Research Article Global Asymptotic Stability of a ...downloads.hindawi.com/journals/jam/2014/395954.pdf · Research Article Global Asymptotic Stability of a Nonautonomous Difference

Research ArticleGlobal Asymptotic Stability of a NonautonomousDifference Equation

Mehmet GuumlmuumlG1 and Oumlzkan Oumlcalan2

1 Department of Mathematics Faculty of Science and Arts Bulent Ecevit University 67100 Zonguldak Turkey2Department of Mathematics Faculty of Science and Arts Afyon Kocatepe University ANS Campus 03200 Afyonkarahisar Turkey

Correspondence should be addressed to Mehmet Gumus mgumuskaraelmasedutr

Received 16 May 2014 Revised 21 July 2014 Accepted 21 July 2014 Published 11 August 2014

Academic Editor Chein-Shan Liu

Copyright copy 2014 M Gumus and O Ocalan 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 the following nonautonomous difference equation 119909119899+1

= (119909119899119909119899minus1

+ 119901119899)(119909119899+ 119909119899minus1) 119899 = 0 1 where 119901

119899gt 0 is a

period-2 sequence and the initial values 119909minus1 1199090isin (0infin) We show that the unique prime period-2 solution of the equation above

is globally asymptotically stable

1 Introduction

In [1] Xianyi and Deming studied the global asymptoticstability of positive solutions of the difference equation

119909119899+1

=

119909119899119909119899minus1

+ 119901

119909119899+ 119909119899minus1

119899 = 0 1 (1)

where 119901 isin [0infin) and the initial conditions are positive realnumbers

The mathematical modeling of a physical physiologicalor economical problem very often leads to difference equa-tions (for partial review of the theory of difference equa-tions and their applications) Moreover a lot of differenceequations with periodic coefficients have been applied inmathematical models in biology In addition between othersin [2ndash10] we can see some more difference equations withperiodic coefficient that have been studied For constantcoefficient see [1 11ndash15] and the references cited therein

In the present paper we consider the following nonau-tonomous difference equation

119909119899+1

=

119909119899119909119899minus1

+ 119901119899

119909119899+ 119909119899minus1

119899 = 0 1 (2)

where 119901119899 is a two periodic sequence of nonnegative real

numbers and the initial conditions 119909minus1 1199090are arbitrary

positive numbersThe autonomous case of (2) is (1)It is natural problem to investigate the behavior of the

solutions of (1) where we replace the constant 119901 by anonnegative periodic sequence 119901

119899 That is we will consider

the recursive sequence (2) where 119901119899 is a two-periodic

sequence of nonnegative real numbersWhat do the solutionsof (2) with positive initial conditions 119909

minus1and 119909

0do

Our aim in this paper is to investigate the question aboveand extend some results obtained in [1] More precisely westudy the existence of a unique positive periodic solutionof (2) and we investigate the boundedness character andthe convergence of the positive solutions to the unique twoperiodic solution of (2) Finally we study the global asyptoticstability of the positive periodic solution (2)

As far as we examine there is no paper dealing with (2)Therefore in this paper we focus on (2) in order to fill thegap

Now assume that

119901119899=

120572 if 119899 is even120573 if 119899 is odd

(3)

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 395954 5 pageshttpdxdoiorg1011552014395954

2 Journal of Applied Mathematics

in (2) Then

1199092119899+1

=

11990921198991199092119899minus1

+ 120572

1199092119899+ 1199092119899minus1

119899 = 0 1

1199092119899+2

=

1199092119899+1

1199092119899+ 120573

1199092119899+1

+ 1199092119899

119899 = 0 1

(4)

where 119909minus1 1199090isin (0infin)

Here we review some results which will be useful in ourinvestigation of the behavior of positive solutions of (2)

Let 119868 be some interval of real numbers and let

119891 119868 times 119868 997888rarr 119868 (5)

be a continuously differentiable functionThen for every pairof initial values 119909

minus1 1199090isin 119868 the difference equation

119909119899+1

= 119891 (119909119899 119909119899minus1

) 119899 = 0 1 (6)

has a unique solution 119909119899

Definition 1 Let (119909 119910) be an equilibrium point of the map

119861 = (119891 119906119899 119906119899minus1

119906119899minus119896

119892 119911119899 119911

119899minus119896) (7)

where 119891 and 119892 are continuously differentiable functions at(119909 119910) The linearized system of

119906119899+1

= 119891 (119906119899 119906119899minus1

119906119899minus119896

119911119899 119911

119899minus119896)

119911119899+1

= 119892 (119906119899 119906119899minus1

119906119899minus119896

119911119899 119911

119899minus119896)

(8)

about the equilibrium point (119909 119910) is V119899+1

= 119861(V119899) = 119861

119895V119899

where

V119899=

(

(

(

(

119906119899

119906119899minus119896

V119899

V119899minus119896

)

)

)

)

(9)

and 119861119895is the Jacobian matrix of system (8) about the

equilibrium point (119909 119910)

Theorem 2 (linearized stability) Consider the following

(1) If both roots of the quadratic equation

1205822

minus 119901120582 minus 119902 = 0 (10)

lie in the open unit disk |120582| lt 1 then the equilibrium 119909

of (6) is locally asymptotically stable(2) If at least one of the roots of (10) has absolute value

greater than one then the equilibrium 119909 of (6) isunstable

(3) A necessary and sufficent condition for both roots of(10) to lie in the open unit disk |120582| lt 1 is

10038161003816100381610038161199011003816100381610038161003816lt 1 minus 119902 lt 2 (11)

(4) A necessary and sufficent condition for both roots of(10) to have absolute value greater than one is

10038161003816100381610038161199021003816100381610038161003816gt 1

10038161003816100381610038161199011003816100381610038161003816lt10038161003816100381610038161 minus 119902

1003816100381610038161003816 (12)

(5) A necessary and sufficent condition for one root of (10)to have absolute value greater than one and for theother to have absolute value less than one is

1199012

+ 4119902 gt 010038161003816100381610038161199011003816100381610038161003816gt10038161003816100381610038161 minus 119902

1003816100381610038161003816 (13)

(6) A necessary and sufficent condition for a root of (10) tohave absolute value equal to one is

10038161003816100381610038161199011003816100381610038161003816=10038161003816100381610038161 minus 119902

1003816100381610038161003816

(14)

or

119902 = minus110038161003816100381610038161199011003816100381610038161003816le 2 (15)

2 Boundedness of Solutions to (2)In this section we mainly investigate the boundednesscharacter of (2) assuming (3) is satisfied

Theorem 3 Let 120572 120573 gt 0 Then every positive solution of (2)is bounded

Proof Suppose for the sake of contadiction that 119909119899 is an

unbounded solution of (2) Then there exist the subse-quences 119909

2119899+1 and 119909

2119899+2 of the solution 119909

119899 such that one

of the following statements is true(i) 1199092119899+1

rarr infin and 1199092119899+2

rarr 119886 (119886 is arbitrary positivereal number) for all 119899 ge 0

(ii) 1199092119899+1

rarr 119887 (119887 is arbitrary positive real number) and1199092119899+2

rarr infin for all 119899 ge 0(iii) 119909

2119899+1rarr infin and 119909

2119899+2rarr infin for all 119899 ge 1

Now we will show that the above statements are neversatisfied

From (2) we have the following inequalities

1199092119899+1

lt 1199092119899minus1

+

120572

1199092119899

for 119899 ge 0

1199092119899+1

lt 1199092119899+

120572

1199092119899minus1

for 119899 ge 0

1199092119899+2

lt 1199092119899+

120573

1199092119899+1

for 119899 ge 0

1199092119899+2

lt 1199092119899+1

+

120573

1199092119899

for 119899 ge 0

(16)

If we get limits on both sides of the above inequalitiesthey contradict cases (i) and (ii) From (16) and using theinduction we obtain that

1199092119899+1

lt 119909minus1+ 120572(

1

1199090

+ sdot sdot sdot +

1

1199092119899

)

1199092119899+2

lt 1199090+ 120573(

1

1199091

+ sdot sdot sdot +

1

1199092119899+1

)

(17)

Journal of Applied Mathematics 3

Also if we consider the limiting case of the above inequalitiesthey contradict case (iii) and complete the proof

3 Periodicity and Stability of Solutions to (2)In this section we show that (2) has the unique two-periodicsolution and the unique prime period-2 solution of thisequation is locally asymptotically stable

Theorem 4 Suppose that 119901119899gt 0 is a two periodic sequence

such that

119901119899+2

= 119901119899 119899 = 0 1 (18)

Then (2) has the unique two-periodic solution

Proof Let 119909119899be a solution of (2) Using (18) 119909

119899is two-

periodic solution if and only if 119909minus1

and 1199090satisfy

119909minus1= 1199091=

1199090119909minus1+ 120572

1199090+ 119909minus1

1199090= 1199092=

11990901199091+ 120573

1199090+ 1199091

(19)

We set 119909minus1= 119909 and 119909

0= 119910 then from (19) we obtain the

system of equations

119909 =

119909119910 + 120572

119909 + 119910

119910 =

119909119910 + 120573

119909 + 119910

(20)

We prove that (2) has a solution (119909 119910) 119909 119910 gt 0 From (20) weget

1199092

+ 119909119910 = 119909119910 + 120572 1199102

+ 119909119910 = 119909119910 + 120573 (21)

It follows that (2) has a prime period-2 solution if and onlyif 120572 120573 isin (0infin) 120572 = 120573 in which case (2) has a unique two-periodic solution

radic120572radic120573 radic120572radic120573 (22)

Theorem 5 Assume that 119901119899gt 0 is a two periodic sequence

and

120572 120573 isin (0infin) (23)

Then (2) has the unique prime period-2 solutionwhich is locallyasymptotically stable

Proof Let 119909119899be a positive solution of (2) and let

radic120572radic120573radic120572radic120573 radic120572radic120573 (24)

be the unique prime period-2 solution of (2) For any 119899 ge 0set

119906119899= 1199092119899minus1

V119899= 1199092119899 (25)

Then (2) is equivalent to the system

119906119899+1

=

119906119899V119899+ 120572

119906119899+ V119899

V119899+1

=

119906119899(V2119899+ 120573) + (120572 + 120573) V

119899

V119899(2119906119899+ V119899) + 120572

(26)

So in order to study (2) we investigate system (26)Let119867 be the function on (0infin)

2 defined by

119867(

119906

V) = (

119906V + 120572119906 + V

119906 (V2 + 120573) + (120572 + 120573) V

V (2119906 + V) + 120572

) = (

119892 (119906 V)ℎ (119906 V)) (27)

Then

(

radic120572

radic120573

) (28)

is a fixed point of 119867 Clearly the unique prime period-2solution is locally asymptotically stable when the eigenvaluesof the jakobianmatrix 119869

119867 evaluated at ( radic120572

radic120573) lie inside the unit

disk We have

119869119867(

radic120572

radic120573

) = (

120597119892

120597119906

(radic120572radic120573)

120597119892

120597V(radic120572radic120573)

120597ℎ

120597119906


120597ℎ

120597V(radic120572radic120573)

) (29)

The linearized system of (26) about (radic120572radic120573) is the system

119911119899+1

= 119861119911119899 (30)

where

119861 = (

120573 minus 120572

120572 + 120573 + 2radic120572120573

0

0

120572 minus 120573

120572 + 120573 + 2radic120572120573

) 119911119899= (

119906119899

V119899

)

(31)

The characteristic equation of 119861 is

1205822

minus (

120573 minus 120572

120572 + 120573 + 2radic120572120573

+

120572 minus 120573

120572 + 120573 + 2radic120572120573

)120582

minus

(120572 minus 120573)2

(120572 + 120573 + 2radic120572120573)

2= 0

(32)

and thus

1205822

minus

(120572 minus 120573)2

(120572 + 120573 + 2radic120572120573)

2= 0 (33)

FromTheorem 2 and (23) all the roots of (33) are of modulusless than 1 So the system (26) is locally asymptotically stableTherefore (2) has the unique prime period-2 solution whichis locally asymptotically stable


4 Global Asymptotic Stability for (2)In this section we give our main result Before doing so inthe next theorem we will provide an alternative proof for theglobal stability of (1) In the proof of the main theorem wewill use a similar technique

Theorem 6 The positive equilibrium point 119909 of (1) is globallyasymptotically stable

Proof We know fromTheorem 3 that every positive solutionof (1) is bounded for119901 isin (0infin) andwe know from [1] that thepositive equilibrium point 119909 of (1) is locally asymptoticallystable for 119901 isin (0infin)

Let 119909119899 be a positive solution of (1) and set

lim sup119909119899= 119870 lim inf 119909

119899= 119896 (34)

Then by the boundedness of 119909119899 119896 119870 are positive real

numbersThus it is easy to see from (1) that

119896 ge

1198962

+ 119901

2119870

119870 le

1198702

+ 119901

2119896

1198962

+ 119901 le 2119870119896 le 1198702

+ 119901

(35)

Thus we have

1198962

le 1198702 (36)

and so

119896 le 119870 (37)

From (37) if 119896 = 119870 the proof is obvious If 119896 lt 119870 this casecontradicts the unique equilibrium solution of (1) which islocally asymptotically stable Consequently it must be119870 = 119896This completes the proof

Now we are ready to give the proof of the main result

Theorem 7 Assume that 119901119899gt 0 is a two-periodic sequence

and 120572 120573 isin (0infin) then (2) has the unique prime period-2solution which is globally asymptotically stable

Proof We must prove that the unique prime period-2 solu-tion of (2) is both locally asymptotically stable and globallyattractive We know from Theorem 5 that the unique primeperiod-2 solution of (2) is locally asymptotically stable for120572 120573 isin (0infin) It remains to verify that every positive solutionof (2) converges the unique prime period-2 solution Since120572 120573 isin (0infin) by Theorem 3 every positive solution of (2) isbounded Let 119909

119899 be a positive solution of (2) and set

lim sup1199092119899+1

= 119872 lim sup1199092119899= 119878

lim inf 1199092119899+1

= 119898 lim inf 1199092119899= 119904

(38)

Then by the boundedness of 119909119899119898119872 119904 and 119878 are positive

real numbers Moreover it is obvious that since 120572 = 120573 thenfrom (2)

lim119899rarrinfin

1199092119899

= lim1199092119899+1

(39)

and it is clear that if lim1199092119899+1

exists then so does lim119899rarrinfin

1199092119899

Thus it is easy to see from (2) that

119898 ge

119898119904 + 120572

119872 + 119878

119872 le

119872119878 + 120572

119898 + 119904

(40)

119904 ge

119898119904 + 120573

119872 + 119878

119878 le

119872119878 + 120573

119898 + 119904

(41)

From (40) we have

119898119872 +119898119878 minus 119898119904 ge 120572 ge 119898119872 + 119904119872 minus119872119878

119898119872 + 119898119878 minus 119898119904 ge 119898119872 + 119904119872 minus119872119878

119878 ge 119904

(42)

From (41) we have

119904119872 + 119904119878 minus 119898119904 ge 120573 ge 119898119878 + 119904119878 minus119872119878

119904119872 + 119904119878 minus 119898119904 ge 119898119878 + 119904119878 minus119872119878

119872 ge 119898

(43)

With cases (42) and (43) if 119904 = 119878 and 119898 = 119872 the proof isobvious If 119904 lt 119878 and119898 lt 119872 this case contradicts the uniqueprime period-2 solution of (2) which is locally asymptoticallystable Consequently it must be 119904 = 119878 and119872 = 119898 The proofis complete

Conflict of Interests

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

Acknowledgments

The authors would like to thank the referees for their helpfulsuggestions This research is supported by TUBITAK andBulent Ecevit University Research Project Coordinatorship

References

[1] L Xianyi and Z Deming ldquoGlobal asymptotic stability in a ratio-nal equationrdquo Journal of Difference Equations and Applicationsvol 9 no 9 pp 833ndash839 2003

[2] RDevault V L Kocic andD Stutson ldquoGlobal behavior of solu-tions of the nonlinear difference equation 119909

119899+1= 119901119899+ 119909119899minus1119909119899rdquo

Journal of Difference Equations and Application vol 11 no 8 pp707ndash719 2005

[3] E A Grove and G Ladas Periodicities in Nonlinear DifferenceEquations Chapman and HallCRC Press 2005

[4] M R S Kulenovic G Ladas and C B Overdeep ldquo On thedynamics of 119909

119899+1= 119901119899+ (119909119899minus1119909119899)rdquo Journal of Difference

Equations and Applications vol 9 no 11 pp 1053ndash1056 2003


[5] M R C Kulenovic G Ladas and C B Overdeep ldquoOn thedynamics of119909

119899+1= 119901119899+(119909119899minus1119909119899)with a period-two coefficientrdquo

Journal of Difference Equations and Applications vol 10 no 10pp 905ndash914 2004

[6] V G Papanicolaou ldquoOn the asymptotic stability of a class oflinear difference equationsrdquoMathematics Magazine vol 69 no1 pp 34ndash43 1996

[7] G Papaschinopoulos C J Schinas and G Stefanidou ldquoOna difference equation with 3-periodic coefficientrdquo Journal ofDifference Equations and Applications vol 11 no 15 pp 1281ndash1287 2005

[8] G Papaschinopoulos and C J Schinas ldquoOn a (k+1)-orderdifference equation with a coefficient of period k+1rdquo Journal ofDifference Equations and Applications vol 11 no 3 pp 215ndash2252005

[9] O Ocalan ldquoAsymptotic behavior of a higher-order recursivesequencerdquo International Journal of Difference Equations vol 7no 2 pp 175ndash180 2012

[10] O Ocalan ldquoDynamics of the difference equation 119909119899+1

= 119901119899+

119909119899minus119896119909119899with a period-two coefficientrdquo Applied Mathematics

and Computation vol 228 pp 31ndash37 2014[11] E M Elabbasy H A El-Metwally and E M Elsayed ldquoGlobal

behavior of the solutions of some difference equationsrdquo Advan-ces in Difference Equations vol 2011 article 28 2011

[12] EM Elsayed andBD Iricanin ldquoOn amax-type and amin-typedifference equationrdquo Applied Mathematics and Computationvol 215 no 2 pp 608ndash614 2009

[13] E M Elsayed ldquoSolutions of rational difference systems of ordertwordquoMathematical andComputerModelling vol 55 no 3-4 pp378ndash384 2012

[14] V L Kocic and G Ladas Global Behavior of Nonlinear Differ-ence Equations of Higher Order with Applications vol 256 ofMathematics and its Applications KluwerAcademicDordrechtThe Netherlands 1993

[15] M R S Kulenovic and G Ladas Dynamics of Second OrderRational Difference Equations With Open Problems and Conjec-tures Chapman amp HallCRC 2001

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Stochastic AnalysisInternational Journal of


in (2) Then

1199092119899+1

=

11990921198991199092119899minus1

+ 120572

1199092119899+ 1199092119899minus1

119899 = 0 1

1199092119899+2

=

1199092119899+1

1199092119899+ 120573

1199092119899+1

+ 1199092119899

119899 = 0 1

(4)

where 119909minus1 1199090isin (0infin)

Here we review some results which will be useful in ourinvestigation of the behavior of positive solutions of (2)

Let 119868 be some interval of real numbers and let

119891 119868 times 119868 997888rarr 119868 (5)

be a continuously differentiable functionThen for every pairof initial values 119909

minus1 1199090isin 119868 the difference equation

119909119899+1

= 119891 (119909119899 119909119899minus1

) 119899 = 0 1 (6)

has a unique solution 119909119899

Definition 1 Let (119909 119910) be an equilibrium point of the map

119861 = (119891 119906119899 119906119899minus1

119906119899minus119896

119892 119911119899 119911

119899minus119896) (7)

where 119891 and 119892 are continuously differentiable functions at(119909 119910) The linearized system of

119906119899+1

= 119891 (119906119899 119906119899minus1

119906119899minus119896

119911119899 119911

119899minus119896)

119911119899+1

= 119892 (119906119899 119906119899minus1

119906119899minus119896

119911119899 119911

119899minus119896)

(8)

about the equilibrium point (119909 119910) is V119899+1

= 119861(V119899) = 119861

119895V119899

where

V119899=

(

(

(

(

119906119899

119906119899minus119896

V119899

V119899minus119896

)

)

)

)

(9)

and 119861119895is the Jacobian matrix of system (8) about the

equilibrium point (119909 119910)

Theorem 2 (linearized stability) Consider the following

(1) If both roots of the quadratic equation

1205822

minus 119901120582 minus 119902 = 0 (10)

lie in the open unit disk |120582| lt 1 then the equilibrium 119909

of (6) is locally asymptotically stable(2) If at least one of the roots of (10) has absolute value

greater than one then the equilibrium 119909 of (6) isunstable

(3) A necessary and sufficent condition for both roots of(10) to lie in the open unit disk |120582| lt 1 is

10038161003816100381610038161199011003816100381610038161003816lt 1 minus 119902 lt 2 (11)

(4) A necessary and sufficent condition for both roots of(10) to have absolute value greater than one is

10038161003816100381610038161199021003816100381610038161003816gt 1

10038161003816100381610038161199011003816100381610038161003816lt10038161003816100381610038161 minus 119902

1003816100381610038161003816 (12)

(5) A necessary and sufficent condition for one root of (10)to have absolute value greater than one and for theother to have absolute value less than one is

1199012

+ 4119902 gt 010038161003816100381610038161199011003816100381610038161003816gt10038161003816100381610038161 minus 119902

1003816100381610038161003816 (13)

(6) A necessary and sufficent condition for a root of (10) tohave absolute value equal to one is

10038161003816100381610038161199011003816100381610038161003816=10038161003816100381610038161 minus 119902

1003816100381610038161003816

(14)

or

119902 = minus110038161003816100381610038161199011003816100381610038161003816le 2 (15)

2 Boundedness of Solutions to (2)In this section we mainly investigate the boundednesscharacter of (2) assuming (3) is satisfied

Theorem 3 Let 120572 120573 gt 0 Then every positive solution of (2)is bounded

Proof Suppose for the sake of contadiction that 119909119899 is an

unbounded solution of (2) Then there exist the subse-quences 119909

2119899+1 and 119909

2119899+2 of the solution 119909

119899 such that one

of the following statements is true(i) 1199092119899+1

rarr infin and 1199092119899+2

rarr 119886 (119886 is arbitrary positivereal number) for all 119899 ge 0

(ii) 1199092119899+1

rarr 119887 (119887 is arbitrary positive real number) and1199092119899+2

rarr infin for all 119899 ge 0(iii) 119909

2119899+1rarr infin and 119909

2119899+2rarr infin for all 119899 ge 1

Now we will show that the above statements are neversatisfied

From (2) we have the following inequalities

1199092119899+1

lt 1199092119899minus1

+

120572

1199092119899

for 119899 ge 0

1199092119899+1

lt 1199092119899+

120572

1199092119899minus1

for 119899 ge 0

1199092119899+2

lt 1199092119899+

120573

1199092119899+1

for 119899 ge 0

1199092119899+2

lt 1199092119899+1

+

120573

1199092119899

for 119899 ge 0

(16)

If we get limits on both sides of the above inequalitiesthey contradict cases (i) and (ii) From (16) and using theinduction we obtain that

1199092119899+1

lt 119909minus1+ 120572(

1

1199090

+ sdot sdot sdot +

1

1199092119899

)

1199092119899+2

lt 1199090+ 120573(

1

1199091

+ sdot sdot sdot +

1

1199092119899+1

)

(17)





such that

119901119899+2

= 119901119899 119899 = 0 1 (18)



119899is two-


and 1199090satisfy

119909minus1= 1199091=

1199090119909minus1+ 120572

1199090+ 119909minus1

1199090= 1199092=

11990901199091+ 120573

1199090+ 1199091

(19)

We set 119909minus1= 119909 and 119909


system of equations

119909 =

119909119910 + 120572

119909 + 119910

119910 =

119909119910 + 120573

119909 + 119910

(20)


1199092

+ 119909119910 = 119909119910 + 120572 1199102

+ 119909119910 = 119909119910 + 120573 (21)




and

120572 120573 isin (0infin) (23)





119906119899= 1199092119899minus1

V119899= 1199092119899 (25)


119906119899+1

=

119906119899V119899+ 120572

119906119899+ V119899

V119899+1

=

119906119899(V2119899+ 120573) + (120572 + 120573) V

119899

V119899(2119906119899+ V119899) + 120572

(26)


2 defined by

119867(

119906

V) = (

119906V + 120572119906 + V

119906 (V2 + 120573) + (120572 + 120573) V

V (2119906 + V) + 120572

) = (

119892 (119906 V)ℎ (119906 V)) (27)

Then

(

radic120572

radic120573

) (28)




disk We have

119869119867(

radic120572

radic120573

) = (

120597119892

120597119906


120597119892

120597V(radic120572radic120573)

120597ℎ

120597119906


120597ℎ

120597V(radic120572radic120573)

) (29)


119911119899+1

= 119861119911119899 (30)

where

119861 = (

120573 minus 120572

120572 + 120573 + 2radic120572120573

0

0

120572 minus 120573

120572 + 120573 + 2radic120572120573

) 119911119899= (

119906119899

V119899

)

(31)


1205822

minus (

120573 minus 120572

120572 + 120573 + 2radic120572120573

+

120572 minus 120573

120572 + 120573 + 2radic120572120573

)120582

minus

(120572 minus 120573)2

(120572 + 120573 + 2radic120572120573)

2= 0

(32)

and thus

1205822

minus

(120572 minus 120573)2

(120572 + 120573 + 2radic120572120573)

2= 0 (33)







lim sup119909119899= 119870 lim inf 119909

119899= 119896 (34)



119896 ge

1198962

+ 119901

2119870

119870 le

1198702

+ 119901

2119896

1198962

+ 119901 le 2119870119896 le 1198702

+ 119901

(35)

Thus we have

1198962

le 1198702 (36)

and so

119896 le 119870 (37)







lim sup1199092119899+1

= 119872 lim sup1199092119899= 119878

lim inf 1199092119899+1

= 119898 lim inf 1199092119899= 119904

(38)



lim119899rarrinfin

1199092119899

= lim1199092119899+1

(39)



1199092119899


119898 ge

119898119904 + 120572

119872 + 119878

119872 le

119872119878 + 120572

119898 + 119904

(40)

119904 ge

119898119904 + 120573

119872 + 119878

119878 le

119872119878 + 120573

119898 + 119904

(41)

From (40) we have

119898119872 +119898119878 minus 119898119904 ge 120572 ge 119898119872 + 119904119872 minus119872119878

119898119872 + 119898119878 minus 119898119904 ge 119898119872 + 119904119872 minus119872119878

119878 ge 119904

(42)

From (41) we have

119904119872 + 119904119878 minus 119898119904 ge 120573 ge 119898119878 + 119904119878 minus119872119878

119904119872 + 119904119878 minus 119898119904 ge 119898119878 + 119904119878 minus119872119878

119872 ge 119898

(43)




Acknowledgments


References



119899+1= 119901119899+ 119909119899minus1119909119899rdquo















= 119901119899+















Volume 2014




Journal of











Journal of


Function Spaces






Algebra













such that

119901119899+2

= 119901119899 119899 = 0 1 (18)



119899is two-


and 1199090satisfy

119909minus1= 1199091=

1199090119909minus1+ 120572

1199090+ 119909minus1

1199090= 1199092=

11990901199091+ 120573

1199090+ 1199091

(19)

We set 119909minus1= 119909 and 119909


system of equations

119909 =

119909119910 + 120572

119909 + 119910

119910 =

119909119910 + 120573

119909 + 119910

(20)


1199092

+ 119909119910 = 119909119910 + 120572 1199102

+ 119909119910 = 119909119910 + 120573 (21)




and

120572 120573 isin (0infin) (23)





119906119899= 1199092119899minus1

V119899= 1199092119899 (25)


119906119899+1

=

119906119899V119899+ 120572

119906119899+ V119899

V119899+1

=

119906119899(V2119899+ 120573) + (120572 + 120573) V

119899

V119899(2119906119899+ V119899) + 120572

(26)


2 defined by

119867(

119906

V) = (

119906V + 120572119906 + V

119906 (V2 + 120573) + (120572 + 120573) V

V (2119906 + V) + 120572

) = (

119892 (119906 V)ℎ (119906 V)) (27)

Then

(

radic120572

radic120573

) (28)




disk We have

119869119867(

radic120572

radic120573

) = (

120597119892

120597119906


120597119892

120597V(radic120572radic120573)

120597ℎ

120597119906


120597ℎ

120597V(radic120572radic120573)

) (29)


119911119899+1

= 119861119911119899 (30)

where

119861 = (

120573 minus 120572

120572 + 120573 + 2radic120572120573

0

0

120572 minus 120573

120572 + 120573 + 2radic120572120573

) 119911119899= (

119906119899

V119899

)

(31)


1205822

minus (

120573 minus 120572

120572 + 120573 + 2radic120572120573

+

120572 minus 120573

120572 + 120573 + 2radic120572120573

)120582

minus

(120572 minus 120573)2

(120572 + 120573 + 2radic120572120573)

2= 0

(32)

and thus

1205822

minus

(120572 minus 120573)2

(120572 + 120573 + 2radic120572120573)

2= 0 (33)







lim sup119909119899= 119870 lim inf 119909

119899= 119896 (34)



119896 ge

1198962

+ 119901

2119870

119870 le

1198702

+ 119901

2119896

1198962

+ 119901 le 2119870119896 le 1198702

+ 119901

(35)

Thus we have

1198962

le 1198702 (36)

and so

119896 le 119870 (37)







lim sup1199092119899+1

= 119872 lim sup1199092119899= 119878

lim inf 1199092119899+1

= 119898 lim inf 1199092119899= 119904

(38)



lim119899rarrinfin

1199092119899

= lim1199092119899+1

(39)



1199092119899


119898 ge

119898119904 + 120572

119872 + 119878

119872 le

119872119878 + 120572

119898 + 119904

(40)

119904 ge

119898119904 + 120573

119872 + 119878

119878 le

119872119878 + 120573

119898 + 119904

(41)

From (40) we have

119898119872 +119898119878 minus 119898119904 ge 120572 ge 119898119872 + 119904119872 minus119872119878

119898119872 + 119898119878 minus 119898119904 ge 119898119872 + 119904119872 minus119872119878

119878 ge 119904

(42)

From (41) we have

119904119872 + 119904119878 minus 119898119904 ge 120573 ge 119898119878 + 119904119878 minus119872119878

119904119872 + 119904119878 minus 119898119904 ge 119898119878 + 119904119878 minus119872119878

119872 ge 119898

(43)




Acknowledgments


References



119899+1= 119901119899+ 119909119899minus1119909119899rdquo















= 119901119899+















Volume 2014




Journal of











Journal of


Function Spaces






Algebra














lim sup119909119899= 119870 lim inf 119909

119899= 119896 (34)



119896 ge

1198962

+ 119901

2119870

119870 le

1198702

+ 119901

2119896

1198962

+ 119901 le 2119870119896 le 1198702

+ 119901

(35)

Thus we have

1198962

le 1198702 (36)

and so

119896 le 119870 (37)







lim sup1199092119899+1

= 119872 lim sup1199092119899= 119878

lim inf 1199092119899+1

= 119898 lim inf 1199092119899= 119904

(38)



lim119899rarrinfin

1199092119899

= lim1199092119899+1

(39)



1199092119899


119898 ge

119898119904 + 120572

119872 + 119878

119872 le

119872119878 + 120572

119898 + 119904

(40)

119904 ge

119898119904 + 120573

119872 + 119878

119878 le

119872119878 + 120573

119898 + 119904

(41)

From (40) we have

119898119872 +119898119878 minus 119898119904 ge 120572 ge 119898119872 + 119904119872 minus119872119878

119898119872 + 119898119878 minus 119898119904 ge 119898119872 + 119904119872 minus119872119878

119878 ge 119904

(42)

From (41) we have

119904119872 + 119904119878 minus 119898119904 ge 120573 ge 119898119878 + 119904119878 minus119872119878

119904119872 + 119904119878 minus 119898119904 ge 119898119878 + 119904119878 minus119872119878

119872 ge 119898

(43)




Acknowledgments


References



119899+1= 119901119899+ 119909119899minus1119909119899rdquo















= 119901119899+















Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















= 119901119899+















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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Research Article Global Asymptotic Stability of a ...downloads.hindawi.com/journals/jam/2014/395954.pdf · Research Article Global Asymptotic Stability of a Nonautonomous Difference