Euler polygons y(ℎ)

(119905) arethe solutions of (8) From (119862cont) (119862lip) and (119862sign) and fromLemma 22 the assumption (D) is satisfied (recall that ℎ issufficiently small) and therefore fromTheorem 9

119910(ℎ)

(119909 119905) le 119872119879

on [1199050 1199050 + 120576] forall119909 isin (119886 119887)Z (57)

But if y(ℎ)

(119905) converge uniformly to u(119905) and 119910(ℎ)

(119909 119905) le 119872119879

on [1199050 1199050 + min120575 120576] for all 119909 isin (119886 119887)Z then there has to be

119906 (119909 119905) le 119872119879

on [1199050 1199050 + min 120575 120576] forall119909 isin (119886 119887)Z (58)

a contradiction with (56)

If the assumption (119862sign) is not satisfied but the nonlinearfunction 119891 satisfies the following(1198621015840

sign) Let 119879 gt 0 be arbitrary and let there exist 119878 ge 119872119879and

119877 le 119898119879such that

119891 (119909 119905 119878) le 0 le 119891 (119909 119905 119877) (59)

for all 119909 isin (119886 119887)Z 119905 isin [0 119879]then we can state the following generalized weak maximumprinciple



[0 119879] Then

119877 le 119906 (119909 119905) le 119878 (60)


Proof The statement can be proved in the similar way asLemma 22 andTheorem 24

As in the previous sections we want to establish thestrong maximum principle First we recall the well-knownGronwallrsquos inequality (see eg [20 Corollary 862])

Lemma 26 Let 120573 119906 [119903 119904] rarr R be continuous functionsand let 119906 be differentiable on (119903 119904) If

1199061015840 (119905) le 120573 (119905) 119906 (119905) for 119905 isin (119903 119904) (61)

then

119906 (119905) le 119906 (119903) 119890int119905

119903120573(120591)119889120591 for 119905 isin [119903 119904] (62)

Further we need the following auxiliary lemma

Lemma 27 Let 119879 gt 0 be arbitrary let 119891 satisfy (119862cont) (119862lip)and (119862


[0 119879] If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) for some 1199090 isin

(119886 119887)Z and 1199050 isin (0 119879] then

119906 (1199090 119905) = 119872119879

(119900119903 119906 (1199090 119905) =119898119879)

forall119905 isin [0 1199050]

(63)


Proof We prove the statement with 119872119879 The case with 119898

119879

is similar First the weak maximum principle (Theorem 24)holds that is 119906(119909 119905) le 119872

119879for all 119909 isin [119886 119887]Z and 119905 isin [0 119879]

Suppose by contradiction that there exists 119905119888

isin (0 1199050] suchthat

119906 (1199090 119905119888) = 119872119879

119906 (1199090 119905) lt 119872119879

on [119905119888minus 120576 119905

119888)

(64)

Without loss of generality let 120576 gt 0 be sufficiently small sothat the function119891(1199090 119905 119906) is uniformly Lipschitz in119906 on [119905

119888minus

120576 119905119888)with Lipschitz constant 119871 gt 0 (follows from (119862lip))With

the help of these facts and also from (119862sign) we can estimate

119906119905(1199090 119905)

= 119896 (119906 (1199090 minus 1 119905) minus 2119906 (1199090 119905) + 119906 (1199090 + 1 119905))

+119891 (1199090 119905 119906 (1199090 119905))

le 119896 (2119872119879minus 2119906 (1199090 119905)) +119891 (1199090 119905 119906 (1199090 119905))

minus119891 (1199090 119905119872119879) +119891 (1199090 119905119872119879

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟le0

le 119896 (2119872119879minus 2119906 (1199090 119905))

+1003816100381610038161003816119891 (1199090 119905 119906 (1199090 119905)) minus119891 (1199090 119905119872119879

)1003816100381610038161003816

le 119896 (2119872119879minus 2119906 (1199090 119905)) + 119871

1003816100381610038161003816119906 (1199090 119905) minus119872119879

1003816100381610038161003816

= 119896 (2119872119879minus 2119906 (1199090 119905)) + 119871 (119872

119879minus119906 (1199090 119905))

= minus (2119896 + 119871) (119906 (1199090 119905) minus119872119879)

(65)

for all 119905 isin (119905119888minus 120576 119905

119888) If we denote 120572 = minus(2119896 + 119871) then the

function 119906(1199090 119905) satisfies 119906119905(1199090 119905) le 120572(119906(1199090 119905) minus119872119879) on (119905

119888minus

120576 119905119888) If we substitute V(119905) = 119906(1199090 119905) minus 119872

119879then the function

V satisfies the following differential inequality

V1015840 (119905) le 120572V (119905) (66)

Therefore Gronwallrsquos inequality (Lemma 26) implies that

V (119905) le V (119905119888minus 120576) 119890120572(119905minus119905119888+120576) (67)

and hence

119906 (1199090 119905) le 119872119879minus (119872

119879minus119906 (1199090 119905119888 minus 120576)) 119890120572(119905minus119905119888+120576)

lt 119872119879

(68)

on [119905119888minus 120576 119905

119888] a contradiction

The strong maximum principle for (36) follows immedi-ately

Theorem 28 Let 119879 gt 0 be arbitrary let 119891 satisfy (119862cont)(119862lip) and (119862

119904119894119892119899) and let 119906 be a solution of (36) defined on

[119886 119887]Z times [0 119879] If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) for some

1199090 isin (119886 119887)Z and 1199050 isin (0 119879] then

119906 (119909 119905) = 119872119879

(or 119906 (119909 119905) =119898119879)

forall119909 isin [119886 119887]Z 119905 isin [0 1199050]

(69)

Proof Lemma 27 yields that 119906(1199090 119905) = 119872119879for all 119905 isin [0 1199050]

If 119906(1199090minus1 119905119888) lt 119872119879(or 119906(1199090+1 119905119888) lt 119872

119879) at some 119905

119888isin [0 1199050]

then applying (119862sign) the following has to be satisfied

119906119905(1199090 119905119888)

= 119896 (119906 (1199090 minus 1 119905119888) minus 2119906 (1199090 119905119888) + 119906 (1199090 + 1 119905

119888))

+119891 (1199090 119905119888 119906 (1199090 119905119888))

lt 119896 (2119872119879minus 2119872

119879) +119891 (1199090 119905119888119872119879

) le 0

(70)

a contradiction with the fact that the function 119906(1199090 119905) isconstant and equal to 119872

119879on [0 1199050] Therefore functions

119906(1199090 minus 1 119905) and 119906(1199090 + 1 119905) are also constant and equal to119872

119879on [0 1199050] Then we can continue inductively in 119909 to the

boundary points 119909 = 119886 or 119909 = 119887 The case with 119898119879is

similar

7 Semidiscrete Reaction-Diffusion EquationGlobal Existence

In this sectionwe combine the local existence and uniquenessand the maximum principle to obtain the global existence ofsolution of (36)

Once again we use known results from the theory ofordinary differential equations First we define the maximalinterval of existence (see [20 Definition 831])

Definition 29 Let g be continuous and let u be a solutionof (37) defined on [0 120578) Then one says [0 120578) is a maximalinterval of existence for u if there does not exist an 1205781 gt 120578and a solution w defined on [0 1205781) such that u(119905) = w(119905) for119905 isin [0 120578)

In the following we apply the extendability theorem (see[20 Theorem 833])

Theorem 30 Let g be continuous and let u be a solution of(37) defined on [0 120596) Then 119906 can be extended to a maximalinterval of existence [0 120578) 0 lt 120578 le infin Furthermore there iseither

120578 = infin

or u (119905)119905rarr120578minus

997888997888997888997888997888rarr infin

(71)

This theorem enables us to conclude with the globalexistence for (36)


Theorem31 Let119891 satisfy (119862cont) (119862lip) and (119862119904119894119892119899) for all119879 gt

0Then (36) has a unique solution defined on [119886 119887]ZtimesR+

0 whichsatisfies

inf119909isin(119886119887)Z 119905isinR

+

0

120593 (119909) 120585119886(119905) 120585

119887(119905) le 119906 (119909 119905)

le sup119909isin(119886119887)Z 119905isinR

+

0

120593 (119909) 120585119886(119905) 120585

119887(119905)

(72)

for all (119909 119905) isin [119886 119887]Z times R+

0

Proof Let us understand problem (36) as the initial valueproblem for the vector ODE (44) From Theorem 18 thereexists a uniquely determined local solution u(119905) From The-orem 30 the solution can be extended to a maximal intervalof existence [0 120578) which is open from right and either

120578 = infin

or u (119905)119905rarr120578minus

997888997888997888997888997888rarr infin

(73)

If u(119905) rarr infin for 119905 rarr 120578minus then for all 119870 gt 0 there haveto exist 119909

119870isin (119886 119887)Z and 119905

119870isin [0 120578) such that

1003816100381610038161003816119906 (119909119870 119905119870)1003816100381610038161003816 gt 119870 (74)

If we put 119870 = max|119898120578| |119872

120578| lt infin (120585

119886 120585

119887are 1198621 functions

on R+

0 and therefore bounded on [0 120578]) then

1003816100381610038161003816119906 (119909119870 119905119870)1003816100381610038161003816 gt max

10038161003816100381610038161003816119898120578

10038161003816100381610038161003816 10038161003816100381610038161003816119872120578

10038161003816100381610038161003816 (75)

a contradiction with the maximum principle in Theorem 24(which holds thanks to (119862cont) (119862lip) and (119862sign))

Therefore there has to be 120578 = infin that is the solution u(119905)is defined on the interval [0infin) and from (119862lip) it has to beunique

Remark 32 All nonlinear functions 119891 listed in Example 12can be considered in Theorems 24 and 31 However it isworth noting that we have additional assumptions on thenonlinearity in the semidiscrete case (conditions (119862cont) and(119862lip)) Thus for non-Lipschitz functions (eg

119891 (119909 119905 119906) =

minus |119906|119901minus1 119906 119906 = 0

0 119906 = 0(76)

with 119901 isin [0 1)) we only get the maximum principles indiscrete case On the other hand in discrete case the validityof maximum principle depends strongly on the interactionbetween the discretization step ℎ and the nonlinearity 119891 (see(12) we illustrate this dependence in detail in Section 8)

Let us finish with the two corollaries that are immediateconsequences of Theorems 24 and 31



(119862cont) (119862lip) and (119862119904119894119892119899

) for all 119879 gt 0 Then the uniquesolution 119906 of (36) is bounded



satisfy (119862cont) (119862lip) and (119862119904119894119892119899) for all 119879 gt 0 Then the unique

solution 119906 of (36) is nonnegative

8 Application Discrete and SemidiscreteNagumo Equation

In this section we apply the results of this paper to themost common nonlinearity occurring in the connectionwith the reaction-diffusion equation the bistabledouble-wellnonlinearity For simplicity we consider only the symmetriccase anduse interval [minus1 1] so that our arguments for positivevalues can be directly reproduced for the negative ones thatis we study

119891 (119909 119905 119906) = 120582119906 (1minus1199062) 120582 isin R (77)

Throughout this section we assume that the initial-boundaryconditions 120593 120585

119886 120585

119887are such that 119898

119879= minus1 and 119872

119879= 1 (or

possibly 119898119879

ge minus1 and 119872119879

le 1) for all 119879 gt 0Starting with the semidiscrete case (36) we observe that

119891 is continuous and locally Lipschitz continuous and satisfies119891(119909 119905 minus1) = 0 = 119891(119909 119905 1) Consequently for any given 120582 isinR we can apply Theorems 24 and 31 to get that there existsa unique solution of the semidiscrete problem (36) such that119906(119909 119905) isin [minus1 1] for all 119909 isin (119886 119887)Z and 119905 isin R+

0 We encounter a more interesting situation if we consider

the problem for the discrete Nagumo equation

119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119896Δ2

119909119909119906 (119909 minus 1 119905)

+ 120582119906 (119909 119905) (1minus1199062(119909 119905))

119906 (119909 0) = 120593 (119909) 119909 isin (119886 119887)Z

119906 (119886 119905) = 120585119886(119905) 119905 isin ℎN0

119906 (119887 119905) = 120585119887(119905) 119905 isin ℎN0

(78)

with 120582 isin R 119909 isin (119886 119887)Z 119905 isin ℎN0 ℎ gt 0 119896 gt 0Let us assume first that 120582 gt 0 We observe that 1198911015840(1) =

minus2120582 Hence the application ofTheorem 9 is restricted to casesfor which the slope of the dashed line in the forbidden area(see Figure 1) given by 2119896 minus 1ℎ (see the assumption (D))satisfies

2119896 minus1ℎ

le minus 2120582 or equivalently

ℎ le1

2 (119896 + 120582)

(79)

Consequently if ℎ le 12(119896 + 120582) we can apply Theorem 9to get that 119906(119909 119905) isin [minus1 1] for all 119909 isin (119886 119887)Z 119905 isin ℎN0 (seeFigure 3(a))

Once ℎ gt 12(119896 + 120582) Theorem 9 is no longer availableand we proceed to Theorem 11 We split our argument intotwo steps

(i) Since 119891(119906) is strictly concave on [0 1] and attains alocal maximum at 119906 = 1radic3 we can for each ℎ gt 0


1minus1minusS

lowast

Slowast

(a) ℎ le 12(119896 + 120582)

1 S

minusSlowast

Slowast

minus1

(b) ℎ isin (12(119896 + 120582) 1(2119896 + 1205822)) (ie 119878 isin (1 119878lowast))

1minus1minusS

lowast

S = Slowast

(c) ℎ = 1(2119896 + 1205822) (ie 119878 = 119878lowast)

1 S

minus1minusSlowast

Slowast

(d) ℎ gt 1(2119896 + 1205822) (ie 119878 gt 119878lowast)

Figure 3 Validity of maximum principles for the discrete Nagumo equation (78) with 119898119879

= minus1 and 119872119879

= 1 If ℎ is small enough we canapplyTheorem 9 see subplot (a) For ℎ isin (12(119896 + 120582) 1(2119896+1205822)] we can get weaker bounds (83) by the application ofTheorem 11 subplots(b) and (c) If ℎ is greater than 1(2119896 + 1205822) our results cannot be applied see subplot (d)

such that 2119896minus1ℎ gt minus2120582 (or equivalently ℎ gt 12(119896+

120582)) find a point 119879 isin (1radic3 1) such that the slope of atangent line of 119891 at 119879 is 2119896 minus 1ℎ One could computethat

119879 = radic 13

minus2119896ℎ minus 13120582ℎ

(80)

(ii) Let us denote by 119878 the 119909-intercept of the tangent lineof119891 at119879 Since the slope of this tangent line is 2119896minus1ℎwe can deduce that

119878 =minus21198793

1 minus 31198792 =2120582ℎradic(13 minus (2119896ℎ minus 1) 3120582ℎ)3

1 minus 2119896ℎ (81)

Given the shape of 119891(119906) for 119906 lt 0 (decreasingconvexly for119906 lt minus1radic3)Theorem 11 could be appliedonce 119891(minus119878) lies below or on the tangent line (cfFigures 2 and 3)We observe the following interestingfact choosing 119878 = 119878lowast = radic2 the tangent line of119891 at 119879 and the function 119891 intersect at the point[minusradic2 120582radic2] for each120582 gt 0 and 119896 gt 0 (see Figure 3(c))

Consequently we can applyTheorem 11 whenever 119878 leradic2 which is equivalent to

ℎ le1

2119896 + 1205822 (82)

If we choose 119878 gt 119878lowast then we can easily observethat 119891(minus119878) lies above the tangent line and thereforeTheorem 11 cannot be applied (see Figure 3(d))

Sincewe intentionally chose a symmetric119891 we can repeatthe same argument on the lower bound of solutions of (78)

If 120582 = 0 problem (78) reduces to the linear case andwe can trivially apply (12) whenever ℎ le 12119896 Finally if120582 lt 0 then the assumption (119863) is satisfied as long as theline (2119896 minus 1ℎ)(119906 minus 1) does not intersect for 119906 lt 0 or istangential to 119891(119906) = 120582119906(1 minus 1199062) One can easily compute thatthe tangential case occurs if 2119896minus1ℎ = 1205824Therefore we canapplyTheorem 9 whenever ℎ le 1(2119896minus1205824) If this conditionis violated we cannot useTheorem 11 since 119891(119906) gt 0 for 119906 gt 1and for each 119878 gt 1 the assumption (1198631015840) does not hold

To sum up depending on values of 120582 and ℎ we obtain thefollowing bounds for the solution of (78)


h

1

2k minus120582

4

Bounds [minus1 1]

1

2k +120582

2

No bounds

Bounds [minusS S]

120582

1

2(k + 120582)

(a)

Bounds [minus1 1]

Bounds [minusS S]

h = ht

2

120582

1

2120582

No bounds

hx

(b)

Figure 4 Bounds of solutions of the discrete Nagumo equation (78) with119898119879

= minus1 and119872119879

= 1 and their dependence on the values of 120582 andℎ (a) and on the values of space and time discretization steps ℎ

119909and ℎ = ℎ

119905for a fixed 120582 gt 0 (b) In the light gray area the bounds follow

fromTheorem 9 In the dark gray area the bounds are implied byTheorem 11 and 119878 is given by (81) In the white area we have no bounds onsolutions

119906 (119909 119905)

isin

[minus1 1] if 120582 le 0 ℎ le1

2119896 minus 1205824

[minus1 1] if 120582 gt 0 ℎ le1

2 (119896 + 120582)

[[

[

minus2120582ℎradic(13 minus (2119896ℎ minus 1) 3120582ℎ)3

1 minus 2119896ℎ2120582ℎradic(13 minus (2119896ℎ minus 1) 3120582ℎ)3

1 minus 2119896ℎ]]

]

if 120582 gt 0 ℎ isin (1

2 (119896 + 120582)

12119896 + 1205822

]

(83)

see Figure 4(a) for the illustrative summary of our results inthis section

Interestingly if we considered general space discretiza-tion step ℎ

119909gt 0 in (78) that is

119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ119905

= 119896119906 (119909 minus ℎ

119909 119905) minus 2119906 (119909 119905) + 119906 (119909 + ℎ

119909 119905)

ℎ2119909

+120582119906 (119909 119905) (1minus1199062(119909 119905))

(84)

one could get the same bounds as in (83) by replacing 119896 with119896ℎ2

119909 The dependence of regions of maximum principlesrsquo

validity on time and space discretization steps ℎ119905and ℎ

119909for

120582 gt 0 is depicted in Figure 4(b) (notice that very small valuesof ℎ

119905are necessary for small ℎ

119909)

9 Final Remarks

In this paper we studied a priori bounds for solutionsof initial-boundary value problems related to discrete andsemidiscrete diffusion Our main motivation for the initial-boundary problems was the direct comparison with theclassical results (Theorems 2 and 3) However note that in

the discrete case the results would be identical if we dealtwith an initial problem on Z On the other hand in thesemidiscrete or classical case even the solutions of lineardiffusion equations are not necessarily bounded (see eg[12])

Similarly the ideas of this paper could be easily extendedto a general reaction-diffusion-type equation (possibly withnonconstant time steps ℎ = ℎ(119905))

119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119886 (119909 119905) 119906 (119909 minus 1 119905)

+ 119887 (119909 119905) 119906 (119909 119905)

+ 119888 (119909 119905) 119906 (119909 + 1 119905)

+ 119891 (119909 119905 119906 (119909 119905))

(85)

For example the weakmaximumprinciple is then valid if ℎ geminus1119887 (replacing ℎ le 12119896) and the following generalization of(119863) holds (we assume that 119887 = minus(119886 + 119888) lt 0)

119898119879(1 minus 119886ℎ minus 119888ℎ) minus (1 + 119887ℎ) 119906

ℎle 119891 (119909 119905 119906)

le119872


ℎ

(86)


Conflict of Interests

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

Acknowledgment

The authors gratefully acknowledge the support by the CzechScience Foundation Grant no GA15-07690S

References

[1] J W Cahn ldquoTheory of crystal growth and interface motion incrystalline materialsrdquo Acta Metallurgica vol 8 no 8 pp 554ndash562 1960

[2] J Bell and C Cosner ldquoThreshold behavior and propagation fornonlinear differential-difference systems motivated by model-ing myelinated axonsrdquo Quarterly of Applied Mathematics vol42 no 1 pp 1ndash14 1984

[3] J P Keener ldquoPropagation and its failure in coupled systems ofdiscrete excitable cellsrdquo SIAM Journal on Applied Mathematicsvol 47 no 3 pp 556ndash572 1987

[4] S-N Chow ldquoLattice dynamical systemsrdquo inDynamical Systemsvol 1822 of Lecture Notes in Mathematics pp 1ndash102 SpringerBerlin Germany 2003

[5] S-N Chow J Mallet-Paret and W Shen ldquoTraveling waves inlattice dynamical systemsrdquo Journal of Differential Equations vol149 no 2 pp 248ndash291 1998

[6] B Zinner ldquoExistence of traveling wavefront solutions for thediscrete Nagumo equationrdquo Journal of Differential Equationsvol 96 no 1 pp 1ndash27 1992

[7] L Xu L Zou Z Chang S Lou X Peng and G Zhang ldquoBifur-cation in a discrete competition systemrdquo Discrete Dynamics inNature and Society vol 2014 Article ID 193143 7 pages 2014

[8] M H Protter and H F Weinberger Maximum Principles inDifferential Equations Springer New York NY USA 1984

[9] J Mawhin H B Thompson and E Tonkes ldquoUniqueness forboundary value problems for second order finite differenceequationsrdquo Journal of Difference Equations andApplications vol10 no 8 pp 749ndash757 2004

[10] P Stehlık and B Thompson ldquoMaximum principles for secondorder dynamic equations on time scalesrdquo Journal of Mathemat-ical Analysis and Applications vol 331 no 2 pp 913ndash926 2007

[11] P Stehlık and J Volek ldquoTransport equation on semidiscretedomains and Poisson-Bernoulli processesrdquo Journal of DifferenceEquations and Applications vol 19 no 3 pp 439ndash456 2013

[12] A Slavık and P Stehlık ldquoDynamic diffusion-type equations ondiscrete-space domainsrdquo Journal of Mathematical Analysis andApplications vol 427 no 1 pp 525ndash545 2015

[13] M Friesl A Slavık and P Stehlık ldquoDiscrete-space partialdynamic equations on time scales and applications to stochasticprocessesrdquoAppliedMathematics Letters vol 37 pp 86ndash90 2014

[14] A Slavık ldquoProduct integration on time scalesrdquo Dynamic Sys-tems and Applications vol 19 no 1 pp 97ndash112 2010

[15] J Volek ldquoMaximum and minimum principles for nonlineartransport equations on discrete-space domainsrdquoElectronic Jour-nal of Differential Equations vol 2014 no 78 pp 1ndash13 2014

[16] M Picone ldquoMaggiorazione degli integrali delle equazioni total-mente paraboliche alle derivate parziali del secondo ordinerdquoAnnali di Matematica Pura ed Applicata vol 7 no 1 pp 145ndash192 1929

[17] E E Levi ldquoSullrsquoequazione del calorerdquo Annali di MatematicaPura ed Applicata vol 14 no 1 pp 187ndash264 1908

[18] L Nirenberg ldquoA strong maximum principle for parabolicequationsrdquo Communications on Pure and Applied Mathematicsvol 6 no 2 pp 167ndash177 1953

[19] H F Weinberger ldquoInvariant sets for weakly coupled parabolicand elliptic systemsrdquo Rendiconti di Matematica vol 8 pp 295ndash310 1975

[20] W G Kelley and A C Peterson The Theory of DifferentialEquations Classical and Qualitative Springer New York NYUSA 2010

[21] E Hairer S P Noslashrsett and G Wanner Solving OrdinaryDifferential Equations I Nonstiff Problems Springer BerlinGermany 2007

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

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Volume 2014

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OptimizationJournal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

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Stochastic AnalysisInternational Journal of


for discrete reaction-diffusion equation (2) and then useEulermethod to show their validity for semidiscrete reaction-diffusion equation (1) Whereas the maximum principles inthe semidiscrete case exhibit similar features to those ofcontinuous reaction-diffusion model (ie they hold undersimilar assumptions) in the discrete case the weakmaximumprinciple holds for a smaller class of functions and the strongmaximum principle is only valid in a weaker sense involvingthe domain of dependence Finally we use the maximumprinciples to get the global existence of solutions of theinitial-boundary problem for the semidiscrete case (1) Allour results are illustrated in detail in Nagumo equationswith a symmetric bistable nonlinearity that is we considerproblems (1)-(2) with 119891(119906) = 120582119906(1 minus 1199062)

Our motivation is twofold First maximum principlescould be used to obtain comparison principles (Protter andWeinberger [8]) which in turn could serve as a valuable toolin the study of traveling waves for example Bell and Cosner[2] Moreover similarly as in the case of (non)existence oftraveling wave solutions for Nagumo equations it has beenshown that discrete and semidiscrete structures influence thevalidity of maximum principles in a significant way Even thesimplest one-dimensional linear problems require additionalassumptions on the step size seeMawhin et al [9] and Stehlıkand Thompson [10] In the case of partial difference andsemidiscrete equations the strong influence of the underlyingstructure on maximum principles has been described in thelinear case for transport equation in Stehlık and Volek [11]and for diffusion-type equations in Slavık and Stehlık [12]and Friesl et al [13] (interestingly the proofs of maximumprinciples in this case are based on product integration seeSlavık [14]) Finally simple maximum principles for nonlin-ear transport equations on semidiscrete domains have beenpresented in Volek [15]

In the classical case maximum principles for diffusion(and parabolic) equations go back to Picone [16] and Levi[17] Strong maximum principles were later established byNirenberg [18] and a survey of various versions and appli-cations could be found in a classical monograph Protter andWeinberger [8]

This paper is segmented in the following way In Sec-tion 2 we briefly summarize results for the classical reaction-diffusion equation Next we prove weak and strong maxi-mum principles for the discrete case (2) (Sections 3 and 4)In the case of the initial-boundary value problem for thesemidiscrete equation (1) we provide local existence results(Section 5) and maximum principles (Section 6) which weconsequently apply to get global existence of solutions inSection 7 Our results are then applied to the Nagumoequation with a symmetric bistable nonlinearity that isproblems (1)-(2) with 119891(119906) = 120582119906(1 minus 1199062) in Section 8

2 Reaction-Diffusion PartialDifferential Equation

In order tomotivate and compare our results for the reaction-diffusionequationsondiscrete-spacedomainswith the classical

reaction-diffusion equation we briefly summarize few basicresults for the following initial-boundary problem

120597119905119906 (119909 119905) = 119896120597

119909119909119906 (119909 119905) + 119891 (119909 119905 119906 (119909 119905))

119909 isin (119886 119887) 119905 isin R+ 119896 gt 0

119906 (119909 0) = 120593 (119909) 119909 isin [119886 119887]

119906 (119886 119905) = 120585119886(119905) 119905 isin R

+

0

119906 (119887 119905) = 120585119887(119905) 119905 isin R

+

0

(3)

where 119891 (119886 119887) times R+ times R rarr R is a reaction function and120593 [119886 119887] rarr R 120585

119886 120585

119887 R+

0 rarr R are initial-boundaryconditions satisfying 120593(119886) = 120585

119886(0) and 120593(119887) = 120585

119887(0)

The following existence and uniqueness result for (3) canbe found for example in [19 page 298]

Theorem 1 Let 119879 gt 0 be arbitrary and let 119891 be uniformlyHolder continuous in 119909 and 119905 and Lipschitz in 119906 for (119909 119905) isin(119886 119887) times (0 119879] Then for all Holder continuous initial-boundaryconditions 120593 120585

119886 120585

119887problem (3) has a unique bounded solution

which is defined on [119886 119887] times [0 119879]

We define the following two numbers for the brevity

119872119879

= max119909isin[119886119887]119905isin[0119879]

120593 (119909) 120585119886(119905) 120585

119887(119905)

119898119879

= min119909isin[119886119887]119905isin[0119879]

120593 (119909) 120585119886(119905) 120585

119887(119905)

(4)

For the linear diffusion equation (ie (3) with119891(119909 119905 119906) equiv0) the maximum principle is proved for example in [8Chapter 31] For the nonlinear problem (3) (ie 119891(119909 119905 119906) =0) the following weak maximum principle holds (see [19Theorem 1])

Theorem 2 Let 119879 gt 0 be arbitrary and let 119891 be uniformlyHolder continuous in 119909 and 119905 and Lipschitz in 119906 for (119909 119905) isin(119886 119887) times (0 119879] and assume that

119891 (119909 119905119872119879) le 0 le 119891 (119909 119905 119898

119879)

forall (119909 119905) isin (119886 119887) times (0 119879] (5)

Let 119906 be a continuous solution of (3) with Holder continuousinitial-boundary conditions 120593 120585

119886 120585

119887 Then

119898119879

le 119906 (119909 119905) le 119872119879

(6)

holds for all (119909 119905) isin [119886 119887] times [0 119879]

Moreover the strong maximum principle also holds (see[19 Theorem 2])

Theorem 3 Let the assumptions ofTheorem 2 be satisfied andlet 119906 be a solution of (3) on [119886 119887] times [0 119879] If 119906(1199090 1199050) = 119872

119879(or

119906(1199090 1199050) = 119898119879) for some (1199090 1199050) isin (119886 119887) times (0 119879] then

119906 (119909 119905) = 119872119879

(119900119903 119906 (119909 119905) =119898119879)

forall (119909 119905) isin [119886 119887] times [0 1199050]

(7)




119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119896Δ2

119909119909119906 (119909 minus 1 119905)

+ 119891 (119909 119905 119906 (119909 119905))


119906 (119909 0) = 120593 (119909) 119909 isin (119886 119887)Z

119906 (119886 119905) = 120585119886(119905) 119905 isin ℎN0

119906 (119887 119905) = 120585119887(119905) 119905 isin ℎN0

(8)


119886 120585










119906 (119909 119905 + ℎ)

=

119906 (119909 119905) + ℎ (119896Δ2119909119909

119906 (119909 minus 1 119905) + 119891 (119909 119905 119906 (119909 119905))) 119909 isin (119886 119887)Z

120585119886(119905 + ℎ) 119909 = 119886

120585119887(119905 + ℎ) 119909 = 119887

(9)


119872119879

= max119909isin(119886119887)Z 119905isin[0119879]ℎN0

120593 (119909) 120585119886(119905) 120585

119887(119905) (10)

119898119879

= min119909isin(119886119887)Z 119905isin[0119879]ℎN0

120593 (119909) 120585119886(119905) 120585

119887(119905) (11)




(119906 minus119898119879) le 119891 (119909 119905 119906) le


(119906 minus119872119879) (12)


and 119906 isin [119898119879119872

119879]



MT

mT

h 997888rarr 0+ f

h 997888rarr 0+

u

f(x t middot)



lt 119872119879 If ℎ gt 12119896 then (119863)





ℎ(119906 minus119898

119879) le 119891 (119909 119905 119906)


ℎ(119906 minus119872

119879) lt 0 for 119906 isin (119898

119879119872

119879)

(13)

a contradiction



119891(119909 119905 119898119879) = 119891(119909 119905119872

119879) = 0 for all 119909 isin (119886 119887)Z and

119905 isin [0 119879]ℎN0


119879and 120585

119886(119905) equiv

120585119887(119905) equiv 119872

119879 From 119891(119909 119905119872


119906 (119909 119905) = 120593 (119909) = 119872119879

for 119905 isin [0 119879]ℎN0

(14)



ℎN0

119898119879

le 119906 (119909 119905) le 119872119879


le 119906 (119909 119905 + ℎ) le 119872119879

(15)





119886(119905 +

ℎ) le 119872119879and 119906(119887 119905 + ℎ) = 120585

119887(119905 + ℎ) le 119872





119906 (119909 119905 + ℎ) = 119906 (119909 119905) + ℎ (119896119906 (119909 minus 1 119905) minus 2119896119906 (119909 119905)

+ 119896119906 (119909 + 1 119905) + 119891 (119909 119905 119906 (119909 119905))) le 2ℎ119896119872119879+ (1

minus 2ℎ119896) 119906 (119909 119905) + ℎ119891 (119909 119905 119906 (119909 119905))

(16)


le 119906(119909 119905) le 119872119879we

get

ℎ119891 (119909 119905 119906 (119909 119905)) le (2ℎ119896 minus 1) (119906 (119909 119905) minus119872119879) (17)

Therefore

119906 (119909 119905 + ℎ) le 2ℎ119896119872119879+ (1minus 2ℎ119896) 119906 (119909 119905)

+ ℎ119891 (119909 119905 119906 (119909 119905))

le 2ℎ119896119872119879+ (1minus 2ℎ119896) 119906 (119909 119905)

+ (2ℎ119896 minus 1) (119906 (119909 119905) minus119872119879) = 119872

119879

(18)



119898119879

le 119906 (119909 119905) le 119872119879

(19)



le 119906(119909 0) le 119872119879for all


ℎN0




119891 (0 119905 0) gt2ℎ119896 minus 1

ℎ(0minus119872

119879) = 0 (20)


119906 (0 1) = 119896119906 (minus1 0) + (1minus 2119896) 119906 (0 0) + 119896119906 (1 0)

+ 119891 (0 0 0) = 119891 (0 0 0) gt 0 = 119872119879

(21)



119879and 119877 le 119898

119879


ℎ(119906 minus119877) le 119891 (119909 119905 119906) le


(119906 minus 119904) (22)

S

R

MT

mT

f

u

f(x t middot)



119906 isin [119877 119878]



119877 le 119906 (119909 119905) le 119878 (23)



119877 le 119898119879

le 119906 (119909 0) le 119872119879

le 119878 forall119909 isin (119886 119887)Z (24)


















(25)


119906 (0 1) = 119906 (0 0) + 119896119906 (minus1 0) minus 2119896119906 (0 0) + 119896119906 (1 0)

+ 119891 (0 0 0) = 119872(26)


119906 (minus2 1) = 119906 (2 1) = 0

119906 (minus1 1) = 119906 (1 1) =119872

2

(27)




plusmn 119895 119895 = 0 1 1199050 minus 119905

ℎ

(28)



plusmn 119895 119895 = 0 1 119905 minus 1199050

ℎ

(29)



[0 119879]ℎN0


[119898119879119872

119879)


(119898119879119872

119879]

(c) 119891(119909 119905119872119879) le 0 and 119891(119909 119905 119898

119879) ge 0



(1) If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) then 119906(119909 119905) =

119872119879(or 119906(119909 119905) = 119898

119879) onD(1199090 1199050)

(2) If 119906(1199090 1199050) lt 119872119879(or 119906(1199090 1199050) gt 119898

119879) then 119906(119909 119905) lt

119872119879(or 119906(119909 119905) gt 119898

119879) onI(1199090 1199050)


119879


119879


119879




+ (1minus 2ℎ119896) 119906 (1199090 1199050 minus ℎ)

+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ))


+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ)) +119872119879

minus119872119879


+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ)) +119872119879

(30)

Thus (11986310158401015840) yields




119879 (32)




119879)

+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ)) +119872119879

(33)





Hence119906 (1199090 1199050) lt 119872

119879 (35)

a contradiction





119906119905(119909 119905) = 119896Δ2

119909119909119906 (119909 minus 1 119905) + 119891 (119909 119905 119906 (119909 119905))

119909 isin (119886 119887)Z 119905 isin R+

0 119896 gt 0

119906 (119909 0) = 120593 (119909) 119909 isin (119886 119887)Z

119906 (119886 119905) = 120585119886(119905) 119905 isin R

+

0

119906 (119887 119905) = 120585119887(119905) 119905 isin R

+

0

(36)

where 119906119905= 120597



119886 120585

119887isin

1198621(R+




u1015840 (119905) = g (119905 u (119905)) 119905 isin R+

0

u (0) = u(37)

where u R+

0 rarr R119873 g R+




119876 = (119905 u) isinR+



1003817100381710038171003817g (1199051 u1) minus g (1199052 u2)1003817100381710038171003817 le 119871

1003817100381710038171003817u1 minus u21003817100381710038171003817 (39)




0times R for all


(119886 119887)Z times R+


0isin (119886 119887)Z 1199050 isin R+

0


119876 (1199090 1199050 119906

0) = (119909

0 119905 119906) isin (119886 119887)Z timesR

+

0

timesR 1003816100381610038161003816119905 minus 119905

0

1003816100381610038161003816 le 1205721003816100381610038161003816119906 minus 119906

0

1003816100381610038161003816 le 120573(40)


1) (119909

0 1199052 119906

2) isin 119876

there is1003816100381610038161003816119891 (1199090 1199051 1199061) minus119891 (1199090 1199052 1199062)

1003816100381610038161003816 le 11987110038161003816100381610038161199061 minus1199062

1003816100381610038161003816 (41)





(((

(

119906(119886 + 1 119905)119906 (119886 + 2 119905)

119906 (119887 minus 2 119905)119906 (119887 minus 1 119905)

)))

)

1015840

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟u1015840(119905)

= 119896((

(


d


))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟A

(((

(

119906(119886 + 1 119905)119906 (119886 + 2 119905)

119906 (119887 minus 2 119905)119906 (119887 minus 1 119905)

)))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟u(119905)

+ 119896(((

(

120585119886(119905)

0

0120585119887(119905)

)))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120585(119905)

+(((

(

119891(119886 + 1 119905 119906 (119886 + 1 119905))119891 (119886 + 2 119905 119906 (119886 + 2 119905))


)))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟f(119905u(119905))

(42)


(

119906(119886 + 1 0)119906 (119886 + 2 0)

119906 (119887 minus 1 0)

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟u(0)

= (

120593(119886 + 1)120593 (119886 + 2)

120593 (119887 minus 1)

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120593

(43)


u1015840 (119905) = 119896Au (119905) + 119896120585 (119905) + f (119905 u (119905)) 119905 isin R+

0

u (0) = 120593(44)










y119894+1 = y

119894+ ℎg (119905

119894 y

119894)

y0 = u (0) = u(45)



y(ℎ)

(119905) = y119894+ (119905 minus 119905

119894) g (119905

119894 y

119894) 119905

119894le 119905 le 119905

119894+1 (46)




119876 = (119905 u) isinR+







119872119879

= max119909isin(119886119887)Z 119905isin[0119879]

120593 (119909) 120585119886(119905) 120585

119887(119905)

119898119879

= min119909isin(119886119887)Z 119905isin[0119879]

120593 (119909) 120585119886(119905) 120585

119887(119905)

(48)



0 and let 119891 satisfy (119862119888119900119899119905

) (119862119897119894119901) and

(119862119904119894119892119899

) 119891(119909 119905119872119879) le 0 le 119891(119909 119905 119898


[0 119879]



(119906 minus119898119879) le 119891 (119909 119905 119906) le


(119906 minus119872119879) (49)


119879]


ℎisin (119886 119887)Z

119905ℎisin [0 119879] and 119906

ℎisin [119898

119879119872

119879] such that

119891 (119909ℎ 119905ℎ 119906

ℎ) gt


(119906ℎminus119872

119879) (50)


119898 119905119898

119906119898)infin119898=1 (we denote 119909

119898= 119909

ℎ119898 119905

119898= 119905

ℎ119898 119906

119898= 119906

ℎ119898) such

thatℎ119898

997888rarr 0+

119891 (119909119898 119905119898 119906

119898) gt

2ℎ119898119896 minus 1ℎ119898

(119906119898

minus119872119879)

(51)




lt119872



infin119897=1 sub

119906119898infin119898=1 such that 119906

119898119897rarr 119872



119898 119905119898 119906



119879119872

119879]


infin119897=1 sub 119906

119898infin119898=1 such

that 119906119898119897



119898119897infin119897=1 sub 119909

119898infin119898=1 such that 119909



119898119897infin119897=1 sub 119905

119898infin119898=1 such that 119905

119898119897rarr


1 minus 2ℎ119898

119896

ℎ119898

ge 119871

119872119879minus119906

119898

le 120573

119909119898

= 119909

10038161003816100381610038161003816119905119898 minus 10038161003816100381610038161003816 lt 120572

(52)


If we put 119909 = 119909119898

= 119905119898

= 119906119898

and = 119872119879



119898

119896

ℎ119898

(119872119879minus ) lt 119891 (119909 )

le 119891 (119909 ) minus 119891 (119909 119872119879)

= 119891 (119909 ) minus 119891 (119909 )

le1003816100381610038161003816119891 (119909 ) minus 119891 (119909 )

1003816100381610038161003816

(53)





[0 119879] Then

119898119879

le 119906 (119909 119905) le 119872119879

(54)






119906 (119909119888 119905119888) gt 119872T (55)


119888) such that


(b) 119906(1199090 1199050) = 119872119879


119906 (1199090 119905) gt 119872119879

on (1199050 1199050 + 120575) (56)







119910(ℎ)

(119909 119905) le 119872119879

on [1199050 1199050 + 120576] forall119909 isin (119886 119887)Z (57)

But if y(ℎ)


(119909 119905) le 119872119879


119906 (119909 119905) le 119872119879





119877 le 119898119879such that

119891 (119909 119905 119878) le 0 le 119891 (119909 119905 119877) (59)




[0 119879] Then

119877 le 119906 (119909 119905) le 119878 (60)





1199061015840 (119905) le 120573 (119905) 119906 (119905) for 119905 isin (119903 119904) (61)

then

119906 (119905) le 119906 (119903) 119890int119905

119903120573(120591)119889120591 for 119905 isin [119903 119904] (62)




[0 119879] If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) for some 1199090 isin

(119886 119887)Z and 1199050 isin (0 119879] then

119906 (1199090 119905) = 119872119879

(119900119903 119906 (1199090 119905) =119898119879)

forall119905 isin [0 1199050]

(63)



119879





119906 (1199090 119905119888) = 119872119879

119906 (1199090 119905) lt 119872119879

on [119905119888minus 120576 119905

119888)

(64)


119888minus



119906119905(1199090 119905)

= 119896 (119906 (1199090 minus 1 119905) minus 2119906 (1199090 119905) + 119906 (1199090 + 1 119905))

+119891 (1199090 119905 119906 (1199090 119905))

le 119896 (2119872119879minus 2119906 (1199090 119905)) +119891 (1199090 119905 119906 (1199090 119905))

minus119891 (1199090 119905119872119879) +119891 (1199090 119905119872119879

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟le0

le 119896 (2119872119879minus 2119906 (1199090 119905))

+1003816100381610038161003816119891 (1199090 119905 119906 (1199090 119905)) minus119891 (1199090 119905119872119879

)1003816100381610038161003816

le 119896 (2119872119879minus 2119906 (1199090 119905)) + 119871

1003816100381610038161003816119906 (1199090 119905) minus119872119879

1003816100381610038161003816

= 119896 (2119872119879minus 2119906 (1199090 119905)) + 119871 (119872

119879minus119906 (1199090 119905))

= minus (2119896 + 119871) (119906 (1199090 119905) minus119872119879)

(65)




119888minus




V1015840 (119905) le 120572V (119905) (66)


V (119905) le V (119905119888minus 120576) 119890120572(119905minus119905119888+120576) (67)

and hence

119906 (1199090 119905) le 119872119879minus (119872


lt 119872119879

(68)

on [119905119888minus 120576 119905





[119886 119887]Z times [0 119879] If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) for some


119906 (119909 119905) = 119872119879

(or 119906 (119909 119905) =119898119879)


(69)


If 119906(1199090minus1 119905119888) lt 119872119879(or 119906(1199090+1 119905119888) lt 119872

119879) at some 119905

119888isin [0 1199050]


119906119905(1199090 119905119888)

= 119896 (119906 (1199090 minus 1 119905119888) minus 2119906 (1199090 119905119888) + 119906 (1199090 + 1 119905

119888))

+119891 (1199090 119905119888 119906 (1199090 119905119888))

lt 119896 (2119872119879minus 2119872

119879) +119891 (1199090 119905119888119872119879

) le 0

(70)






similar







120578 = infin

or u (119905)119905rarr120578minus

997888997888997888997888997888rarr infin

(71)





0 whichsatisfies

inf119909isin(119886119887)Z 119905isinR

+

0

120593 (119909) 120585119886(119905) 120585

119887(119905) le 119906 (119909 119905)


+

0

120593 (119909) 120585119886(119905) 120585

119887(119905)

(72)


0


120578 = infin

or u (119905)119905rarr120578minus

997888997888997888997888997888rarr infin

(73)


119870isin (119886 119887)Z and 119905

119870isin [0 120578) such that

1003816100381610038161003816119906 (119909119870 119905119870)1003816100381610038161003816 gt 119870 (74)

If we put 119870 = max|119898120578| |119872

120578| lt infin (120585

119886 120585


on R+


1003816100381610038161003816119906 (119909119870 119905119870)1003816100381610038161003816 gt max

10038161003816100381610038161003816119898120578

10038161003816100381610038161003816 10038161003816100381610038161003816119872120578

10038161003816100381610038161003816 (75)




119891 (119909 119905 119906) =

minus |119906|119901minus1 119906 119906 = 0

0 119906 = 0(76)





(119862cont) (119862lip) and (119862119904119894119892119899








119891 (119909 119905 119906) = 120582119906 (1minus1199062) 120582 isin R (77)


119886 120585


119879= minus1 and 119872

119879= 1 (or

possibly 119898119879

ge minus1 and 119872119879





119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119896Δ2

119909119909119906 (119909 minus 1 119905)

+ 120582119906 (119909 119905) (1minus1199062(119909 119905))

119906 (119909 0) = 120593 (119909) 119909 isin (119886 119887)Z

119906 (119886 119905) = 120585119886(119905) 119905 isin ℎN0

119906 (119887 119905) = 120585119887(119905) 119905 isin ℎN0

(78)



2119896 minus1ℎ


ℎ le1

2 (119896 + 120582)

(79)





1minus1minusS

lowast

Slowast

(a) ℎ le 12(119896 + 120582)

1 S

minusSlowast

Slowast

minus1


1minus1minusS

lowast

S = Slowast

(c) ℎ = 1(2119896 + 1205822) (ie 119878 = 119878lowast)

1 S

minus1minusSlowast

Slowast

(d) ℎ gt 1(2119896 + 1205822) (ie 119878 gt 119878lowast)


= minus1 and 119872119879




119879 = radic 13


(80)


119878 =minus21198793


1 minus 2119896ℎ (81)



ℎ le1

2119896 + 1205822 (82)






h

1

2k minus120582

4

Bounds [minus1 1]

1

2k +120582

2

No bounds

Bounds [minusS S]

120582

1

2(k + 120582)

(a)

Bounds [minus1 1]

Bounds [minusS S]

h = ht

2

120582

1

2120582

No bounds

hx

(b)


= minus1 and119872119879


119909and ℎ = ℎ



119906 (119909 119905)

isin

[minus1 1] if 120582 le 0 ℎ le1

2119896 minus 1205824

[minus1 1] if 120582 gt 0 ℎ le1

2 (119896 + 120582)

[[

[



1 minus 2119896ℎ]]

]

if 120582 gt 0 ℎ isin (1

2 (119896 + 120582)

12119896 + 1205822

]

(83)



119909gt 0 in (78) that is

119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ119905

= 119896119906 (119909 minus ℎ

119909 119905) minus 2119906 (119909 119905) + 119906 (119909 + ℎ

119909 119905)

ℎ2119909

+120582119906 (119909 119905) (1minus1199062(119909 119905))

(84)




119909for



119909)

9 Final Remarks




119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119886 (119909 119905) 119906 (119909 minus 1 119905)

+ 119887 (119909 119905) 119906 (119909 119905)

+ 119888 (119909 119905) 119906 (119909 + 1 119905)

+ 119891 (119909 119905 119906 (119909 119905))

(85)



ℎle 119891 (119909 119905 119906)

le119872


ℎ

(86)




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119896Δ2

119909119909119906 (119909 minus 1 119905)

+ 119891 (119909 119905 119906 (119909 119905))


119906 (119909 0) = 120593 (119909) 119909 isin (119886 119887)Z

119906 (119886 119905) = 120585119886(119905) 119905 isin ℎN0

119906 (119887 119905) = 120585119887(119905) 119905 isin ℎN0

(8)


119886 120585










119906 (119909 119905 + ℎ)

=

119906 (119909 119905) + ℎ (119896Δ2119909119909

119906 (119909 minus 1 119905) + 119891 (119909 119905 119906 (119909 119905))) 119909 isin (119886 119887)Z

120585119886(119905 + ℎ) 119909 = 119886

120585119887(119905 + ℎ) 119909 = 119887

(9)


119872119879

= max119909isin(119886119887)Z 119905isin[0119879]ℎN0

120593 (119909) 120585119886(119905) 120585

119887(119905) (10)

119898119879

= min119909isin(119886119887)Z 119905isin[0119879]ℎN0

120593 (119909) 120585119886(119905) 120585

119887(119905) (11)




(119906 minus119898119879) le 119891 (119909 119905 119906) le


(119906 minus119872119879) (12)


and 119906 isin [119898119879119872

119879]



MT

mT

h 997888rarr 0+ f

h 997888rarr 0+

u

f(x t middot)



lt 119872119879 If ℎ gt 12119896 then (119863)





ℎ(119906 minus119898

119879) le 119891 (119909 119905 119906)


ℎ(119906 minus119872

119879) lt 0 for 119906 isin (119898

119879119872

119879)

(13)

a contradiction



119891(119909 119905 119898119879) = 119891(119909 119905119872

119879) = 0 for all 119909 isin (119886 119887)Z and

119905 isin [0 119879]ℎN0


119879and 120585

119886(119905) equiv

120585119887(119905) equiv 119872

119879 From 119891(119909 119905119872


119906 (119909 119905) = 120593 (119909) = 119872119879

for 119905 isin [0 119879]ℎN0

(14)



ℎN0

119898119879

le 119906 (119909 119905) le 119872119879


le 119906 (119909 119905 + ℎ) le 119872119879

(15)





119886(119905 +

ℎ) le 119872119879and 119906(119887 119905 + ℎ) = 120585

119887(119905 + ℎ) le 119872





119906 (119909 119905 + ℎ) = 119906 (119909 119905) + ℎ (119896119906 (119909 minus 1 119905) minus 2119896119906 (119909 119905)

+ 119896119906 (119909 + 1 119905) + 119891 (119909 119905 119906 (119909 119905))) le 2ℎ119896119872119879+ (1

minus 2ℎ119896) 119906 (119909 119905) + ℎ119891 (119909 119905 119906 (119909 119905))

(16)


le 119906(119909 119905) le 119872119879we

get

ℎ119891 (119909 119905 119906 (119909 119905)) le (2ℎ119896 minus 1) (119906 (119909 119905) minus119872119879) (17)

Therefore

119906 (119909 119905 + ℎ) le 2ℎ119896119872119879+ (1minus 2ℎ119896) 119906 (119909 119905)

+ ℎ119891 (119909 119905 119906 (119909 119905))

le 2ℎ119896119872119879+ (1minus 2ℎ119896) 119906 (119909 119905)

+ (2ℎ119896 minus 1) (119906 (119909 119905) minus119872119879) = 119872

119879

(18)



119898119879

le 119906 (119909 119905) le 119872119879

(19)



le 119906(119909 0) le 119872119879for all


ℎN0




119891 (0 119905 0) gt2ℎ119896 minus 1

ℎ(0minus119872

119879) = 0 (20)


119906 (0 1) = 119896119906 (minus1 0) + (1minus 2119896) 119906 (0 0) + 119896119906 (1 0)

+ 119891 (0 0 0) = 119891 (0 0 0) gt 0 = 119872119879

(21)



119879and 119877 le 119898

119879


ℎ(119906 minus119877) le 119891 (119909 119905 119906) le


(119906 minus 119904) (22)

S

R

MT

mT

f

u

f(x t middot)



119906 isin [119877 119878]



119877 le 119906 (119909 119905) le 119878 (23)



119877 le 119898119879

le 119906 (119909 0) le 119872119879

le 119878 forall119909 isin (119886 119887)Z (24)


















(25)


119906 (0 1) = 119906 (0 0) + 119896119906 (minus1 0) minus 2119896119906 (0 0) + 119896119906 (1 0)

+ 119891 (0 0 0) = 119872(26)


119906 (minus2 1) = 119906 (2 1) = 0

119906 (minus1 1) = 119906 (1 1) =119872

2

(27)




plusmn 119895 119895 = 0 1 1199050 minus 119905

ℎ

(28)



plusmn 119895 119895 = 0 1 119905 minus 1199050

ℎ

(29)



[0 119879]ℎN0


[119898119879119872

119879)


(119898119879119872

119879]

(c) 119891(119909 119905119872119879) le 0 and 119891(119909 119905 119898

119879) ge 0



(1) If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) then 119906(119909 119905) =

119872119879(or 119906(119909 119905) = 119898

119879) onD(1199090 1199050)

(2) If 119906(1199090 1199050) lt 119872119879(or 119906(1199090 1199050) gt 119898

119879) then 119906(119909 119905) lt

119872119879(or 119906(119909 119905) gt 119898

119879) onI(1199090 1199050)


119879


119879


119879




+ (1minus 2ℎ119896) 119906 (1199090 1199050 minus ℎ)

+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ))


+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ)) +119872119879

minus119872119879


+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ)) +119872119879

(30)

Thus (11986310158401015840) yields




119879 (32)




119879)

+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ)) +119872119879

(33)





Hence119906 (1199090 1199050) lt 119872

119879 (35)

a contradiction





119906119905(119909 119905) = 119896Δ2

119909119909119906 (119909 minus 1 119905) + 119891 (119909 119905 119906 (119909 119905))

119909 isin (119886 119887)Z 119905 isin R+

0 119896 gt 0

119906 (119909 0) = 120593 (119909) 119909 isin (119886 119887)Z

119906 (119886 119905) = 120585119886(119905) 119905 isin R

+

0

119906 (119887 119905) = 120585119887(119905) 119905 isin R

+

0

(36)

where 119906119905= 120597



119886 120585

119887isin

1198621(R+




u1015840 (119905) = g (119905 u (119905)) 119905 isin R+

0

u (0) = u(37)

where u R+

0 rarr R119873 g R+




119876 = (119905 u) isinR+



1003817100381710038171003817g (1199051 u1) minus g (1199052 u2)1003817100381710038171003817 le 119871

1003817100381710038171003817u1 minus u21003817100381710038171003817 (39)




0times R for all


(119886 119887)Z times R+


0isin (119886 119887)Z 1199050 isin R+

0


119876 (1199090 1199050 119906

0) = (119909

0 119905 119906) isin (119886 119887)Z timesR

+

0

timesR 1003816100381610038161003816119905 minus 119905

0

1003816100381610038161003816 le 1205721003816100381610038161003816119906 minus 119906

0

1003816100381610038161003816 le 120573(40)


1) (119909

0 1199052 119906

2) isin 119876

there is1003816100381610038161003816119891 (1199090 1199051 1199061) minus119891 (1199090 1199052 1199062)

1003816100381610038161003816 le 11987110038161003816100381610038161199061 minus1199062

1003816100381610038161003816 (41)





(((

(

119906(119886 + 1 119905)119906 (119886 + 2 119905)

119906 (119887 minus 2 119905)119906 (119887 minus 1 119905)

)))

)

1015840

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟u1015840(119905)

= 119896((

(


d


))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟A

(((

(

119906(119886 + 1 119905)119906 (119886 + 2 119905)

119906 (119887 minus 2 119905)119906 (119887 minus 1 119905)

)))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟u(119905)

+ 119896(((

(

120585119886(119905)

0

0120585119887(119905)

)))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120585(119905)

+(((

(

119891(119886 + 1 119905 119906 (119886 + 1 119905))119891 (119886 + 2 119905 119906 (119886 + 2 119905))


)))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟f(119905u(119905))

(42)


(

119906(119886 + 1 0)119906 (119886 + 2 0)

119906 (119887 minus 1 0)

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟u(0)

= (

120593(119886 + 1)120593 (119886 + 2)

120593 (119887 minus 1)

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120593

(43)


u1015840 (119905) = 119896Au (119905) + 119896120585 (119905) + f (119905 u (119905)) 119905 isin R+

0

u (0) = 120593(44)










y119894+1 = y

119894+ ℎg (119905

119894 y

119894)

y0 = u (0) = u(45)



y(ℎ)

(119905) = y119894+ (119905 minus 119905

119894) g (119905

119894 y

119894) 119905

119894le 119905 le 119905

119894+1 (46)




119876 = (119905 u) isinR+







119872119879

= max119909isin(119886119887)Z 119905isin[0119879]

120593 (119909) 120585119886(119905) 120585

119887(119905)

119898119879

= min119909isin(119886119887)Z 119905isin[0119879]

120593 (119909) 120585119886(119905) 120585

119887(119905)

(48)



0 and let 119891 satisfy (119862119888119900119899119905

) (119862119897119894119901) and

(119862119904119894119892119899

) 119891(119909 119905119872119879) le 0 le 119891(119909 119905 119898


[0 119879]



(119906 minus119898119879) le 119891 (119909 119905 119906) le


(119906 minus119872119879) (49)


119879]


ℎisin (119886 119887)Z

119905ℎisin [0 119879] and 119906

ℎisin [119898

119879119872

119879] such that

119891 (119909ℎ 119905ℎ 119906

ℎ) gt


(119906ℎminus119872

119879) (50)


119898 119905119898

119906119898)infin119898=1 (we denote 119909

119898= 119909

ℎ119898 119905

119898= 119905

ℎ119898 119906

119898= 119906

ℎ119898) such

thatℎ119898

997888rarr 0+

119891 (119909119898 119905119898 119906

119898) gt

2ℎ119898119896 minus 1ℎ119898

(119906119898

minus119872119879)

(51)




lt119872



infin119897=1 sub

119906119898infin119898=1 such that 119906

119898119897rarr 119872



119898 119905119898 119906



119879119872

119879]


infin119897=1 sub 119906

119898infin119898=1 such

that 119906119898119897



119898119897infin119897=1 sub 119909

119898infin119898=1 such that 119909



119898119897infin119897=1 sub 119905

119898infin119898=1 such that 119905

119898119897rarr


1 minus 2ℎ119898

119896

ℎ119898

ge 119871

119872119879minus119906

119898

le 120573

119909119898

= 119909

10038161003816100381610038161003816119905119898 minus 10038161003816100381610038161003816 lt 120572

(52)


If we put 119909 = 119909119898

= 119905119898

= 119906119898

and = 119872119879



119898

119896

ℎ119898

(119872119879minus ) lt 119891 (119909 )

le 119891 (119909 ) minus 119891 (119909 119872119879)

= 119891 (119909 ) minus 119891 (119909 )

le1003816100381610038161003816119891 (119909 ) minus 119891 (119909 )

1003816100381610038161003816

(53)





[0 119879] Then

119898119879

le 119906 (119909 119905) le 119872119879

(54)






119906 (119909119888 119905119888) gt 119872T (55)


119888) such that


(b) 119906(1199090 1199050) = 119872119879


119906 (1199090 119905) gt 119872119879

on (1199050 1199050 + 120575) (56)







119910(ℎ)

(119909 119905) le 119872119879

on [1199050 1199050 + 120576] forall119909 isin (119886 119887)Z (57)

But if y(ℎ)


(119909 119905) le 119872119879


119906 (119909 119905) le 119872119879





119877 le 119898119879such that

119891 (119909 119905 119878) le 0 le 119891 (119909 119905 119877) (59)




[0 119879] Then

119877 le 119906 (119909 119905) le 119878 (60)





1199061015840 (119905) le 120573 (119905) 119906 (119905) for 119905 isin (119903 119904) (61)

then

119906 (119905) le 119906 (119903) 119890int119905

119903120573(120591)119889120591 for 119905 isin [119903 119904] (62)




[0 119879] If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) for some 1199090 isin

(119886 119887)Z and 1199050 isin (0 119879] then

119906 (1199090 119905) = 119872119879

(119900119903 119906 (1199090 119905) =119898119879)

forall119905 isin [0 1199050]

(63)



119879





119906 (1199090 119905119888) = 119872119879

119906 (1199090 119905) lt 119872119879

on [119905119888minus 120576 119905

119888)

(64)


119888minus



119906119905(1199090 119905)

= 119896 (119906 (1199090 minus 1 119905) minus 2119906 (1199090 119905) + 119906 (1199090 + 1 119905))

+119891 (1199090 119905 119906 (1199090 119905))

le 119896 (2119872119879minus 2119906 (1199090 119905)) +119891 (1199090 119905 119906 (1199090 119905))

minus119891 (1199090 119905119872119879) +119891 (1199090 119905119872119879

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟le0

le 119896 (2119872119879minus 2119906 (1199090 119905))

+1003816100381610038161003816119891 (1199090 119905 119906 (1199090 119905)) minus119891 (1199090 119905119872119879

)1003816100381610038161003816

le 119896 (2119872119879minus 2119906 (1199090 119905)) + 119871

1003816100381610038161003816119906 (1199090 119905) minus119872119879

1003816100381610038161003816

= 119896 (2119872119879minus 2119906 (1199090 119905)) + 119871 (119872

119879minus119906 (1199090 119905))

= minus (2119896 + 119871) (119906 (1199090 119905) minus119872119879)

(65)




119888minus




V1015840 (119905) le 120572V (119905) (66)


V (119905) le V (119905119888minus 120576) 119890120572(119905minus119905119888+120576) (67)

and hence

119906 (1199090 119905) le 119872119879minus (119872


lt 119872119879

(68)

on [119905119888minus 120576 119905





[119886 119887]Z times [0 119879] If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) for some


119906 (119909 119905) = 119872119879

(or 119906 (119909 119905) =119898119879)


(69)


If 119906(1199090minus1 119905119888) lt 119872119879(or 119906(1199090+1 119905119888) lt 119872

119879) at some 119905

119888isin [0 1199050]


119906119905(1199090 119905119888)

= 119896 (119906 (1199090 minus 1 119905119888) minus 2119906 (1199090 119905119888) + 119906 (1199090 + 1 119905

119888))

+119891 (1199090 119905119888 119906 (1199090 119905119888))

lt 119896 (2119872119879minus 2119872

119879) +119891 (1199090 119905119888119872119879

) le 0

(70)






similar







120578 = infin

or u (119905)119905rarr120578minus

997888997888997888997888997888rarr infin

(71)





0 whichsatisfies

inf119909isin(119886119887)Z 119905isinR

+

0

120593 (119909) 120585119886(119905) 120585

119887(119905) le 119906 (119909 119905)


+

0

120593 (119909) 120585119886(119905) 120585

119887(119905)

(72)


0


120578 = infin

or u (119905)119905rarr120578minus

997888997888997888997888997888rarr infin

(73)


119870isin (119886 119887)Z and 119905

119870isin [0 120578) such that

1003816100381610038161003816119906 (119909119870 119905119870)1003816100381610038161003816 gt 119870 (74)

If we put 119870 = max|119898120578| |119872

120578| lt infin (120585

119886 120585


on R+


1003816100381610038161003816119906 (119909119870 119905119870)1003816100381610038161003816 gt max

10038161003816100381610038161003816119898120578

10038161003816100381610038161003816 10038161003816100381610038161003816119872120578

10038161003816100381610038161003816 (75)




119891 (119909 119905 119906) =

minus |119906|119901minus1 119906 119906 = 0

0 119906 = 0(76)





(119862cont) (119862lip) and (119862119904119894119892119899








119891 (119909 119905 119906) = 120582119906 (1minus1199062) 120582 isin R (77)


119886 120585


119879= minus1 and 119872

119879= 1 (or

possibly 119898119879

ge minus1 and 119872119879





119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119896Δ2

119909119909119906 (119909 minus 1 119905)

+ 120582119906 (119909 119905) (1minus1199062(119909 119905))

119906 (119909 0) = 120593 (119909) 119909 isin (119886 119887)Z

119906 (119886 119905) = 120585119886(119905) 119905 isin ℎN0

119906 (119887 119905) = 120585119887(119905) 119905 isin ℎN0

(78)



2119896 minus1ℎ


ℎ le1

2 (119896 + 120582)

(79)





1minus1minusS

lowast

Slowast

(a) ℎ le 12(119896 + 120582)

1 S

minusSlowast

Slowast

minus1


1minus1minusS

lowast

S = Slowast

(c) ℎ = 1(2119896 + 1205822) (ie 119878 = 119878lowast)

1 S

minus1minusSlowast

Slowast

(d) ℎ gt 1(2119896 + 1205822) (ie 119878 gt 119878lowast)


= minus1 and 119872119879




119879 = radic 13


(80)


119878 =minus21198793


1 minus 2119896ℎ (81)



ℎ le1

2119896 + 1205822 (82)






h

1

2k minus120582

4

Bounds [minus1 1]

1

2k +120582

2

No bounds

Bounds [minusS S]

120582

1

2(k + 120582)

(a)

Bounds [minus1 1]

Bounds [minusS S]

h = ht

2

120582

1

2120582

No bounds

hx

(b)


= minus1 and119872119879


119909and ℎ = ℎ



119906 (119909 119905)

isin

[minus1 1] if 120582 le 0 ℎ le1

2119896 minus 1205824

[minus1 1] if 120582 gt 0 ℎ le1

2 (119896 + 120582)

[[

[



1 minus 2119896ℎ]]

]

if 120582 gt 0 ℎ isin (1

2 (119896 + 120582)

12119896 + 1205822

]

(83)



119909gt 0 in (78) that is

119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ119905

= 119896119906 (119909 minus ℎ

119909 119905) minus 2119906 (119909 119905) + 119906 (119909 + ℎ

119909 119905)

ℎ2119909

+120582119906 (119909 119905) (1minus1199062(119909 119905))

(84)




119909for



119909)

9 Final Remarks




119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119886 (119909 119905) 119906 (119909 minus 1 119905)

+ 119887 (119909 119905) 119906 (119909 119905)

+ 119888 (119909 119905) 119906 (119909 + 1 119905)

+ 119891 (119909 119905 119906 (119909 119905))

(85)



ℎle 119891 (119909 119905 119906)

le119872


ℎ

(86)




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119906 (119909 119905 + ℎ) = 119906 (119909 119905) + ℎ (119896119906 (119909 minus 1 119905) minus 2119896119906 (119909 119905)

+ 119896119906 (119909 + 1 119905) + 119891 (119909 119905 119906 (119909 119905))) le 2ℎ119896119872119879+ (1

minus 2ℎ119896) 119906 (119909 119905) + ℎ119891 (119909 119905 119906 (119909 119905))

(16)


le 119906(119909 119905) le 119872119879we

get

ℎ119891 (119909 119905 119906 (119909 119905)) le (2ℎ119896 minus 1) (119906 (119909 119905) minus119872119879) (17)

Therefore

119906 (119909 119905 + ℎ) le 2ℎ119896119872119879+ (1minus 2ℎ119896) 119906 (119909 119905)

+ ℎ119891 (119909 119905 119906 (119909 119905))

le 2ℎ119896119872119879+ (1minus 2ℎ119896) 119906 (119909 119905)

+ (2ℎ119896 minus 1) (119906 (119909 119905) minus119872119879) = 119872

119879

(18)



119898119879

le 119906 (119909 119905) le 119872119879

(19)



le 119906(119909 0) le 119872119879for all


ℎN0




119891 (0 119905 0) gt2ℎ119896 minus 1

ℎ(0minus119872

119879) = 0 (20)


119906 (0 1) = 119896119906 (minus1 0) + (1minus 2119896) 119906 (0 0) + 119896119906 (1 0)

+ 119891 (0 0 0) = 119891 (0 0 0) gt 0 = 119872119879

(21)



119879and 119877 le 119898

119879


ℎ(119906 minus119877) le 119891 (119909 119905 119906) le


(119906 minus 119904) (22)

S

R

MT

mT

f

u

f(x t middot)



119906 isin [119877 119878]



119877 le 119906 (119909 119905) le 119878 (23)



119877 le 119898119879

le 119906 (119909 0) le 119872119879

le 119878 forall119909 isin (119886 119887)Z (24)


















(25)


119906 (0 1) = 119906 (0 0) + 119896119906 (minus1 0) minus 2119896119906 (0 0) + 119896119906 (1 0)

+ 119891 (0 0 0) = 119872(26)


119906 (minus2 1) = 119906 (2 1) = 0

119906 (minus1 1) = 119906 (1 1) =119872

2

(27)




plusmn 119895 119895 = 0 1 1199050 minus 119905

ℎ

(28)



plusmn 119895 119895 = 0 1 119905 minus 1199050

ℎ

(29)



[0 119879]ℎN0


[119898119879119872

119879)


(119898119879119872

119879]

(c) 119891(119909 119905119872119879) le 0 and 119891(119909 119905 119898

119879) ge 0



(1) If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) then 119906(119909 119905) =

119872119879(or 119906(119909 119905) = 119898

119879) onD(1199090 1199050)

(2) If 119906(1199090 1199050) lt 119872119879(or 119906(1199090 1199050) gt 119898

119879) then 119906(119909 119905) lt

119872119879(or 119906(119909 119905) gt 119898

119879) onI(1199090 1199050)


119879


119879


119879




+ (1minus 2ℎ119896) 119906 (1199090 1199050 minus ℎ)

+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ))


+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ)) +119872119879

minus119872119879


+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ)) +119872119879

(30)

Thus (11986310158401015840) yields




119879 (32)




119879)

+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ)) +119872119879

(33)





Hence119906 (1199090 1199050) lt 119872

119879 (35)

a contradiction





119906119905(119909 119905) = 119896Δ2

119909119909119906 (119909 minus 1 119905) + 119891 (119909 119905 119906 (119909 119905))

119909 isin (119886 119887)Z 119905 isin R+

0 119896 gt 0

119906 (119909 0) = 120593 (119909) 119909 isin (119886 119887)Z

119906 (119886 119905) = 120585119886(119905) 119905 isin R

+

0

119906 (119887 119905) = 120585119887(119905) 119905 isin R

+

0

(36)

where 119906119905= 120597



119886 120585

119887isin

1198621(R+




u1015840 (119905) = g (119905 u (119905)) 119905 isin R+

0

u (0) = u(37)

where u R+

0 rarr R119873 g R+




119876 = (119905 u) isinR+



1003817100381710038171003817g (1199051 u1) minus g (1199052 u2)1003817100381710038171003817 le 119871

1003817100381710038171003817u1 minus u21003817100381710038171003817 (39)




0times R for all


(119886 119887)Z times R+


0isin (119886 119887)Z 1199050 isin R+

0


119876 (1199090 1199050 119906

0) = (119909

0 119905 119906) isin (119886 119887)Z timesR

+

0

timesR 1003816100381610038161003816119905 minus 119905

0

1003816100381610038161003816 le 1205721003816100381610038161003816119906 minus 119906

0

1003816100381610038161003816 le 120573(40)


1) (119909

0 1199052 119906

2) isin 119876

there is1003816100381610038161003816119891 (1199090 1199051 1199061) minus119891 (1199090 1199052 1199062)

1003816100381610038161003816 le 11987110038161003816100381610038161199061 minus1199062

1003816100381610038161003816 (41)





(((

(

119906(119886 + 1 119905)119906 (119886 + 2 119905)

119906 (119887 minus 2 119905)119906 (119887 minus 1 119905)

)))

)

1015840

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟u1015840(119905)

= 119896((

(


d


))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟A

(((

(

119906(119886 + 1 119905)119906 (119886 + 2 119905)

119906 (119887 minus 2 119905)119906 (119887 minus 1 119905)

)))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟u(119905)

+ 119896(((

(

120585119886(119905)

0

0120585119887(119905)

)))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120585(119905)

+(((

(

119891(119886 + 1 119905 119906 (119886 + 1 119905))119891 (119886 + 2 119905 119906 (119886 + 2 119905))


)))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟f(119905u(119905))

(42)


(

119906(119886 + 1 0)119906 (119886 + 2 0)

119906 (119887 minus 1 0)

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟u(0)

= (

120593(119886 + 1)120593 (119886 + 2)

120593 (119887 minus 1)

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120593

(43)


u1015840 (119905) = 119896Au (119905) + 119896120585 (119905) + f (119905 u (119905)) 119905 isin R+

0

u (0) = 120593(44)










y119894+1 = y

119894+ ℎg (119905

119894 y

119894)

y0 = u (0) = u(45)



y(ℎ)

(119905) = y119894+ (119905 minus 119905

119894) g (119905

119894 y

119894) 119905

119894le 119905 le 119905

119894+1 (46)




119876 = (119905 u) isinR+







119872119879

= max119909isin(119886119887)Z 119905isin[0119879]

120593 (119909) 120585119886(119905) 120585

119887(119905)

119898119879

= min119909isin(119886119887)Z 119905isin[0119879]

120593 (119909) 120585119886(119905) 120585

119887(119905)

(48)



0 and let 119891 satisfy (119862119888119900119899119905

) (119862119897119894119901) and

(119862119904119894119892119899

) 119891(119909 119905119872119879) le 0 le 119891(119909 119905 119898


[0 119879]



(119906 minus119898119879) le 119891 (119909 119905 119906) le


(119906 minus119872119879) (49)


119879]


ℎisin (119886 119887)Z

119905ℎisin [0 119879] and 119906

ℎisin [119898

119879119872

119879] such that

119891 (119909ℎ 119905ℎ 119906

ℎ) gt


(119906ℎminus119872

119879) (50)


119898 119905119898

119906119898)infin119898=1 (we denote 119909

119898= 119909

ℎ119898 119905

119898= 119905

ℎ119898 119906

119898= 119906

ℎ119898) such

thatℎ119898

997888rarr 0+

119891 (119909119898 119905119898 119906

119898) gt

2ℎ119898119896 minus 1ℎ119898

(119906119898

minus119872119879)

(51)




lt119872



infin119897=1 sub

119906119898infin119898=1 such that 119906

119898119897rarr 119872



119898 119905119898 119906



119879119872

119879]


infin119897=1 sub 119906

119898infin119898=1 such

that 119906119898119897



119898119897infin119897=1 sub 119909

119898infin119898=1 such that 119909



119898119897infin119897=1 sub 119905

119898infin119898=1 such that 119905

119898119897rarr


1 minus 2ℎ119898

119896

ℎ119898

ge 119871

119872119879minus119906

119898

le 120573

119909119898

= 119909

10038161003816100381610038161003816119905119898 minus 10038161003816100381610038161003816 lt 120572

(52)


If we put 119909 = 119909119898

= 119905119898

= 119906119898

and = 119872119879



119898

119896

ℎ119898

(119872119879minus ) lt 119891 (119909 )

le 119891 (119909 ) minus 119891 (119909 119872119879)

= 119891 (119909 ) minus 119891 (119909 )

le1003816100381610038161003816119891 (119909 ) minus 119891 (119909 )

1003816100381610038161003816

(53)





[0 119879] Then

119898119879

le 119906 (119909 119905) le 119872119879

(54)






119906 (119909119888 119905119888) gt 119872T (55)


119888) such that


(b) 119906(1199090 1199050) = 119872119879


119906 (1199090 119905) gt 119872119879

on (1199050 1199050 + 120575) (56)







119910(ℎ)

(119909 119905) le 119872119879

on [1199050 1199050 + 120576] forall119909 isin (119886 119887)Z (57)

But if y(ℎ)


(119909 119905) le 119872119879


119906 (119909 119905) le 119872119879





119877 le 119898119879such that

119891 (119909 119905 119878) le 0 le 119891 (119909 119905 119877) (59)




[0 119879] Then

119877 le 119906 (119909 119905) le 119878 (60)





1199061015840 (119905) le 120573 (119905) 119906 (119905) for 119905 isin (119903 119904) (61)

then

119906 (119905) le 119906 (119903) 119890int119905

119903120573(120591)119889120591 for 119905 isin [119903 119904] (62)




[0 119879] If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) for some 1199090 isin

(119886 119887)Z and 1199050 isin (0 119879] then

119906 (1199090 119905) = 119872119879

(119900119903 119906 (1199090 119905) =119898119879)

forall119905 isin [0 1199050]

(63)



119879





119906 (1199090 119905119888) = 119872119879

119906 (1199090 119905) lt 119872119879

on [119905119888minus 120576 119905

119888)

(64)


119888minus



119906119905(1199090 119905)

= 119896 (119906 (1199090 minus 1 119905) minus 2119906 (1199090 119905) + 119906 (1199090 + 1 119905))

+119891 (1199090 119905 119906 (1199090 119905))

le 119896 (2119872119879minus 2119906 (1199090 119905)) +119891 (1199090 119905 119906 (1199090 119905))

minus119891 (1199090 119905119872119879) +119891 (1199090 119905119872119879

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟le0

le 119896 (2119872119879minus 2119906 (1199090 119905))

+1003816100381610038161003816119891 (1199090 119905 119906 (1199090 119905)) minus119891 (1199090 119905119872119879

)1003816100381610038161003816

le 119896 (2119872119879minus 2119906 (1199090 119905)) + 119871

1003816100381610038161003816119906 (1199090 119905) minus119872119879

1003816100381610038161003816

= 119896 (2119872119879minus 2119906 (1199090 119905)) + 119871 (119872

119879minus119906 (1199090 119905))

= minus (2119896 + 119871) (119906 (1199090 119905) minus119872119879)

(65)




119888minus




V1015840 (119905) le 120572V (119905) (66)


V (119905) le V (119905119888minus 120576) 119890120572(119905minus119905119888+120576) (67)

and hence

119906 (1199090 119905) le 119872119879minus (119872


lt 119872119879

(68)

on [119905119888minus 120576 119905





[119886 119887]Z times [0 119879] If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) for some


119906 (119909 119905) = 119872119879

(or 119906 (119909 119905) =119898119879)


(69)


If 119906(1199090minus1 119905119888) lt 119872119879(or 119906(1199090+1 119905119888) lt 119872

119879) at some 119905

119888isin [0 1199050]


119906119905(1199090 119905119888)

= 119896 (119906 (1199090 minus 1 119905119888) minus 2119906 (1199090 119905119888) + 119906 (1199090 + 1 119905

119888))

+119891 (1199090 119905119888 119906 (1199090 119905119888))

lt 119896 (2119872119879minus 2119872

119879) +119891 (1199090 119905119888119872119879

) le 0

(70)






similar







120578 = infin

or u (119905)119905rarr120578minus

997888997888997888997888997888rarr infin

(71)





0 whichsatisfies

inf119909isin(119886119887)Z 119905isinR

+

0

120593 (119909) 120585119886(119905) 120585

119887(119905) le 119906 (119909 119905)


+

0

120593 (119909) 120585119886(119905) 120585

119887(119905)

(72)


0


120578 = infin

or u (119905)119905rarr120578minus

997888997888997888997888997888rarr infin

(73)


119870isin (119886 119887)Z and 119905

119870isin [0 120578) such that

1003816100381610038161003816119906 (119909119870 119905119870)1003816100381610038161003816 gt 119870 (74)

If we put 119870 = max|119898120578| |119872

120578| lt infin (120585

119886 120585


on R+


1003816100381610038161003816119906 (119909119870 119905119870)1003816100381610038161003816 gt max

10038161003816100381610038161003816119898120578

10038161003816100381610038161003816 10038161003816100381610038161003816119872120578

10038161003816100381610038161003816 (75)




119891 (119909 119905 119906) =

minus |119906|119901minus1 119906 119906 = 0

0 119906 = 0(76)





(119862cont) (119862lip) and (119862119904119894119892119899








119891 (119909 119905 119906) = 120582119906 (1minus1199062) 120582 isin R (77)


119886 120585


119879= minus1 and 119872

119879= 1 (or

possibly 119898119879

ge minus1 and 119872119879





119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119896Δ2

119909119909119906 (119909 minus 1 119905)

+ 120582119906 (119909 119905) (1minus1199062(119909 119905))

119906 (119909 0) = 120593 (119909) 119909 isin (119886 119887)Z

119906 (119886 119905) = 120585119886(119905) 119905 isin ℎN0

119906 (119887 119905) = 120585119887(119905) 119905 isin ℎN0

(78)



2119896 minus1ℎ


ℎ le1

2 (119896 + 120582)

(79)





1minus1minusS

lowast

Slowast

(a) ℎ le 12(119896 + 120582)

1 S

minusSlowast

Slowast

minus1


1minus1minusS

lowast

S = Slowast

(c) ℎ = 1(2119896 + 1205822) (ie 119878 = 119878lowast)

1 S

minus1minusSlowast

Slowast

(d) ℎ gt 1(2119896 + 1205822) (ie 119878 gt 119878lowast)


= minus1 and 119872119879




119879 = radic 13


(80)


119878 =minus21198793


1 minus 2119896ℎ (81)



ℎ le1

2119896 + 1205822 (82)






h

1

2k minus120582

4

Bounds [minus1 1]

1

2k +120582

2

No bounds

Bounds [minusS S]

120582

1

2(k + 120582)

(a)

Bounds [minus1 1]

Bounds [minusS S]

h = ht

2

120582

1

2120582

No bounds

hx

(b)


= minus1 and119872119879


119909and ℎ = ℎ



119906 (119909 119905)

isin

[minus1 1] if 120582 le 0 ℎ le1

2119896 minus 1205824

[minus1 1] if 120582 gt 0 ℎ le1

2 (119896 + 120582)

[[

[



1 minus 2119896ℎ]]

]

if 120582 gt 0 ℎ isin (1

2 (119896 + 120582)

12119896 + 1205822

]

(83)



119909gt 0 in (78) that is

119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ119905

= 119896119906 (119909 minus ℎ

119909 119905) minus 2119906 (119909 119905) + 119906 (119909 + ℎ

119909 119905)

ℎ2119909

+120582119906 (119909 119905) (1minus1199062(119909 119905))

(84)




119909for



119909)

9 Final Remarks




119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119886 (119909 119905) 119906 (119909 minus 1 119905)

+ 119887 (119909 119905) 119906 (119909 119905)

+ 119888 (119909 119905) 119906 (119909 + 1 119905)

+ 119891 (119909 119905 119906 (119909 119905))

(85)



ℎle 119891 (119909 119905 119906)

le119872


ℎ

(86)




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















(25)


119906 (0 1) = 119906 (0 0) + 119896119906 (minus1 0) minus 2119896119906 (0 0) + 119896119906 (1 0)

+ 119891 (0 0 0) = 119872(26)


119906 (minus2 1) = 119906 (2 1) = 0

119906 (minus1 1) = 119906 (1 1) =119872

2

(27)




plusmn 119895 119895 = 0 1 1199050 minus 119905

ℎ

(28)



plusmn 119895 119895 = 0 1 119905 minus 1199050

ℎ

(29)



[0 119879]ℎN0


[119898119879119872

119879)


(119898119879119872

119879]

(c) 119891(119909 119905119872119879) le 0 and 119891(119909 119905 119898

119879) ge 0



(1) If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) then 119906(119909 119905) =

119872119879(or 119906(119909 119905) = 119898

119879) onD(1199090 1199050)

(2) If 119906(1199090 1199050) lt 119872119879(or 119906(1199090 1199050) gt 119898

119879) then 119906(119909 119905) lt

119872119879(or 119906(119909 119905) gt 119898

119879) onI(1199090 1199050)


119879


119879


119879




+ (1minus 2ℎ119896) 119906 (1199090 1199050 minus ℎ)

+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ))


+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ)) +119872119879

minus119872119879


+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ)) +119872119879

(30)

Thus (11986310158401015840) yields




119879 (32)




119879)

+ ℎ119891 (1199090 1199050 minus ℎ 119906 (1199090 1199050 minus ℎ)) +119872119879

(33)





Hence119906 (1199090 1199050) lt 119872

119879 (35)

a contradiction





119906119905(119909 119905) = 119896Δ2

119909119909119906 (119909 minus 1 119905) + 119891 (119909 119905 119906 (119909 119905))

119909 isin (119886 119887)Z 119905 isin R+

0 119896 gt 0

119906 (119909 0) = 120593 (119909) 119909 isin (119886 119887)Z

119906 (119886 119905) = 120585119886(119905) 119905 isin R

+

0

119906 (119887 119905) = 120585119887(119905) 119905 isin R

+

0

(36)

where 119906119905= 120597



119886 120585

119887isin

1198621(R+




u1015840 (119905) = g (119905 u (119905)) 119905 isin R+

0

u (0) = u(37)

where u R+

0 rarr R119873 g R+




119876 = (119905 u) isinR+



1003817100381710038171003817g (1199051 u1) minus g (1199052 u2)1003817100381710038171003817 le 119871

1003817100381710038171003817u1 minus u21003817100381710038171003817 (39)




0times R for all


(119886 119887)Z times R+


0isin (119886 119887)Z 1199050 isin R+

0


119876 (1199090 1199050 119906

0) = (119909

0 119905 119906) isin (119886 119887)Z timesR

+

0

timesR 1003816100381610038161003816119905 minus 119905

0

1003816100381610038161003816 le 1205721003816100381610038161003816119906 minus 119906

0

1003816100381610038161003816 le 120573(40)


1) (119909

0 1199052 119906

2) isin 119876

there is1003816100381610038161003816119891 (1199090 1199051 1199061) minus119891 (1199090 1199052 1199062)

1003816100381610038161003816 le 11987110038161003816100381610038161199061 minus1199062

1003816100381610038161003816 (41)





(((

(

119906(119886 + 1 119905)119906 (119886 + 2 119905)

119906 (119887 minus 2 119905)119906 (119887 minus 1 119905)

)))

)

1015840

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟u1015840(119905)

= 119896((

(


d


))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟A

(((

(

119906(119886 + 1 119905)119906 (119886 + 2 119905)

119906 (119887 minus 2 119905)119906 (119887 minus 1 119905)

)))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟u(119905)

+ 119896(((

(

120585119886(119905)

0

0120585119887(119905)

)))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120585(119905)

+(((

(

119891(119886 + 1 119905 119906 (119886 + 1 119905))119891 (119886 + 2 119905 119906 (119886 + 2 119905))


)))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟f(119905u(119905))

(42)


(

119906(119886 + 1 0)119906 (119886 + 2 0)

119906 (119887 minus 1 0)

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟u(0)

= (

120593(119886 + 1)120593 (119886 + 2)

120593 (119887 minus 1)

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120593

(43)


u1015840 (119905) = 119896Au (119905) + 119896120585 (119905) + f (119905 u (119905)) 119905 isin R+

0

u (0) = 120593(44)










y119894+1 = y

119894+ ℎg (119905

119894 y

119894)

y0 = u (0) = u(45)



y(ℎ)

(119905) = y119894+ (119905 minus 119905

119894) g (119905

119894 y

119894) 119905

119894le 119905 le 119905

119894+1 (46)




119876 = (119905 u) isinR+







119872119879

= max119909isin(119886119887)Z 119905isin[0119879]

120593 (119909) 120585119886(119905) 120585

119887(119905)

119898119879

= min119909isin(119886119887)Z 119905isin[0119879]

120593 (119909) 120585119886(119905) 120585

119887(119905)

(48)



0 and let 119891 satisfy (119862119888119900119899119905

) (119862119897119894119901) and

(119862119904119894119892119899

) 119891(119909 119905119872119879) le 0 le 119891(119909 119905 119898


[0 119879]



(119906 minus119898119879) le 119891 (119909 119905 119906) le


(119906 minus119872119879) (49)


119879]


ℎisin (119886 119887)Z

119905ℎisin [0 119879] and 119906

ℎisin [119898

119879119872

119879] such that

119891 (119909ℎ 119905ℎ 119906

ℎ) gt


(119906ℎminus119872

119879) (50)


119898 119905119898

119906119898)infin119898=1 (we denote 119909

119898= 119909

ℎ119898 119905

119898= 119905

ℎ119898 119906

119898= 119906

ℎ119898) such

thatℎ119898

997888rarr 0+

119891 (119909119898 119905119898 119906

119898) gt

2ℎ119898119896 minus 1ℎ119898

(119906119898

minus119872119879)

(51)




lt119872



infin119897=1 sub

119906119898infin119898=1 such that 119906

119898119897rarr 119872



119898 119905119898 119906



119879119872

119879]


infin119897=1 sub 119906

119898infin119898=1 such

that 119906119898119897



119898119897infin119897=1 sub 119909

119898infin119898=1 such that 119909



119898119897infin119897=1 sub 119905

119898infin119898=1 such that 119905

119898119897rarr


1 minus 2ℎ119898

119896

ℎ119898

ge 119871

119872119879minus119906

119898

le 120573

119909119898

= 119909

10038161003816100381610038161003816119905119898 minus 10038161003816100381610038161003816 lt 120572

(52)


If we put 119909 = 119909119898

= 119905119898

= 119906119898

and = 119872119879



119898

119896

ℎ119898

(119872119879minus ) lt 119891 (119909 )

le 119891 (119909 ) minus 119891 (119909 119872119879)

= 119891 (119909 ) minus 119891 (119909 )

le1003816100381610038161003816119891 (119909 ) minus 119891 (119909 )

1003816100381610038161003816

(53)





[0 119879] Then

119898119879

le 119906 (119909 119905) le 119872119879

(54)






119906 (119909119888 119905119888) gt 119872T (55)


119888) such that


(b) 119906(1199090 1199050) = 119872119879


119906 (1199090 119905) gt 119872119879

on (1199050 1199050 + 120575) (56)







119910(ℎ)

(119909 119905) le 119872119879

on [1199050 1199050 + 120576] forall119909 isin (119886 119887)Z (57)

But if y(ℎ)


(119909 119905) le 119872119879


119906 (119909 119905) le 119872119879





119877 le 119898119879such that

119891 (119909 119905 119878) le 0 le 119891 (119909 119905 119877) (59)




[0 119879] Then

119877 le 119906 (119909 119905) le 119878 (60)





1199061015840 (119905) le 120573 (119905) 119906 (119905) for 119905 isin (119903 119904) (61)

then

119906 (119905) le 119906 (119903) 119890int119905

119903120573(120591)119889120591 for 119905 isin [119903 119904] (62)




[0 119879] If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) for some 1199090 isin

(119886 119887)Z and 1199050 isin (0 119879] then

119906 (1199090 119905) = 119872119879

(119900119903 119906 (1199090 119905) =119898119879)

forall119905 isin [0 1199050]

(63)



119879





119906 (1199090 119905119888) = 119872119879

119906 (1199090 119905) lt 119872119879

on [119905119888minus 120576 119905

119888)

(64)


119888minus



119906119905(1199090 119905)

= 119896 (119906 (1199090 minus 1 119905) minus 2119906 (1199090 119905) + 119906 (1199090 + 1 119905))

+119891 (1199090 119905 119906 (1199090 119905))

le 119896 (2119872119879minus 2119906 (1199090 119905)) +119891 (1199090 119905 119906 (1199090 119905))

minus119891 (1199090 119905119872119879) +119891 (1199090 119905119872119879

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟le0

le 119896 (2119872119879minus 2119906 (1199090 119905))

+1003816100381610038161003816119891 (1199090 119905 119906 (1199090 119905)) minus119891 (1199090 119905119872119879

)1003816100381610038161003816

le 119896 (2119872119879minus 2119906 (1199090 119905)) + 119871

1003816100381610038161003816119906 (1199090 119905) minus119872119879

1003816100381610038161003816

= 119896 (2119872119879minus 2119906 (1199090 119905)) + 119871 (119872

119879minus119906 (1199090 119905))

= minus (2119896 + 119871) (119906 (1199090 119905) minus119872119879)

(65)




119888minus




V1015840 (119905) le 120572V (119905) (66)


V (119905) le V (119905119888minus 120576) 119890120572(119905minus119905119888+120576) (67)

and hence

119906 (1199090 119905) le 119872119879minus (119872


lt 119872119879

(68)

on [119905119888minus 120576 119905





[119886 119887]Z times [0 119879] If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) for some


119906 (119909 119905) = 119872119879

(or 119906 (119909 119905) =119898119879)


(69)


If 119906(1199090minus1 119905119888) lt 119872119879(or 119906(1199090+1 119905119888) lt 119872

119879) at some 119905

119888isin [0 1199050]


119906119905(1199090 119905119888)

= 119896 (119906 (1199090 minus 1 119905119888) minus 2119906 (1199090 119905119888) + 119906 (1199090 + 1 119905

119888))

+119891 (1199090 119905119888 119906 (1199090 119905119888))

lt 119896 (2119872119879minus 2119872

119879) +119891 (1199090 119905119888119872119879

) le 0

(70)






similar







120578 = infin

or u (119905)119905rarr120578minus

997888997888997888997888997888rarr infin

(71)





0 whichsatisfies

inf119909isin(119886119887)Z 119905isinR

+

0

120593 (119909) 120585119886(119905) 120585

119887(119905) le 119906 (119909 119905)


+

0

120593 (119909) 120585119886(119905) 120585

119887(119905)

(72)


0


120578 = infin

or u (119905)119905rarr120578minus

997888997888997888997888997888rarr infin

(73)


119870isin (119886 119887)Z and 119905

119870isin [0 120578) such that

1003816100381610038161003816119906 (119909119870 119905119870)1003816100381610038161003816 gt 119870 (74)

If we put 119870 = max|119898120578| |119872

120578| lt infin (120585

119886 120585


on R+


1003816100381610038161003816119906 (119909119870 119905119870)1003816100381610038161003816 gt max

10038161003816100381610038161003816119898120578

10038161003816100381610038161003816 10038161003816100381610038161003816119872120578

10038161003816100381610038161003816 (75)




119891 (119909 119905 119906) =

minus |119906|119901minus1 119906 119906 = 0

0 119906 = 0(76)





(119862cont) (119862lip) and (119862119904119894119892119899








119891 (119909 119905 119906) = 120582119906 (1minus1199062) 120582 isin R (77)


119886 120585


119879= minus1 and 119872

119879= 1 (or

possibly 119898119879

ge minus1 and 119872119879





119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119896Δ2

119909119909119906 (119909 minus 1 119905)

+ 120582119906 (119909 119905) (1minus1199062(119909 119905))

119906 (119909 0) = 120593 (119909) 119909 isin (119886 119887)Z

119906 (119886 119905) = 120585119886(119905) 119905 isin ℎN0

119906 (119887 119905) = 120585119887(119905) 119905 isin ℎN0

(78)



2119896 minus1ℎ


ℎ le1

2 (119896 + 120582)

(79)





1minus1minusS

lowast

Slowast

(a) ℎ le 12(119896 + 120582)

1 S

minusSlowast

Slowast

minus1


1minus1minusS

lowast

S = Slowast

(c) ℎ = 1(2119896 + 1205822) (ie 119878 = 119878lowast)

1 S

minus1minusSlowast

Slowast

(d) ℎ gt 1(2119896 + 1205822) (ie 119878 gt 119878lowast)


= minus1 and 119872119879




119879 = radic 13


(80)


119878 =minus21198793


1 minus 2119896ℎ (81)



ℎ le1

2119896 + 1205822 (82)






h

1

2k minus120582

4

Bounds [minus1 1]

1

2k +120582

2

No bounds

Bounds [minusS S]

120582

1

2(k + 120582)

(a)

Bounds [minus1 1]

Bounds [minusS S]

h = ht

2

120582

1

2120582

No bounds

hx

(b)


= minus1 and119872119879


119909and ℎ = ℎ



119906 (119909 119905)

isin

[minus1 1] if 120582 le 0 ℎ le1

2119896 minus 1205824

[minus1 1] if 120582 gt 0 ℎ le1

2 (119896 + 120582)

[[

[



1 minus 2119896ℎ]]

]

if 120582 gt 0 ℎ isin (1

2 (119896 + 120582)

12119896 + 1205822

]

(83)



119909gt 0 in (78) that is

119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ119905

= 119896119906 (119909 minus ℎ

119909 119905) minus 2119906 (119909 119905) + 119906 (119909 + ℎ

119909 119905)

ℎ2119909

+120582119906 (119909 119905) (1minus1199062(119909 119905))

(84)




119909for



119909)

9 Final Remarks




119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119886 (119909 119905) 119906 (119909 minus 1 119905)

+ 119887 (119909 119905) 119906 (119909 119905)

+ 119888 (119909 119905) 119906 (119909 + 1 119905)

+ 119891 (119909 119905 119906 (119909 119905))

(85)



ℎle 119891 (119909 119905 119906)

le119872


ℎ

(86)




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119906119905(119909 119905) = 119896Δ2

119909119909119906 (119909 minus 1 119905) + 119891 (119909 119905 119906 (119909 119905))

119909 isin (119886 119887)Z 119905 isin R+

0 119896 gt 0

119906 (119909 0) = 120593 (119909) 119909 isin (119886 119887)Z

119906 (119886 119905) = 120585119886(119905) 119905 isin R

+

0

119906 (119887 119905) = 120585119887(119905) 119905 isin R

+

0

(36)

where 119906119905= 120597



119886 120585

119887isin

1198621(R+




u1015840 (119905) = g (119905 u (119905)) 119905 isin R+

0

u (0) = u(37)

where u R+

0 rarr R119873 g R+




119876 = (119905 u) isinR+



1003817100381710038171003817g (1199051 u1) minus g (1199052 u2)1003817100381710038171003817 le 119871

1003817100381710038171003817u1 minus u21003817100381710038171003817 (39)




0times R for all


(119886 119887)Z times R+


0isin (119886 119887)Z 1199050 isin R+

0


119876 (1199090 1199050 119906

0) = (119909

0 119905 119906) isin (119886 119887)Z timesR

+

0

timesR 1003816100381610038161003816119905 minus 119905

0

1003816100381610038161003816 le 1205721003816100381610038161003816119906 minus 119906

0

1003816100381610038161003816 le 120573(40)


1) (119909

0 1199052 119906

2) isin 119876

there is1003816100381610038161003816119891 (1199090 1199051 1199061) minus119891 (1199090 1199052 1199062)

1003816100381610038161003816 le 11987110038161003816100381610038161199061 minus1199062

1003816100381610038161003816 (41)





(((

(

119906(119886 + 1 119905)119906 (119886 + 2 119905)

119906 (119887 minus 2 119905)119906 (119887 minus 1 119905)

)))

)

1015840

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟u1015840(119905)

= 119896((

(


d


))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟A

(((

(

119906(119886 + 1 119905)119906 (119886 + 2 119905)

119906 (119887 minus 2 119905)119906 (119887 minus 1 119905)

)))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟u(119905)

+ 119896(((

(

120585119886(119905)

0

0120585119887(119905)

)))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120585(119905)

+(((

(

119891(119886 + 1 119905 119906 (119886 + 1 119905))119891 (119886 + 2 119905 119906 (119886 + 2 119905))


)))

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟f(119905u(119905))

(42)


(

119906(119886 + 1 0)119906 (119886 + 2 0)

119906 (119887 minus 1 0)

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟u(0)

= (

120593(119886 + 1)120593 (119886 + 2)

120593 (119887 minus 1)

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟120593

(43)


u1015840 (119905) = 119896Au (119905) + 119896120585 (119905) + f (119905 u (119905)) 119905 isin R+

0

u (0) = 120593(44)










y119894+1 = y

119894+ ℎg (119905

119894 y

119894)

y0 = u (0) = u(45)



y(ℎ)

(119905) = y119894+ (119905 minus 119905

119894) g (119905

119894 y

119894) 119905

119894le 119905 le 119905

119894+1 (46)




119876 = (119905 u) isinR+







119872119879

= max119909isin(119886119887)Z 119905isin[0119879]

120593 (119909) 120585119886(119905) 120585

119887(119905)

119898119879

= min119909isin(119886119887)Z 119905isin[0119879]

120593 (119909) 120585119886(119905) 120585

119887(119905)

(48)



0 and let 119891 satisfy (119862119888119900119899119905

) (119862119897119894119901) and

(119862119904119894119892119899

) 119891(119909 119905119872119879) le 0 le 119891(119909 119905 119898


[0 119879]



(119906 minus119898119879) le 119891 (119909 119905 119906) le


(119906 minus119872119879) (49)


119879]


ℎisin (119886 119887)Z

119905ℎisin [0 119879] and 119906

ℎisin [119898

119879119872

119879] such that

119891 (119909ℎ 119905ℎ 119906

ℎ) gt


(119906ℎminus119872

119879) (50)


119898 119905119898

119906119898)infin119898=1 (we denote 119909

119898= 119909

ℎ119898 119905

119898= 119905

ℎ119898 119906

119898= 119906

ℎ119898) such

thatℎ119898

997888rarr 0+

119891 (119909119898 119905119898 119906

119898) gt

2ℎ119898119896 minus 1ℎ119898

(119906119898

minus119872119879)

(51)




lt119872



infin119897=1 sub

119906119898infin119898=1 such that 119906

119898119897rarr 119872



119898 119905119898 119906



119879119872

119879]


infin119897=1 sub 119906

119898infin119898=1 such

that 119906119898119897



119898119897infin119897=1 sub 119909

119898infin119898=1 such that 119909



119898119897infin119897=1 sub 119905

119898infin119898=1 such that 119905

119898119897rarr


1 minus 2ℎ119898

119896

ℎ119898

ge 119871

119872119879minus119906

119898

le 120573

119909119898

= 119909

10038161003816100381610038161003816119905119898 minus 10038161003816100381610038161003816 lt 120572

(52)


If we put 119909 = 119909119898

= 119905119898

= 119906119898

and = 119872119879



119898

119896

ℎ119898

(119872119879minus ) lt 119891 (119909 )

le 119891 (119909 ) minus 119891 (119909 119872119879)

= 119891 (119909 ) minus 119891 (119909 )

le1003816100381610038161003816119891 (119909 ) minus 119891 (119909 )

1003816100381610038161003816

(53)





[0 119879] Then

119898119879

le 119906 (119909 119905) le 119872119879

(54)






119906 (119909119888 119905119888) gt 119872T (55)


119888) such that


(b) 119906(1199090 1199050) = 119872119879


119906 (1199090 119905) gt 119872119879

on (1199050 1199050 + 120575) (56)







119910(ℎ)

(119909 119905) le 119872119879

on [1199050 1199050 + 120576] forall119909 isin (119886 119887)Z (57)

But if y(ℎ)


(119909 119905) le 119872119879


119906 (119909 119905) le 119872119879





119877 le 119898119879such that

119891 (119909 119905 119878) le 0 le 119891 (119909 119905 119877) (59)




[0 119879] Then

119877 le 119906 (119909 119905) le 119878 (60)





1199061015840 (119905) le 120573 (119905) 119906 (119905) for 119905 isin (119903 119904) (61)

then

119906 (119905) le 119906 (119903) 119890int119905

119903120573(120591)119889120591 for 119905 isin [119903 119904] (62)




[0 119879] If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) for some 1199090 isin

(119886 119887)Z and 1199050 isin (0 119879] then

119906 (1199090 119905) = 119872119879

(119900119903 119906 (1199090 119905) =119898119879)

forall119905 isin [0 1199050]

(63)



119879





119906 (1199090 119905119888) = 119872119879

119906 (1199090 119905) lt 119872119879

on [119905119888minus 120576 119905

119888)

(64)


119888minus



119906119905(1199090 119905)

= 119896 (119906 (1199090 minus 1 119905) minus 2119906 (1199090 119905) + 119906 (1199090 + 1 119905))

+119891 (1199090 119905 119906 (1199090 119905))

le 119896 (2119872119879minus 2119906 (1199090 119905)) +119891 (1199090 119905 119906 (1199090 119905))

minus119891 (1199090 119905119872119879) +119891 (1199090 119905119872119879

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟le0

le 119896 (2119872119879minus 2119906 (1199090 119905))

+1003816100381610038161003816119891 (1199090 119905 119906 (1199090 119905)) minus119891 (1199090 119905119872119879

)1003816100381610038161003816

le 119896 (2119872119879minus 2119906 (1199090 119905)) + 119871

1003816100381610038161003816119906 (1199090 119905) minus119872119879

1003816100381610038161003816

= 119896 (2119872119879minus 2119906 (1199090 119905)) + 119871 (119872

119879minus119906 (1199090 119905))

= minus (2119896 + 119871) (119906 (1199090 119905) minus119872119879)

(65)




119888minus




V1015840 (119905) le 120572V (119905) (66)


V (119905) le V (119905119888minus 120576) 119890120572(119905minus119905119888+120576) (67)

and hence

119906 (1199090 119905) le 119872119879minus (119872


lt 119872119879

(68)

on [119905119888minus 120576 119905





[119886 119887]Z times [0 119879] If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) for some


119906 (119909 119905) = 119872119879

(or 119906 (119909 119905) =119898119879)


(69)


If 119906(1199090minus1 119905119888) lt 119872119879(or 119906(1199090+1 119905119888) lt 119872

119879) at some 119905

119888isin [0 1199050]


119906119905(1199090 119905119888)

= 119896 (119906 (1199090 minus 1 119905119888) minus 2119906 (1199090 119905119888) + 119906 (1199090 + 1 119905

119888))

+119891 (1199090 119905119888 119906 (1199090 119905119888))

lt 119896 (2119872119879minus 2119872

119879) +119891 (1199090 119905119888119872119879

) le 0

(70)






similar







120578 = infin

or u (119905)119905rarr120578minus

997888997888997888997888997888rarr infin

(71)





0 whichsatisfies

inf119909isin(119886119887)Z 119905isinR

+

0

120593 (119909) 120585119886(119905) 120585

119887(119905) le 119906 (119909 119905)


+

0

120593 (119909) 120585119886(119905) 120585

119887(119905)

(72)


0


120578 = infin

or u (119905)119905rarr120578minus

997888997888997888997888997888rarr infin

(73)


119870isin (119886 119887)Z and 119905

119870isin [0 120578) such that

1003816100381610038161003816119906 (119909119870 119905119870)1003816100381610038161003816 gt 119870 (74)

If we put 119870 = max|119898120578| |119872

120578| lt infin (120585

119886 120585


on R+


1003816100381610038161003816119906 (119909119870 119905119870)1003816100381610038161003816 gt max

10038161003816100381610038161003816119898120578

10038161003816100381610038161003816 10038161003816100381610038161003816119872120578

10038161003816100381610038161003816 (75)




119891 (119909 119905 119906) =

minus |119906|119901minus1 119906 119906 = 0

0 119906 = 0(76)





(119862cont) (119862lip) and (119862119904119894119892119899








119891 (119909 119905 119906) = 120582119906 (1minus1199062) 120582 isin R (77)


119886 120585


119879= minus1 and 119872

119879= 1 (or

possibly 119898119879

ge minus1 and 119872119879





119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119896Δ2

119909119909119906 (119909 minus 1 119905)

+ 120582119906 (119909 119905) (1minus1199062(119909 119905))

119906 (119909 0) = 120593 (119909) 119909 isin (119886 119887)Z

119906 (119886 119905) = 120585119886(119905) 119905 isin ℎN0

119906 (119887 119905) = 120585119887(119905) 119905 isin ℎN0

(78)



2119896 minus1ℎ


ℎ le1

2 (119896 + 120582)

(79)





1minus1minusS

lowast

Slowast

(a) ℎ le 12(119896 + 120582)

1 S

minusSlowast

Slowast

minus1


1minus1minusS

lowast

S = Slowast

(c) ℎ = 1(2119896 + 1205822) (ie 119878 = 119878lowast)

1 S

minus1minusSlowast

Slowast

(d) ℎ gt 1(2119896 + 1205822) (ie 119878 gt 119878lowast)


= minus1 and 119872119879




119879 = radic 13


(80)


119878 =minus21198793


1 minus 2119896ℎ (81)



ℎ le1

2119896 + 1205822 (82)






h

1

2k minus120582

4

Bounds [minus1 1]

1

2k +120582

2

No bounds

Bounds [minusS S]

120582

1

2(k + 120582)

(a)

Bounds [minus1 1]

Bounds [minusS S]

h = ht

2

120582

1

2120582

No bounds

hx

(b)


= minus1 and119872119879


119909and ℎ = ℎ



119906 (119909 119905)

isin

[minus1 1] if 120582 le 0 ℎ le1

2119896 minus 1205824

[minus1 1] if 120582 gt 0 ℎ le1

2 (119896 + 120582)

[[

[



1 minus 2119896ℎ]]

]

if 120582 gt 0 ℎ isin (1

2 (119896 + 120582)

12119896 + 1205822

]

(83)



119909gt 0 in (78) that is

119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ119905

= 119896119906 (119909 minus ℎ

119909 119905) minus 2119906 (119909 119905) + 119906 (119909 + ℎ

119909 119905)

ℎ2119909

+120582119906 (119909 119905) (1minus1199062(119909 119905))

(84)




119909for



119909)

9 Final Remarks




119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119886 (119909 119905) 119906 (119909 minus 1 119905)

+ 119887 (119909 119905) 119906 (119909 119905)

+ 119888 (119909 119905) 119906 (119909 + 1 119905)

+ 119891 (119909 119905 119906 (119909 119905))

(85)



ℎle 119891 (119909 119905 119906)

le119872


ℎ

(86)




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















y119894+1 = y

119894+ ℎg (119905

119894 y

119894)

y0 = u (0) = u(45)



y(ℎ)

(119905) = y119894+ (119905 minus 119905

119894) g (119905

119894 y

119894) 119905

119894le 119905 le 119905

119894+1 (46)




119876 = (119905 u) isinR+







119872119879

= max119909isin(119886119887)Z 119905isin[0119879]

120593 (119909) 120585119886(119905) 120585

119887(119905)

119898119879

= min119909isin(119886119887)Z 119905isin[0119879]

120593 (119909) 120585119886(119905) 120585

119887(119905)

(48)



0 and let 119891 satisfy (119862119888119900119899119905

) (119862119897119894119901) and

(119862119904119894119892119899

) 119891(119909 119905119872119879) le 0 le 119891(119909 119905 119898


[0 119879]



(119906 minus119898119879) le 119891 (119909 119905 119906) le


(119906 minus119872119879) (49)


119879]


ℎisin (119886 119887)Z

119905ℎisin [0 119879] and 119906

ℎisin [119898

119879119872

119879] such that

119891 (119909ℎ 119905ℎ 119906

ℎ) gt


(119906ℎminus119872

119879) (50)


119898 119905119898

119906119898)infin119898=1 (we denote 119909

119898= 119909

ℎ119898 119905

119898= 119905

ℎ119898 119906

119898= 119906

ℎ119898) such

thatℎ119898

997888rarr 0+

119891 (119909119898 119905119898 119906

119898) gt

2ℎ119898119896 minus 1ℎ119898

(119906119898

minus119872119879)

(51)




lt119872



infin119897=1 sub

119906119898infin119898=1 such that 119906

119898119897rarr 119872



119898 119905119898 119906



119879119872

119879]


infin119897=1 sub 119906

119898infin119898=1 such

that 119906119898119897



119898119897infin119897=1 sub 119909

119898infin119898=1 such that 119909



119898119897infin119897=1 sub 119905

119898infin119898=1 such that 119905

119898119897rarr


1 minus 2ℎ119898

119896

ℎ119898

ge 119871

119872119879minus119906

119898

le 120573

119909119898

= 119909

10038161003816100381610038161003816119905119898 minus 10038161003816100381610038161003816 lt 120572

(52)


If we put 119909 = 119909119898

= 119905119898

= 119906119898

and = 119872119879



119898

119896

ℎ119898

(119872119879minus ) lt 119891 (119909 )

le 119891 (119909 ) minus 119891 (119909 119872119879)

= 119891 (119909 ) minus 119891 (119909 )

le1003816100381610038161003816119891 (119909 ) minus 119891 (119909 )

1003816100381610038161003816

(53)





[0 119879] Then

119898119879

le 119906 (119909 119905) le 119872119879

(54)






119906 (119909119888 119905119888) gt 119872T (55)


119888) such that


(b) 119906(1199090 1199050) = 119872119879


119906 (1199090 119905) gt 119872119879

on (1199050 1199050 + 120575) (56)







119910(ℎ)

(119909 119905) le 119872119879

on [1199050 1199050 + 120576] forall119909 isin (119886 119887)Z (57)

But if y(ℎ)


(119909 119905) le 119872119879


119906 (119909 119905) le 119872119879





119877 le 119898119879such that

119891 (119909 119905 119878) le 0 le 119891 (119909 119905 119877) (59)




[0 119879] Then

119877 le 119906 (119909 119905) le 119878 (60)





1199061015840 (119905) le 120573 (119905) 119906 (119905) for 119905 isin (119903 119904) (61)

then

119906 (119905) le 119906 (119903) 119890int119905

119903120573(120591)119889120591 for 119905 isin [119903 119904] (62)




[0 119879] If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) for some 1199090 isin

(119886 119887)Z and 1199050 isin (0 119879] then

119906 (1199090 119905) = 119872119879

(119900119903 119906 (1199090 119905) =119898119879)

forall119905 isin [0 1199050]

(63)



119879





119906 (1199090 119905119888) = 119872119879

119906 (1199090 119905) lt 119872119879

on [119905119888minus 120576 119905

119888)

(64)


119888minus



119906119905(1199090 119905)

= 119896 (119906 (1199090 minus 1 119905) minus 2119906 (1199090 119905) + 119906 (1199090 + 1 119905))

+119891 (1199090 119905 119906 (1199090 119905))

le 119896 (2119872119879minus 2119906 (1199090 119905)) +119891 (1199090 119905 119906 (1199090 119905))

minus119891 (1199090 119905119872119879) +119891 (1199090 119905119872119879

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟le0

le 119896 (2119872119879minus 2119906 (1199090 119905))

+1003816100381610038161003816119891 (1199090 119905 119906 (1199090 119905)) minus119891 (1199090 119905119872119879

)1003816100381610038161003816

le 119896 (2119872119879minus 2119906 (1199090 119905)) + 119871

1003816100381610038161003816119906 (1199090 119905) minus119872119879

1003816100381610038161003816

= 119896 (2119872119879minus 2119906 (1199090 119905)) + 119871 (119872

119879minus119906 (1199090 119905))

= minus (2119896 + 119871) (119906 (1199090 119905) minus119872119879)

(65)




119888minus




V1015840 (119905) le 120572V (119905) (66)


V (119905) le V (119905119888minus 120576) 119890120572(119905minus119905119888+120576) (67)

and hence

119906 (1199090 119905) le 119872119879minus (119872


lt 119872119879

(68)

on [119905119888minus 120576 119905





[119886 119887]Z times [0 119879] If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) for some


119906 (119909 119905) = 119872119879

(or 119906 (119909 119905) =119898119879)


(69)


If 119906(1199090minus1 119905119888) lt 119872119879(or 119906(1199090+1 119905119888) lt 119872

119879) at some 119905

119888isin [0 1199050]


119906119905(1199090 119905119888)

= 119896 (119906 (1199090 minus 1 119905119888) minus 2119906 (1199090 119905119888) + 119906 (1199090 + 1 119905

119888))

+119891 (1199090 119905119888 119906 (1199090 119905119888))

lt 119896 (2119872119879minus 2119872

119879) +119891 (1199090 119905119888119872119879

) le 0

(70)






similar







120578 = infin

or u (119905)119905rarr120578minus

997888997888997888997888997888rarr infin

(71)





0 whichsatisfies

inf119909isin(119886119887)Z 119905isinR

+

0

120593 (119909) 120585119886(119905) 120585

119887(119905) le 119906 (119909 119905)


+

0

120593 (119909) 120585119886(119905) 120585

119887(119905)

(72)


0


120578 = infin

or u (119905)119905rarr120578minus

997888997888997888997888997888rarr infin

(73)


119870isin (119886 119887)Z and 119905

119870isin [0 120578) such that

1003816100381610038161003816119906 (119909119870 119905119870)1003816100381610038161003816 gt 119870 (74)

If we put 119870 = max|119898120578| |119872

120578| lt infin (120585

119886 120585


on R+


1003816100381610038161003816119906 (119909119870 119905119870)1003816100381610038161003816 gt max

10038161003816100381610038161003816119898120578

10038161003816100381610038161003816 10038161003816100381610038161003816119872120578

10038161003816100381610038161003816 (75)




119891 (119909 119905 119906) =

minus |119906|119901minus1 119906 119906 = 0

0 119906 = 0(76)





(119862cont) (119862lip) and (119862119904119894119892119899








119891 (119909 119905 119906) = 120582119906 (1minus1199062) 120582 isin R (77)


119886 120585


119879= minus1 and 119872

119879= 1 (or

possibly 119898119879

ge minus1 and 119872119879





119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119896Δ2

119909119909119906 (119909 minus 1 119905)

+ 120582119906 (119909 119905) (1minus1199062(119909 119905))

119906 (119909 0) = 120593 (119909) 119909 isin (119886 119887)Z

119906 (119886 119905) = 120585119886(119905) 119905 isin ℎN0

119906 (119887 119905) = 120585119887(119905) 119905 isin ℎN0

(78)



2119896 minus1ℎ


ℎ le1

2 (119896 + 120582)

(79)





1minus1minusS

lowast

Slowast

(a) ℎ le 12(119896 + 120582)

1 S

minusSlowast

Slowast

minus1


1minus1minusS

lowast

S = Slowast

(c) ℎ = 1(2119896 + 1205822) (ie 119878 = 119878lowast)

1 S

minus1minusSlowast

Slowast

(d) ℎ gt 1(2119896 + 1205822) (ie 119878 gt 119878lowast)


= minus1 and 119872119879




119879 = radic 13


(80)


119878 =minus21198793


1 minus 2119896ℎ (81)



ℎ le1

2119896 + 1205822 (82)






h

1

2k minus120582

4

Bounds [minus1 1]

1

2k +120582

2

No bounds

Bounds [minusS S]

120582

1

2(k + 120582)

(a)

Bounds [minus1 1]

Bounds [minusS S]

h = ht

2

120582

1

2120582

No bounds

hx

(b)


= minus1 and119872119879


119909and ℎ = ℎ



119906 (119909 119905)

isin

[minus1 1] if 120582 le 0 ℎ le1

2119896 minus 1205824

[minus1 1] if 120582 gt 0 ℎ le1

2 (119896 + 120582)

[[

[



1 minus 2119896ℎ]]

]

if 120582 gt 0 ℎ isin (1

2 (119896 + 120582)

12119896 + 1205822

]

(83)



119909gt 0 in (78) that is

119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ119905

= 119896119906 (119909 minus ℎ

119909 119905) minus 2119906 (119909 119905) + 119906 (119909 + ℎ

119909 119905)

ℎ2119909

+120582119906 (119909 119905) (1minus1199062(119909 119905))

(84)




119909for



119909)

9 Final Remarks




119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119886 (119909 119905) 119906 (119909 minus 1 119905)

+ 119887 (119909 119905) 119906 (119909 119905)

+ 119888 (119909 119905) 119906 (119909 + 1 119905)

+ 119891 (119909 119905 119906 (119909 119905))

(85)



ℎle 119891 (119909 119905 119906)

le119872


ℎ

(86)




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










If we put 119909 = 119909119898

= 119905119898

= 119906119898

and = 119872119879



119898

119896

ℎ119898

(119872119879minus ) lt 119891 (119909 )

le 119891 (119909 ) minus 119891 (119909 119872119879)

= 119891 (119909 ) minus 119891 (119909 )

le1003816100381610038161003816119891 (119909 ) minus 119891 (119909 )

1003816100381610038161003816

(53)





[0 119879] Then

119898119879

le 119906 (119909 119905) le 119872119879

(54)






119906 (119909119888 119905119888) gt 119872T (55)


119888) such that


(b) 119906(1199090 1199050) = 119872119879


119906 (1199090 119905) gt 119872119879

on (1199050 1199050 + 120575) (56)







119910(ℎ)

(119909 119905) le 119872119879

on [1199050 1199050 + 120576] forall119909 isin (119886 119887)Z (57)

But if y(ℎ)


(119909 119905) le 119872119879


119906 (119909 119905) le 119872119879





119877 le 119898119879such that

119891 (119909 119905 119878) le 0 le 119891 (119909 119905 119877) (59)




[0 119879] Then

119877 le 119906 (119909 119905) le 119878 (60)





1199061015840 (119905) le 120573 (119905) 119906 (119905) for 119905 isin (119903 119904) (61)

then

119906 (119905) le 119906 (119903) 119890int119905

119903120573(120591)119889120591 for 119905 isin [119903 119904] (62)




[0 119879] If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) for some 1199090 isin

(119886 119887)Z and 1199050 isin (0 119879] then

119906 (1199090 119905) = 119872119879

(119900119903 119906 (1199090 119905) =119898119879)

forall119905 isin [0 1199050]

(63)



119879





119906 (1199090 119905119888) = 119872119879

119906 (1199090 119905) lt 119872119879

on [119905119888minus 120576 119905

119888)

(64)


119888minus



119906119905(1199090 119905)

= 119896 (119906 (1199090 minus 1 119905) minus 2119906 (1199090 119905) + 119906 (1199090 + 1 119905))

+119891 (1199090 119905 119906 (1199090 119905))

le 119896 (2119872119879minus 2119906 (1199090 119905)) +119891 (1199090 119905 119906 (1199090 119905))

minus119891 (1199090 119905119872119879) +119891 (1199090 119905119872119879

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟le0

le 119896 (2119872119879minus 2119906 (1199090 119905))

+1003816100381610038161003816119891 (1199090 119905 119906 (1199090 119905)) minus119891 (1199090 119905119872119879

)1003816100381610038161003816

le 119896 (2119872119879minus 2119906 (1199090 119905)) + 119871

1003816100381610038161003816119906 (1199090 119905) minus119872119879

1003816100381610038161003816

= 119896 (2119872119879minus 2119906 (1199090 119905)) + 119871 (119872

119879minus119906 (1199090 119905))

= minus (2119896 + 119871) (119906 (1199090 119905) minus119872119879)

(65)




119888minus




V1015840 (119905) le 120572V (119905) (66)


V (119905) le V (119905119888minus 120576) 119890120572(119905minus119905119888+120576) (67)

and hence

119906 (1199090 119905) le 119872119879minus (119872


lt 119872119879

(68)

on [119905119888minus 120576 119905





[119886 119887]Z times [0 119879] If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) for some


119906 (119909 119905) = 119872119879

(or 119906 (119909 119905) =119898119879)


(69)


If 119906(1199090minus1 119905119888) lt 119872119879(or 119906(1199090+1 119905119888) lt 119872

119879) at some 119905

119888isin [0 1199050]


119906119905(1199090 119905119888)

= 119896 (119906 (1199090 minus 1 119905119888) minus 2119906 (1199090 119905119888) + 119906 (1199090 + 1 119905

119888))

+119891 (1199090 119905119888 119906 (1199090 119905119888))

lt 119896 (2119872119879minus 2119872

119879) +119891 (1199090 119905119888119872119879

) le 0

(70)






similar







120578 = infin

or u (119905)119905rarr120578minus

997888997888997888997888997888rarr infin

(71)





0 whichsatisfies

inf119909isin(119886119887)Z 119905isinR

+

0

120593 (119909) 120585119886(119905) 120585

119887(119905) le 119906 (119909 119905)


+

0

120593 (119909) 120585119886(119905) 120585

119887(119905)

(72)


0


120578 = infin

or u (119905)119905rarr120578minus

997888997888997888997888997888rarr infin

(73)


119870isin (119886 119887)Z and 119905

119870isin [0 120578) such that

1003816100381610038161003816119906 (119909119870 119905119870)1003816100381610038161003816 gt 119870 (74)

If we put 119870 = max|119898120578| |119872

120578| lt infin (120585

119886 120585


on R+


1003816100381610038161003816119906 (119909119870 119905119870)1003816100381610038161003816 gt max

10038161003816100381610038161003816119898120578

10038161003816100381610038161003816 10038161003816100381610038161003816119872120578

10038161003816100381610038161003816 (75)




119891 (119909 119905 119906) =

minus |119906|119901minus1 119906 119906 = 0

0 119906 = 0(76)





(119862cont) (119862lip) and (119862119904119894119892119899








119891 (119909 119905 119906) = 120582119906 (1minus1199062) 120582 isin R (77)


119886 120585


119879= minus1 and 119872

119879= 1 (or

possibly 119898119879

ge minus1 and 119872119879





119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119896Δ2

119909119909119906 (119909 minus 1 119905)

+ 120582119906 (119909 119905) (1minus1199062(119909 119905))

119906 (119909 0) = 120593 (119909) 119909 isin (119886 119887)Z

119906 (119886 119905) = 120585119886(119905) 119905 isin ℎN0

119906 (119887 119905) = 120585119887(119905) 119905 isin ℎN0

(78)



2119896 minus1ℎ


ℎ le1

2 (119896 + 120582)

(79)





1minus1minusS

lowast

Slowast

(a) ℎ le 12(119896 + 120582)

1 S

minusSlowast

Slowast

minus1


1minus1minusS

lowast

S = Slowast

(c) ℎ = 1(2119896 + 1205822) (ie 119878 = 119878lowast)

1 S

minus1minusSlowast

Slowast

(d) ℎ gt 1(2119896 + 1205822) (ie 119878 gt 119878lowast)


= minus1 and 119872119879




119879 = radic 13


(80)


119878 =minus21198793


1 minus 2119896ℎ (81)



ℎ le1

2119896 + 1205822 (82)






h

1

2k minus120582

4

Bounds [minus1 1]

1

2k +120582

2

No bounds

Bounds [minusS S]

120582

1

2(k + 120582)

(a)

Bounds [minus1 1]

Bounds [minusS S]

h = ht

2

120582

1

2120582

No bounds

hx

(b)


= minus1 and119872119879


119909and ℎ = ℎ



119906 (119909 119905)

isin

[minus1 1] if 120582 le 0 ℎ le1

2119896 minus 1205824

[minus1 1] if 120582 gt 0 ℎ le1

2 (119896 + 120582)

[[

[



1 minus 2119896ℎ]]

]

if 120582 gt 0 ℎ isin (1

2 (119896 + 120582)

12119896 + 1205822

]

(83)



119909gt 0 in (78) that is

119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ119905

= 119896119906 (119909 minus ℎ

119909 119905) minus 2119906 (119909 119905) + 119906 (119909 + ℎ

119909 119905)

ℎ2119909

+120582119906 (119909 119905) (1minus1199062(119909 119905))

(84)




119909for



119909)

9 Final Remarks




119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119886 (119909 119905) 119906 (119909 minus 1 119905)

+ 119887 (119909 119905) 119906 (119909 119905)

+ 119888 (119909 119905) 119906 (119909 + 1 119905)

+ 119891 (119909 119905 119906 (119909 119905))

(85)



ℎle 119891 (119909 119905 119906)

le119872


ℎ

(86)




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119879





119906 (1199090 119905119888) = 119872119879

119906 (1199090 119905) lt 119872119879

on [119905119888minus 120576 119905

119888)

(64)


119888minus



119906119905(1199090 119905)

= 119896 (119906 (1199090 minus 1 119905) minus 2119906 (1199090 119905) + 119906 (1199090 + 1 119905))

+119891 (1199090 119905 119906 (1199090 119905))

le 119896 (2119872119879minus 2119906 (1199090 119905)) +119891 (1199090 119905 119906 (1199090 119905))

minus119891 (1199090 119905119872119879) +119891 (1199090 119905119872119879

)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟le0

le 119896 (2119872119879minus 2119906 (1199090 119905))

+1003816100381610038161003816119891 (1199090 119905 119906 (1199090 119905)) minus119891 (1199090 119905119872119879

)1003816100381610038161003816

le 119896 (2119872119879minus 2119906 (1199090 119905)) + 119871

1003816100381610038161003816119906 (1199090 119905) minus119872119879

1003816100381610038161003816

= 119896 (2119872119879minus 2119906 (1199090 119905)) + 119871 (119872

119879minus119906 (1199090 119905))

= minus (2119896 + 119871) (119906 (1199090 119905) minus119872119879)

(65)




119888minus




V1015840 (119905) le 120572V (119905) (66)


V (119905) le V (119905119888minus 120576) 119890120572(119905minus119905119888+120576) (67)

and hence

119906 (1199090 119905) le 119872119879minus (119872


lt 119872119879

(68)

on [119905119888minus 120576 119905





[119886 119887]Z times [0 119879] If 119906(1199090 1199050) = 119872119879(or 119906(1199090 1199050) = 119898

119879) for some


119906 (119909 119905) = 119872119879

(or 119906 (119909 119905) =119898119879)


(69)


If 119906(1199090minus1 119905119888) lt 119872119879(or 119906(1199090+1 119905119888) lt 119872

119879) at some 119905

119888isin [0 1199050]


119906119905(1199090 119905119888)

= 119896 (119906 (1199090 minus 1 119905119888) minus 2119906 (1199090 119905119888) + 119906 (1199090 + 1 119905

119888))

+119891 (1199090 119905119888 119906 (1199090 119905119888))

lt 119896 (2119872119879minus 2119872

119879) +119891 (1199090 119905119888119872119879

) le 0

(70)






similar







120578 = infin

or u (119905)119905rarr120578minus

997888997888997888997888997888rarr infin

(71)





0 whichsatisfies

inf119909isin(119886119887)Z 119905isinR

+

0

120593 (119909) 120585119886(119905) 120585

119887(119905) le 119906 (119909 119905)


+

0

120593 (119909) 120585119886(119905) 120585

119887(119905)

(72)


0


120578 = infin

or u (119905)119905rarr120578minus

997888997888997888997888997888rarr infin

(73)


119870isin (119886 119887)Z and 119905

119870isin [0 120578) such that

1003816100381610038161003816119906 (119909119870 119905119870)1003816100381610038161003816 gt 119870 (74)

If we put 119870 = max|119898120578| |119872

120578| lt infin (120585

119886 120585


on R+


1003816100381610038161003816119906 (119909119870 119905119870)1003816100381610038161003816 gt max

10038161003816100381610038161003816119898120578

10038161003816100381610038161003816 10038161003816100381610038161003816119872120578

10038161003816100381610038161003816 (75)




119891 (119909 119905 119906) =

minus |119906|119901minus1 119906 119906 = 0

0 119906 = 0(76)





(119862cont) (119862lip) and (119862119904119894119892119899








119891 (119909 119905 119906) = 120582119906 (1minus1199062) 120582 isin R (77)


119886 120585


119879= minus1 and 119872

119879= 1 (or

possibly 119898119879

ge minus1 and 119872119879





119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119896Δ2

119909119909119906 (119909 minus 1 119905)

+ 120582119906 (119909 119905) (1minus1199062(119909 119905))

119906 (119909 0) = 120593 (119909) 119909 isin (119886 119887)Z

119906 (119886 119905) = 120585119886(119905) 119905 isin ℎN0

119906 (119887 119905) = 120585119887(119905) 119905 isin ℎN0

(78)



2119896 minus1ℎ


ℎ le1

2 (119896 + 120582)

(79)





1minus1minusS

lowast

Slowast

(a) ℎ le 12(119896 + 120582)

1 S

minusSlowast

Slowast

minus1


1minus1minusS

lowast

S = Slowast

(c) ℎ = 1(2119896 + 1205822) (ie 119878 = 119878lowast)

1 S

minus1minusSlowast

Slowast

(d) ℎ gt 1(2119896 + 1205822) (ie 119878 gt 119878lowast)


= minus1 and 119872119879




119879 = radic 13


(80)


119878 =minus21198793


1 minus 2119896ℎ (81)



ℎ le1

2119896 + 1205822 (82)






h

1

2k minus120582

4

Bounds [minus1 1]

1

2k +120582

2

No bounds

Bounds [minusS S]

120582

1

2(k + 120582)

(a)

Bounds [minus1 1]

Bounds [minusS S]

h = ht

2

120582

1

2120582

No bounds

hx

(b)


= minus1 and119872119879


119909and ℎ = ℎ



119906 (119909 119905)

isin

[minus1 1] if 120582 le 0 ℎ le1

2119896 minus 1205824

[minus1 1] if 120582 gt 0 ℎ le1

2 (119896 + 120582)

[[

[



1 minus 2119896ℎ]]

]

if 120582 gt 0 ℎ isin (1

2 (119896 + 120582)

12119896 + 1205822

]

(83)



119909gt 0 in (78) that is

119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ119905

= 119896119906 (119909 minus ℎ

119909 119905) minus 2119906 (119909 119905) + 119906 (119909 + ℎ

119909 119905)

ℎ2119909

+120582119906 (119909 119905) (1minus1199062(119909 119905))

(84)




119909for



119909)

9 Final Remarks




119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119886 (119909 119905) 119906 (119909 minus 1 119905)

+ 119887 (119909 119905) 119906 (119909 119905)

+ 119888 (119909 119905) 119906 (119909 + 1 119905)

+ 119891 (119909 119905 119906 (119909 119905))

(85)



ℎle 119891 (119909 119905 119906)

le119872


ℎ

(86)




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0 whichsatisfies

inf119909isin(119886119887)Z 119905isinR

+

0

120593 (119909) 120585119886(119905) 120585

119887(119905) le 119906 (119909 119905)


+

0

120593 (119909) 120585119886(119905) 120585

119887(119905)

(72)


0


120578 = infin

or u (119905)119905rarr120578minus

997888997888997888997888997888rarr infin

(73)


119870isin (119886 119887)Z and 119905

119870isin [0 120578) such that

1003816100381610038161003816119906 (119909119870 119905119870)1003816100381610038161003816 gt 119870 (74)

If we put 119870 = max|119898120578| |119872

120578| lt infin (120585

119886 120585


on R+


1003816100381610038161003816119906 (119909119870 119905119870)1003816100381610038161003816 gt max

10038161003816100381610038161003816119898120578

10038161003816100381610038161003816 10038161003816100381610038161003816119872120578

10038161003816100381610038161003816 (75)




119891 (119909 119905 119906) =

minus |119906|119901minus1 119906 119906 = 0

0 119906 = 0(76)





(119862cont) (119862lip) and (119862119904119894119892119899








119891 (119909 119905 119906) = 120582119906 (1minus1199062) 120582 isin R (77)


119886 120585


119879= minus1 and 119872

119879= 1 (or

possibly 119898119879

ge minus1 and 119872119879





119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119896Δ2

119909119909119906 (119909 minus 1 119905)

+ 120582119906 (119909 119905) (1minus1199062(119909 119905))

119906 (119909 0) = 120593 (119909) 119909 isin (119886 119887)Z

119906 (119886 119905) = 120585119886(119905) 119905 isin ℎN0

119906 (119887 119905) = 120585119887(119905) 119905 isin ℎN0

(78)



2119896 minus1ℎ


ℎ le1

2 (119896 + 120582)

(79)





1minus1minusS

lowast

Slowast

(a) ℎ le 12(119896 + 120582)

1 S

minusSlowast

Slowast

minus1


1minus1minusS

lowast

S = Slowast

(c) ℎ = 1(2119896 + 1205822) (ie 119878 = 119878lowast)

1 S

minus1minusSlowast

Slowast

(d) ℎ gt 1(2119896 + 1205822) (ie 119878 gt 119878lowast)


= minus1 and 119872119879




119879 = radic 13


(80)


119878 =minus21198793


1 minus 2119896ℎ (81)



ℎ le1

2119896 + 1205822 (82)






h

1

2k minus120582

4

Bounds [minus1 1]

1

2k +120582

2

No bounds

Bounds [minusS S]

120582

1

2(k + 120582)

(a)

Bounds [minus1 1]

Bounds [minusS S]

h = ht

2

120582

1

2120582

No bounds

hx

(b)


= minus1 and119872119879


119909and ℎ = ℎ



119906 (119909 119905)

isin

[minus1 1] if 120582 le 0 ℎ le1

2119896 minus 1205824

[minus1 1] if 120582 gt 0 ℎ le1

2 (119896 + 120582)

[[

[



1 minus 2119896ℎ]]

]

if 120582 gt 0 ℎ isin (1

2 (119896 + 120582)

12119896 + 1205822

]

(83)



119909gt 0 in (78) that is

119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ119905

= 119896119906 (119909 minus ℎ

119909 119905) minus 2119906 (119909 119905) + 119906 (119909 + ℎ

119909 119905)

ℎ2119909

+120582119906 (119909 119905) (1minus1199062(119909 119905))

(84)




119909for



119909)

9 Final Remarks




119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119886 (119909 119905) 119906 (119909 minus 1 119905)

+ 119887 (119909 119905) 119906 (119909 119905)

+ 119888 (119909 119905) 119906 (119909 + 1 119905)

+ 119891 (119909 119905 119906 (119909 119905))

(85)



ℎle 119891 (119909 119905 119906)

le119872


ℎ

(86)




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1minus1minusS

lowast

Slowast

(a) ℎ le 12(119896 + 120582)

1 S

minusSlowast

Slowast

minus1


1minus1minusS

lowast

S = Slowast

(c) ℎ = 1(2119896 + 1205822) (ie 119878 = 119878lowast)

1 S

minus1minusSlowast

Slowast

(d) ℎ gt 1(2119896 + 1205822) (ie 119878 gt 119878lowast)


= minus1 and 119872119879




119879 = radic 13


(80)


119878 =minus21198793


1 minus 2119896ℎ (81)



ℎ le1

2119896 + 1205822 (82)






h

1

2k minus120582

4

Bounds [minus1 1]

1

2k +120582

2

No bounds

Bounds [minusS S]

120582

1

2(k + 120582)

(a)

Bounds [minus1 1]

Bounds [minusS S]

h = ht

2

120582

1

2120582

No bounds

hx

(b)


= minus1 and119872119879


119909and ℎ = ℎ



119906 (119909 119905)

isin

[minus1 1] if 120582 le 0 ℎ le1

2119896 minus 1205824

[minus1 1] if 120582 gt 0 ℎ le1

2 (119896 + 120582)

[[

[



1 minus 2119896ℎ]]

]

if 120582 gt 0 ℎ isin (1

2 (119896 + 120582)

12119896 + 1205822

]

(83)



119909gt 0 in (78) that is

119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ119905

= 119896119906 (119909 minus ℎ

119909 119905) minus 2119906 (119909 119905) + 119906 (119909 + ℎ

119909 119905)

ℎ2119909

+120582119906 (119909 119905) (1minus1199062(119909 119905))

(84)




119909for



119909)

9 Final Remarks




119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119886 (119909 119905) 119906 (119909 minus 1 119905)

+ 119887 (119909 119905) 119906 (119909 119905)

+ 119888 (119909 119905) 119906 (119909 + 1 119905)

+ 119891 (119909 119905 119906 (119909 119905))

(85)



ℎle 119891 (119909 119905 119906)

le119872


ℎ

(86)




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










h

1

2k minus120582

4

Bounds [minus1 1]

1

2k +120582

2

No bounds

Bounds [minusS S]

120582

1

2(k + 120582)

(a)

Bounds [minus1 1]

Bounds [minusS S]

h = ht

2

120582

1

2120582

No bounds

hx

(b)


= minus1 and119872119879


119909and ℎ = ℎ



119906 (119909 119905)

isin

[minus1 1] if 120582 le 0 ℎ le1

2119896 minus 1205824

[minus1 1] if 120582 gt 0 ℎ le1

2 (119896 + 120582)

[[

[



1 minus 2119896ℎ]]

]

if 120582 gt 0 ℎ isin (1

2 (119896 + 120582)

12119896 + 1205822

]

(83)



119909gt 0 in (78) that is

119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ119905

= 119896119906 (119909 minus ℎ

119909 119905) minus 2119906 (119909 119905) + 119906 (119909 + ℎ

119909 119905)

ℎ2119909

+120582119906 (119909 119905) (1minus1199062(119909 119905))

(84)




119909for



119909)

9 Final Remarks




119906 (119909 119905 + ℎ) minus 119906 (119909 119905)

ℎ= 119886 (119909 119905) 119906 (119909 minus 1 119905)

+ 119887 (119909 119905) 119906 (119909 119905)

+ 119888 (119909 119905) 119906 (119909 + 1 119905)

+ 119891 (119909 119905 119906 (119909 119905))

(85)



ℎle 119891 (119909 119905 119906)

le119872


ℎ

(86)




Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgment


References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Maximum Principles for Discrete and