Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 968541 5 pageshttpdxdoiorg1011552013968541

Research ArticleExistence of Periodic Solutions to Multidelay FunctionalDifferential Equations of Second Order

Cemil Tunccedil and Ramazan Yazgan

Department of Mathematics Faculty of Sciences Yuzuncu Yıl University 65080 Van Turkey

Correspondence should be addressed to Cemil Tunc cemtuncyahoocom

Received 23 August 2013 Accepted 3 October 2013

Academic Editor S A Mohiuddine

Copyright copy 2013 C Tunc and R Yazgan 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

Using Lyapunov-Krasovskii functional approach we establish a new result to guarantee the existence of periodic solutions of acertain multidelay nonlinear functional differential equation of second order By this work we extend and improve some earlierresult in the literature

1 Introduction

It is well known that the problem of the existence of periodicsolutions of retarded functional differential equations ofsecond order is not only very important in the backgroundapplications but also of considerable significance in theoryof differential equations Besides the scope of retardedfunctional differential equations is very general For exam-ple it contains ordinary differential equations differential-difference equations integrodifferential equations and so onThe motivation of this paper is that in recent years the studyof the existence of periodic solutions to various kinds ofretarded functional differential equations of second order hasbecome one of the most attractive topics in the literatureEspecially by using the famous continuation theorem ofdegree theory (see Gaines and Mawhin [1]) many authorshave made a lot of interesting contributions to the topic forretarded functional differential equations of second orderHere we would not like to give the details of these works

On the other hand amongst the achieved excellentresults one of them is the famous Yoshizawarsquos theorem [2]for existence of periodic solutions of retarded functionaldifferential equations which has vital influence and has beenwidely used in the literature This theorem has also beengenerally one of the best results in the literature from the pasttill now It should be noted that in 1994 Zhao et al [3] provedfour sufficiency theorems on the existence of periodic solu-tions for a class of retarded functional differentials to have the

existence of an 120596-periodic solution By this work the authorsproved that their theorems are better than Yoshizawarsquos [2]periodic solutions theorem primarily by removing restric-tions of the size of the constant delay ℎ An example ofapplication was given at the end of the paper Namely inthe same paper the authors applied the following TheoremA to discuss the existence of an 120596-periodic solution of thenonlinear delay differential equation of the second order

11990910158401015840

(119905) + 1198861199091015840

(119905) + 119892 (119909 (119905 minus 120591)) = 119901 (119905) (1)

where 119901(119905) is an external force 119892(119909(119905 minus 120591)) is a delayedrestoring force the delay 120591 is a positive constant and thefriction is proportional to the velocity and 119886 is a positiveconstant It should be noted that a feedback system withfriction proportional to velocity an external force 119901(119905) anda delayed restoring force 119892(119909(119905 minus 120591)) (120591 gt 0) may be writtenas (1) (see Burton [4])

Consider the following general nonautonomous delaydifferential equation

= 119865 (119905 119909119905) 119909

119905= 119909 (119905 + 120579)

minus ℎ le 120579 le 0 119905 ge 0

(2)

where 119865 R times 119862 rarr R119899 119862 = 119862([minusℎ 0]R119899) ℎ is a positiveconstant and we suppose that 119865 is continuous 120596-periodicand takes closed bounded sets into bounded sets of R119899 andsuch that solutions of initial value problems are unique ℎ

2 Abstract and Applied Analysis

can be either larger than 120596 or equal to or smaller than 120596Here (119862 sdot ) is the Banach space of continuous function 120601

[minusℎ 0] rarr R119899 with supremum norm ℎ gt 0 119862119867is the open

119867-ball in 119862 119862119867

= 120601 isin 119862([minusℎ 0]R119899) 120601 lt 119867 Standardexistence theory see Burton [4] shows that if 120601 isin 119862

119867and

119905 ge 0 then there is at least one continuous solution 119909(119905 1199050 120601)

such that on [1199050 1199050+ 120572) satisfying (2) for 119905 gt 119905

0 119909119905(119905 120601) = 120601

and 120572 is a positive constant If there is a closed subset 119861 sub 119862119867

such that the solution remains in 119861 then 120572 = infin Further thesymbol | sdot | will denote a convenient norm in R119899 with |119909|=max1le119894le119899

|119909119894| Let us assume that119862(119905) = 120601 [119905minus120572 119905] rarr R119899 |

120601 is continuous and 120601119905denotes the 120601 in the particular 119862(119905)

and that 120601119905 = max

119905minus120572le119904le119905|120601(119905)| Finally by the periodicity

we mean that that there is an 120596 gt 0 such that 119865(119905 120601) is 120596-periodic in the sense that if 119909(119905) is a solution of (2) so is119909(119905 + 120596)

Definition 1 Solutions of (2) are uniform bounded at 119905 = 0 iffor each 119861

1there exists 119861

2such that [120601 isin 119862 120601 lt 119861

1 119905 ge 0]

imply that |119909(119905 0 120601)| lt 1198612(see Burton [4])

Definition 2 Solutions of (2) are uniform ultimate boundedfor bound 119861 at 119905 = 0 if for each 119861

3gt 0 there exists a 119870 gt 0

such that [120601 isin 119862 120601 lt 1198613 119905 ge 119870] imply that |119909(119905 0 120601)| lt 119861

(see Burton [4])

The first theorem given in Zhao et al [3] is the following

Theorem A If the solutions of (2) are ultimately bounded bythe bound 119861 then

(i) equation (2) has an120596-periodic solution and is boundedby 119861

(ii) if (2) is autonomous then (2) has an equilibriumsolution and is bounded by 119861 (see Zhao et al [3])

Regarding (1) Zhao et al [3] proved the following theoremas example of application

Theorem B Assume that the following conditions hold

(1) 119901(119905) is an 120596-periodic continuous function 119892 is acontinuous differentiable function

(2) lim|119909|rarrinfin

119892(119909) sgn119909 = infin and there is a bounded setΩ containing the origin such that |1198921015840(119909)| le 119888 onΩ

119888Ω119888is the complement of the set Ω

(3) 120591119888 lt 119886

Then (1) has an 120596-periodic motion

Moreover when 119901(119905) = 119870 119870 is a constant under theabove conditions (1) has a constant motion 119909 = 119888

0 and the

constant 1198880satisfies 119892(119888

0) = 119870 In fact from the condition

(2) there is a bounded setΩ1containing the origin such that

|1198921015840(119909)| le 119888 and int

119909

0119892(119904)119889119904 gt 0 on Ω

119888

1 Ω1198881is the complement

of the set Ω1

In this paper we consider the following nonlinear differ-ential equation of second order withmultiple constant delays120591119894(gt 0)

11990910158401015840

(119905) + 119891 (119909 (119905) 1199091015840

(119905)) + 119892 (119909 (119905) 1199091015840

(119905)) 1199091015840

(119905) 1199091015840

(119905)

+ ℎ (119909 (119905)) +

119899

sum

119894=1

119892119894(119909 (119905 minus 120591

119894)) = 119901 (119905)

(3)

where 120591119894are fixed constants delay with 119905 minus 120591

119894ge 0 the primes

in (3) denote differentiation with respect to 119905 isin R 119891 119892 ℎ 119892119894

and 119901 are continuous functions in their respective argumentsonR2R2RR andR respectively and also depend onlyon the arguments displayed explicitlyThe continuity of thesefunctions is a sufficient condition for existence of the solutionof (3) It is also assumed as basic that the functions 119891 119892 ℎand 119892

119894satisfy a Lipschitz condition in 119909 119909

1015840 119909(119905 minus 120591

1) 119909(119905 minus

1205912) 119909(119905 minus 120591

119899) By this assumption the uniqueness of

solutions of (3) is guaranteedThe derivatives 119889119892119894119889119909 equiv 119892

1015840

119894(119909)

exist and are continuous It should be noted that throughoutthe paper sometimes 119909(119905) and 119910(119905) are abbreviated as 119909 and119910 respectively

We write (3) in system form as follows

1199091015840= 119910

1199101015840= minus 119891 (119909 119910) + 119892 (119909 119910) 119910 119910 minus ℎ (119909) minus

119899

sum

119894=1

119892119894(119909)

+

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 + 119901 (119905)

(4)

where 1198921015840

119894(119909(119905 + 119904)) = 119889119892

119894119889119909

It is clear that (3) is a particular case of (2) It shouldbe noted that the reason or the motivation for taking intoconsideration (3) comes from the followingmodified Lienardtype equation of the form

11990910158401015840

(119905) + 119891 (119909 (119905)) + 119892 (119909 (119905)) 1199091015840

(119905) 1199091015840

(119905) + ℎ (119909 (119905)) = 119890 (119905)

(5)

These types of equations have great applications in theory andapplications of the differential equations Therefore till nowthe qualitative behaviors the stability boundedness globalexistence existence of periodic solutions and so forth ofthese type differential equations have been studied by manyresearchers and the researches on these topics are still beingdone in the literature For example we refer the readers to thebooks ofAhmad andRao [5] Burton [4] Gaines andMawhin[1] and the papers of Constantin [6] Graef [7] Huang andYu[8] Jin [9] Liu andHuang [10]NapolesValdes [11] Qian [12]Tunc [13ndash21] C Tunc and E Tunc [22] Zhao et al [3] Zhou[23] and the references cited in these works

We here give certain sufficient conditions to guaranteethe existence of an 120596-periodic solution of (3) This paper isinspired by the mentioned papers and that in the literatureOur aim is to generalize and improve the application given in[3] for (3) This paper has also a contribution to the inves-tigation of the qualitative behaviors of retarded functional

Abstract and Applied Analysis 3

differential equations of second order and itmay be useful forresearchers whowork on the abovementioned topics Finallywithout using the famous continuation theorem of degreetheory which belongs to Gaines and Mawhin [1] we provethe following main result This case makes the topic of thispaper interesting

2 Main Result

Our main result is the following

Theorem 3 We assume that there are positive constants 119886 119886119887 120591 and 119888

119894such that the following conditions hold

(i) 119886 ge 119891(119909 119910) + 119892(119909 119910)119910 ge 119886

(ii) lim|119909|rarrinfin

ℎ(119909) sgn119909 = infin and there is a bounded setΩ containing the origin such that |ℎ1015840(119909)| le 119887 onΩ

119888Ω119888is the complement of the set Ω

(iii) lim|119909|rarrinfin

119892119894(119909) sgn119909 = infin and there is a bounded set

Ω containing the origin such that |1198921015840119894(119909)| le 119888

119894onΩ119888Ω119888

is the complement of the set Ω

(iv) 119901(119905) is an 120596-periodic continuous function

If 120591sum119899

119894=1119888119894lt 119886 then (3) has an 120596-periodic solution

Moreover 119901(119905) = 119870 119870-constant under the above con-ditions (3) has a constant motion 119909 = 119888

0 and the constant 119888

0

satisfies ℎ(1198880) + 119892119894(1198880) = 119870

Proof Define the Lyapunov-Krasovskii functional 119881 =

119881(119909119905 119910119905)

119881 =

1

2

1199102+ int

119909

0

ℎ (119904) 119889119904 +

119899

sum

119894=1

int

119909

0

119892119894(119904) 119889119904

+

119899

sum

119894=1

120582119894int

0

minus120591119894

int

119905

119905+119904

1199102

(120579) 119889120579 119889119904

(6)

where 120582119894are some positive constants to be determined later

in the proofEvaluating the time derivative of 119881 along system (4) we

get

= minus 119891 (119909 119910) + 119892 (119909 119910) 119910 1199102

+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904

+ 119910119901 (119905) +

119899

sum

119894=1

120582119894int

0

minus120591119894

(1199102

(119905) minus 1199102

(119905 + 119904)) 119889119904

= minus 119891 (119909 119910) + 119892 (119909 119910) 119910 1199102

+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904

+ 119910119901 (119905) +

119899

sum

119894=1

(120582119894120591119894) 1199102

minus

119899

sum

119894=1

120582119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904

(7)

By noting the assumption |1198921015840

119894(119909)| le 119888

119894of the theorem and

the estimate 2|120572120574| le 1205722+ 1205742 one can obtain the following

estimates

119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904

le

119899

sum

119894=1

int

0

minus120591119894

100381610038161003816100381610038161198921015840

119894(119909 (119905 + 119904))

10038161003816100381610038161003816

1003816100381610038161003816119910 (119905)

1003816100381610038161003816

1003816100381610038161003816119910 (119905 + 119904)

1003816100381610038161003816119889119904

le

1

2

119899

sum

119894=1

int

0

minus120591119894

119888119894(1199102

(119905) + 1199102

(119905 + 119904)) 119889119904

le

1

2

119899

sum

119894=1

(119888119894120591119894) 1199102+

1

2

119899

sum

119894=1

119888119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904

(8)

Then it follows that

le minus 119891 (119909 119910) + 119892 (119909 119910) 119910 1199102+ 119910119901 (119905)

+

1

2

119899

sum

119894=1

(119888119894+ 2120582119894) 1205911198941199102

+

1

2

119899

sum

119894=1

(119888119894minus 2120582119894) int

0

minus120591119894

1199102

(119905 + 119904) 119889119904

(9)

Let 120582119894= 1198881198942 and 120591 = max120591

1 120591

119899 In fact these choices

imply that

le minus119891 (119909 119910) + 119892 (119909 119910) 119910 minus 120591

119899

sum

119894=1

119888119894minus

1003816100381610038161003816119901 (119905)

1003816100381610038161003816

10038161003816100381610038161199101003816100381610038161003816

minus1

1199102

le minus(119886 minus 120591

119899

sum

119894=1

119888119894minus

1003816100381610038161003816119901 (119905)

1003816100381610038161003816

10038161003816100381610038161199101003816100381610038161003816

minus1

)1199102

(10)

In view of the continuity and periodicity of the function119901 and the assumption 119886 minus 120591sum

119899

119894=1119888119894gt 0 it follows that there is

a bounded set Ω2supe Ω1with Ω

2containing the origin and a

positive constant 120583 such that

120583 le 119886 minus 119888120591 minus

1003816100381610038161003816119901 (119905)

1003816100381610038161003816

10038161003816100381610038161199101003816100381610038161003816

for R times Ω119888

2 (11)

Therefore we can write

le minus1205831199102 for (119905 119909 119910) isin R times Ω

119888

2times Ω119888

2 (12)

From the last estimate we can arrive that the 119910-coor-dinate of the solutions of system (4) is ultimately bounded

4 Abstract and Applied Analysis

for a positive constant 120573 On the other hand since 119901 is acontinuous periodic function if |119910| le 120573 and119881

1= 119881+119910 then

subject to the assumptions of the theorem it can be easily seenthat there is a constant 119870

1gt 0 onR times Ω

119888

2such that

1= + 119910

1015840

= minus 119891 (119909 119910) + 119892 (119909 119910) 119910 1199102

+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 + 119910119901 (119905)

+

119899

sum

119894=1

(120582119894120591119894) 1199102minus

119899

sum

119894=1

120582119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904

minus 119891 (119909 119910) + 119892 (119909 119910) 119910 119910 minus ℎ (119909) minus

119899

sum

119894=1

119892119894(119909)

+

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 + 119901 (119905)

le minus1198861199102+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904

+ 119910119901 (119905) +

119899

sum

119894=1

(120582119894120591119894) 1199102

minus

119899

sum

119894=1

120582119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904 + 11988610038161003816100381610038161199101003816100381610038161003816minus ℎ (119909) minus

119899

sum

119894=1

119892119894(119909)

+

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 + 119901 (119905)

le minusℎ (119909) minus

119899

sum

119894=1

119892119894(119909) + 119870

1

(13)

Since ℎ(119909) rarr infin and 119892119894(119909) rarr infin as 119909 rarr infin then it can

be chosen a positive constant 1198611such that

1le minus05 for 119909 ge 119861

1 (14)

Therefore we can conclude that there is a positive constant1205721such that the 119909-coordinate of the solutions of system (4)

satisfies 119909 le 1205721for |119910| le 120573

In a similar manner if |119910| le 120573 and 1198812

= 119881 minus 119910 thensubject to the assumptions of the theorem it can be easilyfollowed by the time derivative of the functional119881

2that there

is a constant 1198702gt 0 onR times Ω

119888

2such that

2= minus 119910

1015840

= minus 119891 (119909 119910) + 119892 (119909 119910) 119910 1199102

+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 + 119910119901 (119905)

+

119899

sum

119894=1

(120582119894120591119894) 1199102minus

119899

sum

119894=1

120582119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904

+ 119891 (119909 119910) + 119892 (119909 119910) 119910 119910 + ℎ (119909) +

119899

sum

119894=1

119892119894(119909)

minus

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 minus 119901 (119905)

le minus 1198861199102+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904

+ 119910119901 (119905) +

119899

sum

119894=1

(120582119894120591119894) 1199102minus

119899

sum

119894=1

120582119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904

+ 11988610038161003816100381610038161199101003816100381610038161003816+ ℎ (119909) +

119899

sum

119894=1

119892119894(119909)

minus

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 minus 119901 (119905)

le ℎ (119909) +

119899

sum

119894=1

119892119894(119909) + 119870

2

(15)

Since ℎ(119909) rarr minusinfin and 119892119894(119909) rarr minusinfin as 119909 rarr minusinfin then

it can be chosen a positive constant 1198612such that

2le minus05 for 119909 le minus119861

2 (16)

Then we can conclude that there is a positive constant1205722such that the 119909-coordinate of the solutions of system (4)

satisfies 119909 ge minus1205722for |119910| le 120573 On gathering the above whole

discussion one can see that the solutions of system (4) areultimately boundedTherefore (3) has an 120596-periodic motion(solution) When 119901(119905) = 119870 119870-constant (3) has a constantmotion 119909 = 119888

0 From (3) it can be seen that the constant 119909 =

1198880is given by ℎ(119888

0) + 119892119894(1198880) = 119870

Conflict of Interests

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

References

[1] R E Gaines and J L Mawhin Coincidence Degree andNonlinear Differential Equations vol 568 of Lecture Notes inMathematics Springer Berlin Germany 1977

[2] T Yoshizawa ldquoStability theory by Liapunovrsquos second methodrdquoPublications of the Mathematical Society of Japan 9 TheMathematical Society of Japan Tokyo Japan 1966

[3] J M Zhao K L Huang andQ S Lu ldquoThe existence of periodicsolutions for a class of functional-differential equations andtheir applicationrdquo Applied Mathematics and Mechanics EnglishEdition vol 15 no 1 pp 49ndash59 1994 translated from AppliedMathematics and Mechanics vol 15 no 1 pp 49ndash58 1994

Abstract and Applied Analysis 5

[4] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations Dover Mineola NY USA2005 Corrected Version of the 1985 Original

[5] S Ahmad and M R RaoTheory of Ordinary differential Equa-tions With Applications in Biology and Engineering AffiliatedEast-West Press New Delhi India 1999

[6] A Constantin ldquoA note on a second-order nonlinear differentialsystemrdquo Glasgow Mathematical Journal vol 42 no 2 pp 195ndash199 2000

[7] J R Graef ldquoOn the generalized Lienard equation with negativedampingrdquo Journal of Differential Equations vol 12 pp 34ndash621972

[8] L H Huang and J S Yu ldquoOn boundedness of solutions ofgeneralized Lienardrsquos system and its applicationrdquo Annals ofDifferential Equations vol 9 no 3 pp 311ndash318 1993

[9] Z Jin ldquoBoundedness and convergence of solutions of a second-order nonlinear differential systemrdquo Journal of MathematicalAnalysis and Applications vol 256 no 2 pp 360ndash374 2001

[10] B Liu and L Huang ldquoBoundedness of solutions for a class ofretarded Lienard equationrdquo Journal of Mathematical Analysisand Applications vol 286 no 2 pp 422ndash434 2003

[11] J E Napoles Valdes ldquoBoundedness and global asymptoticstability of the forced Lienard equationrdquo Revista de la UnionMatematica Argentina vol 41 no 4 pp 47ndash59 2000

[12] C X Qian ldquoBoundedness and asymptotic behaviour of solu-tions of a second-order nonlinear systemrdquo The Bulletin of theLondon Mathematical Society vol 24 no 3 pp 281ndash288 1992

[13] C Tunc ldquoA note on the bounded solutionsrdquoAppliedMathemat-ics and Information Sciences vol 8 no 1 pp 393ndash399 2014

[14] C Tunc ldquoA note on boundedness of solutions to a class of non-autonomous differential equations of second orderrdquo ApplicableAnalysis and Discrete Mathematics vol 4 no 2 pp 361ndash3722010

[15] C Tunc ldquoBoundedness results for solutions of certain nonlineardifferential equations of second orderrdquo Journal of the IndonesianMathematical Society vol 16 no 2 pp 115ndash126 2010

[16] C Tunc ldquoUniformly stability and boundedness of solutions ofsecond order nonlinear delay differential equationsrdquo Appliedand Computational Mathematics vol 10 no 3 pp 449ndash4622011

[17] C Tunc ldquoOn the boundedness of solutions of a non-auton-omous differential equation of second orderrdquo Sarajevo Journalof Mathematics vol 7(19) no 1 pp 19ndash29 2011

[18] C Tunc ldquoStability and uniform boundedness results for non-autonomous Lienard-type equations with a variable deviatingargumentrdquoActaMathematica Vietnamica vol 37 no 3 pp 311ndash325 2012

[19] C Tunc ldquoOn the stability and boundedness of solutions of aclass of nonautonomous differential equations of second orderwith multiple deviating argumentsrdquoAfrikaMatematika vol 23no 2 pp 249ndash259 2012

[20] C Tunc ldquoStability to vector Lienard equation with constantdeviating argumentrdquo Nonlinear Dynamics vol 73 no 3 pp1245ndash1251 2013

[21] C Tunc ldquoNew results on the existence of periodic solutionsfor Rayleigh equation with state-dependent delayrdquo Journal ofMathematical and Fundamental Sciences vol 45 no 2 pp 154ndash162 2013

[22] C Tunc and E Tunc ldquoOn the asymptotic behavior of solutionsof certain second-order differential equationsrdquo Journal of theFranklin Institute vol 344 no 5 pp 391ndash398 2007

[23] J Zhou ldquoNecessary and sufficient conditions for boundednessand convergence of a second-order nonlinear differential sys-temrdquo Acta Mathematica Sinica Chinese Series vol 43 no 3 pp415ndash420 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

can be either larger than 120596 or equal to or smaller than 120596Here (119862 sdot ) is the Banach space of continuous function 120601

[minusℎ 0] rarr R119899 with supremum norm ℎ gt 0 119862119867is the open

119867-ball in 119862 119862119867

= 120601 isin 119862([minusℎ 0]R119899) 120601 lt 119867 Standardexistence theory see Burton [4] shows that if 120601 isin 119862

119867and

119905 ge 0 then there is at least one continuous solution 119909(119905 1199050 120601)

such that on [1199050 1199050+ 120572) satisfying (2) for 119905 gt 119905

0 119909119905(119905 120601) = 120601

and 120572 is a positive constant If there is a closed subset 119861 sub 119862119867

such that the solution remains in 119861 then 120572 = infin Further thesymbol | sdot | will denote a convenient norm in R119899 with |119909|=max1le119894le119899

|119909119894| Let us assume that119862(119905) = 120601 [119905minus120572 119905] rarr R119899 |

120601 is continuous and 120601119905denotes the 120601 in the particular 119862(119905)

and that 120601119905 = max

119905minus120572le119904le119905|120601(119905)| Finally by the periodicity

we mean that that there is an 120596 gt 0 such that 119865(119905 120601) is 120596-periodic in the sense that if 119909(119905) is a solution of (2) so is119909(119905 + 120596)

Definition 1 Solutions of (2) are uniform bounded at 119905 = 0 iffor each 119861

1there exists 119861

2such that [120601 isin 119862 120601 lt 119861

1 119905 ge 0]

imply that |119909(119905 0 120601)| lt 1198612(see Burton [4])

Definition 2 Solutions of (2) are uniform ultimate boundedfor bound 119861 at 119905 = 0 if for each 119861

3gt 0 there exists a 119870 gt 0

such that [120601 isin 119862 120601 lt 1198613 119905 ge 119870] imply that |119909(119905 0 120601)| lt 119861

(see Burton [4])

The first theorem given in Zhao et al [3] is the following

Theorem A If the solutions of (2) are ultimately bounded bythe bound 119861 then

(i) equation (2) has an120596-periodic solution and is boundedby 119861

(ii) if (2) is autonomous then (2) has an equilibriumsolution and is bounded by 119861 (see Zhao et al [3])

Regarding (1) Zhao et al [3] proved the following theoremas example of application

Theorem B Assume that the following conditions hold

(1) 119901(119905) is an 120596-periodic continuous function 119892 is acontinuous differentiable function

(2) lim|119909|rarrinfin

119892(119909) sgn119909 = infin and there is a bounded setΩ containing the origin such that |1198921015840(119909)| le 119888 onΩ

119888Ω119888is the complement of the set Ω

(3) 120591119888 lt 119886

Then (1) has an 120596-periodic motion

Moreover when 119901(119905) = 119870 119870 is a constant under theabove conditions (1) has a constant motion 119909 = 119888

0 and the

constant 1198880satisfies 119892(119888

0) = 119870 In fact from the condition

(2) there is a bounded setΩ1containing the origin such that

|1198921015840(119909)| le 119888 and int

119909

0119892(119904)119889119904 gt 0 on Ω

119888

1 Ω1198881is the complement

of the set Ω1

In this paper we consider the following nonlinear differ-ential equation of second order withmultiple constant delays120591119894(gt 0)

11990910158401015840

(119905) + 119891 (119909 (119905) 1199091015840

(119905)) + 119892 (119909 (119905) 1199091015840

(119905)) 1199091015840

(119905) 1199091015840

(119905)

+ ℎ (119909 (119905)) +

119899

sum

119894=1

119892119894(119909 (119905 minus 120591

119894)) = 119901 (119905)

(3)

where 120591119894are fixed constants delay with 119905 minus 120591

119894ge 0 the primes

in (3) denote differentiation with respect to 119905 isin R 119891 119892 ℎ 119892119894

and 119901 are continuous functions in their respective argumentsonR2R2RR andR respectively and also depend onlyon the arguments displayed explicitlyThe continuity of thesefunctions is a sufficient condition for existence of the solutionof (3) It is also assumed as basic that the functions 119891 119892 ℎand 119892

119894satisfy a Lipschitz condition in 119909 119909

1015840 119909(119905 minus 120591

1) 119909(119905 minus

1205912) 119909(119905 minus 120591

119899) By this assumption the uniqueness of

solutions of (3) is guaranteedThe derivatives 119889119892119894119889119909 equiv 119892

1015840

119894(119909)

exist and are continuous It should be noted that throughoutthe paper sometimes 119909(119905) and 119910(119905) are abbreviated as 119909 and119910 respectively

We write (3) in system form as follows

1199091015840= 119910

1199101015840= minus 119891 (119909 119910) + 119892 (119909 119910) 119910 119910 minus ℎ (119909) minus

119899

sum

119894=1

119892119894(119909)

+

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 + 119901 (119905)

(4)

where 1198921015840

119894(119909(119905 + 119904)) = 119889119892

119894119889119909

It is clear that (3) is a particular case of (2) It shouldbe noted that the reason or the motivation for taking intoconsideration (3) comes from the followingmodified Lienardtype equation of the form

11990910158401015840

(119905) + 119891 (119909 (119905)) + 119892 (119909 (119905)) 1199091015840

(119905) 1199091015840

(119905) + ℎ (119909 (119905)) = 119890 (119905)

(5)

These types of equations have great applications in theory andapplications of the differential equations Therefore till nowthe qualitative behaviors the stability boundedness globalexistence existence of periodic solutions and so forth ofthese type differential equations have been studied by manyresearchers and the researches on these topics are still beingdone in the literature For example we refer the readers to thebooks ofAhmad andRao [5] Burton [4] Gaines andMawhin[1] and the papers of Constantin [6] Graef [7] Huang andYu[8] Jin [9] Liu andHuang [10]NapolesValdes [11] Qian [12]Tunc [13ndash21] C Tunc and E Tunc [22] Zhao et al [3] Zhou[23] and the references cited in these works

We here give certain sufficient conditions to guaranteethe existence of an 120596-periodic solution of (3) This paper isinspired by the mentioned papers and that in the literatureOur aim is to generalize and improve the application given in[3] for (3) This paper has also a contribution to the inves-tigation of the qualitative behaviors of retarded functional

Abstract and Applied Analysis 3

differential equations of second order and itmay be useful forresearchers whowork on the abovementioned topics Finallywithout using the famous continuation theorem of degreetheory which belongs to Gaines and Mawhin [1] we provethe following main result This case makes the topic of thispaper interesting

2 Main Result

Our main result is the following

Theorem 3 We assume that there are positive constants 119886 119886119887 120591 and 119888

119894such that the following conditions hold

(i) 119886 ge 119891(119909 119910) + 119892(119909 119910)119910 ge 119886

(ii) lim|119909|rarrinfin

ℎ(119909) sgn119909 = infin and there is a bounded setΩ containing the origin such that |ℎ1015840(119909)| le 119887 onΩ

119888Ω119888is the complement of the set Ω

(iii) lim|119909|rarrinfin

119892119894(119909) sgn119909 = infin and there is a bounded set

Ω containing the origin such that |1198921015840119894(119909)| le 119888

119894onΩ119888Ω119888

is the complement of the set Ω

(iv) 119901(119905) is an 120596-periodic continuous function

If 120591sum119899

119894=1119888119894lt 119886 then (3) has an 120596-periodic solution

Moreover 119901(119905) = 119870 119870-constant under the above con-ditions (3) has a constant motion 119909 = 119888

0 and the constant 119888

0

satisfies ℎ(1198880) + 119892119894(1198880) = 119870

Proof Define the Lyapunov-Krasovskii functional 119881 =

119881(119909119905 119910119905)

119881 =

1

2

1199102+ int

119909

0

ℎ (119904) 119889119904 +

119899

sum

119894=1

int

119909

0

119892119894(119904) 119889119904

+

119899

sum

119894=1

120582119894int

0

minus120591119894

int

119905

119905+119904

1199102

(120579) 119889120579 119889119904

(6)

where 120582119894are some positive constants to be determined later

in the proofEvaluating the time derivative of 119881 along system (4) we

get

= minus 119891 (119909 119910) + 119892 (119909 119910) 119910 1199102

+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904

+ 119910119901 (119905) +

119899

sum

119894=1

120582119894int

0

minus120591119894

(1199102

(119905) minus 1199102

(119905 + 119904)) 119889119904

= minus 119891 (119909 119910) + 119892 (119909 119910) 119910 1199102

+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904

+ 119910119901 (119905) +

119899

sum

119894=1

(120582119894120591119894) 1199102

minus

119899

sum

119894=1

120582119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904

(7)

By noting the assumption |1198921015840

119894(119909)| le 119888

119894of the theorem and

the estimate 2|120572120574| le 1205722+ 1205742 one can obtain the following

estimates

119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904

le

119899

sum

119894=1

int

0

minus120591119894

100381610038161003816100381610038161198921015840

119894(119909 (119905 + 119904))

10038161003816100381610038161003816

1003816100381610038161003816119910 (119905)

1003816100381610038161003816

1003816100381610038161003816119910 (119905 + 119904)

1003816100381610038161003816119889119904

le

1

2

119899

sum

119894=1

int

0

minus120591119894

119888119894(1199102

(119905) + 1199102

(119905 + 119904)) 119889119904

le

1

2

119899

sum

119894=1

(119888119894120591119894) 1199102+

1

2

119899

sum

119894=1

119888119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904

(8)

Then it follows that

le minus 119891 (119909 119910) + 119892 (119909 119910) 119910 1199102+ 119910119901 (119905)

+

1

2

119899

sum

119894=1

(119888119894+ 2120582119894) 1205911198941199102

+

1

2

119899

sum

119894=1

(119888119894minus 2120582119894) int

0

minus120591119894

1199102

(119905 + 119904) 119889119904

(9)

Let 120582119894= 1198881198942 and 120591 = max120591

1 120591

119899 In fact these choices

imply that

le minus119891 (119909 119910) + 119892 (119909 119910) 119910 minus 120591

119899

sum

119894=1

119888119894minus

1003816100381610038161003816119901 (119905)

1003816100381610038161003816

10038161003816100381610038161199101003816100381610038161003816

minus1

1199102

le minus(119886 minus 120591

119899

sum

119894=1

119888119894minus

1003816100381610038161003816119901 (119905)

1003816100381610038161003816

10038161003816100381610038161199101003816100381610038161003816

minus1

)1199102

(10)

In view of the continuity and periodicity of the function119901 and the assumption 119886 minus 120591sum

119899

119894=1119888119894gt 0 it follows that there is

a bounded set Ω2supe Ω1with Ω

2containing the origin and a

positive constant 120583 such that

120583 le 119886 minus 119888120591 minus

1003816100381610038161003816119901 (119905)

1003816100381610038161003816

10038161003816100381610038161199101003816100381610038161003816

for R times Ω119888

2 (11)

Therefore we can write

le minus1205831199102 for (119905 119909 119910) isin R times Ω

119888

2times Ω119888

2 (12)

From the last estimate we can arrive that the 119910-coor-dinate of the solutions of system (4) is ultimately bounded

4 Abstract and Applied Analysis

for a positive constant 120573 On the other hand since 119901 is acontinuous periodic function if |119910| le 120573 and119881

1= 119881+119910 then

subject to the assumptions of the theorem it can be easily seenthat there is a constant 119870

1gt 0 onR times Ω

119888

2such that

1= + 119910

1015840

= minus 119891 (119909 119910) + 119892 (119909 119910) 119910 1199102

+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 + 119910119901 (119905)

+

119899

sum

119894=1

(120582119894120591119894) 1199102minus

119899

sum

119894=1

120582119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904

minus 119891 (119909 119910) + 119892 (119909 119910) 119910 119910 minus ℎ (119909) minus

119899

sum

119894=1

119892119894(119909)

+

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 + 119901 (119905)

le minus1198861199102+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904

+ 119910119901 (119905) +

119899

sum

119894=1

(120582119894120591119894) 1199102

minus

119899

sum

119894=1

120582119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904 + 11988610038161003816100381610038161199101003816100381610038161003816minus ℎ (119909) minus

119899

sum

119894=1

119892119894(119909)

+

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 + 119901 (119905)

le minusℎ (119909) minus

119899

sum

119894=1

119892119894(119909) + 119870

1

(13)

Since ℎ(119909) rarr infin and 119892119894(119909) rarr infin as 119909 rarr infin then it can

be chosen a positive constant 1198611such that

1le minus05 for 119909 ge 119861

1 (14)

Therefore we can conclude that there is a positive constant1205721such that the 119909-coordinate of the solutions of system (4)

satisfies 119909 le 1205721for |119910| le 120573

In a similar manner if |119910| le 120573 and 1198812

= 119881 minus 119910 thensubject to the assumptions of the theorem it can be easilyfollowed by the time derivative of the functional119881

2that there

is a constant 1198702gt 0 onR times Ω

119888

2such that

2= minus 119910

1015840

= minus 119891 (119909 119910) + 119892 (119909 119910) 119910 1199102

+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 + 119910119901 (119905)

+

119899

sum

119894=1

(120582119894120591119894) 1199102minus

119899

sum

119894=1

120582119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904

+ 119891 (119909 119910) + 119892 (119909 119910) 119910 119910 + ℎ (119909) +

119899

sum

119894=1

119892119894(119909)

minus

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 minus 119901 (119905)

le minus 1198861199102+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904

+ 119910119901 (119905) +

119899

sum

119894=1

(120582119894120591119894) 1199102minus

119899

sum

119894=1

120582119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904

+ 11988610038161003816100381610038161199101003816100381610038161003816+ ℎ (119909) +

119899

sum

119894=1

119892119894(119909)

minus

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 minus 119901 (119905)

le ℎ (119909) +

119899

sum

119894=1

119892119894(119909) + 119870

2

(15)

Since ℎ(119909) rarr minusinfin and 119892119894(119909) rarr minusinfin as 119909 rarr minusinfin then

it can be chosen a positive constant 1198612such that

2le minus05 for 119909 le minus119861

2 (16)

Then we can conclude that there is a positive constant1205722such that the 119909-coordinate of the solutions of system (4)

satisfies 119909 ge minus1205722for |119910| le 120573 On gathering the above whole

discussion one can see that the solutions of system (4) areultimately boundedTherefore (3) has an 120596-periodic motion(solution) When 119901(119905) = 119870 119870-constant (3) has a constantmotion 119909 = 119888

0 From (3) it can be seen that the constant 119909 =

1198880is given by ℎ(119888

0) + 119892119894(1198880) = 119870

Conflict of Interests

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

References

[1] R E Gaines and J L Mawhin Coincidence Degree andNonlinear Differential Equations vol 568 of Lecture Notes inMathematics Springer Berlin Germany 1977

[2] T Yoshizawa ldquoStability theory by Liapunovrsquos second methodrdquoPublications of the Mathematical Society of Japan 9 TheMathematical Society of Japan Tokyo Japan 1966

[3] J M Zhao K L Huang andQ S Lu ldquoThe existence of periodicsolutions for a class of functional-differential equations andtheir applicationrdquo Applied Mathematics and Mechanics EnglishEdition vol 15 no 1 pp 49ndash59 1994 translated from AppliedMathematics and Mechanics vol 15 no 1 pp 49ndash58 1994

Abstract and Applied Analysis 5

[4] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations Dover Mineola NY USA2005 Corrected Version of the 1985 Original

[5] S Ahmad and M R RaoTheory of Ordinary differential Equa-tions With Applications in Biology and Engineering AffiliatedEast-West Press New Delhi India 1999

[6] A Constantin ldquoA note on a second-order nonlinear differentialsystemrdquo Glasgow Mathematical Journal vol 42 no 2 pp 195ndash199 2000

[7] J R Graef ldquoOn the generalized Lienard equation with negativedampingrdquo Journal of Differential Equations vol 12 pp 34ndash621972

[8] L H Huang and J S Yu ldquoOn boundedness of solutions ofgeneralized Lienardrsquos system and its applicationrdquo Annals ofDifferential Equations vol 9 no 3 pp 311ndash318 1993

[9] Z Jin ldquoBoundedness and convergence of solutions of a second-order nonlinear differential systemrdquo Journal of MathematicalAnalysis and Applications vol 256 no 2 pp 360ndash374 2001

[10] B Liu and L Huang ldquoBoundedness of solutions for a class ofretarded Lienard equationrdquo Journal of Mathematical Analysisand Applications vol 286 no 2 pp 422ndash434 2003

[11] J E Napoles Valdes ldquoBoundedness and global asymptoticstability of the forced Lienard equationrdquo Revista de la UnionMatematica Argentina vol 41 no 4 pp 47ndash59 2000

[12] C X Qian ldquoBoundedness and asymptotic behaviour of solu-tions of a second-order nonlinear systemrdquo The Bulletin of theLondon Mathematical Society vol 24 no 3 pp 281ndash288 1992

[13] C Tunc ldquoA note on the bounded solutionsrdquoAppliedMathemat-ics and Information Sciences vol 8 no 1 pp 393ndash399 2014

[14] C Tunc ldquoA note on boundedness of solutions to a class of non-autonomous differential equations of second orderrdquo ApplicableAnalysis and Discrete Mathematics vol 4 no 2 pp 361ndash3722010

[15] C Tunc ldquoBoundedness results for solutions of certain nonlineardifferential equations of second orderrdquo Journal of the IndonesianMathematical Society vol 16 no 2 pp 115ndash126 2010

[16] C Tunc ldquoUniformly stability and boundedness of solutions ofsecond order nonlinear delay differential equationsrdquo Appliedand Computational Mathematics vol 10 no 3 pp 449ndash4622011

[17] C Tunc ldquoOn the boundedness of solutions of a non-auton-omous differential equation of second orderrdquo Sarajevo Journalof Mathematics vol 7(19) no 1 pp 19ndash29 2011

[18] C Tunc ldquoStability and uniform boundedness results for non-autonomous Lienard-type equations with a variable deviatingargumentrdquoActaMathematica Vietnamica vol 37 no 3 pp 311ndash325 2012

[19] C Tunc ldquoOn the stability and boundedness of solutions of aclass of nonautonomous differential equations of second orderwith multiple deviating argumentsrdquoAfrikaMatematika vol 23no 2 pp 249ndash259 2012

[20] C Tunc ldquoStability to vector Lienard equation with constantdeviating argumentrdquo Nonlinear Dynamics vol 73 no 3 pp1245ndash1251 2013

[21] C Tunc ldquoNew results on the existence of periodic solutionsfor Rayleigh equation with state-dependent delayrdquo Journal ofMathematical and Fundamental Sciences vol 45 no 2 pp 154ndash162 2013

[22] C Tunc and E Tunc ldquoOn the asymptotic behavior of solutionsof certain second-order differential equationsrdquo Journal of theFranklin Institute vol 344 no 5 pp 391ndash398 2007

[23] J Zhou ldquoNecessary and sufficient conditions for boundednessand convergence of a second-order nonlinear differential sys-temrdquo Acta Mathematica Sinica Chinese Series vol 43 no 3 pp415ndash420 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

differential equations of second order and itmay be useful forresearchers whowork on the abovementioned topics Finallywithout using the famous continuation theorem of degreetheory which belongs to Gaines and Mawhin [1] we provethe following main result This case makes the topic of thispaper interesting

2 Main Result

Our main result is the following

Theorem 3 We assume that there are positive constants 119886 119886119887 120591 and 119888

119894such that the following conditions hold

(i) 119886 ge 119891(119909 119910) + 119892(119909 119910)119910 ge 119886

(ii) lim|119909|rarrinfin

ℎ(119909) sgn119909 = infin and there is a bounded setΩ containing the origin such that |ℎ1015840(119909)| le 119887 onΩ

119888Ω119888is the complement of the set Ω

(iii) lim|119909|rarrinfin

119892119894(119909) sgn119909 = infin and there is a bounded set

Ω containing the origin such that |1198921015840119894(119909)| le 119888

119894onΩ119888Ω119888

is the complement of the set Ω

(iv) 119901(119905) is an 120596-periodic continuous function

If 120591sum119899

119894=1119888119894lt 119886 then (3) has an 120596-periodic solution

Moreover 119901(119905) = 119870 119870-constant under the above con-ditions (3) has a constant motion 119909 = 119888

0 and the constant 119888

0

satisfies ℎ(1198880) + 119892119894(1198880) = 119870

Proof Define the Lyapunov-Krasovskii functional 119881 =

119881(119909119905 119910119905)

119881 =

1

2

1199102+ int

119909

0

ℎ (119904) 119889119904 +

119899

sum

119894=1

int

119909

0

119892119894(119904) 119889119904

+

119899

sum

119894=1

120582119894int

0

minus120591119894

int

119905

119905+119904

1199102

(120579) 119889120579 119889119904

(6)

where 120582119894are some positive constants to be determined later

in the proofEvaluating the time derivative of 119881 along system (4) we

get

= minus 119891 (119909 119910) + 119892 (119909 119910) 119910 1199102

+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904

+ 119910119901 (119905) +

119899

sum

119894=1

120582119894int

0

minus120591119894

(1199102

(119905) minus 1199102

(119905 + 119904)) 119889119904

= minus 119891 (119909 119910) + 119892 (119909 119910) 119910 1199102

+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904

+ 119910119901 (119905) +

119899

sum

119894=1

(120582119894120591119894) 1199102

minus

119899

sum

119894=1

120582119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904

(7)

By noting the assumption |1198921015840

119894(119909)| le 119888

119894of the theorem and

the estimate 2|120572120574| le 1205722+ 1205742 one can obtain the following

estimates

119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904

le

119899

sum

119894=1

int

0

minus120591119894

100381610038161003816100381610038161198921015840

119894(119909 (119905 + 119904))

10038161003816100381610038161003816

1003816100381610038161003816119910 (119905)

1003816100381610038161003816

1003816100381610038161003816119910 (119905 + 119904)

1003816100381610038161003816119889119904

le

1

2

119899

sum

119894=1

int

0

minus120591119894

119888119894(1199102

(119905) + 1199102

(119905 + 119904)) 119889119904

le

1

2

119899

sum

119894=1

(119888119894120591119894) 1199102+

1

2

119899

sum

119894=1

119888119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904

(8)

Then it follows that

le minus 119891 (119909 119910) + 119892 (119909 119910) 119910 1199102+ 119910119901 (119905)

+

1

2

119899

sum

119894=1

(119888119894+ 2120582119894) 1205911198941199102

+

1

2

119899

sum

119894=1

(119888119894minus 2120582119894) int

0

minus120591119894

1199102

(119905 + 119904) 119889119904

(9)

Let 120582119894= 1198881198942 and 120591 = max120591

1 120591

119899 In fact these choices

imply that

le minus119891 (119909 119910) + 119892 (119909 119910) 119910 minus 120591

119899

sum

119894=1

119888119894minus

1003816100381610038161003816119901 (119905)

1003816100381610038161003816

10038161003816100381610038161199101003816100381610038161003816

minus1

1199102

le minus(119886 minus 120591

119899

sum

119894=1

119888119894minus

1003816100381610038161003816119901 (119905)

1003816100381610038161003816

10038161003816100381610038161199101003816100381610038161003816

minus1

)1199102

(10)

In view of the continuity and periodicity of the function119901 and the assumption 119886 minus 120591sum

119899

119894=1119888119894gt 0 it follows that there is

a bounded set Ω2supe Ω1with Ω

2containing the origin and a

positive constant 120583 such that

120583 le 119886 minus 119888120591 minus

1003816100381610038161003816119901 (119905)

1003816100381610038161003816

10038161003816100381610038161199101003816100381610038161003816

for R times Ω119888

2 (11)

Therefore we can write

le minus1205831199102 for (119905 119909 119910) isin R times Ω

119888

2times Ω119888

2 (12)

From the last estimate we can arrive that the 119910-coor-dinate of the solutions of system (4) is ultimately bounded

4 Abstract and Applied Analysis

for a positive constant 120573 On the other hand since 119901 is acontinuous periodic function if |119910| le 120573 and119881

1= 119881+119910 then

subject to the assumptions of the theorem it can be easily seenthat there is a constant 119870

1gt 0 onR times Ω

119888

2such that

1= + 119910

1015840

= minus 119891 (119909 119910) + 119892 (119909 119910) 119910 1199102

+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 + 119910119901 (119905)

+

119899

sum

119894=1

(120582119894120591119894) 1199102minus

119899

sum

119894=1

120582119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904

minus 119891 (119909 119910) + 119892 (119909 119910) 119910 119910 minus ℎ (119909) minus

119899

sum

119894=1

119892119894(119909)

+

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 + 119901 (119905)

le minus1198861199102+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904

+ 119910119901 (119905) +

119899

sum

119894=1

(120582119894120591119894) 1199102

minus

119899

sum

119894=1

120582119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904 + 11988610038161003816100381610038161199101003816100381610038161003816minus ℎ (119909) minus

119899

sum

119894=1

119892119894(119909)

+

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 + 119901 (119905)

le minusℎ (119909) minus

119899

sum

119894=1

119892119894(119909) + 119870

1

(13)

Since ℎ(119909) rarr infin and 119892119894(119909) rarr infin as 119909 rarr infin then it can

be chosen a positive constant 1198611such that

1le minus05 for 119909 ge 119861

1 (14)

Therefore we can conclude that there is a positive constant1205721such that the 119909-coordinate of the solutions of system (4)

satisfies 119909 le 1205721for |119910| le 120573

In a similar manner if |119910| le 120573 and 1198812

= 119881 minus 119910 thensubject to the assumptions of the theorem it can be easilyfollowed by the time derivative of the functional119881

2that there

is a constant 1198702gt 0 onR times Ω

119888

2such that

2= minus 119910

1015840

= minus 119891 (119909 119910) + 119892 (119909 119910) 119910 1199102

+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 + 119910119901 (119905)

+

119899

sum

119894=1

(120582119894120591119894) 1199102minus

119899

sum

119894=1

120582119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904

+ 119891 (119909 119910) + 119892 (119909 119910) 119910 119910 + ℎ (119909) +

119899

sum

119894=1

119892119894(119909)

minus

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 minus 119901 (119905)

le minus 1198861199102+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904

+ 119910119901 (119905) +

119899

sum

119894=1

(120582119894120591119894) 1199102minus

119899

sum

119894=1

120582119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904

+ 11988610038161003816100381610038161199101003816100381610038161003816+ ℎ (119909) +

119899

sum

119894=1

119892119894(119909)

minus

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 minus 119901 (119905)

le ℎ (119909) +

119899

sum

119894=1

119892119894(119909) + 119870

2

(15)

Since ℎ(119909) rarr minusinfin and 119892119894(119909) rarr minusinfin as 119909 rarr minusinfin then

it can be chosen a positive constant 1198612such that

2le minus05 for 119909 le minus119861

2 (16)

Then we can conclude that there is a positive constant1205722such that the 119909-coordinate of the solutions of system (4)

satisfies 119909 ge minus1205722for |119910| le 120573 On gathering the above whole

discussion one can see that the solutions of system (4) areultimately boundedTherefore (3) has an 120596-periodic motion(solution) When 119901(119905) = 119870 119870-constant (3) has a constantmotion 119909 = 119888

0 From (3) it can be seen that the constant 119909 =

1198880is given by ℎ(119888

0) + 119892119894(1198880) = 119870

Conflict of Interests

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

References

[1] R E Gaines and J L Mawhin Coincidence Degree andNonlinear Differential Equations vol 568 of Lecture Notes inMathematics Springer Berlin Germany 1977

[2] T Yoshizawa ldquoStability theory by Liapunovrsquos second methodrdquoPublications of the Mathematical Society of Japan 9 TheMathematical Society of Japan Tokyo Japan 1966

[3] J M Zhao K L Huang andQ S Lu ldquoThe existence of periodicsolutions for a class of functional-differential equations andtheir applicationrdquo Applied Mathematics and Mechanics EnglishEdition vol 15 no 1 pp 49ndash59 1994 translated from AppliedMathematics and Mechanics vol 15 no 1 pp 49ndash58 1994

Abstract and Applied Analysis 5

[4] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations Dover Mineola NY USA2005 Corrected Version of the 1985 Original

[5] S Ahmad and M R RaoTheory of Ordinary differential Equa-tions With Applications in Biology and Engineering AffiliatedEast-West Press New Delhi India 1999

[6] A Constantin ldquoA note on a second-order nonlinear differentialsystemrdquo Glasgow Mathematical Journal vol 42 no 2 pp 195ndash199 2000

[7] J R Graef ldquoOn the generalized Lienard equation with negativedampingrdquo Journal of Differential Equations vol 12 pp 34ndash621972

[8] L H Huang and J S Yu ldquoOn boundedness of solutions ofgeneralized Lienardrsquos system and its applicationrdquo Annals ofDifferential Equations vol 9 no 3 pp 311ndash318 1993

[9] Z Jin ldquoBoundedness and convergence of solutions of a second-order nonlinear differential systemrdquo Journal of MathematicalAnalysis and Applications vol 256 no 2 pp 360ndash374 2001

[10] B Liu and L Huang ldquoBoundedness of solutions for a class ofretarded Lienard equationrdquo Journal of Mathematical Analysisand Applications vol 286 no 2 pp 422ndash434 2003

[11] J E Napoles Valdes ldquoBoundedness and global asymptoticstability of the forced Lienard equationrdquo Revista de la UnionMatematica Argentina vol 41 no 4 pp 47ndash59 2000

[12] C X Qian ldquoBoundedness and asymptotic behaviour of solu-tions of a second-order nonlinear systemrdquo The Bulletin of theLondon Mathematical Society vol 24 no 3 pp 281ndash288 1992

[13] C Tunc ldquoA note on the bounded solutionsrdquoAppliedMathemat-ics and Information Sciences vol 8 no 1 pp 393ndash399 2014

[14] C Tunc ldquoA note on boundedness of solutions to a class of non-autonomous differential equations of second orderrdquo ApplicableAnalysis and Discrete Mathematics vol 4 no 2 pp 361ndash3722010

[15] C Tunc ldquoBoundedness results for solutions of certain nonlineardifferential equations of second orderrdquo Journal of the IndonesianMathematical Society vol 16 no 2 pp 115ndash126 2010

[16] C Tunc ldquoUniformly stability and boundedness of solutions ofsecond order nonlinear delay differential equationsrdquo Appliedand Computational Mathematics vol 10 no 3 pp 449ndash4622011

[17] C Tunc ldquoOn the boundedness of solutions of a non-auton-omous differential equation of second orderrdquo Sarajevo Journalof Mathematics vol 7(19) no 1 pp 19ndash29 2011

[18] C Tunc ldquoStability and uniform boundedness results for non-autonomous Lienard-type equations with a variable deviatingargumentrdquoActaMathematica Vietnamica vol 37 no 3 pp 311ndash325 2012

[19] C Tunc ldquoOn the stability and boundedness of solutions of aclass of nonautonomous differential equations of second orderwith multiple deviating argumentsrdquoAfrikaMatematika vol 23no 2 pp 249ndash259 2012

[20] C Tunc ldquoStability to vector Lienard equation with constantdeviating argumentrdquo Nonlinear Dynamics vol 73 no 3 pp1245ndash1251 2013

[21] C Tunc ldquoNew results on the existence of periodic solutionsfor Rayleigh equation with state-dependent delayrdquo Journal ofMathematical and Fundamental Sciences vol 45 no 2 pp 154ndash162 2013

[22] C Tunc and E Tunc ldquoOn the asymptotic behavior of solutionsof certain second-order differential equationsrdquo Journal of theFranklin Institute vol 344 no 5 pp 391ndash398 2007

[23] J Zhou ldquoNecessary and sufficient conditions for boundednessand convergence of a second-order nonlinear differential sys-temrdquo Acta Mathematica Sinica Chinese Series vol 43 no 3 pp415ndash420 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

for a positive constant 120573 On the other hand since 119901 is acontinuous periodic function if |119910| le 120573 and119881

1= 119881+119910 then

subject to the assumptions of the theorem it can be easily seenthat there is a constant 119870

1gt 0 onR times Ω

119888

2such that

1= + 119910

1015840

= minus 119891 (119909 119910) + 119892 (119909 119910) 119910 1199102

+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 + 119910119901 (119905)

+

119899

sum

119894=1

(120582119894120591119894) 1199102minus

119899

sum

119894=1

120582119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904

minus 119891 (119909 119910) + 119892 (119909 119910) 119910 119910 minus ℎ (119909) minus

119899

sum

119894=1

119892119894(119909)

+

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 + 119901 (119905)

le minus1198861199102+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904

+ 119910119901 (119905) +

119899

sum

119894=1

(120582119894120591119894) 1199102

minus

119899

sum

119894=1

120582119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904 + 11988610038161003816100381610038161199101003816100381610038161003816minus ℎ (119909) minus

119899

sum

119894=1

119892119894(119909)

+

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 + 119901 (119905)

le minusℎ (119909) minus

119899

sum

119894=1

119892119894(119909) + 119870

1

(13)

Since ℎ(119909) rarr infin and 119892119894(119909) rarr infin as 119909 rarr infin then it can

be chosen a positive constant 1198611such that

1le minus05 for 119909 ge 119861

1 (14)

Therefore we can conclude that there is a positive constant1205721such that the 119909-coordinate of the solutions of system (4)

satisfies 119909 le 1205721for |119910| le 120573

In a similar manner if |119910| le 120573 and 1198812

= 119881 minus 119910 thensubject to the assumptions of the theorem it can be easilyfollowed by the time derivative of the functional119881

2that there

is a constant 1198702gt 0 onR times Ω

119888

2such that

2= minus 119910

1015840

= minus 119891 (119909 119910) + 119892 (119909 119910) 119910 1199102

+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 + 119910119901 (119905)

+

119899

sum

119894=1

(120582119894120591119894) 1199102minus

119899

sum

119894=1

120582119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904

+ 119891 (119909 119910) + 119892 (119909 119910) 119910 119910 + ℎ (119909) +

119899

sum

119894=1

119892119894(119909)

minus

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 minus 119901 (119905)

le minus 1198861199102+ 119910

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904

+ 119910119901 (119905) +

119899

sum

119894=1

(120582119894120591119894) 1199102minus

119899

sum

119894=1

120582119894int

0

minus120591119894

1199102

(119905 + 119904) 119889119904

+ 11988610038161003816100381610038161199101003816100381610038161003816+ ℎ (119909) +

119899

sum

119894=1

119892119894(119909)

minus

119899

sum

119894=1

int

0

minus120591119894

1198921015840

119894(119909 (119905 + 119904)) 119910 (119905 + 119904) 119889119904 minus 119901 (119905)

le ℎ (119909) +

119899

sum

119894=1

119892119894(119909) + 119870

2

(15)

Since ℎ(119909) rarr minusinfin and 119892119894(119909) rarr minusinfin as 119909 rarr minusinfin then

it can be chosen a positive constant 1198612such that

2le minus05 for 119909 le minus119861

2 (16)

Then we can conclude that there is a positive constant1205722such that the 119909-coordinate of the solutions of system (4)

satisfies 119909 ge minus1205722for |119910| le 120573 On gathering the above whole

discussion one can see that the solutions of system (4) areultimately boundedTherefore (3) has an 120596-periodic motion(solution) When 119901(119905) = 119870 119870-constant (3) has a constantmotion 119909 = 119888

0 From (3) it can be seen that the constant 119909 =

1198880is given by ℎ(119888

0) + 119892119894(1198880) = 119870

Conflict of Interests

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

References

[1] R E Gaines and J L Mawhin Coincidence Degree andNonlinear Differential Equations vol 568 of Lecture Notes inMathematics Springer Berlin Germany 1977

[2] T Yoshizawa ldquoStability theory by Liapunovrsquos second methodrdquoPublications of the Mathematical Society of Japan 9 TheMathematical Society of Japan Tokyo Japan 1966

[3] J M Zhao K L Huang andQ S Lu ldquoThe existence of periodicsolutions for a class of functional-differential equations andtheir applicationrdquo Applied Mathematics and Mechanics EnglishEdition vol 15 no 1 pp 49ndash59 1994 translated from AppliedMathematics and Mechanics vol 15 no 1 pp 49ndash58 1994

Abstract and Applied Analysis 5

[4] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations Dover Mineola NY USA2005 Corrected Version of the 1985 Original

[5] S Ahmad and M R RaoTheory of Ordinary differential Equa-tions With Applications in Biology and Engineering AffiliatedEast-West Press New Delhi India 1999

[6] A Constantin ldquoA note on a second-order nonlinear differentialsystemrdquo Glasgow Mathematical Journal vol 42 no 2 pp 195ndash199 2000

[7] J R Graef ldquoOn the generalized Lienard equation with negativedampingrdquo Journal of Differential Equations vol 12 pp 34ndash621972

[8] L H Huang and J S Yu ldquoOn boundedness of solutions ofgeneralized Lienardrsquos system and its applicationrdquo Annals ofDifferential Equations vol 9 no 3 pp 311ndash318 1993

[9] Z Jin ldquoBoundedness and convergence of solutions of a second-order nonlinear differential systemrdquo Journal of MathematicalAnalysis and Applications vol 256 no 2 pp 360ndash374 2001

[10] B Liu and L Huang ldquoBoundedness of solutions for a class ofretarded Lienard equationrdquo Journal of Mathematical Analysisand Applications vol 286 no 2 pp 422ndash434 2003

[11] J E Napoles Valdes ldquoBoundedness and global asymptoticstability of the forced Lienard equationrdquo Revista de la UnionMatematica Argentina vol 41 no 4 pp 47ndash59 2000

[12] C X Qian ldquoBoundedness and asymptotic behaviour of solu-tions of a second-order nonlinear systemrdquo The Bulletin of theLondon Mathematical Society vol 24 no 3 pp 281ndash288 1992

[13] C Tunc ldquoA note on the bounded solutionsrdquoAppliedMathemat-ics and Information Sciences vol 8 no 1 pp 393ndash399 2014

[14] C Tunc ldquoA note on boundedness of solutions to a class of non-autonomous differential equations of second orderrdquo ApplicableAnalysis and Discrete Mathematics vol 4 no 2 pp 361ndash3722010

[15] C Tunc ldquoBoundedness results for solutions of certain nonlineardifferential equations of second orderrdquo Journal of the IndonesianMathematical Society vol 16 no 2 pp 115ndash126 2010

[16] C Tunc ldquoUniformly stability and boundedness of solutions ofsecond order nonlinear delay differential equationsrdquo Appliedand Computational Mathematics vol 10 no 3 pp 449ndash4622011

[17] C Tunc ldquoOn the boundedness of solutions of a non-auton-omous differential equation of second orderrdquo Sarajevo Journalof Mathematics vol 7(19) no 1 pp 19ndash29 2011

[18] C Tunc ldquoStability and uniform boundedness results for non-autonomous Lienard-type equations with a variable deviatingargumentrdquoActaMathematica Vietnamica vol 37 no 3 pp 311ndash325 2012

[19] C Tunc ldquoOn the stability and boundedness of solutions of aclass of nonautonomous differential equations of second orderwith multiple deviating argumentsrdquoAfrikaMatematika vol 23no 2 pp 249ndash259 2012

[20] C Tunc ldquoStability to vector Lienard equation with constantdeviating argumentrdquo Nonlinear Dynamics vol 73 no 3 pp1245ndash1251 2013

[21] C Tunc ldquoNew results on the existence of periodic solutionsfor Rayleigh equation with state-dependent delayrdquo Journal ofMathematical and Fundamental Sciences vol 45 no 2 pp 154ndash162 2013

[22] C Tunc and E Tunc ldquoOn the asymptotic behavior of solutionsof certain second-order differential equationsrdquo Journal of theFranklin Institute vol 344 no 5 pp 391ndash398 2007

[23] J Zhou ldquoNecessary and sufficient conditions for boundednessand convergence of a second-order nonlinear differential sys-temrdquo Acta Mathematica Sinica Chinese Series vol 43 no 3 pp415ndash420 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

[4] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations Dover Mineola NY USA2005 Corrected Version of the 1985 Original

[5] S Ahmad and M R RaoTheory of Ordinary differential Equa-tions With Applications in Biology and Engineering AffiliatedEast-West Press New Delhi India 1999

[6] A Constantin ldquoA note on a second-order nonlinear differentialsystemrdquo Glasgow Mathematical Journal vol 42 no 2 pp 195ndash199 2000

[7] J R Graef ldquoOn the generalized Lienard equation with negativedampingrdquo Journal of Differential Equations vol 12 pp 34ndash621972

[8] L H Huang and J S Yu ldquoOn boundedness of solutions ofgeneralized Lienardrsquos system and its applicationrdquo Annals ofDifferential Equations vol 9 no 3 pp 311ndash318 1993

[9] Z Jin ldquoBoundedness and convergence of solutions of a second-order nonlinear differential systemrdquo Journal of MathematicalAnalysis and Applications vol 256 no 2 pp 360ndash374 2001

[10] B Liu and L Huang ldquoBoundedness of solutions for a class ofretarded Lienard equationrdquo Journal of Mathematical Analysisand Applications vol 286 no 2 pp 422ndash434 2003

[11] J E Napoles Valdes ldquoBoundedness and global asymptoticstability of the forced Lienard equationrdquo Revista de la UnionMatematica Argentina vol 41 no 4 pp 47ndash59 2000

[12] C X Qian ldquoBoundedness and asymptotic behaviour of solu-tions of a second-order nonlinear systemrdquo The Bulletin of theLondon Mathematical Society vol 24 no 3 pp 281ndash288 1992

[13] C Tunc ldquoA note on the bounded solutionsrdquoAppliedMathemat-ics and Information Sciences vol 8 no 1 pp 393ndash399 2014

[14] C Tunc ldquoA note on boundedness of solutions to a class of non-autonomous differential equations of second orderrdquo ApplicableAnalysis and Discrete Mathematics vol 4 no 2 pp 361ndash3722010

[15] C Tunc ldquoBoundedness results for solutions of certain nonlineardifferential equations of second orderrdquo Journal of the IndonesianMathematical Society vol 16 no 2 pp 115ndash126 2010

[16] C Tunc ldquoUniformly stability and boundedness of solutions ofsecond order nonlinear delay differential equationsrdquo Appliedand Computational Mathematics vol 10 no 3 pp 449ndash4622011

[17] C Tunc ldquoOn the boundedness of solutions of a non-auton-omous differential equation of second orderrdquo Sarajevo Journalof Mathematics vol 7(19) no 1 pp 19ndash29 2011

[18] C Tunc ldquoStability and uniform boundedness results for non-autonomous Lienard-type equations with a variable deviatingargumentrdquoActaMathematica Vietnamica vol 37 no 3 pp 311ndash325 2012

[19] C Tunc ldquoOn the stability and boundedness of solutions of aclass of nonautonomous differential equations of second orderwith multiple deviating argumentsrdquoAfrikaMatematika vol 23no 2 pp 249ndash259 2012

[20] C Tunc ldquoStability to vector Lienard equation with constantdeviating argumentrdquo Nonlinear Dynamics vol 73 no 3 pp1245ndash1251 2013

[21] C Tunc ldquoNew results on the existence of periodic solutionsfor Rayleigh equation with state-dependent delayrdquo Journal ofMathematical and Fundamental Sciences vol 45 no 2 pp 154ndash162 2013

[22] C Tunc and E Tunc ldquoOn the asymptotic behavior of solutionsof certain second-order differential equationsrdquo Journal of theFranklin Institute vol 344 no 5 pp 391ndash398 2007

[23] J Zhou ldquoNecessary and sufficient conditions for boundednessand convergence of a second-order nonlinear differential sys-temrdquo Acta Mathematica Sinica Chinese Series vol 43 no 3 pp415ndash420 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of