Research ArticleStability and Boundedness of Solutions to a CertainSecond-Order Nonautonomous Stochastic Differential Equation

A T Ademola1 S Moyo2 B S Ogundare1 M O Ogundiran1 and O A Adesina1

1Research Group in Differential Equations and Applications (RGDEA) Department of Mathematics Obafemi Awolowo UniversityIle-Ife 220005 Nigeria2Institute for Systems Science amp Research and Postgraduate Support Directorate Durban University of TechnologyDurban 4000 South Africa

Correspondence should be addressed to O A Adesina oadesinaoauifeedung

Received 28 July 2016 Revised 9 November 2016 Accepted 27 November 2016

Academic Editor Ying Hu

Copyright copy 2016 A T Ademola et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper focuses on stability and boundedness of certain nonlinear nonautonomous second-order stochastic differentialequations Lyapunovrsquos second method is employed by constructing a suitable complete Lyapunov function and is used to obtaincriteria on the nonlinear functions that guarantee stability and boundedness of solutions Our results are new in fact accordingto our observations from the relevant literature this is the first attempt on stability and boundedness of solutions of second-ordernonlinear nonautonomous stochastic differential equations Finally examples together with their numerical simulations are givento authenticate and affirm the correctness of the obtained results

1 Introduction

Differential equations of second-order have generated a greatdeal of applications in various fields of science and technol-ogy such as biology chemistry physics mechanics controltechnology communication network automatic regulationeconomy and ecology to mention few In addition thestudy of problems that involve the behaviour of solutions ofordinary differential equations (ODE) delay or functionaldifferential equations (DDE) and stochastic differentialequations (SDE) has been dealt with by many outstandingauthors see for instance Arnold [1] Burton [2 3] Hale[4] Oksendal [5] Shaikihet [6] and Yoshizawa [7 8] whichcontain the background to the study and the expositorypapers of Abou-El-Ela et al [9 10] Ademola et al [11 12]Alaba and Ogundare [13] Burton and Hatvani [14] Cahlonand Schmidt [15] Caraballo et al [16] Domoshnitsky [17]Gikhman and Skorokhod [18 19] Grigoryan [20] Ivanov etal [21] Jedrzejewski and Brochard [22] Jin and Zengrong[23] Kolarova [24] Kolmanovskii and Shaikhet [25 26]Kroopnick [27] Liu and Raffoul [28] Mao [29] Ogundareet al [30ndash32] Raffoul [33] Rezaeyan and Farnoosh [34]Tunc [35ndash43] Wang and Zhu [44] Xianfeng and Wei [45]

Yenicerioglu [46 47] Yoshizawa [48] Zhu et al [49] and thereferences cited therein

The authors in [18 19] investigated the second-orderlinear scalar equations of the form

11988410158401015840119905 + (119886 (119905) + 119887 (119905) 120578119894) 119905 = 0 119905 ge 1199050 (1)

where 119894 is a general disturbance process (the derivativeof a martingale) In [11 12] the authors discussed stabilityboundedness and periodic solutions to the following second-order ordinary and delay differential equations

[120601 (119909 (119905)) 1199091015840 (119905)]1015840 + 119892 (119905 119909 (119905) 1199091015840 (119905)) 1199091015840 (119905)+ 120593 (119905) ℎ (119909 (119905)) = 119901 (119905 119909 (119905) 1199091015840 (119905)) (2)

11990910158401015840 (119905)+ 120601 (119905) 119891 (119909 (119905) 119909 (119905 minus 120591 (119905)) 1199091015840 (119905) 1199091015840 (119905 minus 120591 (119905)))+ 119892 (119909 (119905 minus 120591 (119905))) = 119901 (119905 119909 (119905) 1199091015840 (119905))

(3)

Hindawi Publishing CorporationInternational Journal of AnalysisVolume 2016 Article ID 2012315 11 pageshttpdxdoiorg10115520162012315

2 International Journal of Analysis

respectively where 119891 119892 119901 ℎ 120601 and 120593 are continuousfunctions in their respective arguments In their contribu-tions the authors in [9 10] investigated asymptotic stabilityand boundedness of solutions of the following second-orderstochastic delay differential equations

11990910158401015840 (119905) + 1198861199091015840 (119905) + 119887119909 (119905 minus ℎ) + 120590119909 (119905) 1205961015840 (119905) = 0 (4)

11990910158401015840 (119905) + 1198861199091015840 (119905) + 119891 (119909 (119905 minus ℎ)) + 120590119909 (119905 minus 120591) 1205961015840 (119905) = 0 (5)

11990910158401015840 (119905) + 119892 (1199091015840 (119905)) + 119887119909 (119905 minus ℎ) + 120590119909 (119905) 1205961015840 (119905)= 119901 (119905 119909 (119905) 1199091015840 (119905) 1199091015840 (119905 minus ℎ)) (6)

respectively where 119886 119887 and 120590 are positive constants ℎ 120591 aredelay constants 119891 119892 and 119901 are continuous functions in theirrespective arguments and 119908(119905) isin R119898 is an 119898-dimensionalstandard Brownian motion defined on the probability space(also called Wiener process) Recently in 2016 the authors in[43] discussed global existence and boundedness of solutionsof a certain nonlinear integrodifferential equation of second-order with multiple deviating arguments

[119901 (119909 (119905)) 1199091015840 (119905)]1015840 + 119886 (119905) 119891 (119905 119909 (119905) 1199091015840 (119905)) 1199091015840 (119905)+ 119887 (119905) 119892 (119905 1199091015840 (119905)) + 119899sum

119894=1119888119894 (119905) ℎ119894 (119909 (119905 minus 120591119894))

= int1199050119888 (119905 119904) 1199091015840 (119904) 119889119904

(7)

where 120591119894 (119894 = 1 2 119899) are positive constants 119886 119887 and 119888 aredefined on R+ and 119891 119892 ℎ and 119901 are continuous functionsdefined in their respective arguments

Although second-order stochastic delay differential equa-tions have started receiving attention of authors according toour observation from relevant literature there is no previousliterature available on the stability and boundedness of solu-tions of second-order nonlinear nonautonomous stochasticdifferential equation The aim of this paper is to bridgethis gap Consider the following second-order nonlinearnonautonomous stochastic differential equation

11990910158401015840 (119905) + 119892 (119909 (119905) 1199091015840 (119905)) 1199091015840 (119905) + 119891 (119909 (119905))+ 120590119909 (119905) 1205961015840 (119905) = 119901 (119905 119909 (119905) 1199091015840 (119905)) (8)

where 120590 is a positive constant the functions 119892 119891 and 119901are continuous in their respective arguments on R2R andR+ times R2 respectively with R fl (minusinfininfin) R+ fl [0infin)and 120596 (a standard Wiener process representing the noise) isdefined on R Furthermore it is assumed that the continuityof the functions 119892 119891 and 119901 is sufficient for the existence ofsolutions and the local Lipschitz condition for (8) to havea unique continuous solution denoted by (119909(119905) 119910(119905)) Theprimes denote differentiationwith respect to the independent

variable 119905 isin R+ If 1199091015840(119905) = 119910(119905) then (8) is equivalent to thesystem

1199091015840 (119905) = 119910 (119905) 1199101015840 (119905) = 119901 (119905 119909 (119905) 119910 (119905)) minus 119891 (119909) minus 119892 (119909 (119905) 119910 (119905)) 119910 (119905)

minus 120590119909 (119905) 1205961015840 (119905) (9)

where the derivative of the function 119891 (ie 1198911015840) exists and iscontinuous for all 119909 Despite the applicability of these classesof equations there is no previous result on nonautonomoussecond-order nonlinear stochastic differential equation (8)The motivation for this investigation comes from the worksin [9ndash12 18 19] If 120590 = 0 in (8) then we have a generalsecond-order nonlinear ordinary differential equation whichhas been discussed extensively in relevant literature Theremaining parts of this paper are organized as follows InSection 2 we give the preliminary results on stochasticdifferential equations Main results and their proofs arepresented in Section 3 while examples and simulation ofsolutions are given in Section 4 to validate our results

2 Preliminary Results

Let (ΩF F119905119905gt0P) be a complete probability space witha filtration F119905119905gt0 satisfying the usual conditions (ie it isright continuous and F0 contains allP-null sets) Let119861(119905) =(1198611(119905) 119861119898(119905))119879 be an 119898-dimensional Brownian motiondefined on the probability space Let |sdot| denotes the Euclideannorm inR119899 If119860 is a vector ormatrix its transpose is denotedby 119860119879 If 119860 is a matrix its trace norm is denoted by

|119860| = radictrace (119860119879119860) (10)

For more exposition in this regard see Mao [29] and Arnold[1] Now let us consider a nonautonomous 119899-dimensionalstochastic differential equation

119889119883 (119905) = 119865 (119905 119883 (119905)) 119889119905 + 119866 (119905 119883 (119905)) 119889119861 (119905) (11)

on 119905 gt 0 with initial value 119883(0) = 1198830 isin R119899 Here 119865 R+ timesR119899 rarr R119899 and119866 R+timesR119899 rarr R119899times119898 aremeasurable functionsSuppose that both 119865 and 119866 are sufficiently smooth for (11) tohave a unique continuous solution on 119905 ge 0 which is denotedby119883(119905 1198830) if X(0) = 0 Assume further that

119865 (119905 0) = 119866 (119905 0) = 0 (12)

for all 119905 ge 0 Then the stochastic differential equation (11)admits zero solution119883(119905 0) equiv 0Definition 1 (see [1]) The zero solution of the stochasticdifferential equation (11) is said to be stochastically stable orstable in probability if for every pair of 120598 isin (0 1) and 119903 gt 0there exists a 1205750 = 1205750(120598 119903) gt 0 such that

Pr 1003816100381610038161003816119883 (119905 1198830)1003816100381610038161003816 lt 119903 forall119905 ge 0 ge 1 minus 120598whenever 100381610038161003816100381611988301003816100381610038161003816 lt 1205750 (13)

Otherwise it is said to be stochastically unstable

International Journal of Analysis 3

Definition 2 (see [1]) The zero solution of the stochasticdifferential equation (11) is said to be stochastically asymptot-ically stable if it is stochastically stable and in addition if forevery 120598 isin (0 1) and 119903 gt 0 there exists a 120575 = 120575(120598) gt 0 suchthat

Pr lim119905rarrinfin

119883(119905 1198830) = 0 ge 1 minus 120598 whenever 100381610038161003816100381611988301003816100381610038161003816 lt 120575 (14)

Definition 3 A solution 119883(1199050 1198830) of the stochastic differ-ential equation (11) is said to be stochastically bounded orbounded in probability if it satisfies

1198641198830 1003817100381710038171003817119883 (119905 1198830)1003817100381710038171003817 le 119862 (1199050 100381710038171003817100381711988301003817100381710038171003817) forall119905 ge 1199050 (15)

where 1198641198830 denotes the expectation operator with respect tothe probability law associated with1198830 119862 R+ timesR119899 andR+ isa constant depending on 1199050 and1198830Definition 4 The solutions 119883(1199050 1198830) of the stochastic dif-ferential equation (11) are said to be uniformly stochasticallybounded if 119862 in inequality (15) is independent of 1199050

For ℎ gt 0 let 119880ℎ = 119883 isin R119899 |119883| lt ℎ sub R119899 andlet 11986212(119880ℎ times R+R+) denote the family of all nonnegativefunctions119881(119905 119883(119905)) (Lyapunov function) defined onR+times119880ℎwhich are twice continuously differentiable in 119883 and once in119905 By Itorsquos formula we have

119889119881 (119905 119883 (119905)) = 119871119881 (119905 119883 (119905)) 119889119905+ 119881119909 (119905 119883 (119905)) 119866 (119905 119883 (119905)) 119889119861 (119905) (16)

where

119871119881 (119905 119883 (119905))= 120597119881 (119905 119883 (119905))120597119905 + 120597119881 (119905 119883 (119905))120597119909119894 119865 (119905 119883 (119905))+ 12 trace [119866119879 (119905 119883 (119905)) 119881119909119909 (119905 119883 (119905)) 119866 (119905 119883 (119905))]

(17)

Furthermore

119881119909119909 (119905 119883 (119905)) = (1205972119881 (119905 119883 (119905))120597119909119894120597119909119895 )119899times119899

119894 119895 = 1 119899 (18)

In this study we will use the diffusion operator 119871119881(119905 119883(119905))defined in (17) to replace 1198811015840(119905 119883(119905)) = (119889119889119905)119881(119905 119883(119905))Wenow present the basic results that will be used in the proofsof the main results

Lemma 5 (see [1]) Assume that there exist 119881 isin 11986212(R+ times119880ℎR+) and 120601 isin K such that

(i) 119881(119905 0) = 0(ii) 119881(119905 119883(119905)) gt 120601(119883(119905))(iii) 119871119881(119905 119883(119905)) le 0 for all (119905 119883) isin R+ times 119880ℎ

Then the zero solution of stochastic differential equation (11) isstochastically stable

Lemma 6 (see [1]) Suppose that there exist 119881 isin 11986212(R+ times119880ℎR+) and 1206010 1206011 1206012 isin K such that

(i) 119881(119905 0) = 0(ii) 1206010(119883(119905)) le 119881(119905 119883(119905)) le 1206011(119883(119905)) 1206010(119903) rarr infin as119903 rarr infin(iii) 119871119881(119905 119883(119905)) le minus1206012(119883(119905)) for all (119905 119883) isin R+ times 119880ℎ

Then the zero solution of stochastic differential equation (11) isuniformly stochastically asymptotically stable in the large

Assumption 7 (see [28 33]) Let 119881 isin 11986212(R+ times R119899R+)and suppose that for any solutions 119883(1199050 1198830) of stochasticdifferential equation (11) and for any fixed 0 le 1199050 le 119879 lt infinwe have

1198641198830 int11987911990501198812119909119894 (119905 119883 (119905)) 1198662119894119896 (119905 119883 (119905)) 119889119905 lt infin

1 le 119894 le 119899 1 le 119896 le 119898 (19)

Assumption 8 (see [28 33]) A special case of the generalcondition (19) is the following condition Assume that thereexits a function 120590(119905) such that10038161003816100381610038161003816119881119909119894 (119905 119883 (119905)) 119866119894119896 (119905 119883 (119905))10038161003816100381610038161003816 lt 120590 (119905)

119883 isin R119899 1 le 119894 le 119899 1 le 119896 le 119898 (20)

and for any fixed 0 le 1199050 le 119879 lt infin

int11987911990501205902 (119905) 119889119905 lt infin (21)

Lemma 9 (see [28 33]) Assume there exists a Lyapunov func-tion 119881(119905 119883(119905)) isin 11986212(R+ times R119899R+) satisfying Assumption 7such that for all (119905 119883) isin R+ timesR119899

(i) 119883(119905)119901 le 119881(119905 119883(119905)) le 119883(119905)119902(ii) 119871119881(119905 119883(119905)) le minus120572(119905)119883(119905)119903 + 120573(119905)(iii) 119881(119905 119883(119905)) minus 119881119903119902(119905 119883(119905)) le 120574

where 120572 120573 isin 119862(R+R+) 119901 119902 and 119903 are positive constants119901 ge 1 and 120574 is a nonnegative constant Then all solutions ofthe stochastic differential equation (11) satisfy

1198641198830 1003817100381710038171003817119883 (119905 1198830)1003817100381710038171003817 le 119881 (1199050 1198830) 119890minusint1199051199050 120572(119904)119889119904+ int1199051199050(120574120572 (119906) + 120573 (119906)) 119890minusint119905119906 120572(119904)1198891199041198891199061119901

(22)

for all 119905 ge 1199050Lemma 10 (see [28 33]) Assume there exists a Lyapunovfunction 119881(119905 119883(119905)) isin 11986212(R+ times R119899R+) satisfying Assump-tion 7 such that for all (119905 119883) isin R+ timesR119899

(i) 119883(119905)119901 le 119881(119905 119883(119905))(ii) 119871119881(119905 119883(119905)) le minus120572(119905)119881119902(119905 119883(119905)) + 120573(119905)(iii) 119881(119905 119883(119905)) minus 119881119902(119905 119883(119905)) le 120574

4 International Journal of Analysis

where 120572 120573 isin 119862(R+R+) 119901 119902 are positive constants 119901 ge 1 and120574 is a nonnegative constant Then all solutions of the stochasticdifferential equation (11) satisfy (22) for all 119905 ge 1199050Corollary 11 (see [28 33]) (i) Assume that hypotheses (i) to(iii) of Lemma 9 hold In addition

int1199051199050(120574120572 (119906) + 120573 (119906)) 119890minusint119905119906 120572(119904)119889119904119889119906 le 119872 forall119905 ge 1199050 ge 0 (23)

for some positive constant 119872 then all solutions of stochasticdifferential equation (11) are uniformly stochastically bounded

(ii) Assume that hypotheses (i) to (iii) of Lemma 10 holdIf condition (23) is satisfied then all solutions of the stochasticdifferential equation (11) are stochastically bounded

3 Main Results

Let (119909(119905) 119910(119905)) be any solution of the stochastic differentialequation (9) the main tool employed in the proofs ofour results is the continuously differentiable function 119881 =119881(119905 119909(119905) 119910(119905)) defined as

2119881 = 11988721199092 + 1198871199102 + 2119909119891 (119909) + (119886119909 + 119910)2 (24)

where 119886 and 119887 are positive constants and the function 119891 is asdefined in Section 1

Theorem 12 Suppose that 119886 119887 120590 and 1198720 are positiveconstants such that

(i) 119886 le 119892(119909 119910) for all 119909 and 119910(ii) 119887119909 le 119891(119909) le 119861119909 for all 119909 = 0 and 1205902 lt 2119886119887(119887 + 1)minus1(iii) |119901(119905 119909 119910)| le 1198720 for all 119905 ge 0 119909 and 119910

Then solution (119909(119905) 119910(119905)) of the stochastic differential equation(9) is uniformly stochastically bounded

Remark 13 We note the following

(i) Whenever the functions 119892(119909 1199091015840) = 119886 119891(119909) = 119887119909 and1205961015840 = 119901(119905 119909 1199091015840) = 0 then the stochastic differentialequation (8) becomes a second-order linear ordinarydifferential equation

11990910158401015840 + 1198861199091015840 + 119887119909 = 0 (25)

and conditions (i) to (iii) of Theorem 12 reduce toRouth Hurwitz criteria 119886 gt 0 and 119887 gt 0 forthe asymptotic stability of the second-order lineardifferential equation (25)

(ii) The term 120590119909(119905)1205961015840(119905) in the stochastic differentialequation (8) is an extension of the ordinary casediscussed recently by authors in [11 18 23 31 32 35ndash37 40]

We shall now state and prove a result that will be used inthe proofs of our results

Lemma 14 Under the hypotheses of Theorem 12 there existpositive constants 1198630 = 1198630(119886 119887) and 1198631 = 1198631(119886 119887 119861) suchthat

1198630 (1199092 (119905) + 1199102 (119905)) le 119881 (119905 119909 (119905) 119910 (119905))le 1198631 (1199092 (119905) + 1199102 (119905)) (26)

for all 119905 ge 0 119909 and 119910 In addition there exist positive constants1198632 = 1198632(119886 119887 120590) and 1198633 = 1198633(119886 119887) such that119871119881 (119905 119909 (119905) 119910 (119905))

le minus1198632 (1199092 (119905) + 1199102 (119905))+ 1198633 (|119909 (119905)| + 1003816100381610038161003816119910 (119905)1003816100381610038161003816) 1003816100381610038161003816119901 (119905 119909 (119905) 119910 (119905))1003816100381610038161003816

(27)

for all 119905 ge 0 119909 and 119910Proof Let (119909(119905) 119910(119905)) be any solution of the stochastic differ-ential equation (9) since 119883 = (119909 119910) isin R2 it follows from(24) that

119881 (119905 0 0) = 0 (28)

for all 119905 ge 0Moreover from (24) and the fact that 119891(119909) ge 119886119909for all 119909 = 0 there exists a positive constant 1205750 such that

119881 (119905 119883) ge 1205750 (1199092 + 1199102) (29)

for all 119905 ge 0 119909 and 119910 where1205750 fl min 1198872 + 2119887 +min 119886 1 119887 +min 119886 1 (30)

It is clear from inequality (29) that

119881 (119905 119883) = 0 lArrrArr 1199092 + 1199102 = 0119881 (119905 119883) gt 0 lArrrArr 1199092 + 1199102 = 0 (31)

119881 (119905 119883) 997888rarr +infin as 1199092 + 1199102 997888rarr infin (32)

Furthermore since 119891(119909) le 119861119909 for all 119909 = 0 it follows from(24) that there exists a positive constant 1205751 such that

119881 (119905 119883) le 1205751 (1199092 + 1199102) (33)

for all 119905 ge 0 119909 and 119910 where1205751 fl max 1198872 + 2119861 +max 119886 1 119887 +max 119886 1 (34)

From inequalities (29) and (33) we have

1205750 (1199092 + 1199102) le 119881 (119905 119883) le 1205751 (1199092 + 1199102) (35)

for all 119905 ge 0 119909 and 119910 It is not difficult to see that estimates(35) satisfy inequalities (26) of Lemma 14 with 1205750 and 1205751equivalent to1198630 and1198631 respectively

International Journal of Analysis 5

Moreover applying Itorsquos formula in (24) using system (9)we find that

119871119881 (119905 119883) = 12 [119886119891 (119909)119909 minus 121205902 (119887 + 1)] 1199092minus 12 [(119887 + 1) 119892 (119909 119910) minus 119886] 1199102 minus119882119894+ [119886119909 + (119887 + 1) 119910] 119901 (119905 119909 119910)

(119894 = 1 2)

(36)

where

1198821 fl 14 [119886119891 (119909)119909 minus 121205902 (119887 + 1)] 1199092+ 4 [119886119892 (119909 119910) minus (1198862 + 1198872)] 119909119910+ [(119887 + 1) 119892 (119909 119910) minus 119886] 1199102

1198822 fl 14 [119886119891 (119909)119909 minus 121205902 (119887 + 1)] 1199092

+ 4 [119886119891 (119909)119909 minus 1198911015840 (119909)] 119909119910+ [(119887 + 1) 119892 (119909 119910) minus 119886] 1199102

(37)

It is clear from the inequalities

4 [119886119892 (119909 119910) minus (1198862 + 1198872)]2lt [119886119891 (119909)119909 minus 121205902 (119887 + 1)] [(119887 + 1) 119892 (119909 119910) minus 119886]

4 [119886119891 (119909)119909 minus 1198911015840 (119909)]lt [119886119891 (119909)119909 minus 121205902 (119887 + 1)] [(119887 + 1) 119892 (119909 119910) minus 119886]

(38)

that

1198821 = 1198822 ge [[radic119886119891 (119909)119909 minus 121205902 (119887 + 1) |119909|

minus radic(119887 + 1) 119892 (119909 119910) minus 119886 10038161003816100381610038161199101003816100381610038161003816]]2

ge 0(39)

for all 119909 and 119910 Using inequality (39) and hypotheses (i) and(ii) ofTheorem 12 in (36) there exist positive constants 1205752 and1205753 such that

119871119881 (119905 119883) le minus1205752 (1199092 + 1199102)+ 1205753 (|119909| + 10038161003816100381610038161199101003816100381610038161003816) 1003816100381610038161003816119901 (119905 119909 119910)1003816100381610038161003816 (40)

for all 119905 ge 0 119909 and 119910 where1205752 fl 12 min 119886119887 minus 121205902 (119887 + 1) 119886119887 1205753 fl max 119886 119887 + 1 (41)

Inequality (40) satisfies inequality (27) with 1205752 and 1205753equivalent to 1198632 and 1198633 respectively This completes theproof of Lemma 14

Proof ofTheorem 12 Let (119909(119905) 119910(119905)) be any solution of system(9) From inequality (40) and assumption (iii) ofTheorem 12we have

119871119881 (119905 119883)le minus121205752 (1199092 + 1199102)minus 1212057521198720 [(|119909| minus 120575minus12 1205753)2 + (10038161003816100381610038161199101003816100381610038161003816 minus 120575minus12 1205753)2]+1198720120575minus12 12057523

(42)

for 119905 ge 0 119909 and 119910 Since 1205752 1205753 and1198720 are positives and(|119909| minus 120575minus12 1205753)2 + (10038161003816100381610038161199101003816100381610038161003816 minus 120575minus12 1205753)2 ge 0 (43)

for all 119909 and 119910 there exist positive constants 1205754 and 1205755 suchthat

119871119881 (119905 119883) le minus1205754 (1199092 + 1199102) + 1205755 (44)

for all 119905 ge 0 119909 119910 where 1205754 fl (12)1205752 and 1205755 fl 1198720120575minus12 12057523 Hence condition (ii) of Lemma 9 is satisfied with 120572(119905) fl 1205754119903 fl 2 and 120573(119905) fl 1205755 Also from inequality (35) hypotheses(i) and (iii) of Lemma 9 hold with 119901 = 119902 = 2 so that 120574 = 0

Furthermore from inequality (23) we have

int1199051199050[(120574120572 (119906) + 120573 (119906)) 119890minus1205754 int119905119906 120572(119904)119889119904] 119889119906= int11990511990501205755119890minus1205754 int119905119906 119889119904119889119906 = 120575minus14 1205755 [1 minus 119890minus1205754(119905minus1199050)]

le 120575minus14 1205755(45)

for all 119905 ge 1199050 ge 0 Inequality (45) satisfies estimate (23) with119872 fl 120575minus14 1205755 = 21198720120575minus22 12057523 gt 0 Moreover from (9) and (24)there exists a positive constant 1205756 such that

10038161003816100381610038161003816119881119909119894 (119905 119883)119866119894119896 (119905 119883)10038161003816100381610038161003816le 12120590 [(2119886 + 119887 + 1) 1199092 + (119887 + 1) 1199102]le 1205756 (1199092 + 1199102) fl 120582 (119905)

(46)

where

1205756 fl 12120590max 2119886 + 119887 + 1 119887 + 1 (47)

6 International Journal of Analysis

Also

int119879119905012057526 (1199092 (119905) + 1199102 (119905))2 119889119905 lt infin (48)

for any fixed 0 le 1199050 le 119879 lt infin Thus from inequalities (46)and (48) estimates (20) and (21) hold respectively Finallyfrom inequalities (33) and (45) we have

1198641198830 1003817100381710038171003817119883 (119905 1198830)1003817100381710038171003817 le (120575111988320 + 21198720120575minus22 12057523)12 (49)

for all 119905 ge 1199050 ge 0 where1198830 fl (11990920 +11991020) and119862 fl 1205751Thus thesolutions (119909(119905) 119910(119905)) of the stochastic differential equation (9)are uniformly stochastically bounded

Theorem 15 If assumptions of Theorem 12 hold then thesolution (119909(119905) 119910(119905)) of the stochastic differential equation (9)is stochastically bounded

Proof Suppose that (119909(119905) 119910(119905)) is any solution of the stochas-tic differential equation (9) From inequalities (33) and (44)there exists a positive constant 1205757 such that

119871119881 (119905 119883) le minus1205757119881 (119905 119883) + 1205755 (50)

for all 119905 ge 0 119909 and119910 where1205757 fl 120575minus11 1205754Hence from inequal-ities (29) and (50) hypotheses of Lemma 10 hold Moreoverfrom inequalities (45) (46) (48) and (49) assumption (ii)of Corollary 11 holds Thus by Corollary 11 all solutionsof the stochastic differential equation (9) are stochasticallybounded This completes the proof of Theorem 15

Next we shall discuss the stability of the trivial solutionof the stochastic differential equation (8) Suppose that119901(119905 119909 1199091015840) = 0 (8) specializes to

11990910158401015840 (119905) + 119892 (119909 (119905) 1199091015840 (119905)) 1199091015840 (119905) + 119891 (119909 (119905))+ 120590119909 (119905) 1205961015840 (119905) = 0 (51)

Equation (51) has the following equivalent system

1199091015840 (119905) = 119910 (119905) 1199101015840 (119905) = minus119891 (119909) minus 119892 (119909 (119905) 119910 (119905)) 119910 (119905) minus 120590119909 (119905) 1205961015840 (119905) (52)

where the functions 119891 119892 and 120596 are defined in Section 1

Theorem 16 If assumptions (i) and (ii) of Theorem 12 holdthen the trivial solution of the stochastic differential equation(52) is stochastically stable

Proof Let (119909(119905) 119910(119905)) be any solution of the stochastic differ-ential equation (52) From equation (28) and estimate (29)assumptions (i) and (ii) of Lemma 5 hold so that the function119881(119905 119883) is positive definite Furthermore using Itorsquos formulaalong the solution path of (52) we obtain

119871119881 (119905 119883) le minus1205752 (1199092 (119905) + 1199102 (119905)) le 0 (53)

for all 119905 ge 0 119909 and 119910 where 1205752 is defined in (40)Inequality (53) satisfies hypothesis (iii) of Lemma 5 henceby Lemma 5 the trivial solution of the stochastic differentialequation (52) is stochastically stableThis completes the proofof Theorem 16

Theorem 17 If assumptions (i) and (ii) of Theorem 12 holdthen the trivial solution of the stochastic differential equation(52) is not only uniformly stochastically asymptotically stablebut also uniformly stochastically asymptotically stable in thelarge

Proof Let (119909(119905) 119910(119905)) be any solution of the stochastic differ-ential equation (52) In view of (28) and estimate (29) thefunction 119881(119905 119883) is positive definite Furthermore estimate(32) and inequality (33) show that the function 119881(119905 119883) isradially unbounded and decrescent respectively It followsfrom (28) estimate (32) inequality (35) and the first inequal-ity in (53) that all assumptions of Lemma 6 hold Thus byLemma 6 the trivial solution of the stochastic differentialequation (52) is uniformly stochastically asymptotically stablein the large If estimate (32) is omitted then the trivial solutionof the stochastic differential equation (52) is uniformlystochastically asymptotically stable This completes the proofof Theorem 17

Next if the function 119901(119905 119909 1199091015840) is replaced by 119901(119905) isin119862(R+R+) we have the following special case11990910158401015840 (119905) + 119892 (119909 (119905) 1199091015840 (119905)) 1199091015840 (119905) + 119891 (119909 (119905))

+ 120590119909 (119905) 1205961015840 (119905) = 119901 (119905) (54)

of (8) Equation (54) has the following equivalent system

1199091015840 (119905) = 119910 (119905) 1199101015840 (119905) = 119901 (119905) minus 119891 (119909) minus 119892 (119909 (119905) 119910 (119905)) 119910 (119905)

minus 120590119909 (119905) 1205961015840 (119905) (55)

with the following result

Corollary 18 If assumptions (i) and (ii) of Theorem 12 holdand hypothesis (iii) is replaced by the boundedness of thefunction 119901(119905) then the solutions (119909(119905) 119910(119905)) of the stochasticdifferential equation (55) are not only stochastically boundedbut also uniformly stochastically bounded

Proof Theproof of Corollary 18 is similar to the proof ofThe-orems 12 and 15This completes the proof of Corollary 18

4 Examples

In this section we shall present two examples to illustrate theapplications of the results we obtained in the previous section

Example 1 Consider the second-order nonlinear nonau-tonomous stochastic differential equation

11990910158401015840 + (3 + 10038161003816100381610038161003816cos (1199091199091015840)10038161003816100381610038161003816) 1199091015840 + 119909 + sin119909 + 011199091205961015840 (119905)= (1 + 2119905 + 10038161003816100381610038161003816119909119909101584010038161003816100381610038161003816)minus1

(56)

International Journal of Analysis 7

Equation (56) is equivalent to system

1199091015840 = 1199101199101015840 = (1 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816)minus1 minus (119909 + sin119909)

minus [3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816] 119910 minus 011199091205961015840 (119905) (57)

Now from systems (9) and (57) we have the followingrelations

(i) The function

119892 (119909 119910) fl 3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816 (58)

Noting that 1003816100381610038161003816cos (119909119910)1003816100381610038161003816 ge 0 (59)

for all 119909 and 119910 it follows that119892 (119909 119910) = 3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816 ge 119886 = 3 (60)

for all 119909 and 119910The behaviour of the function 119892(119909 119910)is shown below in Figure 1

(ii) The function

119891 (119909) fl 119909 + sin119909 (61)

Since

minus02 le 119865 (119909) = sin119909119909 le 1 (62)

for all 119909 = 0 then we have

1 = 119887 le 119891 (119909)119909 = 1 + sin119909119909 le 119861 = 2 (63)

for all 119909 = 0 and since 120590 fl 01 it follows that1205902 lt 2119886119887(119887 + 1)minus1 implies that 0 lt 299 The function119891(119909)119909 and its bounds are shown in Figure 2(iii) The function

119901 (119905 119909 119910) fl 11 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816 (64)

Clearly

1003816100381610038161003816119901 (119905 119909 119910)1003816100381610038161003816 = 11 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816 le 1 = 1198720 (65)

for all 119905 ge 0 119909 and 119910Now from items (i) (ii) above and (24) the continuouslydifferentiable function 119881(119905 119883) used for system (57) is

2119881 (119905 119883) = 31199092 + 1199102 + (3119909 + 119910)2 (66)

Different views of the function119881(119905 119883) are shown in Figure 3From (66) it is not difficult to show that

(1199092 + 1199102) le 119881 (119905 119883) le 3 (1199092 + 1199102) (67)

Figure 1 Behaviour of the function 119892(119909 119910)

minus6120587 minus4120587 minus2120587 612058741205872120587

ge minus0225

25

2

15

1

05

minus05

minus1

F(x) f(x)x

f(x)x = 1 + sin(x)x

F(x) = sin(x)xx

b = 076

F(x)

Figure 2 Bounds on the function 119891(119909)119909

for all 119905 ge 0 119909 and 119910 From (35) and (67) we have 1205750 = 11205751 = 3 119901 = 2 and 119902 = 2 and thus inequalities (67) satisfycondition (i) of Lemma 9 Also from the first inequality in(67) we have

119881 (119905 119883) 997888rarr +infin as 1199092 + 1199102 997888rarr infin (68)

Estimate (68) verifies (32) (ie the function 119881(119905 119883) definedby (66) is radially unbounded) Next applying Itorsquos formulain (66) using system (57) we find that

119871119881 (119905 119883) = 12119909119910 + 31199102 minus 119909 (3119909 + 2119910) (1 + sin119909119909 )minus 119910 (3119909 + 2119910) (3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816) + 11001199092minus 11990910 (3119909 + 2119910)+ (3119909 + 2119910) (1 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816)minus1

(69)

Using the estimates in items (i) to (iii) of Example 1 and theinequality 211990911199092 le 11990921 + 11990922 in (69) we obtain

119871119881 (119905 119883) le minus29 (1199092 + 1199102) + 3 (|119909| + 10038161003816100381610038161199101003816100381610038161003816) (70)

for all 119905 ge 0 119909 and 119910 Inequality (70) satisfies inequality (40)where 1205752 = 29 and 1205753 = 3 Since

(|119909| minus 105)2 + (10038161003816100381610038161199101003816100381610038161003816 minus 105)2 ge 0 (71)

for all 119909 and 119910 it follows from inequality (70) that

119871119881 (119905 119883) le minus145 (1199092 + 1199102) + 32 (72)

8 International Journal of Analysis

Figure 3 The behaviour of the function 119881(119905 119883)

for all 119905 ge 0 119909 and 119910 Inequality (72) satisfies assumption(ii) of Lemma 9 and estimate (44) with 120572(119905) = 1205754 = 145 and120573(119905) = 1205755 = 32 Since 119903 = 119901 = 119902 = 2 it follows that 120574 = 0 sothat assumption (iii) of Lemma 9 holds In addition

int1199051199050[(120574120572 (119906) + 120573 (119906)) 119890minusint119905119906120572 (119904) 119889119904] 119889119906 le 16 (73)

for all 119905 ge 1199050 ge 0 Estimate (73) satisfies (23) and (45) with119872 = 26 Furthermore

119881119909119894 (119905 119883)119866119894119896 (119905 119883) = minus 110 (31199092 + 2119909119910) (74)

and10038161003816100381610038161003816119881119909119894 (119905 119883)119866119894119896 (119905 119883)10038161003816100381610038161003816 le 25 (1199092 + 1199102) (75)

for all 119905 ge 0 119909 and119910 Inequality (75) satisfies inequalities (20)and (21) with

120582 (119905) = 25 (1199092 + 1199102) (76)

Hence by Corollary 11 (i) all solutions of stochastic differen-tial equation (57) are uniformly stochastically bounded

Example 2 If 119901(119905 119909 1199091015840) = 119901(119905 119909 119910) = 0 in (56) and system(57) we have the following stochastic differential equation

11990910158401015840 + (3 + 10038161003816100381610038161003816cos (1199091199091015840)10038161003816100381610038161003816) 1199091015840 + 119909 + sin119909 + 011199091205961015840 (119905)= 0 (77)

Equation (77) is equivalent to system

1199091015840 = 1199101199101015840 = minus (119909 + sin119909) minus [3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816] 119910 minus 011199091205961015840 (119905) (78)

Now from systems (52) and (78) items (i) and (ii) of Example 1hold Also equations (66) (67) and estimate (68) hold thatis

2119881 (119905 119883) = 31199092 + 1199102 + (3119909 + 119910)2 119881 (119905 0) = 0 forall119905 ge 0

(1199092 + 1199102) le 119881 (119905 119883) le 3 (1199092 + 1199102) forall119905 ge 0 119909 119910119881 (119905 119883) 997888rarr +infin as 1199092 + 1199102 997888rarr infin

(79)

112

1416

182

0500

10001500

0

02

04

06

08

1

t

x(t) y(t)

Figure 4 Graph of solutions of (56) in 3D

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(b)

Figure 5

Furthermore application of Itorsquos formula in (66) and usingsystem (78) yield

119871119881 (119905 119883) le minus29 (1199092 + 1199102) (80)

for all 119905 ge 0 119909 119910 and thus

119871119881 (119905 119883) le 0 (81)

International Journal of Analysis 9

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

000200400600801012

t

minus004

minus002

x(t)y(t)

x(t)

y(t)

times1011

(b)

Figure 6

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

000200400600801012

t

minus002

minus008

minus006

minus004

times1011

x(t)y(t)

x(t)

y(t)

(b)

Figure 7

for all 119905 ge 0 119909 and 119910 Moreover from (79) and (80)all assumptions of Theorem 17 and Lemma 6 are satisfiedThus by Lemma 6 the trivial solution of system (78) is notonly uniformly stochastically asymptotically stable but alsouniformly stochastically asymptotically stable in the largeFinally from (79) and (81) the function 119881(119905 119883) is positivedefinite and

119871119881 (119905 119883) le 0 forall (119905 119883) isin R+ timesR

2 (82)

Hence assumptions of Theorem 17 and Lemma 5 hold byTheorem 17 and Lemma 5 the trivial solution of system (78)is stochastically stable

Simulation of Solutions In what follows we shall nowsimulate the solutions of (56) (resp system (57)) and (78)(resp system (79)) Our approach depends on the Euler-Maruyama method which enables us to get approximatenumerical solution for the considered systems It will be seenfrom our figures that the simulated solutions are bounded

which justifies our given results For instance when 120590 = 01the numerical solutions of (56) in three-dimensional spaceare shown in Figure 4 If we vary the value of the noise inthe numerical solution (119909(119905) 119910(119905)) of system (57) as 120590 = 01and 120590 = 10 we have Figures 5(a) and 5(b) respectively Itcan be seen that when the noise is increased the stochasticitybecomes more pronounced The behaviour of the numericalsolution (119909(119905) 119910(119905)) of system (57) when 120590 = 05 and 120590 = 20is shown in Figures 6(a) and 6(b) respectivelyThe behaviourof the numerical solution (119909(119905) 119910(119905)) of system (57) for 120590 = 0and 120590 = 50 is shown in Figures 7(a) and 7(b) respectivelyFor the case of (78) Figure 8 shows the closeness of thesolution (119909(119905)) and the perturbed solution (119909120598(119905)) for a verylarge 119905 which implies asymptotic stability in the large for theconsidered SDE

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

10 International Journal of Analysis

0 1 2 3 4 5 6 7 8 9 100

0005

001

0015

002

0025

t times1011

x(t)x120576(t)

x(t)x120576(t)

Figure 8 Graph of solutions of (78)

References

[1] L Arnold Stochastic Differential Equations Theory and Appli-cations John Wiley amp Sons 1974

[2] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations vol 178 of Mathematics inScience and Engineering Academic Press Inc Orlando FlaUSA 1985

[3] T A Burton Volterra Integral and Differential EquationsAcademic Press New York NY USA 1983

[4] J K HaleTheory of Functional Differential Equations SpringerNew York NY USA 1977

[5] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer 2000

[6] L Shaikihet Lyapunov Functionals and Stability of StochasticFunctional Differential Equations Springer International 2013

[7] T Yoshizawa StabilityTheory and Existence of Periodic Solutionsand almost Periodic Solutions Spriger New York NY USA1975

[8] T Yoshizawa StabilityTheory by Liapunovrsquos SecondMethodTheMathematical Society of Japan 1966

[9] A M A Abou-El-Ela A I Sadek and A M Mahmoud ldquoOnthe stability of solutions for certain second-order stochasticdelay differential equationsrdquo Differential Equations and ControlProcesses no 2 pp 1ndash13 2015

[10] A M Abou-El-Ela A I Sadek A M Mahmoud and R OTaie ldquoOn the stochastic stability and boundedness of solutionsfor stochastic delay differential equation of the second orderrdquoChinese Journal of Mathematics vol 2015 Article ID 358936 8pages 2015

[11] A T Ademola Boundedness and Stability of Solutions to CertainSecond Order Differential Equations Differential Equations andControl Processes 2015

[12] A T Ademola B S Ogundare M O Ogundiran and O AAdesina ldquoPeriodicity stability and boundedness of solutions tocertain second order delay differential equationsrdquo InternationalJournal of Differential Equations vol 2016 Article ID 284370910 pages 2016

[13] J G Alaba and B S Ogundare ldquoOn stability and bound-edness properties of solutions of certain second order non-autonomous nonlinear ordinary differential equationrdquo Kragu-jevac Journal of Mathematics vol 39 no 2 pp 255ndash266 2015

[14] T A Burton and L Hatvani ldquoAsymptotic stability of secondorder ordinary functional and partial differential equationsrdquoJournal of Mathematical Analysis and Applications vol 176 no1 pp 261ndash281 1993

[15] B Cahlon andD Schmidt ldquoStability criteria for certain second-order delay differential equations with mixed coefficientsrdquoJournal of Computational and AppliedMathematics vol 170 no1 pp 79ndash102 2004

[16] T Caraballo M A Diop and A S Ndoye ldquoFixed points andexponential stability for stochastic partial integro-differentialequations with delaysrdquo Advances in Dynamical Systems andApplications vol 9 no 2 pp 133ndash147 2014

[17] A Domoshnitsky ldquoNonoscillation maximum principles andexponential stability of secondorder delay differential equationswithout damping termrdquo Domoshnitsky Journal of Inequalitiesand Applications vol 2014 article 361 2014

[18] I I Gikhman and A V Skorokhod Stochastische Differential-gleichungen Akademie Berlin Germany 1971 (Russian)

[19] I I Gikhman On the Stability of the Solutions of StochasticDifferential Equations Predelrsquonyye Teoremy i StatisticheskiyeVyvody Tashkent Uzbekistan 1966

[20] G A Grigoryan ldquoBoundedness and stability criteria for linearordinary differential equations of the second orderrdquo RussianMathematics vol 57 no 12 pp 8ndash15 2013

[21] A F Ivanov Y I Kazmerchuk and A V Swishchuk ldquoThe-ory stochastic stability and applications of stochastic delaydifferential equations a survey of recent resultsrdquo in DifferentialEquations and Dynamical Systems vol 11 no 1 2003

[22] F Jedrzejewski and D Brochard ldquoLyapounv exponents andstability stochastic dynamical structuresrdquo 2000

[23] Z Jin and L Zengrong ldquoOn the global asymptotic behaviorof solutions to a non autonomous generalized Lienard systemrdquoJournal of Mathematical Research and Exposition vol 21 no 3pp 410ndash414 2001

[24] E Kolarova ldquoAn application of stochastic integral equations toelectrical networksrdquo Acta Electrotechnica et Informatica vol 8no 3 pp 14ndash17 2008

[25] V B Kolmanovskii and L E Shaikhet ldquoA method for con-structing Lyapunov functionals for stochastic systems with aftereffectrdquo Differentsialrsquonye Uravneniya vol 29 no 11 pp 1909ndash2022 1993

[26] V Kolmanovskii and L Shaikhet ldquoConstruction of Lyapunovfunctionals for stochastic hereditary systems a survey of somerecent resultsrdquo Mathematical and Computer Modelling vol 36no 6 pp 691ndash716 2002

[27] A J Kroopnick ldquoBounded solutions to 11990910158401015840 + 119902(119905)119887(119909) = 119891(119905)rdquoInternational Journal of Mathematical Education in Science andTechnology vol 41 no 6 pp 829ndash836 2010

[28] R Liu and Y Raffoul ldquoBoundedness and exponential stabilityof highly nonlinear stochastic differential equationsrdquo ElectronicJournal of Differential Equations vol 2009 no 143 pp 1ndash102009

[29] X Mao ldquoSome contributions to stochastic asymptotic stabilityand boundedness via multiple Lyapunov functionsrdquo Journal ofMathematical Analysis and Applications vol 260 no 2 pp 325ndash340 2001

[30] B S Ogundare A T Ademola M O Ogundiran and O AAdesina ldquoOn the qualitative behaviour of solutions to certainsecond order nonlinear differential equation with delayrdquoAnnalidellrsquoUniversitarsquo di Ferrara 2016

International Journal of Analysis 11

[31] B S Ogundare and A U Afuwape ldquoBoundedness and stabilityproperties of solutions of generalized Lienard equationrdquo KochiJournal of Mathematics vol 9 pp 97ndash108 2014

[32] B S Ogundare and G E Okecha ldquoBoundedness periodicityand stability of solutions to x(t) + g(x) + b(t)h(x) = p(t x x)rdquoMathematical Sciences Research Journal vol 11 no 5 pp 432ndash443 2007

[33] Y N Raffoul ldquoBoundedness and exponential asymptotic sta-bility in dynamical systems with applications to nonlineardifferential equations with unbounded termsrdquo Advances inDynamical Systems and Applications vol 2 no 1 pp 107ndash1212007

[34] R Rezaeyan and R Farnoosh ldquoStochastic differential equationsand application of the Kalman-Bucy filter in the modeling ofRC circuitrdquo Applied Mathematical Sciences vol 4 no 21-24 pp1119ndash1127 2010

[35] C Tunc ldquoA note on the stability and boundedness of non-autonomous differential equations of second order with avariable deviating argumentrdquo Afrika Matematika vol 25 no 2pp 417ndash425 2014

[36] C Tunc ldquoA note on the bounded solutions to 11990910158401015840 + 119888(119905 119909 1199091015840 ) +119902(119905)119887(119909) = 119891(119905)rdquo Applied Mathematics amp Information Sciencesvol 8 no 1 pp 393ndash399 2014

[37] C Tunc ldquoBoundedness analysis for certain two-dimensionaldifferential systems via a Lyapunov approachrdquo Bulletin Mathe-matique de la Societe des Sciences Mathematiques de Roumanievol 53 no 1 pp 61ndash68 2010

[38] 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

[39] C Tunc ldquoStability and boundedness in multi delay vectorLienard equationrdquo Filomat vol 27 no 3 pp 435ndash445 2013

[40] C Tunc ldquoStability and boundedness of solutions of non-autonomous differential equations of second orderrdquo Journalof Computational Analysis and Applications vol 13 no 6 pp1067ndash1074 2011

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

[42] 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

[43] C Tunc and T Ayhan ldquoGlobal existence and boundedness ofsolutions of a certain nonlinear integro-differential equationof second order with multiple deviating argumentsrdquo Journal ofInequalities and Applications vol 2016 article no 46 2016

[44] F Wang and H Zhu ldquoExistence uniqueness and stabilityof periodic solutions of a duffing equation under periodicand anti-periodic eigenvalues conditionsrdquo Taiwanese Journal ofMathematics vol 19 no 5 pp 1457ndash1468 2015

[45] Z Xianfeng and J Wei ldquoStability and boundedness of aretarded Lienard-type equationrdquo Chinese Quarterly Journal ofMathematics vol 18 no 1 pp 7ndash12 2003

[46] A F Yenicerioglu ldquoThe behavior of solutions of second orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 332 no 2 pp 1278ndash1290 2007

[47] A F Yenicerioglu ldquoStability properties of second order delayintegro-differential equationsrdquo Computers and Mathematicswith Applications vol 56 no 12 pp 3109ndash3117 2008

[48] T Yoshizawa ldquoLiapunovrsquos function and boundedness of solu-tionsrdquo Funkcialaj Ekvacioj vol 2 pp 71ndash103 1958

[49] W Zhu J Huang X Ruan and Z Zhao ldquoExponential stabilityof stochastic differential equation with mixed delayrdquo Journal ofAppliedMathematics vol 2014 Article ID 187037 11 pages 2014

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 International Journal of Analysis

respectively where 119891 119892 119901 ℎ 120601 and 120593 are continuousfunctions in their respective arguments In their contribu-tions the authors in [9 10] investigated asymptotic stabilityand boundedness of solutions of the following second-orderstochastic delay differential equations

11990910158401015840 (119905) + 1198861199091015840 (119905) + 119887119909 (119905 minus ℎ) + 120590119909 (119905) 1205961015840 (119905) = 0 (4)

11990910158401015840 (119905) + 1198861199091015840 (119905) + 119891 (119909 (119905 minus ℎ)) + 120590119909 (119905 minus 120591) 1205961015840 (119905) = 0 (5)

11990910158401015840 (119905) + 119892 (1199091015840 (119905)) + 119887119909 (119905 minus ℎ) + 120590119909 (119905) 1205961015840 (119905)= 119901 (119905 119909 (119905) 1199091015840 (119905) 1199091015840 (119905 minus ℎ)) (6)

respectively where 119886 119887 and 120590 are positive constants ℎ 120591 aredelay constants 119891 119892 and 119901 are continuous functions in theirrespective arguments and 119908(119905) isin R119898 is an 119898-dimensionalstandard Brownian motion defined on the probability space(also called Wiener process) Recently in 2016 the authors in[43] discussed global existence and boundedness of solutionsof a certain nonlinear integrodifferential equation of second-order with multiple deviating arguments

[119901 (119909 (119905)) 1199091015840 (119905)]1015840 + 119886 (119905) 119891 (119905 119909 (119905) 1199091015840 (119905)) 1199091015840 (119905)+ 119887 (119905) 119892 (119905 1199091015840 (119905)) + 119899sum

119894=1119888119894 (119905) ℎ119894 (119909 (119905 minus 120591119894))

= int1199050119888 (119905 119904) 1199091015840 (119904) 119889119904

(7)

where 120591119894 (119894 = 1 2 119899) are positive constants 119886 119887 and 119888 aredefined on R+ and 119891 119892 ℎ and 119901 are continuous functionsdefined in their respective arguments

Although second-order stochastic delay differential equa-tions have started receiving attention of authors according toour observation from relevant literature there is no previousliterature available on the stability and boundedness of solu-tions of second-order nonlinear nonautonomous stochasticdifferential equation The aim of this paper is to bridgethis gap Consider the following second-order nonlinearnonautonomous stochastic differential equation

11990910158401015840 (119905) + 119892 (119909 (119905) 1199091015840 (119905)) 1199091015840 (119905) + 119891 (119909 (119905))+ 120590119909 (119905) 1205961015840 (119905) = 119901 (119905 119909 (119905) 1199091015840 (119905)) (8)

where 120590 is a positive constant the functions 119892 119891 and 119901are continuous in their respective arguments on R2R andR+ times R2 respectively with R fl (minusinfininfin) R+ fl [0infin)and 120596 (a standard Wiener process representing the noise) isdefined on R Furthermore it is assumed that the continuityof the functions 119892 119891 and 119901 is sufficient for the existence ofsolutions and the local Lipschitz condition for (8) to havea unique continuous solution denoted by (119909(119905) 119910(119905)) Theprimes denote differentiationwith respect to the independent

variable 119905 isin R+ If 1199091015840(119905) = 119910(119905) then (8) is equivalent to thesystem

1199091015840 (119905) = 119910 (119905) 1199101015840 (119905) = 119901 (119905 119909 (119905) 119910 (119905)) minus 119891 (119909) minus 119892 (119909 (119905) 119910 (119905)) 119910 (119905)

minus 120590119909 (119905) 1205961015840 (119905) (9)

where the derivative of the function 119891 (ie 1198911015840) exists and iscontinuous for all 119909 Despite the applicability of these classesof equations there is no previous result on nonautonomoussecond-order nonlinear stochastic differential equation (8)The motivation for this investigation comes from the worksin [9ndash12 18 19] If 120590 = 0 in (8) then we have a generalsecond-order nonlinear ordinary differential equation whichhas been discussed extensively in relevant literature Theremaining parts of this paper are organized as follows InSection 2 we give the preliminary results on stochasticdifferential equations Main results and their proofs arepresented in Section 3 while examples and simulation ofsolutions are given in Section 4 to validate our results

2 Preliminary Results

Let (ΩF F119905119905gt0P) be a complete probability space witha filtration F119905119905gt0 satisfying the usual conditions (ie it isright continuous and F0 contains allP-null sets) Let119861(119905) =(1198611(119905) 119861119898(119905))119879 be an 119898-dimensional Brownian motiondefined on the probability space Let |sdot| denotes the Euclideannorm inR119899 If119860 is a vector ormatrix its transpose is denotedby 119860119879 If 119860 is a matrix its trace norm is denoted by

|119860| = radictrace (119860119879119860) (10)

For more exposition in this regard see Mao [29] and Arnold[1] Now let us consider a nonautonomous 119899-dimensionalstochastic differential equation

119889119883 (119905) = 119865 (119905 119883 (119905)) 119889119905 + 119866 (119905 119883 (119905)) 119889119861 (119905) (11)

on 119905 gt 0 with initial value 119883(0) = 1198830 isin R119899 Here 119865 R+ timesR119899 rarr R119899 and119866 R+timesR119899 rarr R119899times119898 aremeasurable functionsSuppose that both 119865 and 119866 are sufficiently smooth for (11) tohave a unique continuous solution on 119905 ge 0 which is denotedby119883(119905 1198830) if X(0) = 0 Assume further that

119865 (119905 0) = 119866 (119905 0) = 0 (12)

for all 119905 ge 0 Then the stochastic differential equation (11)admits zero solution119883(119905 0) equiv 0Definition 1 (see [1]) The zero solution of the stochasticdifferential equation (11) is said to be stochastically stable orstable in probability if for every pair of 120598 isin (0 1) and 119903 gt 0there exists a 1205750 = 1205750(120598 119903) gt 0 such that

Pr 1003816100381610038161003816119883 (119905 1198830)1003816100381610038161003816 lt 119903 forall119905 ge 0 ge 1 minus 120598whenever 100381610038161003816100381611988301003816100381610038161003816 lt 1205750 (13)

Otherwise it is said to be stochastically unstable

International Journal of Analysis 3

Definition 2 (see [1]) The zero solution of the stochasticdifferential equation (11) is said to be stochastically asymptot-ically stable if it is stochastically stable and in addition if forevery 120598 isin (0 1) and 119903 gt 0 there exists a 120575 = 120575(120598) gt 0 suchthat

Pr lim119905rarrinfin

119883(119905 1198830) = 0 ge 1 minus 120598 whenever 100381610038161003816100381611988301003816100381610038161003816 lt 120575 (14)

Definition 3 A solution 119883(1199050 1198830) of the stochastic differ-ential equation (11) is said to be stochastically bounded orbounded in probability if it satisfies

1198641198830 1003817100381710038171003817119883 (119905 1198830)1003817100381710038171003817 le 119862 (1199050 100381710038171003817100381711988301003817100381710038171003817) forall119905 ge 1199050 (15)

where 1198641198830 denotes the expectation operator with respect tothe probability law associated with1198830 119862 R+ timesR119899 andR+ isa constant depending on 1199050 and1198830Definition 4 The solutions 119883(1199050 1198830) of the stochastic dif-ferential equation (11) are said to be uniformly stochasticallybounded if 119862 in inequality (15) is independent of 1199050

For ℎ gt 0 let 119880ℎ = 119883 isin R119899 |119883| lt ℎ sub R119899 andlet 11986212(119880ℎ times R+R+) denote the family of all nonnegativefunctions119881(119905 119883(119905)) (Lyapunov function) defined onR+times119880ℎwhich are twice continuously differentiable in 119883 and once in119905 By Itorsquos formula we have

119889119881 (119905 119883 (119905)) = 119871119881 (119905 119883 (119905)) 119889119905+ 119881119909 (119905 119883 (119905)) 119866 (119905 119883 (119905)) 119889119861 (119905) (16)

where

119871119881 (119905 119883 (119905))= 120597119881 (119905 119883 (119905))120597119905 + 120597119881 (119905 119883 (119905))120597119909119894 119865 (119905 119883 (119905))+ 12 trace [119866119879 (119905 119883 (119905)) 119881119909119909 (119905 119883 (119905)) 119866 (119905 119883 (119905))]

(17)

Furthermore

119881119909119909 (119905 119883 (119905)) = (1205972119881 (119905 119883 (119905))120597119909119894120597119909119895 )119899times119899

119894 119895 = 1 119899 (18)

In this study we will use the diffusion operator 119871119881(119905 119883(119905))defined in (17) to replace 1198811015840(119905 119883(119905)) = (119889119889119905)119881(119905 119883(119905))Wenow present the basic results that will be used in the proofsof the main results

Lemma 5 (see [1]) Assume that there exist 119881 isin 11986212(R+ times119880ℎR+) and 120601 isin K such that

(i) 119881(119905 0) = 0(ii) 119881(119905 119883(119905)) gt 120601(119883(119905))(iii) 119871119881(119905 119883(119905)) le 0 for all (119905 119883) isin R+ times 119880ℎ

Then the zero solution of stochastic differential equation (11) isstochastically stable

Lemma 6 (see [1]) Suppose that there exist 119881 isin 11986212(R+ times119880ℎR+) and 1206010 1206011 1206012 isin K such that

(i) 119881(119905 0) = 0(ii) 1206010(119883(119905)) le 119881(119905 119883(119905)) le 1206011(119883(119905)) 1206010(119903) rarr infin as119903 rarr infin(iii) 119871119881(119905 119883(119905)) le minus1206012(119883(119905)) for all (119905 119883) isin R+ times 119880ℎ

Then the zero solution of stochastic differential equation (11) isuniformly stochastically asymptotically stable in the large

Assumption 7 (see [28 33]) Let 119881 isin 11986212(R+ times R119899R+)and suppose that for any solutions 119883(1199050 1198830) of stochasticdifferential equation (11) and for any fixed 0 le 1199050 le 119879 lt infinwe have

1198641198830 int11987911990501198812119909119894 (119905 119883 (119905)) 1198662119894119896 (119905 119883 (119905)) 119889119905 lt infin

1 le 119894 le 119899 1 le 119896 le 119898 (19)

Assumption 8 (see [28 33]) A special case of the generalcondition (19) is the following condition Assume that thereexits a function 120590(119905) such that10038161003816100381610038161003816119881119909119894 (119905 119883 (119905)) 119866119894119896 (119905 119883 (119905))10038161003816100381610038161003816 lt 120590 (119905)

119883 isin R119899 1 le 119894 le 119899 1 le 119896 le 119898 (20)

and for any fixed 0 le 1199050 le 119879 lt infin

int11987911990501205902 (119905) 119889119905 lt infin (21)

Lemma 9 (see [28 33]) Assume there exists a Lyapunov func-tion 119881(119905 119883(119905)) isin 11986212(R+ times R119899R+) satisfying Assumption 7such that for all (119905 119883) isin R+ timesR119899

(i) 119883(119905)119901 le 119881(119905 119883(119905)) le 119883(119905)119902(ii) 119871119881(119905 119883(119905)) le minus120572(119905)119883(119905)119903 + 120573(119905)(iii) 119881(119905 119883(119905)) minus 119881119903119902(119905 119883(119905)) le 120574

where 120572 120573 isin 119862(R+R+) 119901 119902 and 119903 are positive constants119901 ge 1 and 120574 is a nonnegative constant Then all solutions ofthe stochastic differential equation (11) satisfy

1198641198830 1003817100381710038171003817119883 (119905 1198830)1003817100381710038171003817 le 119881 (1199050 1198830) 119890minusint1199051199050 120572(119904)119889119904+ int1199051199050(120574120572 (119906) + 120573 (119906)) 119890minusint119905119906 120572(119904)1198891199041198891199061119901

(22)

for all 119905 ge 1199050Lemma 10 (see [28 33]) Assume there exists a Lyapunovfunction 119881(119905 119883(119905)) isin 11986212(R+ times R119899R+) satisfying Assump-tion 7 such that for all (119905 119883) isin R+ timesR119899

(i) 119883(119905)119901 le 119881(119905 119883(119905))(ii) 119871119881(119905 119883(119905)) le minus120572(119905)119881119902(119905 119883(119905)) + 120573(119905)(iii) 119881(119905 119883(119905)) minus 119881119902(119905 119883(119905)) le 120574

4 International Journal of Analysis

where 120572 120573 isin 119862(R+R+) 119901 119902 are positive constants 119901 ge 1 and120574 is a nonnegative constant Then all solutions of the stochasticdifferential equation (11) satisfy (22) for all 119905 ge 1199050Corollary 11 (see [28 33]) (i) Assume that hypotheses (i) to(iii) of Lemma 9 hold In addition

int1199051199050(120574120572 (119906) + 120573 (119906)) 119890minusint119905119906 120572(119904)119889119904119889119906 le 119872 forall119905 ge 1199050 ge 0 (23)

for some positive constant 119872 then all solutions of stochasticdifferential equation (11) are uniformly stochastically bounded

(ii) Assume that hypotheses (i) to (iii) of Lemma 10 holdIf condition (23) is satisfied then all solutions of the stochasticdifferential equation (11) are stochastically bounded

3 Main Results

Let (119909(119905) 119910(119905)) be any solution of the stochastic differentialequation (9) the main tool employed in the proofs ofour results is the continuously differentiable function 119881 =119881(119905 119909(119905) 119910(119905)) defined as

2119881 = 11988721199092 + 1198871199102 + 2119909119891 (119909) + (119886119909 + 119910)2 (24)

where 119886 and 119887 are positive constants and the function 119891 is asdefined in Section 1

Theorem 12 Suppose that 119886 119887 120590 and 1198720 are positiveconstants such that

(i) 119886 le 119892(119909 119910) for all 119909 and 119910(ii) 119887119909 le 119891(119909) le 119861119909 for all 119909 = 0 and 1205902 lt 2119886119887(119887 + 1)minus1(iii) |119901(119905 119909 119910)| le 1198720 for all 119905 ge 0 119909 and 119910

Then solution (119909(119905) 119910(119905)) of the stochastic differential equation(9) is uniformly stochastically bounded

Remark 13 We note the following

(i) Whenever the functions 119892(119909 1199091015840) = 119886 119891(119909) = 119887119909 and1205961015840 = 119901(119905 119909 1199091015840) = 0 then the stochastic differentialequation (8) becomes a second-order linear ordinarydifferential equation

11990910158401015840 + 1198861199091015840 + 119887119909 = 0 (25)

and conditions (i) to (iii) of Theorem 12 reduce toRouth Hurwitz criteria 119886 gt 0 and 119887 gt 0 forthe asymptotic stability of the second-order lineardifferential equation (25)

(ii) The term 120590119909(119905)1205961015840(119905) in the stochastic differentialequation (8) is an extension of the ordinary casediscussed recently by authors in [11 18 23 31 32 35ndash37 40]

We shall now state and prove a result that will be used inthe proofs of our results

Lemma 14 Under the hypotheses of Theorem 12 there existpositive constants 1198630 = 1198630(119886 119887) and 1198631 = 1198631(119886 119887 119861) suchthat

1198630 (1199092 (119905) + 1199102 (119905)) le 119881 (119905 119909 (119905) 119910 (119905))le 1198631 (1199092 (119905) + 1199102 (119905)) (26)

for all 119905 ge 0 119909 and 119910 In addition there exist positive constants1198632 = 1198632(119886 119887 120590) and 1198633 = 1198633(119886 119887) such that119871119881 (119905 119909 (119905) 119910 (119905))

le minus1198632 (1199092 (119905) + 1199102 (119905))+ 1198633 (|119909 (119905)| + 1003816100381610038161003816119910 (119905)1003816100381610038161003816) 1003816100381610038161003816119901 (119905 119909 (119905) 119910 (119905))1003816100381610038161003816

(27)

for all 119905 ge 0 119909 and 119910Proof Let (119909(119905) 119910(119905)) be any solution of the stochastic differ-ential equation (9) since 119883 = (119909 119910) isin R2 it follows from(24) that

119881 (119905 0 0) = 0 (28)

for all 119905 ge 0Moreover from (24) and the fact that 119891(119909) ge 119886119909for all 119909 = 0 there exists a positive constant 1205750 such that

119881 (119905 119883) ge 1205750 (1199092 + 1199102) (29)

for all 119905 ge 0 119909 and 119910 where1205750 fl min 1198872 + 2119887 +min 119886 1 119887 +min 119886 1 (30)

It is clear from inequality (29) that

119881 (119905 119883) = 0 lArrrArr 1199092 + 1199102 = 0119881 (119905 119883) gt 0 lArrrArr 1199092 + 1199102 = 0 (31)

119881 (119905 119883) 997888rarr +infin as 1199092 + 1199102 997888rarr infin (32)

Furthermore since 119891(119909) le 119861119909 for all 119909 = 0 it follows from(24) that there exists a positive constant 1205751 such that

119881 (119905 119883) le 1205751 (1199092 + 1199102) (33)

for all 119905 ge 0 119909 and 119910 where1205751 fl max 1198872 + 2119861 +max 119886 1 119887 +max 119886 1 (34)

From inequalities (29) and (33) we have

1205750 (1199092 + 1199102) le 119881 (119905 119883) le 1205751 (1199092 + 1199102) (35)

for all 119905 ge 0 119909 and 119910 It is not difficult to see that estimates(35) satisfy inequalities (26) of Lemma 14 with 1205750 and 1205751equivalent to1198630 and1198631 respectively

International Journal of Analysis 5

Moreover applying Itorsquos formula in (24) using system (9)we find that

119871119881 (119905 119883) = 12 [119886119891 (119909)119909 minus 121205902 (119887 + 1)] 1199092minus 12 [(119887 + 1) 119892 (119909 119910) minus 119886] 1199102 minus119882119894+ [119886119909 + (119887 + 1) 119910] 119901 (119905 119909 119910)

(119894 = 1 2)

(36)

where

1198821 fl 14 [119886119891 (119909)119909 minus 121205902 (119887 + 1)] 1199092+ 4 [119886119892 (119909 119910) minus (1198862 + 1198872)] 119909119910+ [(119887 + 1) 119892 (119909 119910) minus 119886] 1199102

1198822 fl 14 [119886119891 (119909)119909 minus 121205902 (119887 + 1)] 1199092

+ 4 [119886119891 (119909)119909 minus 1198911015840 (119909)] 119909119910+ [(119887 + 1) 119892 (119909 119910) minus 119886] 1199102

(37)

It is clear from the inequalities

4 [119886119892 (119909 119910) minus (1198862 + 1198872)]2lt [119886119891 (119909)119909 minus 121205902 (119887 + 1)] [(119887 + 1) 119892 (119909 119910) minus 119886]

4 [119886119891 (119909)119909 minus 1198911015840 (119909)]lt [119886119891 (119909)119909 minus 121205902 (119887 + 1)] [(119887 + 1) 119892 (119909 119910) minus 119886]

(38)

that

1198821 = 1198822 ge [[radic119886119891 (119909)119909 minus 121205902 (119887 + 1) |119909|

minus radic(119887 + 1) 119892 (119909 119910) minus 119886 10038161003816100381610038161199101003816100381610038161003816]]2

ge 0(39)

for all 119909 and 119910 Using inequality (39) and hypotheses (i) and(ii) ofTheorem 12 in (36) there exist positive constants 1205752 and1205753 such that

119871119881 (119905 119883) le minus1205752 (1199092 + 1199102)+ 1205753 (|119909| + 10038161003816100381610038161199101003816100381610038161003816) 1003816100381610038161003816119901 (119905 119909 119910)1003816100381610038161003816 (40)

for all 119905 ge 0 119909 and 119910 where1205752 fl 12 min 119886119887 minus 121205902 (119887 + 1) 119886119887 1205753 fl max 119886 119887 + 1 (41)

Inequality (40) satisfies inequality (27) with 1205752 and 1205753equivalent to 1198632 and 1198633 respectively This completes theproof of Lemma 14

Proof ofTheorem 12 Let (119909(119905) 119910(119905)) be any solution of system(9) From inequality (40) and assumption (iii) ofTheorem 12we have

119871119881 (119905 119883)le minus121205752 (1199092 + 1199102)minus 1212057521198720 [(|119909| minus 120575minus12 1205753)2 + (10038161003816100381610038161199101003816100381610038161003816 minus 120575minus12 1205753)2]+1198720120575minus12 12057523

(42)

for 119905 ge 0 119909 and 119910 Since 1205752 1205753 and1198720 are positives and(|119909| minus 120575minus12 1205753)2 + (10038161003816100381610038161199101003816100381610038161003816 minus 120575minus12 1205753)2 ge 0 (43)

for all 119909 and 119910 there exist positive constants 1205754 and 1205755 suchthat

119871119881 (119905 119883) le minus1205754 (1199092 + 1199102) + 1205755 (44)

for all 119905 ge 0 119909 119910 where 1205754 fl (12)1205752 and 1205755 fl 1198720120575minus12 12057523 Hence condition (ii) of Lemma 9 is satisfied with 120572(119905) fl 1205754119903 fl 2 and 120573(119905) fl 1205755 Also from inequality (35) hypotheses(i) and (iii) of Lemma 9 hold with 119901 = 119902 = 2 so that 120574 = 0

Furthermore from inequality (23) we have

int1199051199050[(120574120572 (119906) + 120573 (119906)) 119890minus1205754 int119905119906 120572(119904)119889119904] 119889119906= int11990511990501205755119890minus1205754 int119905119906 119889119904119889119906 = 120575minus14 1205755 [1 minus 119890minus1205754(119905minus1199050)]

le 120575minus14 1205755(45)

for all 119905 ge 1199050 ge 0 Inequality (45) satisfies estimate (23) with119872 fl 120575minus14 1205755 = 21198720120575minus22 12057523 gt 0 Moreover from (9) and (24)there exists a positive constant 1205756 such that

10038161003816100381610038161003816119881119909119894 (119905 119883)119866119894119896 (119905 119883)10038161003816100381610038161003816le 12120590 [(2119886 + 119887 + 1) 1199092 + (119887 + 1) 1199102]le 1205756 (1199092 + 1199102) fl 120582 (119905)

(46)

where

1205756 fl 12120590max 2119886 + 119887 + 1 119887 + 1 (47)

6 International Journal of Analysis

Also

int119879119905012057526 (1199092 (119905) + 1199102 (119905))2 119889119905 lt infin (48)

for any fixed 0 le 1199050 le 119879 lt infin Thus from inequalities (46)and (48) estimates (20) and (21) hold respectively Finallyfrom inequalities (33) and (45) we have

1198641198830 1003817100381710038171003817119883 (119905 1198830)1003817100381710038171003817 le (120575111988320 + 21198720120575minus22 12057523)12 (49)

for all 119905 ge 1199050 ge 0 where1198830 fl (11990920 +11991020) and119862 fl 1205751Thus thesolutions (119909(119905) 119910(119905)) of the stochastic differential equation (9)are uniformly stochastically bounded

Theorem 15 If assumptions of Theorem 12 hold then thesolution (119909(119905) 119910(119905)) of the stochastic differential equation (9)is stochastically bounded

Proof Suppose that (119909(119905) 119910(119905)) is any solution of the stochas-tic differential equation (9) From inequalities (33) and (44)there exists a positive constant 1205757 such that

119871119881 (119905 119883) le minus1205757119881 (119905 119883) + 1205755 (50)

for all 119905 ge 0 119909 and119910 where1205757 fl 120575minus11 1205754Hence from inequal-ities (29) and (50) hypotheses of Lemma 10 hold Moreoverfrom inequalities (45) (46) (48) and (49) assumption (ii)of Corollary 11 holds Thus by Corollary 11 all solutionsof the stochastic differential equation (9) are stochasticallybounded This completes the proof of Theorem 15

Next we shall discuss the stability of the trivial solutionof the stochastic differential equation (8) Suppose that119901(119905 119909 1199091015840) = 0 (8) specializes to

11990910158401015840 (119905) + 119892 (119909 (119905) 1199091015840 (119905)) 1199091015840 (119905) + 119891 (119909 (119905))+ 120590119909 (119905) 1205961015840 (119905) = 0 (51)

Equation (51) has the following equivalent system

1199091015840 (119905) = 119910 (119905) 1199101015840 (119905) = minus119891 (119909) minus 119892 (119909 (119905) 119910 (119905)) 119910 (119905) minus 120590119909 (119905) 1205961015840 (119905) (52)

where the functions 119891 119892 and 120596 are defined in Section 1

Theorem 16 If assumptions (i) and (ii) of Theorem 12 holdthen the trivial solution of the stochastic differential equation(52) is stochastically stable

Proof Let (119909(119905) 119910(119905)) be any solution of the stochastic differ-ential equation (52) From equation (28) and estimate (29)assumptions (i) and (ii) of Lemma 5 hold so that the function119881(119905 119883) is positive definite Furthermore using Itorsquos formulaalong the solution path of (52) we obtain

119871119881 (119905 119883) le minus1205752 (1199092 (119905) + 1199102 (119905)) le 0 (53)

for all 119905 ge 0 119909 and 119910 where 1205752 is defined in (40)Inequality (53) satisfies hypothesis (iii) of Lemma 5 henceby Lemma 5 the trivial solution of the stochastic differentialequation (52) is stochastically stableThis completes the proofof Theorem 16

Theorem 17 If assumptions (i) and (ii) of Theorem 12 holdthen the trivial solution of the stochastic differential equation(52) is not only uniformly stochastically asymptotically stablebut also uniformly stochastically asymptotically stable in thelarge

Proof Let (119909(119905) 119910(119905)) be any solution of the stochastic differ-ential equation (52) In view of (28) and estimate (29) thefunction 119881(119905 119883) is positive definite Furthermore estimate(32) and inequality (33) show that the function 119881(119905 119883) isradially unbounded and decrescent respectively It followsfrom (28) estimate (32) inequality (35) and the first inequal-ity in (53) that all assumptions of Lemma 6 hold Thus byLemma 6 the trivial solution of the stochastic differentialequation (52) is uniformly stochastically asymptotically stablein the large If estimate (32) is omitted then the trivial solutionof the stochastic differential equation (52) is uniformlystochastically asymptotically stable This completes the proofof Theorem 17

Next if the function 119901(119905 119909 1199091015840) is replaced by 119901(119905) isin119862(R+R+) we have the following special case11990910158401015840 (119905) + 119892 (119909 (119905) 1199091015840 (119905)) 1199091015840 (119905) + 119891 (119909 (119905))

+ 120590119909 (119905) 1205961015840 (119905) = 119901 (119905) (54)

of (8) Equation (54) has the following equivalent system

1199091015840 (119905) = 119910 (119905) 1199101015840 (119905) = 119901 (119905) minus 119891 (119909) minus 119892 (119909 (119905) 119910 (119905)) 119910 (119905)

minus 120590119909 (119905) 1205961015840 (119905) (55)

with the following result

Corollary 18 If assumptions (i) and (ii) of Theorem 12 holdand hypothesis (iii) is replaced by the boundedness of thefunction 119901(119905) then the solutions (119909(119905) 119910(119905)) of the stochasticdifferential equation (55) are not only stochastically boundedbut also uniformly stochastically bounded

Proof Theproof of Corollary 18 is similar to the proof ofThe-orems 12 and 15This completes the proof of Corollary 18

4 Examples

In this section we shall present two examples to illustrate theapplications of the results we obtained in the previous section

Example 1 Consider the second-order nonlinear nonau-tonomous stochastic differential equation

11990910158401015840 + (3 + 10038161003816100381610038161003816cos (1199091199091015840)10038161003816100381610038161003816) 1199091015840 + 119909 + sin119909 + 011199091205961015840 (119905)= (1 + 2119905 + 10038161003816100381610038161003816119909119909101584010038161003816100381610038161003816)minus1

(56)

International Journal of Analysis 7

Equation (56) is equivalent to system

1199091015840 = 1199101199101015840 = (1 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816)minus1 minus (119909 + sin119909)

minus [3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816] 119910 minus 011199091205961015840 (119905) (57)

Now from systems (9) and (57) we have the followingrelations

(i) The function

119892 (119909 119910) fl 3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816 (58)

Noting that 1003816100381610038161003816cos (119909119910)1003816100381610038161003816 ge 0 (59)

for all 119909 and 119910 it follows that119892 (119909 119910) = 3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816 ge 119886 = 3 (60)

for all 119909 and 119910The behaviour of the function 119892(119909 119910)is shown below in Figure 1

(ii) The function

119891 (119909) fl 119909 + sin119909 (61)

Since

minus02 le 119865 (119909) = sin119909119909 le 1 (62)

for all 119909 = 0 then we have

1 = 119887 le 119891 (119909)119909 = 1 + sin119909119909 le 119861 = 2 (63)

for all 119909 = 0 and since 120590 fl 01 it follows that1205902 lt 2119886119887(119887 + 1)minus1 implies that 0 lt 299 The function119891(119909)119909 and its bounds are shown in Figure 2(iii) The function

119901 (119905 119909 119910) fl 11 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816 (64)

Clearly

1003816100381610038161003816119901 (119905 119909 119910)1003816100381610038161003816 = 11 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816 le 1 = 1198720 (65)

for all 119905 ge 0 119909 and 119910Now from items (i) (ii) above and (24) the continuouslydifferentiable function 119881(119905 119883) used for system (57) is

2119881 (119905 119883) = 31199092 + 1199102 + (3119909 + 119910)2 (66)

Different views of the function119881(119905 119883) are shown in Figure 3From (66) it is not difficult to show that

(1199092 + 1199102) le 119881 (119905 119883) le 3 (1199092 + 1199102) (67)

Figure 1 Behaviour of the function 119892(119909 119910)

minus6120587 minus4120587 minus2120587 612058741205872120587

ge minus0225

25

2

15

1

05

minus05

minus1

F(x) f(x)x

f(x)x = 1 + sin(x)x

F(x) = sin(x)xx

b = 076

F(x)

Figure 2 Bounds on the function 119891(119909)119909

for all 119905 ge 0 119909 and 119910 From (35) and (67) we have 1205750 = 11205751 = 3 119901 = 2 and 119902 = 2 and thus inequalities (67) satisfycondition (i) of Lemma 9 Also from the first inequality in(67) we have

119881 (119905 119883) 997888rarr +infin as 1199092 + 1199102 997888rarr infin (68)

Estimate (68) verifies (32) (ie the function 119881(119905 119883) definedby (66) is radially unbounded) Next applying Itorsquos formulain (66) using system (57) we find that

119871119881 (119905 119883) = 12119909119910 + 31199102 minus 119909 (3119909 + 2119910) (1 + sin119909119909 )minus 119910 (3119909 + 2119910) (3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816) + 11001199092minus 11990910 (3119909 + 2119910)+ (3119909 + 2119910) (1 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816)minus1

(69)

Using the estimates in items (i) to (iii) of Example 1 and theinequality 211990911199092 le 11990921 + 11990922 in (69) we obtain

119871119881 (119905 119883) le minus29 (1199092 + 1199102) + 3 (|119909| + 10038161003816100381610038161199101003816100381610038161003816) (70)

for all 119905 ge 0 119909 and 119910 Inequality (70) satisfies inequality (40)where 1205752 = 29 and 1205753 = 3 Since

(|119909| minus 105)2 + (10038161003816100381610038161199101003816100381610038161003816 minus 105)2 ge 0 (71)

for all 119909 and 119910 it follows from inequality (70) that

119871119881 (119905 119883) le minus145 (1199092 + 1199102) + 32 (72)

8 International Journal of Analysis

Figure 3 The behaviour of the function 119881(119905 119883)

for all 119905 ge 0 119909 and 119910 Inequality (72) satisfies assumption(ii) of Lemma 9 and estimate (44) with 120572(119905) = 1205754 = 145 and120573(119905) = 1205755 = 32 Since 119903 = 119901 = 119902 = 2 it follows that 120574 = 0 sothat assumption (iii) of Lemma 9 holds In addition

int1199051199050[(120574120572 (119906) + 120573 (119906)) 119890minusint119905119906120572 (119904) 119889119904] 119889119906 le 16 (73)

for all 119905 ge 1199050 ge 0 Estimate (73) satisfies (23) and (45) with119872 = 26 Furthermore

119881119909119894 (119905 119883)119866119894119896 (119905 119883) = minus 110 (31199092 + 2119909119910) (74)

and10038161003816100381610038161003816119881119909119894 (119905 119883)119866119894119896 (119905 119883)10038161003816100381610038161003816 le 25 (1199092 + 1199102) (75)

for all 119905 ge 0 119909 and119910 Inequality (75) satisfies inequalities (20)and (21) with

120582 (119905) = 25 (1199092 + 1199102) (76)

Hence by Corollary 11 (i) all solutions of stochastic differen-tial equation (57) are uniformly stochastically bounded

Example 2 If 119901(119905 119909 1199091015840) = 119901(119905 119909 119910) = 0 in (56) and system(57) we have the following stochastic differential equation

11990910158401015840 + (3 + 10038161003816100381610038161003816cos (1199091199091015840)10038161003816100381610038161003816) 1199091015840 + 119909 + sin119909 + 011199091205961015840 (119905)= 0 (77)

Equation (77) is equivalent to system

1199091015840 = 1199101199101015840 = minus (119909 + sin119909) minus [3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816] 119910 minus 011199091205961015840 (119905) (78)

Now from systems (52) and (78) items (i) and (ii) of Example 1hold Also equations (66) (67) and estimate (68) hold thatis

2119881 (119905 119883) = 31199092 + 1199102 + (3119909 + 119910)2 119881 (119905 0) = 0 forall119905 ge 0

(1199092 + 1199102) le 119881 (119905 119883) le 3 (1199092 + 1199102) forall119905 ge 0 119909 119910119881 (119905 119883) 997888rarr +infin as 1199092 + 1199102 997888rarr infin

(79)

112

1416

182

0500

10001500

0

02

04

06

08

1

t

x(t) y(t)

Figure 4 Graph of solutions of (56) in 3D

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(b)

Figure 5

Furthermore application of Itorsquos formula in (66) and usingsystem (78) yield

119871119881 (119905 119883) le minus29 (1199092 + 1199102) (80)

for all 119905 ge 0 119909 119910 and thus

119871119881 (119905 119883) le 0 (81)

International Journal of Analysis 9

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

000200400600801012

t

minus004

minus002

x(t)y(t)

x(t)

y(t)

times1011

(b)

Figure 6

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

000200400600801012

t

minus002

minus008

minus006

minus004

times1011

x(t)y(t)

x(t)

y(t)

(b)

Figure 7

for all 119905 ge 0 119909 and 119910 Moreover from (79) and (80)all assumptions of Theorem 17 and Lemma 6 are satisfiedThus by Lemma 6 the trivial solution of system (78) is notonly uniformly stochastically asymptotically stable but alsouniformly stochastically asymptotically stable in the largeFinally from (79) and (81) the function 119881(119905 119883) is positivedefinite and

119871119881 (119905 119883) le 0 forall (119905 119883) isin R+ timesR

2 (82)

Hence assumptions of Theorem 17 and Lemma 5 hold byTheorem 17 and Lemma 5 the trivial solution of system (78)is stochastically stable

Simulation of Solutions In what follows we shall nowsimulate the solutions of (56) (resp system (57)) and (78)(resp system (79)) Our approach depends on the Euler-Maruyama method which enables us to get approximatenumerical solution for the considered systems It will be seenfrom our figures that the simulated solutions are bounded

which justifies our given results For instance when 120590 = 01the numerical solutions of (56) in three-dimensional spaceare shown in Figure 4 If we vary the value of the noise inthe numerical solution (119909(119905) 119910(119905)) of system (57) as 120590 = 01and 120590 = 10 we have Figures 5(a) and 5(b) respectively Itcan be seen that when the noise is increased the stochasticitybecomes more pronounced The behaviour of the numericalsolution (119909(119905) 119910(119905)) of system (57) when 120590 = 05 and 120590 = 20is shown in Figures 6(a) and 6(b) respectivelyThe behaviourof the numerical solution (119909(119905) 119910(119905)) of system (57) for 120590 = 0and 120590 = 50 is shown in Figures 7(a) and 7(b) respectivelyFor the case of (78) Figure 8 shows the closeness of thesolution (119909(119905)) and the perturbed solution (119909120598(119905)) for a verylarge 119905 which implies asymptotic stability in the large for theconsidered SDE

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

10 International Journal of Analysis

0 1 2 3 4 5 6 7 8 9 100

0005

001

0015

002

0025

t times1011

x(t)x120576(t)

x(t)x120576(t)

Figure 8 Graph of solutions of (78)

References

[1] L Arnold Stochastic Differential Equations Theory and Appli-cations John Wiley amp Sons 1974

[2] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations vol 178 of Mathematics inScience and Engineering Academic Press Inc Orlando FlaUSA 1985

[3] T A Burton Volterra Integral and Differential EquationsAcademic Press New York NY USA 1983

[4] J K HaleTheory of Functional Differential Equations SpringerNew York NY USA 1977

[5] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer 2000

[6] L Shaikihet Lyapunov Functionals and Stability of StochasticFunctional Differential Equations Springer International 2013

[7] T Yoshizawa StabilityTheory and Existence of Periodic Solutionsand almost Periodic Solutions Spriger New York NY USA1975

[8] T Yoshizawa StabilityTheory by Liapunovrsquos SecondMethodTheMathematical Society of Japan 1966

[9] A M A Abou-El-Ela A I Sadek and A M Mahmoud ldquoOnthe stability of solutions for certain second-order stochasticdelay differential equationsrdquo Differential Equations and ControlProcesses no 2 pp 1ndash13 2015

[10] A M Abou-El-Ela A I Sadek A M Mahmoud and R OTaie ldquoOn the stochastic stability and boundedness of solutionsfor stochastic delay differential equation of the second orderrdquoChinese Journal of Mathematics vol 2015 Article ID 358936 8pages 2015

[11] A T Ademola Boundedness and Stability of Solutions to CertainSecond Order Differential Equations Differential Equations andControl Processes 2015

[12] A T Ademola B S Ogundare M O Ogundiran and O AAdesina ldquoPeriodicity stability and boundedness of solutions tocertain second order delay differential equationsrdquo InternationalJournal of Differential Equations vol 2016 Article ID 284370910 pages 2016

[13] J G Alaba and B S Ogundare ldquoOn stability and bound-edness properties of solutions of certain second order non-autonomous nonlinear ordinary differential equationrdquo Kragu-jevac Journal of Mathematics vol 39 no 2 pp 255ndash266 2015

[14] T A Burton and L Hatvani ldquoAsymptotic stability of secondorder ordinary functional and partial differential equationsrdquoJournal of Mathematical Analysis and Applications vol 176 no1 pp 261ndash281 1993

[15] B Cahlon andD Schmidt ldquoStability criteria for certain second-order delay differential equations with mixed coefficientsrdquoJournal of Computational and AppliedMathematics vol 170 no1 pp 79ndash102 2004

[16] T Caraballo M A Diop and A S Ndoye ldquoFixed points andexponential stability for stochastic partial integro-differentialequations with delaysrdquo Advances in Dynamical Systems andApplications vol 9 no 2 pp 133ndash147 2014

[17] A Domoshnitsky ldquoNonoscillation maximum principles andexponential stability of secondorder delay differential equationswithout damping termrdquo Domoshnitsky Journal of Inequalitiesand Applications vol 2014 article 361 2014

[18] I I Gikhman and A V Skorokhod Stochastische Differential-gleichungen Akademie Berlin Germany 1971 (Russian)

[19] I I Gikhman On the Stability of the Solutions of StochasticDifferential Equations Predelrsquonyye Teoremy i StatisticheskiyeVyvody Tashkent Uzbekistan 1966

[20] G A Grigoryan ldquoBoundedness and stability criteria for linearordinary differential equations of the second orderrdquo RussianMathematics vol 57 no 12 pp 8ndash15 2013

[21] A F Ivanov Y I Kazmerchuk and A V Swishchuk ldquoThe-ory stochastic stability and applications of stochastic delaydifferential equations a survey of recent resultsrdquo in DifferentialEquations and Dynamical Systems vol 11 no 1 2003

[22] F Jedrzejewski and D Brochard ldquoLyapounv exponents andstability stochastic dynamical structuresrdquo 2000

[23] Z Jin and L Zengrong ldquoOn the global asymptotic behaviorof solutions to a non autonomous generalized Lienard systemrdquoJournal of Mathematical Research and Exposition vol 21 no 3pp 410ndash414 2001

[24] E Kolarova ldquoAn application of stochastic integral equations toelectrical networksrdquo Acta Electrotechnica et Informatica vol 8no 3 pp 14ndash17 2008

[25] V B Kolmanovskii and L E Shaikhet ldquoA method for con-structing Lyapunov functionals for stochastic systems with aftereffectrdquo Differentsialrsquonye Uravneniya vol 29 no 11 pp 1909ndash2022 1993

[26] V Kolmanovskii and L Shaikhet ldquoConstruction of Lyapunovfunctionals for stochastic hereditary systems a survey of somerecent resultsrdquo Mathematical and Computer Modelling vol 36no 6 pp 691ndash716 2002

[27] A J Kroopnick ldquoBounded solutions to 11990910158401015840 + 119902(119905)119887(119909) = 119891(119905)rdquoInternational Journal of Mathematical Education in Science andTechnology vol 41 no 6 pp 829ndash836 2010

[28] R Liu and Y Raffoul ldquoBoundedness and exponential stabilityof highly nonlinear stochastic differential equationsrdquo ElectronicJournal of Differential Equations vol 2009 no 143 pp 1ndash102009

[29] X Mao ldquoSome contributions to stochastic asymptotic stabilityand boundedness via multiple Lyapunov functionsrdquo Journal ofMathematical Analysis and Applications vol 260 no 2 pp 325ndash340 2001

[30] B S Ogundare A T Ademola M O Ogundiran and O AAdesina ldquoOn the qualitative behaviour of solutions to certainsecond order nonlinear differential equation with delayrdquoAnnalidellrsquoUniversitarsquo di Ferrara 2016

International Journal of Analysis 11

[31] B S Ogundare and A U Afuwape ldquoBoundedness and stabilityproperties of solutions of generalized Lienard equationrdquo KochiJournal of Mathematics vol 9 pp 97ndash108 2014

[32] B S Ogundare and G E Okecha ldquoBoundedness periodicityand stability of solutions to x(t) + g(x) + b(t)h(x) = p(t x x)rdquoMathematical Sciences Research Journal vol 11 no 5 pp 432ndash443 2007

[33] Y N Raffoul ldquoBoundedness and exponential asymptotic sta-bility in dynamical systems with applications to nonlineardifferential equations with unbounded termsrdquo Advances inDynamical Systems and Applications vol 2 no 1 pp 107ndash1212007

[34] R Rezaeyan and R Farnoosh ldquoStochastic differential equationsand application of the Kalman-Bucy filter in the modeling ofRC circuitrdquo Applied Mathematical Sciences vol 4 no 21-24 pp1119ndash1127 2010

[35] C Tunc ldquoA note on the stability and boundedness of non-autonomous differential equations of second order with avariable deviating argumentrdquo Afrika Matematika vol 25 no 2pp 417ndash425 2014

[36] C Tunc ldquoA note on the bounded solutions to 11990910158401015840 + 119888(119905 119909 1199091015840 ) +119902(119905)119887(119909) = 119891(119905)rdquo Applied Mathematics amp Information Sciencesvol 8 no 1 pp 393ndash399 2014

[37] C Tunc ldquoBoundedness analysis for certain two-dimensionaldifferential systems via a Lyapunov approachrdquo Bulletin Mathe-matique de la Societe des Sciences Mathematiques de Roumanievol 53 no 1 pp 61ndash68 2010

[38] 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

[39] C Tunc ldquoStability and boundedness in multi delay vectorLienard equationrdquo Filomat vol 27 no 3 pp 435ndash445 2013

[40] C Tunc ldquoStability and boundedness of solutions of non-autonomous differential equations of second orderrdquo Journalof Computational Analysis and Applications vol 13 no 6 pp1067ndash1074 2011

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

[42] 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

[43] C Tunc and T Ayhan ldquoGlobal existence and boundedness ofsolutions of a certain nonlinear integro-differential equationof second order with multiple deviating argumentsrdquo Journal ofInequalities and Applications vol 2016 article no 46 2016

[44] F Wang and H Zhu ldquoExistence uniqueness and stabilityof periodic solutions of a duffing equation under periodicand anti-periodic eigenvalues conditionsrdquo Taiwanese Journal ofMathematics vol 19 no 5 pp 1457ndash1468 2015

[45] Z Xianfeng and J Wei ldquoStability and boundedness of aretarded Lienard-type equationrdquo Chinese Quarterly Journal ofMathematics vol 18 no 1 pp 7ndash12 2003

[46] A F Yenicerioglu ldquoThe behavior of solutions of second orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 332 no 2 pp 1278ndash1290 2007

[47] A F Yenicerioglu ldquoStability properties of second order delayintegro-differential equationsrdquo Computers and Mathematicswith Applications vol 56 no 12 pp 3109ndash3117 2008

[48] T Yoshizawa ldquoLiapunovrsquos function and boundedness of solu-tionsrdquo Funkcialaj Ekvacioj vol 2 pp 71ndash103 1958

[49] W Zhu J Huang X Ruan and Z Zhao ldquoExponential stabilityof stochastic differential equation with mixed delayrdquo Journal ofAppliedMathematics vol 2014 Article ID 187037 11 pages 2014

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Stochastic AnalysisInternational Journal of

International Journal of Analysis 3

Definition 2 (see [1]) The zero solution of the stochasticdifferential equation (11) is said to be stochastically asymptot-ically stable if it is stochastically stable and in addition if forevery 120598 isin (0 1) and 119903 gt 0 there exists a 120575 = 120575(120598) gt 0 suchthat

Pr lim119905rarrinfin

119883(119905 1198830) = 0 ge 1 minus 120598 whenever 100381610038161003816100381611988301003816100381610038161003816 lt 120575 (14)

Definition 3 A solution 119883(1199050 1198830) of the stochastic differ-ential equation (11) is said to be stochastically bounded orbounded in probability if it satisfies

1198641198830 1003817100381710038171003817119883 (119905 1198830)1003817100381710038171003817 le 119862 (1199050 100381710038171003817100381711988301003817100381710038171003817) forall119905 ge 1199050 (15)

where 1198641198830 denotes the expectation operator with respect tothe probability law associated with1198830 119862 R+ timesR119899 andR+ isa constant depending on 1199050 and1198830Definition 4 The solutions 119883(1199050 1198830) of the stochastic dif-ferential equation (11) are said to be uniformly stochasticallybounded if 119862 in inequality (15) is independent of 1199050

For ℎ gt 0 let 119880ℎ = 119883 isin R119899 |119883| lt ℎ sub R119899 andlet 11986212(119880ℎ times R+R+) denote the family of all nonnegativefunctions119881(119905 119883(119905)) (Lyapunov function) defined onR+times119880ℎwhich are twice continuously differentiable in 119883 and once in119905 By Itorsquos formula we have

119889119881 (119905 119883 (119905)) = 119871119881 (119905 119883 (119905)) 119889119905+ 119881119909 (119905 119883 (119905)) 119866 (119905 119883 (119905)) 119889119861 (119905) (16)

where

119871119881 (119905 119883 (119905))= 120597119881 (119905 119883 (119905))120597119905 + 120597119881 (119905 119883 (119905))120597119909119894 119865 (119905 119883 (119905))+ 12 trace [119866119879 (119905 119883 (119905)) 119881119909119909 (119905 119883 (119905)) 119866 (119905 119883 (119905))]

(17)

Furthermore

119881119909119909 (119905 119883 (119905)) = (1205972119881 (119905 119883 (119905))120597119909119894120597119909119895 )119899times119899

119894 119895 = 1 119899 (18)

In this study we will use the diffusion operator 119871119881(119905 119883(119905))defined in (17) to replace 1198811015840(119905 119883(119905)) = (119889119889119905)119881(119905 119883(119905))Wenow present the basic results that will be used in the proofsof the main results

Lemma 5 (see [1]) Assume that there exist 119881 isin 11986212(R+ times119880ℎR+) and 120601 isin K such that

(i) 119881(119905 0) = 0(ii) 119881(119905 119883(119905)) gt 120601(119883(119905))(iii) 119871119881(119905 119883(119905)) le 0 for all (119905 119883) isin R+ times 119880ℎ

Then the zero solution of stochastic differential equation (11) isstochastically stable

Lemma 6 (see [1]) Suppose that there exist 119881 isin 11986212(R+ times119880ℎR+) and 1206010 1206011 1206012 isin K such that

(i) 119881(119905 0) = 0(ii) 1206010(119883(119905)) le 119881(119905 119883(119905)) le 1206011(119883(119905)) 1206010(119903) rarr infin as119903 rarr infin(iii) 119871119881(119905 119883(119905)) le minus1206012(119883(119905)) for all (119905 119883) isin R+ times 119880ℎ

Then the zero solution of stochastic differential equation (11) isuniformly stochastically asymptotically stable in the large

Assumption 7 (see [28 33]) Let 119881 isin 11986212(R+ times R119899R+)and suppose that for any solutions 119883(1199050 1198830) of stochasticdifferential equation (11) and for any fixed 0 le 1199050 le 119879 lt infinwe have

1198641198830 int11987911990501198812119909119894 (119905 119883 (119905)) 1198662119894119896 (119905 119883 (119905)) 119889119905 lt infin

1 le 119894 le 119899 1 le 119896 le 119898 (19)

Assumption 8 (see [28 33]) A special case of the generalcondition (19) is the following condition Assume that thereexits a function 120590(119905) such that10038161003816100381610038161003816119881119909119894 (119905 119883 (119905)) 119866119894119896 (119905 119883 (119905))10038161003816100381610038161003816 lt 120590 (119905)

119883 isin R119899 1 le 119894 le 119899 1 le 119896 le 119898 (20)

and for any fixed 0 le 1199050 le 119879 lt infin

int11987911990501205902 (119905) 119889119905 lt infin (21)

Lemma 9 (see [28 33]) Assume there exists a Lyapunov func-tion 119881(119905 119883(119905)) isin 11986212(R+ times R119899R+) satisfying Assumption 7such that for all (119905 119883) isin R+ timesR119899

(i) 119883(119905)119901 le 119881(119905 119883(119905)) le 119883(119905)119902(ii) 119871119881(119905 119883(119905)) le minus120572(119905)119883(119905)119903 + 120573(119905)(iii) 119881(119905 119883(119905)) minus 119881119903119902(119905 119883(119905)) le 120574

where 120572 120573 isin 119862(R+R+) 119901 119902 and 119903 are positive constants119901 ge 1 and 120574 is a nonnegative constant Then all solutions ofthe stochastic differential equation (11) satisfy

1198641198830 1003817100381710038171003817119883 (119905 1198830)1003817100381710038171003817 le 119881 (1199050 1198830) 119890minusint1199051199050 120572(119904)119889119904+ int1199051199050(120574120572 (119906) + 120573 (119906)) 119890minusint119905119906 120572(119904)1198891199041198891199061119901

(22)

for all 119905 ge 1199050Lemma 10 (see [28 33]) Assume there exists a Lyapunovfunction 119881(119905 119883(119905)) isin 11986212(R+ times R119899R+) satisfying Assump-tion 7 such that for all (119905 119883) isin R+ timesR119899

(i) 119883(119905)119901 le 119881(119905 119883(119905))(ii) 119871119881(119905 119883(119905)) le minus120572(119905)119881119902(119905 119883(119905)) + 120573(119905)(iii) 119881(119905 119883(119905)) minus 119881119902(119905 119883(119905)) le 120574

4 International Journal of Analysis

where 120572 120573 isin 119862(R+R+) 119901 119902 are positive constants 119901 ge 1 and120574 is a nonnegative constant Then all solutions of the stochasticdifferential equation (11) satisfy (22) for all 119905 ge 1199050Corollary 11 (see [28 33]) (i) Assume that hypotheses (i) to(iii) of Lemma 9 hold In addition

int1199051199050(120574120572 (119906) + 120573 (119906)) 119890minusint119905119906 120572(119904)119889119904119889119906 le 119872 forall119905 ge 1199050 ge 0 (23)

for some positive constant 119872 then all solutions of stochasticdifferential equation (11) are uniformly stochastically bounded

(ii) Assume that hypotheses (i) to (iii) of Lemma 10 holdIf condition (23) is satisfied then all solutions of the stochasticdifferential equation (11) are stochastically bounded

3 Main Results

Let (119909(119905) 119910(119905)) be any solution of the stochastic differentialequation (9) the main tool employed in the proofs ofour results is the continuously differentiable function 119881 =119881(119905 119909(119905) 119910(119905)) defined as

2119881 = 11988721199092 + 1198871199102 + 2119909119891 (119909) + (119886119909 + 119910)2 (24)

where 119886 and 119887 are positive constants and the function 119891 is asdefined in Section 1

Theorem 12 Suppose that 119886 119887 120590 and 1198720 are positiveconstants such that

(i) 119886 le 119892(119909 119910) for all 119909 and 119910(ii) 119887119909 le 119891(119909) le 119861119909 for all 119909 = 0 and 1205902 lt 2119886119887(119887 + 1)minus1(iii) |119901(119905 119909 119910)| le 1198720 for all 119905 ge 0 119909 and 119910

Then solution (119909(119905) 119910(119905)) of the stochastic differential equation(9) is uniformly stochastically bounded

Remark 13 We note the following

(i) Whenever the functions 119892(119909 1199091015840) = 119886 119891(119909) = 119887119909 and1205961015840 = 119901(119905 119909 1199091015840) = 0 then the stochastic differentialequation (8) becomes a second-order linear ordinarydifferential equation

11990910158401015840 + 1198861199091015840 + 119887119909 = 0 (25)

and conditions (i) to (iii) of Theorem 12 reduce toRouth Hurwitz criteria 119886 gt 0 and 119887 gt 0 forthe asymptotic stability of the second-order lineardifferential equation (25)

(ii) The term 120590119909(119905)1205961015840(119905) in the stochastic differentialequation (8) is an extension of the ordinary casediscussed recently by authors in [11 18 23 31 32 35ndash37 40]

We shall now state and prove a result that will be used inthe proofs of our results

Lemma 14 Under the hypotheses of Theorem 12 there existpositive constants 1198630 = 1198630(119886 119887) and 1198631 = 1198631(119886 119887 119861) suchthat

1198630 (1199092 (119905) + 1199102 (119905)) le 119881 (119905 119909 (119905) 119910 (119905))le 1198631 (1199092 (119905) + 1199102 (119905)) (26)

for all 119905 ge 0 119909 and 119910 In addition there exist positive constants1198632 = 1198632(119886 119887 120590) and 1198633 = 1198633(119886 119887) such that119871119881 (119905 119909 (119905) 119910 (119905))

le minus1198632 (1199092 (119905) + 1199102 (119905))+ 1198633 (|119909 (119905)| + 1003816100381610038161003816119910 (119905)1003816100381610038161003816) 1003816100381610038161003816119901 (119905 119909 (119905) 119910 (119905))1003816100381610038161003816

(27)

for all 119905 ge 0 119909 and 119910Proof Let (119909(119905) 119910(119905)) be any solution of the stochastic differ-ential equation (9) since 119883 = (119909 119910) isin R2 it follows from(24) that

119881 (119905 0 0) = 0 (28)

for all 119905 ge 0Moreover from (24) and the fact that 119891(119909) ge 119886119909for all 119909 = 0 there exists a positive constant 1205750 such that

119881 (119905 119883) ge 1205750 (1199092 + 1199102) (29)

for all 119905 ge 0 119909 and 119910 where1205750 fl min 1198872 + 2119887 +min 119886 1 119887 +min 119886 1 (30)

It is clear from inequality (29) that

119881 (119905 119883) = 0 lArrrArr 1199092 + 1199102 = 0119881 (119905 119883) gt 0 lArrrArr 1199092 + 1199102 = 0 (31)

119881 (119905 119883) 997888rarr +infin as 1199092 + 1199102 997888rarr infin (32)

Furthermore since 119891(119909) le 119861119909 for all 119909 = 0 it follows from(24) that there exists a positive constant 1205751 such that

119881 (119905 119883) le 1205751 (1199092 + 1199102) (33)

for all 119905 ge 0 119909 and 119910 where1205751 fl max 1198872 + 2119861 +max 119886 1 119887 +max 119886 1 (34)

From inequalities (29) and (33) we have

1205750 (1199092 + 1199102) le 119881 (119905 119883) le 1205751 (1199092 + 1199102) (35)

for all 119905 ge 0 119909 and 119910 It is not difficult to see that estimates(35) satisfy inequalities (26) of Lemma 14 with 1205750 and 1205751equivalent to1198630 and1198631 respectively

International Journal of Analysis 5

Moreover applying Itorsquos formula in (24) using system (9)we find that

119871119881 (119905 119883) = 12 [119886119891 (119909)119909 minus 121205902 (119887 + 1)] 1199092minus 12 [(119887 + 1) 119892 (119909 119910) minus 119886] 1199102 minus119882119894+ [119886119909 + (119887 + 1) 119910] 119901 (119905 119909 119910)

(119894 = 1 2)

(36)

where

1198821 fl 14 [119886119891 (119909)119909 minus 121205902 (119887 + 1)] 1199092+ 4 [119886119892 (119909 119910) minus (1198862 + 1198872)] 119909119910+ [(119887 + 1) 119892 (119909 119910) minus 119886] 1199102

1198822 fl 14 [119886119891 (119909)119909 minus 121205902 (119887 + 1)] 1199092

+ 4 [119886119891 (119909)119909 minus 1198911015840 (119909)] 119909119910+ [(119887 + 1) 119892 (119909 119910) minus 119886] 1199102

(37)

It is clear from the inequalities

4 [119886119892 (119909 119910) minus (1198862 + 1198872)]2lt [119886119891 (119909)119909 minus 121205902 (119887 + 1)] [(119887 + 1) 119892 (119909 119910) minus 119886]

4 [119886119891 (119909)119909 minus 1198911015840 (119909)]lt [119886119891 (119909)119909 minus 121205902 (119887 + 1)] [(119887 + 1) 119892 (119909 119910) minus 119886]

(38)

that

1198821 = 1198822 ge [[radic119886119891 (119909)119909 minus 121205902 (119887 + 1) |119909|

minus radic(119887 + 1) 119892 (119909 119910) minus 119886 10038161003816100381610038161199101003816100381610038161003816]]2

ge 0(39)

for all 119909 and 119910 Using inequality (39) and hypotheses (i) and(ii) ofTheorem 12 in (36) there exist positive constants 1205752 and1205753 such that

119871119881 (119905 119883) le minus1205752 (1199092 + 1199102)+ 1205753 (|119909| + 10038161003816100381610038161199101003816100381610038161003816) 1003816100381610038161003816119901 (119905 119909 119910)1003816100381610038161003816 (40)

for all 119905 ge 0 119909 and 119910 where1205752 fl 12 min 119886119887 minus 121205902 (119887 + 1) 119886119887 1205753 fl max 119886 119887 + 1 (41)

Inequality (40) satisfies inequality (27) with 1205752 and 1205753equivalent to 1198632 and 1198633 respectively This completes theproof of Lemma 14

Proof ofTheorem 12 Let (119909(119905) 119910(119905)) be any solution of system(9) From inequality (40) and assumption (iii) ofTheorem 12we have

119871119881 (119905 119883)le minus121205752 (1199092 + 1199102)minus 1212057521198720 [(|119909| minus 120575minus12 1205753)2 + (10038161003816100381610038161199101003816100381610038161003816 minus 120575minus12 1205753)2]+1198720120575minus12 12057523

(42)

for 119905 ge 0 119909 and 119910 Since 1205752 1205753 and1198720 are positives and(|119909| minus 120575minus12 1205753)2 + (10038161003816100381610038161199101003816100381610038161003816 minus 120575minus12 1205753)2 ge 0 (43)

for all 119909 and 119910 there exist positive constants 1205754 and 1205755 suchthat

119871119881 (119905 119883) le minus1205754 (1199092 + 1199102) + 1205755 (44)

for all 119905 ge 0 119909 119910 where 1205754 fl (12)1205752 and 1205755 fl 1198720120575minus12 12057523 Hence condition (ii) of Lemma 9 is satisfied with 120572(119905) fl 1205754119903 fl 2 and 120573(119905) fl 1205755 Also from inequality (35) hypotheses(i) and (iii) of Lemma 9 hold with 119901 = 119902 = 2 so that 120574 = 0

Furthermore from inequality (23) we have

int1199051199050[(120574120572 (119906) + 120573 (119906)) 119890minus1205754 int119905119906 120572(119904)119889119904] 119889119906= int11990511990501205755119890minus1205754 int119905119906 119889119904119889119906 = 120575minus14 1205755 [1 minus 119890minus1205754(119905minus1199050)]

le 120575minus14 1205755(45)

for all 119905 ge 1199050 ge 0 Inequality (45) satisfies estimate (23) with119872 fl 120575minus14 1205755 = 21198720120575minus22 12057523 gt 0 Moreover from (9) and (24)there exists a positive constant 1205756 such that

10038161003816100381610038161003816119881119909119894 (119905 119883)119866119894119896 (119905 119883)10038161003816100381610038161003816le 12120590 [(2119886 + 119887 + 1) 1199092 + (119887 + 1) 1199102]le 1205756 (1199092 + 1199102) fl 120582 (119905)

(46)

where

1205756 fl 12120590max 2119886 + 119887 + 1 119887 + 1 (47)

6 International Journal of Analysis

Also

int119879119905012057526 (1199092 (119905) + 1199102 (119905))2 119889119905 lt infin (48)

for any fixed 0 le 1199050 le 119879 lt infin Thus from inequalities (46)and (48) estimates (20) and (21) hold respectively Finallyfrom inequalities (33) and (45) we have

1198641198830 1003817100381710038171003817119883 (119905 1198830)1003817100381710038171003817 le (120575111988320 + 21198720120575minus22 12057523)12 (49)

for all 119905 ge 1199050 ge 0 where1198830 fl (11990920 +11991020) and119862 fl 1205751Thus thesolutions (119909(119905) 119910(119905)) of the stochastic differential equation (9)are uniformly stochastically bounded

Theorem 15 If assumptions of Theorem 12 hold then thesolution (119909(119905) 119910(119905)) of the stochastic differential equation (9)is stochastically bounded

Proof Suppose that (119909(119905) 119910(119905)) is any solution of the stochas-tic differential equation (9) From inequalities (33) and (44)there exists a positive constant 1205757 such that

119871119881 (119905 119883) le minus1205757119881 (119905 119883) + 1205755 (50)

for all 119905 ge 0 119909 and119910 where1205757 fl 120575minus11 1205754Hence from inequal-ities (29) and (50) hypotheses of Lemma 10 hold Moreoverfrom inequalities (45) (46) (48) and (49) assumption (ii)of Corollary 11 holds Thus by Corollary 11 all solutionsof the stochastic differential equation (9) are stochasticallybounded This completes the proof of Theorem 15

Next we shall discuss the stability of the trivial solutionof the stochastic differential equation (8) Suppose that119901(119905 119909 1199091015840) = 0 (8) specializes to

11990910158401015840 (119905) + 119892 (119909 (119905) 1199091015840 (119905)) 1199091015840 (119905) + 119891 (119909 (119905))+ 120590119909 (119905) 1205961015840 (119905) = 0 (51)

Equation (51) has the following equivalent system

1199091015840 (119905) = 119910 (119905) 1199101015840 (119905) = minus119891 (119909) minus 119892 (119909 (119905) 119910 (119905)) 119910 (119905) minus 120590119909 (119905) 1205961015840 (119905) (52)

where the functions 119891 119892 and 120596 are defined in Section 1

Theorem 16 If assumptions (i) and (ii) of Theorem 12 holdthen the trivial solution of the stochastic differential equation(52) is stochastically stable

Proof Let (119909(119905) 119910(119905)) be any solution of the stochastic differ-ential equation (52) From equation (28) and estimate (29)assumptions (i) and (ii) of Lemma 5 hold so that the function119881(119905 119883) is positive definite Furthermore using Itorsquos formulaalong the solution path of (52) we obtain

119871119881 (119905 119883) le minus1205752 (1199092 (119905) + 1199102 (119905)) le 0 (53)

for all 119905 ge 0 119909 and 119910 where 1205752 is defined in (40)Inequality (53) satisfies hypothesis (iii) of Lemma 5 henceby Lemma 5 the trivial solution of the stochastic differentialequation (52) is stochastically stableThis completes the proofof Theorem 16

Theorem 17 If assumptions (i) and (ii) of Theorem 12 holdthen the trivial solution of the stochastic differential equation(52) is not only uniformly stochastically asymptotically stablebut also uniformly stochastically asymptotically stable in thelarge

Proof Let (119909(119905) 119910(119905)) be any solution of the stochastic differ-ential equation (52) In view of (28) and estimate (29) thefunction 119881(119905 119883) is positive definite Furthermore estimate(32) and inequality (33) show that the function 119881(119905 119883) isradially unbounded and decrescent respectively It followsfrom (28) estimate (32) inequality (35) and the first inequal-ity in (53) that all assumptions of Lemma 6 hold Thus byLemma 6 the trivial solution of the stochastic differentialequation (52) is uniformly stochastically asymptotically stablein the large If estimate (32) is omitted then the trivial solutionof the stochastic differential equation (52) is uniformlystochastically asymptotically stable This completes the proofof Theorem 17

Next if the function 119901(119905 119909 1199091015840) is replaced by 119901(119905) isin119862(R+R+) we have the following special case11990910158401015840 (119905) + 119892 (119909 (119905) 1199091015840 (119905)) 1199091015840 (119905) + 119891 (119909 (119905))

+ 120590119909 (119905) 1205961015840 (119905) = 119901 (119905) (54)

of (8) Equation (54) has the following equivalent system

1199091015840 (119905) = 119910 (119905) 1199101015840 (119905) = 119901 (119905) minus 119891 (119909) minus 119892 (119909 (119905) 119910 (119905)) 119910 (119905)

minus 120590119909 (119905) 1205961015840 (119905) (55)

with the following result

Corollary 18 If assumptions (i) and (ii) of Theorem 12 holdand hypothesis (iii) is replaced by the boundedness of thefunction 119901(119905) then the solutions (119909(119905) 119910(119905)) of the stochasticdifferential equation (55) are not only stochastically boundedbut also uniformly stochastically bounded

Proof Theproof of Corollary 18 is similar to the proof ofThe-orems 12 and 15This completes the proof of Corollary 18

4 Examples

In this section we shall present two examples to illustrate theapplications of the results we obtained in the previous section

Example 1 Consider the second-order nonlinear nonau-tonomous stochastic differential equation

11990910158401015840 + (3 + 10038161003816100381610038161003816cos (1199091199091015840)10038161003816100381610038161003816) 1199091015840 + 119909 + sin119909 + 011199091205961015840 (119905)= (1 + 2119905 + 10038161003816100381610038161003816119909119909101584010038161003816100381610038161003816)minus1

(56)

International Journal of Analysis 7

Equation (56) is equivalent to system

1199091015840 = 1199101199101015840 = (1 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816)minus1 minus (119909 + sin119909)

minus [3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816] 119910 minus 011199091205961015840 (119905) (57)

Now from systems (9) and (57) we have the followingrelations

(i) The function

119892 (119909 119910) fl 3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816 (58)

Noting that 1003816100381610038161003816cos (119909119910)1003816100381610038161003816 ge 0 (59)

for all 119909 and 119910 it follows that119892 (119909 119910) = 3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816 ge 119886 = 3 (60)

for all 119909 and 119910The behaviour of the function 119892(119909 119910)is shown below in Figure 1

(ii) The function

119891 (119909) fl 119909 + sin119909 (61)

Since

minus02 le 119865 (119909) = sin119909119909 le 1 (62)

for all 119909 = 0 then we have

1 = 119887 le 119891 (119909)119909 = 1 + sin119909119909 le 119861 = 2 (63)

for all 119909 = 0 and since 120590 fl 01 it follows that1205902 lt 2119886119887(119887 + 1)minus1 implies that 0 lt 299 The function119891(119909)119909 and its bounds are shown in Figure 2(iii) The function

119901 (119905 119909 119910) fl 11 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816 (64)

Clearly

1003816100381610038161003816119901 (119905 119909 119910)1003816100381610038161003816 = 11 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816 le 1 = 1198720 (65)

for all 119905 ge 0 119909 and 119910Now from items (i) (ii) above and (24) the continuouslydifferentiable function 119881(119905 119883) used for system (57) is

2119881 (119905 119883) = 31199092 + 1199102 + (3119909 + 119910)2 (66)

Different views of the function119881(119905 119883) are shown in Figure 3From (66) it is not difficult to show that

(1199092 + 1199102) le 119881 (119905 119883) le 3 (1199092 + 1199102) (67)

Figure 1 Behaviour of the function 119892(119909 119910)

minus6120587 minus4120587 minus2120587 612058741205872120587

ge minus0225

25

2

15

1

05

minus05

minus1

F(x) f(x)x

f(x)x = 1 + sin(x)x

F(x) = sin(x)xx

b = 076

F(x)

Figure 2 Bounds on the function 119891(119909)119909

for all 119905 ge 0 119909 and 119910 From (35) and (67) we have 1205750 = 11205751 = 3 119901 = 2 and 119902 = 2 and thus inequalities (67) satisfycondition (i) of Lemma 9 Also from the first inequality in(67) we have

119881 (119905 119883) 997888rarr +infin as 1199092 + 1199102 997888rarr infin (68)

Estimate (68) verifies (32) (ie the function 119881(119905 119883) definedby (66) is radially unbounded) Next applying Itorsquos formulain (66) using system (57) we find that

119871119881 (119905 119883) = 12119909119910 + 31199102 minus 119909 (3119909 + 2119910) (1 + sin119909119909 )minus 119910 (3119909 + 2119910) (3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816) + 11001199092minus 11990910 (3119909 + 2119910)+ (3119909 + 2119910) (1 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816)minus1

(69)

Using the estimates in items (i) to (iii) of Example 1 and theinequality 211990911199092 le 11990921 + 11990922 in (69) we obtain

119871119881 (119905 119883) le minus29 (1199092 + 1199102) + 3 (|119909| + 10038161003816100381610038161199101003816100381610038161003816) (70)

for all 119905 ge 0 119909 and 119910 Inequality (70) satisfies inequality (40)where 1205752 = 29 and 1205753 = 3 Since

(|119909| minus 105)2 + (10038161003816100381610038161199101003816100381610038161003816 minus 105)2 ge 0 (71)

for all 119909 and 119910 it follows from inequality (70) that

119871119881 (119905 119883) le minus145 (1199092 + 1199102) + 32 (72)

8 International Journal of Analysis

Figure 3 The behaviour of the function 119881(119905 119883)

for all 119905 ge 0 119909 and 119910 Inequality (72) satisfies assumption(ii) of Lemma 9 and estimate (44) with 120572(119905) = 1205754 = 145 and120573(119905) = 1205755 = 32 Since 119903 = 119901 = 119902 = 2 it follows that 120574 = 0 sothat assumption (iii) of Lemma 9 holds In addition

int1199051199050[(120574120572 (119906) + 120573 (119906)) 119890minusint119905119906120572 (119904) 119889119904] 119889119906 le 16 (73)

for all 119905 ge 1199050 ge 0 Estimate (73) satisfies (23) and (45) with119872 = 26 Furthermore

119881119909119894 (119905 119883)119866119894119896 (119905 119883) = minus 110 (31199092 + 2119909119910) (74)

and10038161003816100381610038161003816119881119909119894 (119905 119883)119866119894119896 (119905 119883)10038161003816100381610038161003816 le 25 (1199092 + 1199102) (75)

for all 119905 ge 0 119909 and119910 Inequality (75) satisfies inequalities (20)and (21) with

120582 (119905) = 25 (1199092 + 1199102) (76)

Hence by Corollary 11 (i) all solutions of stochastic differen-tial equation (57) are uniformly stochastically bounded

Example 2 If 119901(119905 119909 1199091015840) = 119901(119905 119909 119910) = 0 in (56) and system(57) we have the following stochastic differential equation

11990910158401015840 + (3 + 10038161003816100381610038161003816cos (1199091199091015840)10038161003816100381610038161003816) 1199091015840 + 119909 + sin119909 + 011199091205961015840 (119905)= 0 (77)

Equation (77) is equivalent to system

1199091015840 = 1199101199101015840 = minus (119909 + sin119909) minus [3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816] 119910 minus 011199091205961015840 (119905) (78)

Now from systems (52) and (78) items (i) and (ii) of Example 1hold Also equations (66) (67) and estimate (68) hold thatis

2119881 (119905 119883) = 31199092 + 1199102 + (3119909 + 119910)2 119881 (119905 0) = 0 forall119905 ge 0

(1199092 + 1199102) le 119881 (119905 119883) le 3 (1199092 + 1199102) forall119905 ge 0 119909 119910119881 (119905 119883) 997888rarr +infin as 1199092 + 1199102 997888rarr infin

(79)

112

1416

182

0500

10001500

0

02

04

06

08

1

t

x(t) y(t)

Figure 4 Graph of solutions of (56) in 3D

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(b)

Figure 5

Furthermore application of Itorsquos formula in (66) and usingsystem (78) yield

119871119881 (119905 119883) le minus29 (1199092 + 1199102) (80)

for all 119905 ge 0 119909 119910 and thus

119871119881 (119905 119883) le 0 (81)

International Journal of Analysis 9

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

000200400600801012

t

minus004

minus002

x(t)y(t)

x(t)

y(t)

times1011

(b)

Figure 6

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

000200400600801012

t

minus002

minus008

minus006

minus004

times1011

x(t)y(t)

x(t)

y(t)

(b)

Figure 7

for all 119905 ge 0 119909 and 119910 Moreover from (79) and (80)all assumptions of Theorem 17 and Lemma 6 are satisfiedThus by Lemma 6 the trivial solution of system (78) is notonly uniformly stochastically asymptotically stable but alsouniformly stochastically asymptotically stable in the largeFinally from (79) and (81) the function 119881(119905 119883) is positivedefinite and

119871119881 (119905 119883) le 0 forall (119905 119883) isin R+ timesR

2 (82)

Hence assumptions of Theorem 17 and Lemma 5 hold byTheorem 17 and Lemma 5 the trivial solution of system (78)is stochastically stable

Simulation of Solutions In what follows we shall nowsimulate the solutions of (56) (resp system (57)) and (78)(resp system (79)) Our approach depends on the Euler-Maruyama method which enables us to get approximatenumerical solution for the considered systems It will be seenfrom our figures that the simulated solutions are bounded

which justifies our given results For instance when 120590 = 01the numerical solutions of (56) in three-dimensional spaceare shown in Figure 4 If we vary the value of the noise inthe numerical solution (119909(119905) 119910(119905)) of system (57) as 120590 = 01and 120590 = 10 we have Figures 5(a) and 5(b) respectively Itcan be seen that when the noise is increased the stochasticitybecomes more pronounced The behaviour of the numericalsolution (119909(119905) 119910(119905)) of system (57) when 120590 = 05 and 120590 = 20is shown in Figures 6(a) and 6(b) respectivelyThe behaviourof the numerical solution (119909(119905) 119910(119905)) of system (57) for 120590 = 0and 120590 = 50 is shown in Figures 7(a) and 7(b) respectivelyFor the case of (78) Figure 8 shows the closeness of thesolution (119909(119905)) and the perturbed solution (119909120598(119905)) for a verylarge 119905 which implies asymptotic stability in the large for theconsidered SDE

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

10 International Journal of Analysis

0 1 2 3 4 5 6 7 8 9 100

0005

001

0015

002

0025

t times1011

x(t)x120576(t)

x(t)x120576(t)

Figure 8 Graph of solutions of (78)

References

[1] L Arnold Stochastic Differential Equations Theory and Appli-cations John Wiley amp Sons 1974

[2] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations vol 178 of Mathematics inScience and Engineering Academic Press Inc Orlando FlaUSA 1985

[3] T A Burton Volterra Integral and Differential EquationsAcademic Press New York NY USA 1983

[4] J K HaleTheory of Functional Differential Equations SpringerNew York NY USA 1977

[5] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer 2000

[6] L Shaikihet Lyapunov Functionals and Stability of StochasticFunctional Differential Equations Springer International 2013

[7] T Yoshizawa StabilityTheory and Existence of Periodic Solutionsand almost Periodic Solutions Spriger New York NY USA1975

[8] T Yoshizawa StabilityTheory by Liapunovrsquos SecondMethodTheMathematical Society of Japan 1966

[9] A M A Abou-El-Ela A I Sadek and A M Mahmoud ldquoOnthe stability of solutions for certain second-order stochasticdelay differential equationsrdquo Differential Equations and ControlProcesses no 2 pp 1ndash13 2015

[10] A M Abou-El-Ela A I Sadek A M Mahmoud and R OTaie ldquoOn the stochastic stability and boundedness of solutionsfor stochastic delay differential equation of the second orderrdquoChinese Journal of Mathematics vol 2015 Article ID 358936 8pages 2015

[11] A T Ademola Boundedness and Stability of Solutions to CertainSecond Order Differential Equations Differential Equations andControl Processes 2015

[12] A T Ademola B S Ogundare M O Ogundiran and O AAdesina ldquoPeriodicity stability and boundedness of solutions tocertain second order delay differential equationsrdquo InternationalJournal of Differential Equations vol 2016 Article ID 284370910 pages 2016

[13] J G Alaba and B S Ogundare ldquoOn stability and bound-edness properties of solutions of certain second order non-autonomous nonlinear ordinary differential equationrdquo Kragu-jevac Journal of Mathematics vol 39 no 2 pp 255ndash266 2015

[14] T A Burton and L Hatvani ldquoAsymptotic stability of secondorder ordinary functional and partial differential equationsrdquoJournal of Mathematical Analysis and Applications vol 176 no1 pp 261ndash281 1993

[15] B Cahlon andD Schmidt ldquoStability criteria for certain second-order delay differential equations with mixed coefficientsrdquoJournal of Computational and AppliedMathematics vol 170 no1 pp 79ndash102 2004

[16] T Caraballo M A Diop and A S Ndoye ldquoFixed points andexponential stability for stochastic partial integro-differentialequations with delaysrdquo Advances in Dynamical Systems andApplications vol 9 no 2 pp 133ndash147 2014

[17] A Domoshnitsky ldquoNonoscillation maximum principles andexponential stability of secondorder delay differential equationswithout damping termrdquo Domoshnitsky Journal of Inequalitiesand Applications vol 2014 article 361 2014

[18] I I Gikhman and A V Skorokhod Stochastische Differential-gleichungen Akademie Berlin Germany 1971 (Russian)

[19] I I Gikhman On the Stability of the Solutions of StochasticDifferential Equations Predelrsquonyye Teoremy i StatisticheskiyeVyvody Tashkent Uzbekistan 1966

[20] G A Grigoryan ldquoBoundedness and stability criteria for linearordinary differential equations of the second orderrdquo RussianMathematics vol 57 no 12 pp 8ndash15 2013

[21] A F Ivanov Y I Kazmerchuk and A V Swishchuk ldquoThe-ory stochastic stability and applications of stochastic delaydifferential equations a survey of recent resultsrdquo in DifferentialEquations and Dynamical Systems vol 11 no 1 2003

[22] F Jedrzejewski and D Brochard ldquoLyapounv exponents andstability stochastic dynamical structuresrdquo 2000

[23] Z Jin and L Zengrong ldquoOn the global asymptotic behaviorof solutions to a non autonomous generalized Lienard systemrdquoJournal of Mathematical Research and Exposition vol 21 no 3pp 410ndash414 2001

[24] E Kolarova ldquoAn application of stochastic integral equations toelectrical networksrdquo Acta Electrotechnica et Informatica vol 8no 3 pp 14ndash17 2008

[25] V B Kolmanovskii and L E Shaikhet ldquoA method for con-structing Lyapunov functionals for stochastic systems with aftereffectrdquo Differentsialrsquonye Uravneniya vol 29 no 11 pp 1909ndash2022 1993

[26] V Kolmanovskii and L Shaikhet ldquoConstruction of Lyapunovfunctionals for stochastic hereditary systems a survey of somerecent resultsrdquo Mathematical and Computer Modelling vol 36no 6 pp 691ndash716 2002

[27] A J Kroopnick ldquoBounded solutions to 11990910158401015840 + 119902(119905)119887(119909) = 119891(119905)rdquoInternational Journal of Mathematical Education in Science andTechnology vol 41 no 6 pp 829ndash836 2010

[28] R Liu and Y Raffoul ldquoBoundedness and exponential stabilityof highly nonlinear stochastic differential equationsrdquo ElectronicJournal of Differential Equations vol 2009 no 143 pp 1ndash102009

[29] X Mao ldquoSome contributions to stochastic asymptotic stabilityand boundedness via multiple Lyapunov functionsrdquo Journal ofMathematical Analysis and Applications vol 260 no 2 pp 325ndash340 2001

[30] B S Ogundare A T Ademola M O Ogundiran and O AAdesina ldquoOn the qualitative behaviour of solutions to certainsecond order nonlinear differential equation with delayrdquoAnnalidellrsquoUniversitarsquo di Ferrara 2016

International Journal of Analysis 11

[31] B S Ogundare and A U Afuwape ldquoBoundedness and stabilityproperties of solutions of generalized Lienard equationrdquo KochiJournal of Mathematics vol 9 pp 97ndash108 2014

[32] B S Ogundare and G E Okecha ldquoBoundedness periodicityand stability of solutions to x(t) + g(x) + b(t)h(x) = p(t x x)rdquoMathematical Sciences Research Journal vol 11 no 5 pp 432ndash443 2007

[33] Y N Raffoul ldquoBoundedness and exponential asymptotic sta-bility in dynamical systems with applications to nonlineardifferential equations with unbounded termsrdquo Advances inDynamical Systems and Applications vol 2 no 1 pp 107ndash1212007

[34] R Rezaeyan and R Farnoosh ldquoStochastic differential equationsand application of the Kalman-Bucy filter in the modeling ofRC circuitrdquo Applied Mathematical Sciences vol 4 no 21-24 pp1119ndash1127 2010

[35] C Tunc ldquoA note on the stability and boundedness of non-autonomous differential equations of second order with avariable deviating argumentrdquo Afrika Matematika vol 25 no 2pp 417ndash425 2014

[36] C Tunc ldquoA note on the bounded solutions to 11990910158401015840 + 119888(119905 119909 1199091015840 ) +119902(119905)119887(119909) = 119891(119905)rdquo Applied Mathematics amp Information Sciencesvol 8 no 1 pp 393ndash399 2014

[37] C Tunc ldquoBoundedness analysis for certain two-dimensionaldifferential systems via a Lyapunov approachrdquo Bulletin Mathe-matique de la Societe des Sciences Mathematiques de Roumanievol 53 no 1 pp 61ndash68 2010

[38] 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

[39] C Tunc ldquoStability and boundedness in multi delay vectorLienard equationrdquo Filomat vol 27 no 3 pp 435ndash445 2013

[40] C Tunc ldquoStability and boundedness of solutions of non-autonomous differential equations of second orderrdquo Journalof Computational Analysis and Applications vol 13 no 6 pp1067ndash1074 2011

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

[42] 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

[43] C Tunc and T Ayhan ldquoGlobal existence and boundedness ofsolutions of a certain nonlinear integro-differential equationof second order with multiple deviating argumentsrdquo Journal ofInequalities and Applications vol 2016 article no 46 2016

[44] F Wang and H Zhu ldquoExistence uniqueness and stabilityof periodic solutions of a duffing equation under periodicand anti-periodic eigenvalues conditionsrdquo Taiwanese Journal ofMathematics vol 19 no 5 pp 1457ndash1468 2015

[45] Z Xianfeng and J Wei ldquoStability and boundedness of aretarded Lienard-type equationrdquo Chinese Quarterly Journal ofMathematics vol 18 no 1 pp 7ndash12 2003

[46] A F Yenicerioglu ldquoThe behavior of solutions of second orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 332 no 2 pp 1278ndash1290 2007

[47] A F Yenicerioglu ldquoStability properties of second order delayintegro-differential equationsrdquo Computers and Mathematicswith Applications vol 56 no 12 pp 3109ndash3117 2008

[48] T Yoshizawa ldquoLiapunovrsquos function and boundedness of solu-tionsrdquo Funkcialaj Ekvacioj vol 2 pp 71ndash103 1958

[49] W Zhu J Huang X Ruan and Z Zhao ldquoExponential stabilityof stochastic differential equation with mixed delayrdquo Journal ofAppliedMathematics vol 2014 Article ID 187037 11 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Journal of Analysis

where 120572 120573 isin 119862(R+R+) 119901 119902 are positive constants 119901 ge 1 and120574 is a nonnegative constant Then all solutions of the stochasticdifferential equation (11) satisfy (22) for all 119905 ge 1199050Corollary 11 (see [28 33]) (i) Assume that hypotheses (i) to(iii) of Lemma 9 hold In addition

int1199051199050(120574120572 (119906) + 120573 (119906)) 119890minusint119905119906 120572(119904)119889119904119889119906 le 119872 forall119905 ge 1199050 ge 0 (23)

for some positive constant 119872 then all solutions of stochasticdifferential equation (11) are uniformly stochastically bounded

(ii) Assume that hypotheses (i) to (iii) of Lemma 10 holdIf condition (23) is satisfied then all solutions of the stochasticdifferential equation (11) are stochastically bounded

3 Main Results

Let (119909(119905) 119910(119905)) be any solution of the stochastic differentialequation (9) the main tool employed in the proofs ofour results is the continuously differentiable function 119881 =119881(119905 119909(119905) 119910(119905)) defined as

2119881 = 11988721199092 + 1198871199102 + 2119909119891 (119909) + (119886119909 + 119910)2 (24)

where 119886 and 119887 are positive constants and the function 119891 is asdefined in Section 1

Theorem 12 Suppose that 119886 119887 120590 and 1198720 are positiveconstants such that

(i) 119886 le 119892(119909 119910) for all 119909 and 119910(ii) 119887119909 le 119891(119909) le 119861119909 for all 119909 = 0 and 1205902 lt 2119886119887(119887 + 1)minus1(iii) |119901(119905 119909 119910)| le 1198720 for all 119905 ge 0 119909 and 119910

Then solution (119909(119905) 119910(119905)) of the stochastic differential equation(9) is uniformly stochastically bounded

Remark 13 We note the following

(i) Whenever the functions 119892(119909 1199091015840) = 119886 119891(119909) = 119887119909 and1205961015840 = 119901(119905 119909 1199091015840) = 0 then the stochastic differentialequation (8) becomes a second-order linear ordinarydifferential equation

11990910158401015840 + 1198861199091015840 + 119887119909 = 0 (25)

and conditions (i) to (iii) of Theorem 12 reduce toRouth Hurwitz criteria 119886 gt 0 and 119887 gt 0 forthe asymptotic stability of the second-order lineardifferential equation (25)

(ii) The term 120590119909(119905)1205961015840(119905) in the stochastic differentialequation (8) is an extension of the ordinary casediscussed recently by authors in [11 18 23 31 32 35ndash37 40]

We shall now state and prove a result that will be used inthe proofs of our results

Lemma 14 Under the hypotheses of Theorem 12 there existpositive constants 1198630 = 1198630(119886 119887) and 1198631 = 1198631(119886 119887 119861) suchthat

1198630 (1199092 (119905) + 1199102 (119905)) le 119881 (119905 119909 (119905) 119910 (119905))le 1198631 (1199092 (119905) + 1199102 (119905)) (26)

for all 119905 ge 0 119909 and 119910 In addition there exist positive constants1198632 = 1198632(119886 119887 120590) and 1198633 = 1198633(119886 119887) such that119871119881 (119905 119909 (119905) 119910 (119905))

le minus1198632 (1199092 (119905) + 1199102 (119905))+ 1198633 (|119909 (119905)| + 1003816100381610038161003816119910 (119905)1003816100381610038161003816) 1003816100381610038161003816119901 (119905 119909 (119905) 119910 (119905))1003816100381610038161003816

(27)

for all 119905 ge 0 119909 and 119910Proof Let (119909(119905) 119910(119905)) be any solution of the stochastic differ-ential equation (9) since 119883 = (119909 119910) isin R2 it follows from(24) that

119881 (119905 0 0) = 0 (28)

for all 119905 ge 0Moreover from (24) and the fact that 119891(119909) ge 119886119909for all 119909 = 0 there exists a positive constant 1205750 such that

119881 (119905 119883) ge 1205750 (1199092 + 1199102) (29)

for all 119905 ge 0 119909 and 119910 where1205750 fl min 1198872 + 2119887 +min 119886 1 119887 +min 119886 1 (30)

It is clear from inequality (29) that

119881 (119905 119883) = 0 lArrrArr 1199092 + 1199102 = 0119881 (119905 119883) gt 0 lArrrArr 1199092 + 1199102 = 0 (31)

119881 (119905 119883) 997888rarr +infin as 1199092 + 1199102 997888rarr infin (32)

Furthermore since 119891(119909) le 119861119909 for all 119909 = 0 it follows from(24) that there exists a positive constant 1205751 such that

119881 (119905 119883) le 1205751 (1199092 + 1199102) (33)

for all 119905 ge 0 119909 and 119910 where1205751 fl max 1198872 + 2119861 +max 119886 1 119887 +max 119886 1 (34)

From inequalities (29) and (33) we have

1205750 (1199092 + 1199102) le 119881 (119905 119883) le 1205751 (1199092 + 1199102) (35)

for all 119905 ge 0 119909 and 119910 It is not difficult to see that estimates(35) satisfy inequalities (26) of Lemma 14 with 1205750 and 1205751equivalent to1198630 and1198631 respectively

International Journal of Analysis 5

Moreover applying Itorsquos formula in (24) using system (9)we find that

119871119881 (119905 119883) = 12 [119886119891 (119909)119909 minus 121205902 (119887 + 1)] 1199092minus 12 [(119887 + 1) 119892 (119909 119910) minus 119886] 1199102 minus119882119894+ [119886119909 + (119887 + 1) 119910] 119901 (119905 119909 119910)

(119894 = 1 2)

(36)

where

1198821 fl 14 [119886119891 (119909)119909 minus 121205902 (119887 + 1)] 1199092+ 4 [119886119892 (119909 119910) minus (1198862 + 1198872)] 119909119910+ [(119887 + 1) 119892 (119909 119910) minus 119886] 1199102

1198822 fl 14 [119886119891 (119909)119909 minus 121205902 (119887 + 1)] 1199092

+ 4 [119886119891 (119909)119909 minus 1198911015840 (119909)] 119909119910+ [(119887 + 1) 119892 (119909 119910) minus 119886] 1199102

(37)

It is clear from the inequalities

4 [119886119892 (119909 119910) minus (1198862 + 1198872)]2lt [119886119891 (119909)119909 minus 121205902 (119887 + 1)] [(119887 + 1) 119892 (119909 119910) minus 119886]

4 [119886119891 (119909)119909 minus 1198911015840 (119909)]lt [119886119891 (119909)119909 minus 121205902 (119887 + 1)] [(119887 + 1) 119892 (119909 119910) minus 119886]

(38)

that

1198821 = 1198822 ge [[radic119886119891 (119909)119909 minus 121205902 (119887 + 1) |119909|

minus radic(119887 + 1) 119892 (119909 119910) minus 119886 10038161003816100381610038161199101003816100381610038161003816]]2

ge 0(39)

for all 119909 and 119910 Using inequality (39) and hypotheses (i) and(ii) ofTheorem 12 in (36) there exist positive constants 1205752 and1205753 such that

119871119881 (119905 119883) le minus1205752 (1199092 + 1199102)+ 1205753 (|119909| + 10038161003816100381610038161199101003816100381610038161003816) 1003816100381610038161003816119901 (119905 119909 119910)1003816100381610038161003816 (40)

for all 119905 ge 0 119909 and 119910 where1205752 fl 12 min 119886119887 minus 121205902 (119887 + 1) 119886119887 1205753 fl max 119886 119887 + 1 (41)

Inequality (40) satisfies inequality (27) with 1205752 and 1205753equivalent to 1198632 and 1198633 respectively This completes theproof of Lemma 14

Proof ofTheorem 12 Let (119909(119905) 119910(119905)) be any solution of system(9) From inequality (40) and assumption (iii) ofTheorem 12we have

119871119881 (119905 119883)le minus121205752 (1199092 + 1199102)minus 1212057521198720 [(|119909| minus 120575minus12 1205753)2 + (10038161003816100381610038161199101003816100381610038161003816 minus 120575minus12 1205753)2]+1198720120575minus12 12057523

(42)

for 119905 ge 0 119909 and 119910 Since 1205752 1205753 and1198720 are positives and(|119909| minus 120575minus12 1205753)2 + (10038161003816100381610038161199101003816100381610038161003816 minus 120575minus12 1205753)2 ge 0 (43)

for all 119909 and 119910 there exist positive constants 1205754 and 1205755 suchthat

119871119881 (119905 119883) le minus1205754 (1199092 + 1199102) + 1205755 (44)

for all 119905 ge 0 119909 119910 where 1205754 fl (12)1205752 and 1205755 fl 1198720120575minus12 12057523 Hence condition (ii) of Lemma 9 is satisfied with 120572(119905) fl 1205754119903 fl 2 and 120573(119905) fl 1205755 Also from inequality (35) hypotheses(i) and (iii) of Lemma 9 hold with 119901 = 119902 = 2 so that 120574 = 0

Furthermore from inequality (23) we have

int1199051199050[(120574120572 (119906) + 120573 (119906)) 119890minus1205754 int119905119906 120572(119904)119889119904] 119889119906= int11990511990501205755119890minus1205754 int119905119906 119889119904119889119906 = 120575minus14 1205755 [1 minus 119890minus1205754(119905minus1199050)]

le 120575minus14 1205755(45)

for all 119905 ge 1199050 ge 0 Inequality (45) satisfies estimate (23) with119872 fl 120575minus14 1205755 = 21198720120575minus22 12057523 gt 0 Moreover from (9) and (24)there exists a positive constant 1205756 such that

10038161003816100381610038161003816119881119909119894 (119905 119883)119866119894119896 (119905 119883)10038161003816100381610038161003816le 12120590 [(2119886 + 119887 + 1) 1199092 + (119887 + 1) 1199102]le 1205756 (1199092 + 1199102) fl 120582 (119905)

(46)

where

1205756 fl 12120590max 2119886 + 119887 + 1 119887 + 1 (47)

6 International Journal of Analysis

Also

int119879119905012057526 (1199092 (119905) + 1199102 (119905))2 119889119905 lt infin (48)

for any fixed 0 le 1199050 le 119879 lt infin Thus from inequalities (46)and (48) estimates (20) and (21) hold respectively Finallyfrom inequalities (33) and (45) we have

1198641198830 1003817100381710038171003817119883 (119905 1198830)1003817100381710038171003817 le (120575111988320 + 21198720120575minus22 12057523)12 (49)

for all 119905 ge 1199050 ge 0 where1198830 fl (11990920 +11991020) and119862 fl 1205751Thus thesolutions (119909(119905) 119910(119905)) of the stochastic differential equation (9)are uniformly stochastically bounded

Theorem 15 If assumptions of Theorem 12 hold then thesolution (119909(119905) 119910(119905)) of the stochastic differential equation (9)is stochastically bounded

Proof Suppose that (119909(119905) 119910(119905)) is any solution of the stochas-tic differential equation (9) From inequalities (33) and (44)there exists a positive constant 1205757 such that

119871119881 (119905 119883) le minus1205757119881 (119905 119883) + 1205755 (50)

for all 119905 ge 0 119909 and119910 where1205757 fl 120575minus11 1205754Hence from inequal-ities (29) and (50) hypotheses of Lemma 10 hold Moreoverfrom inequalities (45) (46) (48) and (49) assumption (ii)of Corollary 11 holds Thus by Corollary 11 all solutionsof the stochastic differential equation (9) are stochasticallybounded This completes the proof of Theorem 15

Next we shall discuss the stability of the trivial solutionof the stochastic differential equation (8) Suppose that119901(119905 119909 1199091015840) = 0 (8) specializes to

11990910158401015840 (119905) + 119892 (119909 (119905) 1199091015840 (119905)) 1199091015840 (119905) + 119891 (119909 (119905))+ 120590119909 (119905) 1205961015840 (119905) = 0 (51)

Equation (51) has the following equivalent system

1199091015840 (119905) = 119910 (119905) 1199101015840 (119905) = minus119891 (119909) minus 119892 (119909 (119905) 119910 (119905)) 119910 (119905) minus 120590119909 (119905) 1205961015840 (119905) (52)

where the functions 119891 119892 and 120596 are defined in Section 1

Theorem 16 If assumptions (i) and (ii) of Theorem 12 holdthen the trivial solution of the stochastic differential equation(52) is stochastically stable

Proof Let (119909(119905) 119910(119905)) be any solution of the stochastic differ-ential equation (52) From equation (28) and estimate (29)assumptions (i) and (ii) of Lemma 5 hold so that the function119881(119905 119883) is positive definite Furthermore using Itorsquos formulaalong the solution path of (52) we obtain

119871119881 (119905 119883) le minus1205752 (1199092 (119905) + 1199102 (119905)) le 0 (53)

for all 119905 ge 0 119909 and 119910 where 1205752 is defined in (40)Inequality (53) satisfies hypothesis (iii) of Lemma 5 henceby Lemma 5 the trivial solution of the stochastic differentialequation (52) is stochastically stableThis completes the proofof Theorem 16

Theorem 17 If assumptions (i) and (ii) of Theorem 12 holdthen the trivial solution of the stochastic differential equation(52) is not only uniformly stochastically asymptotically stablebut also uniformly stochastically asymptotically stable in thelarge

Proof Let (119909(119905) 119910(119905)) be any solution of the stochastic differ-ential equation (52) In view of (28) and estimate (29) thefunction 119881(119905 119883) is positive definite Furthermore estimate(32) and inequality (33) show that the function 119881(119905 119883) isradially unbounded and decrescent respectively It followsfrom (28) estimate (32) inequality (35) and the first inequal-ity in (53) that all assumptions of Lemma 6 hold Thus byLemma 6 the trivial solution of the stochastic differentialequation (52) is uniformly stochastically asymptotically stablein the large If estimate (32) is omitted then the trivial solutionof the stochastic differential equation (52) is uniformlystochastically asymptotically stable This completes the proofof Theorem 17

Next if the function 119901(119905 119909 1199091015840) is replaced by 119901(119905) isin119862(R+R+) we have the following special case11990910158401015840 (119905) + 119892 (119909 (119905) 1199091015840 (119905)) 1199091015840 (119905) + 119891 (119909 (119905))

+ 120590119909 (119905) 1205961015840 (119905) = 119901 (119905) (54)

of (8) Equation (54) has the following equivalent system

1199091015840 (119905) = 119910 (119905) 1199101015840 (119905) = 119901 (119905) minus 119891 (119909) minus 119892 (119909 (119905) 119910 (119905)) 119910 (119905)

minus 120590119909 (119905) 1205961015840 (119905) (55)

with the following result

Corollary 18 If assumptions (i) and (ii) of Theorem 12 holdand hypothesis (iii) is replaced by the boundedness of thefunction 119901(119905) then the solutions (119909(119905) 119910(119905)) of the stochasticdifferential equation (55) are not only stochastically boundedbut also uniformly stochastically bounded

Proof Theproof of Corollary 18 is similar to the proof ofThe-orems 12 and 15This completes the proof of Corollary 18

4 Examples

In this section we shall present two examples to illustrate theapplications of the results we obtained in the previous section

Example 1 Consider the second-order nonlinear nonau-tonomous stochastic differential equation

11990910158401015840 + (3 + 10038161003816100381610038161003816cos (1199091199091015840)10038161003816100381610038161003816) 1199091015840 + 119909 + sin119909 + 011199091205961015840 (119905)= (1 + 2119905 + 10038161003816100381610038161003816119909119909101584010038161003816100381610038161003816)minus1

(56)

International Journal of Analysis 7

Equation (56) is equivalent to system

1199091015840 = 1199101199101015840 = (1 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816)minus1 minus (119909 + sin119909)

minus [3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816] 119910 minus 011199091205961015840 (119905) (57)

Now from systems (9) and (57) we have the followingrelations

(i) The function

119892 (119909 119910) fl 3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816 (58)

Noting that 1003816100381610038161003816cos (119909119910)1003816100381610038161003816 ge 0 (59)

for all 119909 and 119910 it follows that119892 (119909 119910) = 3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816 ge 119886 = 3 (60)

for all 119909 and 119910The behaviour of the function 119892(119909 119910)is shown below in Figure 1

(ii) The function

119891 (119909) fl 119909 + sin119909 (61)

Since

minus02 le 119865 (119909) = sin119909119909 le 1 (62)

for all 119909 = 0 then we have

1 = 119887 le 119891 (119909)119909 = 1 + sin119909119909 le 119861 = 2 (63)

for all 119909 = 0 and since 120590 fl 01 it follows that1205902 lt 2119886119887(119887 + 1)minus1 implies that 0 lt 299 The function119891(119909)119909 and its bounds are shown in Figure 2(iii) The function

119901 (119905 119909 119910) fl 11 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816 (64)

Clearly

1003816100381610038161003816119901 (119905 119909 119910)1003816100381610038161003816 = 11 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816 le 1 = 1198720 (65)

for all 119905 ge 0 119909 and 119910Now from items (i) (ii) above and (24) the continuouslydifferentiable function 119881(119905 119883) used for system (57) is

2119881 (119905 119883) = 31199092 + 1199102 + (3119909 + 119910)2 (66)

Different views of the function119881(119905 119883) are shown in Figure 3From (66) it is not difficult to show that

(1199092 + 1199102) le 119881 (119905 119883) le 3 (1199092 + 1199102) (67)

Figure 1 Behaviour of the function 119892(119909 119910)

minus6120587 minus4120587 minus2120587 612058741205872120587

ge minus0225

25

2

15

1

05

minus05

minus1

F(x) f(x)x

f(x)x = 1 + sin(x)x

F(x) = sin(x)xx

b = 076

F(x)

Figure 2 Bounds on the function 119891(119909)119909

for all 119905 ge 0 119909 and 119910 From (35) and (67) we have 1205750 = 11205751 = 3 119901 = 2 and 119902 = 2 and thus inequalities (67) satisfycondition (i) of Lemma 9 Also from the first inequality in(67) we have

119881 (119905 119883) 997888rarr +infin as 1199092 + 1199102 997888rarr infin (68)

Estimate (68) verifies (32) (ie the function 119881(119905 119883) definedby (66) is radially unbounded) Next applying Itorsquos formulain (66) using system (57) we find that

119871119881 (119905 119883) = 12119909119910 + 31199102 minus 119909 (3119909 + 2119910) (1 + sin119909119909 )minus 119910 (3119909 + 2119910) (3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816) + 11001199092minus 11990910 (3119909 + 2119910)+ (3119909 + 2119910) (1 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816)minus1

(69)

Using the estimates in items (i) to (iii) of Example 1 and theinequality 211990911199092 le 11990921 + 11990922 in (69) we obtain

119871119881 (119905 119883) le minus29 (1199092 + 1199102) + 3 (|119909| + 10038161003816100381610038161199101003816100381610038161003816) (70)

for all 119905 ge 0 119909 and 119910 Inequality (70) satisfies inequality (40)where 1205752 = 29 and 1205753 = 3 Since

(|119909| minus 105)2 + (10038161003816100381610038161199101003816100381610038161003816 minus 105)2 ge 0 (71)

for all 119909 and 119910 it follows from inequality (70) that

119871119881 (119905 119883) le minus145 (1199092 + 1199102) + 32 (72)

8 International Journal of Analysis

Figure 3 The behaviour of the function 119881(119905 119883)

for all 119905 ge 0 119909 and 119910 Inequality (72) satisfies assumption(ii) of Lemma 9 and estimate (44) with 120572(119905) = 1205754 = 145 and120573(119905) = 1205755 = 32 Since 119903 = 119901 = 119902 = 2 it follows that 120574 = 0 sothat assumption (iii) of Lemma 9 holds In addition

int1199051199050[(120574120572 (119906) + 120573 (119906)) 119890minusint119905119906120572 (119904) 119889119904] 119889119906 le 16 (73)

for all 119905 ge 1199050 ge 0 Estimate (73) satisfies (23) and (45) with119872 = 26 Furthermore

119881119909119894 (119905 119883)119866119894119896 (119905 119883) = minus 110 (31199092 + 2119909119910) (74)

and10038161003816100381610038161003816119881119909119894 (119905 119883)119866119894119896 (119905 119883)10038161003816100381610038161003816 le 25 (1199092 + 1199102) (75)

for all 119905 ge 0 119909 and119910 Inequality (75) satisfies inequalities (20)and (21) with

120582 (119905) = 25 (1199092 + 1199102) (76)

Hence by Corollary 11 (i) all solutions of stochastic differen-tial equation (57) are uniformly stochastically bounded

Example 2 If 119901(119905 119909 1199091015840) = 119901(119905 119909 119910) = 0 in (56) and system(57) we have the following stochastic differential equation

11990910158401015840 + (3 + 10038161003816100381610038161003816cos (1199091199091015840)10038161003816100381610038161003816) 1199091015840 + 119909 + sin119909 + 011199091205961015840 (119905)= 0 (77)

Equation (77) is equivalent to system

1199091015840 = 1199101199101015840 = minus (119909 + sin119909) minus [3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816] 119910 minus 011199091205961015840 (119905) (78)

Now from systems (52) and (78) items (i) and (ii) of Example 1hold Also equations (66) (67) and estimate (68) hold thatis

2119881 (119905 119883) = 31199092 + 1199102 + (3119909 + 119910)2 119881 (119905 0) = 0 forall119905 ge 0

(1199092 + 1199102) le 119881 (119905 119883) le 3 (1199092 + 1199102) forall119905 ge 0 119909 119910119881 (119905 119883) 997888rarr +infin as 1199092 + 1199102 997888rarr infin

(79)

112

1416

182

0500

10001500

0

02

04

06

08

1

t

x(t) y(t)

Figure 4 Graph of solutions of (56) in 3D

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(b)

Figure 5

Furthermore application of Itorsquos formula in (66) and usingsystem (78) yield

119871119881 (119905 119883) le minus29 (1199092 + 1199102) (80)

for all 119905 ge 0 119909 119910 and thus

119871119881 (119905 119883) le 0 (81)

International Journal of Analysis 9

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

000200400600801012

t

minus004

minus002

x(t)y(t)

x(t)

y(t)

times1011

(b)

Figure 6

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

000200400600801012

t

minus002

minus008

minus006

minus004

times1011

x(t)y(t)

x(t)

y(t)

(b)

Figure 7

for all 119905 ge 0 119909 and 119910 Moreover from (79) and (80)all assumptions of Theorem 17 and Lemma 6 are satisfiedThus by Lemma 6 the trivial solution of system (78) is notonly uniformly stochastically asymptotically stable but alsouniformly stochastically asymptotically stable in the largeFinally from (79) and (81) the function 119881(119905 119883) is positivedefinite and

119871119881 (119905 119883) le 0 forall (119905 119883) isin R+ timesR

2 (82)

Hence assumptions of Theorem 17 and Lemma 5 hold byTheorem 17 and Lemma 5 the trivial solution of system (78)is stochastically stable

Simulation of Solutions In what follows we shall nowsimulate the solutions of (56) (resp system (57)) and (78)(resp system (79)) Our approach depends on the Euler-Maruyama method which enables us to get approximatenumerical solution for the considered systems It will be seenfrom our figures that the simulated solutions are bounded

which justifies our given results For instance when 120590 = 01the numerical solutions of (56) in three-dimensional spaceare shown in Figure 4 If we vary the value of the noise inthe numerical solution (119909(119905) 119910(119905)) of system (57) as 120590 = 01and 120590 = 10 we have Figures 5(a) and 5(b) respectively Itcan be seen that when the noise is increased the stochasticitybecomes more pronounced The behaviour of the numericalsolution (119909(119905) 119910(119905)) of system (57) when 120590 = 05 and 120590 = 20is shown in Figures 6(a) and 6(b) respectivelyThe behaviourof the numerical solution (119909(119905) 119910(119905)) of system (57) for 120590 = 0and 120590 = 50 is shown in Figures 7(a) and 7(b) respectivelyFor the case of (78) Figure 8 shows the closeness of thesolution (119909(119905)) and the perturbed solution (119909120598(119905)) for a verylarge 119905 which implies asymptotic stability in the large for theconsidered SDE

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

10 International Journal of Analysis

0 1 2 3 4 5 6 7 8 9 100

0005

001

0015

002

0025

t times1011

x(t)x120576(t)

x(t)x120576(t)

Figure 8 Graph of solutions of (78)

References

[1] L Arnold Stochastic Differential Equations Theory and Appli-cations John Wiley amp Sons 1974

[2] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations vol 178 of Mathematics inScience and Engineering Academic Press Inc Orlando FlaUSA 1985

[3] T A Burton Volterra Integral and Differential EquationsAcademic Press New York NY USA 1983

[4] J K HaleTheory of Functional Differential Equations SpringerNew York NY USA 1977

[5] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer 2000

[6] L Shaikihet Lyapunov Functionals and Stability of StochasticFunctional Differential Equations Springer International 2013

[7] T Yoshizawa StabilityTheory and Existence of Periodic Solutionsand almost Periodic Solutions Spriger New York NY USA1975

[8] T Yoshizawa StabilityTheory by Liapunovrsquos SecondMethodTheMathematical Society of Japan 1966

[9] A M A Abou-El-Ela A I Sadek and A M Mahmoud ldquoOnthe stability of solutions for certain second-order stochasticdelay differential equationsrdquo Differential Equations and ControlProcesses no 2 pp 1ndash13 2015

[10] A M Abou-El-Ela A I Sadek A M Mahmoud and R OTaie ldquoOn the stochastic stability and boundedness of solutionsfor stochastic delay differential equation of the second orderrdquoChinese Journal of Mathematics vol 2015 Article ID 358936 8pages 2015

[11] A T Ademola Boundedness and Stability of Solutions to CertainSecond Order Differential Equations Differential Equations andControl Processes 2015

[12] A T Ademola B S Ogundare M O Ogundiran and O AAdesina ldquoPeriodicity stability and boundedness of solutions tocertain second order delay differential equationsrdquo InternationalJournal of Differential Equations vol 2016 Article ID 284370910 pages 2016

[13] J G Alaba and B S Ogundare ldquoOn stability and bound-edness properties of solutions of certain second order non-autonomous nonlinear ordinary differential equationrdquo Kragu-jevac Journal of Mathematics vol 39 no 2 pp 255ndash266 2015

[14] T A Burton and L Hatvani ldquoAsymptotic stability of secondorder ordinary functional and partial differential equationsrdquoJournal of Mathematical Analysis and Applications vol 176 no1 pp 261ndash281 1993

[15] B Cahlon andD Schmidt ldquoStability criteria for certain second-order delay differential equations with mixed coefficientsrdquoJournal of Computational and AppliedMathematics vol 170 no1 pp 79ndash102 2004

[16] T Caraballo M A Diop and A S Ndoye ldquoFixed points andexponential stability for stochastic partial integro-differentialequations with delaysrdquo Advances in Dynamical Systems andApplications vol 9 no 2 pp 133ndash147 2014

[17] A Domoshnitsky ldquoNonoscillation maximum principles andexponential stability of secondorder delay differential equationswithout damping termrdquo Domoshnitsky Journal of Inequalitiesand Applications vol 2014 article 361 2014

[18] I I Gikhman and A V Skorokhod Stochastische Differential-gleichungen Akademie Berlin Germany 1971 (Russian)

[19] I I Gikhman On the Stability of the Solutions of StochasticDifferential Equations Predelrsquonyye Teoremy i StatisticheskiyeVyvody Tashkent Uzbekistan 1966

[20] G A Grigoryan ldquoBoundedness and stability criteria for linearordinary differential equations of the second orderrdquo RussianMathematics vol 57 no 12 pp 8ndash15 2013

[21] A F Ivanov Y I Kazmerchuk and A V Swishchuk ldquoThe-ory stochastic stability and applications of stochastic delaydifferential equations a survey of recent resultsrdquo in DifferentialEquations and Dynamical Systems vol 11 no 1 2003

[22] F Jedrzejewski and D Brochard ldquoLyapounv exponents andstability stochastic dynamical structuresrdquo 2000

[23] Z Jin and L Zengrong ldquoOn the global asymptotic behaviorof solutions to a non autonomous generalized Lienard systemrdquoJournal of Mathematical Research and Exposition vol 21 no 3pp 410ndash414 2001

[24] E Kolarova ldquoAn application of stochastic integral equations toelectrical networksrdquo Acta Electrotechnica et Informatica vol 8no 3 pp 14ndash17 2008

[25] V B Kolmanovskii and L E Shaikhet ldquoA method for con-structing Lyapunov functionals for stochastic systems with aftereffectrdquo Differentsialrsquonye Uravneniya vol 29 no 11 pp 1909ndash2022 1993

[26] V Kolmanovskii and L Shaikhet ldquoConstruction of Lyapunovfunctionals for stochastic hereditary systems a survey of somerecent resultsrdquo Mathematical and Computer Modelling vol 36no 6 pp 691ndash716 2002

[27] A J Kroopnick ldquoBounded solutions to 11990910158401015840 + 119902(119905)119887(119909) = 119891(119905)rdquoInternational Journal of Mathematical Education in Science andTechnology vol 41 no 6 pp 829ndash836 2010

[28] R Liu and Y Raffoul ldquoBoundedness and exponential stabilityof highly nonlinear stochastic differential equationsrdquo ElectronicJournal of Differential Equations vol 2009 no 143 pp 1ndash102009

[29] X Mao ldquoSome contributions to stochastic asymptotic stabilityand boundedness via multiple Lyapunov functionsrdquo Journal ofMathematical Analysis and Applications vol 260 no 2 pp 325ndash340 2001

[30] B S Ogundare A T Ademola M O Ogundiran and O AAdesina ldquoOn the qualitative behaviour of solutions to certainsecond order nonlinear differential equation with delayrdquoAnnalidellrsquoUniversitarsquo di Ferrara 2016

International Journal of Analysis 11

[31] B S Ogundare and A U Afuwape ldquoBoundedness and stabilityproperties of solutions of generalized Lienard equationrdquo KochiJournal of Mathematics vol 9 pp 97ndash108 2014

[32] B S Ogundare and G E Okecha ldquoBoundedness periodicityand stability of solutions to x(t) + g(x) + b(t)h(x) = p(t x x)rdquoMathematical Sciences Research Journal vol 11 no 5 pp 432ndash443 2007

[33] Y N Raffoul ldquoBoundedness and exponential asymptotic sta-bility in dynamical systems with applications to nonlineardifferential equations with unbounded termsrdquo Advances inDynamical Systems and Applications vol 2 no 1 pp 107ndash1212007

[34] R Rezaeyan and R Farnoosh ldquoStochastic differential equationsand application of the Kalman-Bucy filter in the modeling ofRC circuitrdquo Applied Mathematical Sciences vol 4 no 21-24 pp1119ndash1127 2010

[35] C Tunc ldquoA note on the stability and boundedness of non-autonomous differential equations of second order with avariable deviating argumentrdquo Afrika Matematika vol 25 no 2pp 417ndash425 2014

[36] C Tunc ldquoA note on the bounded solutions to 11990910158401015840 + 119888(119905 119909 1199091015840 ) +119902(119905)119887(119909) = 119891(119905)rdquo Applied Mathematics amp Information Sciencesvol 8 no 1 pp 393ndash399 2014

[37] C Tunc ldquoBoundedness analysis for certain two-dimensionaldifferential systems via a Lyapunov approachrdquo Bulletin Mathe-matique de la Societe des Sciences Mathematiques de Roumanievol 53 no 1 pp 61ndash68 2010

[38] 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

[39] C Tunc ldquoStability and boundedness in multi delay vectorLienard equationrdquo Filomat vol 27 no 3 pp 435ndash445 2013

[40] C Tunc ldquoStability and boundedness of solutions of non-autonomous differential equations of second orderrdquo Journalof Computational Analysis and Applications vol 13 no 6 pp1067ndash1074 2011

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

[42] 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

[43] C Tunc and T Ayhan ldquoGlobal existence and boundedness ofsolutions of a certain nonlinear integro-differential equationof second order with multiple deviating argumentsrdquo Journal ofInequalities and Applications vol 2016 article no 46 2016

[44] F Wang and H Zhu ldquoExistence uniqueness and stabilityof periodic solutions of a duffing equation under periodicand anti-periodic eigenvalues conditionsrdquo Taiwanese Journal ofMathematics vol 19 no 5 pp 1457ndash1468 2015

[45] Z Xianfeng and J Wei ldquoStability and boundedness of aretarded Lienard-type equationrdquo Chinese Quarterly Journal ofMathematics vol 18 no 1 pp 7ndash12 2003

[46] A F Yenicerioglu ldquoThe behavior of solutions of second orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 332 no 2 pp 1278ndash1290 2007

[47] A F Yenicerioglu ldquoStability properties of second order delayintegro-differential equationsrdquo Computers and Mathematicswith Applications vol 56 no 12 pp 3109ndash3117 2008

[48] T Yoshizawa ldquoLiapunovrsquos function and boundedness of solu-tionsrdquo Funkcialaj Ekvacioj vol 2 pp 71ndash103 1958

[49] W Zhu J Huang X Ruan and Z Zhao ldquoExponential stabilityof stochastic differential equation with mixed delayrdquo Journal ofAppliedMathematics vol 2014 Article ID 187037 11 pages 2014

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Analysis 5

Moreover applying Itorsquos formula in (24) using system (9)we find that

119871119881 (119905 119883) = 12 [119886119891 (119909)119909 minus 121205902 (119887 + 1)] 1199092minus 12 [(119887 + 1) 119892 (119909 119910) minus 119886] 1199102 minus119882119894+ [119886119909 + (119887 + 1) 119910] 119901 (119905 119909 119910)

(119894 = 1 2)

(36)

where

1198821 fl 14 [119886119891 (119909)119909 minus 121205902 (119887 + 1)] 1199092+ 4 [119886119892 (119909 119910) minus (1198862 + 1198872)] 119909119910+ [(119887 + 1) 119892 (119909 119910) minus 119886] 1199102

1198822 fl 14 [119886119891 (119909)119909 minus 121205902 (119887 + 1)] 1199092

+ 4 [119886119891 (119909)119909 minus 1198911015840 (119909)] 119909119910+ [(119887 + 1) 119892 (119909 119910) minus 119886] 1199102

(37)

It is clear from the inequalities

4 [119886119892 (119909 119910) minus (1198862 + 1198872)]2lt [119886119891 (119909)119909 minus 121205902 (119887 + 1)] [(119887 + 1) 119892 (119909 119910) minus 119886]

4 [119886119891 (119909)119909 minus 1198911015840 (119909)]lt [119886119891 (119909)119909 minus 121205902 (119887 + 1)] [(119887 + 1) 119892 (119909 119910) minus 119886]

(38)

that

1198821 = 1198822 ge [[radic119886119891 (119909)119909 minus 121205902 (119887 + 1) |119909|

minus radic(119887 + 1) 119892 (119909 119910) minus 119886 10038161003816100381610038161199101003816100381610038161003816]]2

ge 0(39)

for all 119909 and 119910 Using inequality (39) and hypotheses (i) and(ii) ofTheorem 12 in (36) there exist positive constants 1205752 and1205753 such that

119871119881 (119905 119883) le minus1205752 (1199092 + 1199102)+ 1205753 (|119909| + 10038161003816100381610038161199101003816100381610038161003816) 1003816100381610038161003816119901 (119905 119909 119910)1003816100381610038161003816 (40)

for all 119905 ge 0 119909 and 119910 where1205752 fl 12 min 119886119887 minus 121205902 (119887 + 1) 119886119887 1205753 fl max 119886 119887 + 1 (41)

Inequality (40) satisfies inequality (27) with 1205752 and 1205753equivalent to 1198632 and 1198633 respectively This completes theproof of Lemma 14

Proof ofTheorem 12 Let (119909(119905) 119910(119905)) be any solution of system(9) From inequality (40) and assumption (iii) ofTheorem 12we have

119871119881 (119905 119883)le minus121205752 (1199092 + 1199102)minus 1212057521198720 [(|119909| minus 120575minus12 1205753)2 + (10038161003816100381610038161199101003816100381610038161003816 minus 120575minus12 1205753)2]+1198720120575minus12 12057523

(42)

for 119905 ge 0 119909 and 119910 Since 1205752 1205753 and1198720 are positives and(|119909| minus 120575minus12 1205753)2 + (10038161003816100381610038161199101003816100381610038161003816 minus 120575minus12 1205753)2 ge 0 (43)

for all 119909 and 119910 there exist positive constants 1205754 and 1205755 suchthat

119871119881 (119905 119883) le minus1205754 (1199092 + 1199102) + 1205755 (44)

for all 119905 ge 0 119909 119910 where 1205754 fl (12)1205752 and 1205755 fl 1198720120575minus12 12057523 Hence condition (ii) of Lemma 9 is satisfied with 120572(119905) fl 1205754119903 fl 2 and 120573(119905) fl 1205755 Also from inequality (35) hypotheses(i) and (iii) of Lemma 9 hold with 119901 = 119902 = 2 so that 120574 = 0

Furthermore from inequality (23) we have

int1199051199050[(120574120572 (119906) + 120573 (119906)) 119890minus1205754 int119905119906 120572(119904)119889119904] 119889119906= int11990511990501205755119890minus1205754 int119905119906 119889119904119889119906 = 120575minus14 1205755 [1 minus 119890minus1205754(119905minus1199050)]

le 120575minus14 1205755(45)

for all 119905 ge 1199050 ge 0 Inequality (45) satisfies estimate (23) with119872 fl 120575minus14 1205755 = 21198720120575minus22 12057523 gt 0 Moreover from (9) and (24)there exists a positive constant 1205756 such that

10038161003816100381610038161003816119881119909119894 (119905 119883)119866119894119896 (119905 119883)10038161003816100381610038161003816le 12120590 [(2119886 + 119887 + 1) 1199092 + (119887 + 1) 1199102]le 1205756 (1199092 + 1199102) fl 120582 (119905)

(46)

where

1205756 fl 12120590max 2119886 + 119887 + 1 119887 + 1 (47)

6 International Journal of Analysis

Also

int119879119905012057526 (1199092 (119905) + 1199102 (119905))2 119889119905 lt infin (48)

for any fixed 0 le 1199050 le 119879 lt infin Thus from inequalities (46)and (48) estimates (20) and (21) hold respectively Finallyfrom inequalities (33) and (45) we have

1198641198830 1003817100381710038171003817119883 (119905 1198830)1003817100381710038171003817 le (120575111988320 + 21198720120575minus22 12057523)12 (49)

for all 119905 ge 1199050 ge 0 where1198830 fl (11990920 +11991020) and119862 fl 1205751Thus thesolutions (119909(119905) 119910(119905)) of the stochastic differential equation (9)are uniformly stochastically bounded

Theorem 15 If assumptions of Theorem 12 hold then thesolution (119909(119905) 119910(119905)) of the stochastic differential equation (9)is stochastically bounded

Proof Suppose that (119909(119905) 119910(119905)) is any solution of the stochas-tic differential equation (9) From inequalities (33) and (44)there exists a positive constant 1205757 such that

119871119881 (119905 119883) le minus1205757119881 (119905 119883) + 1205755 (50)

for all 119905 ge 0 119909 and119910 where1205757 fl 120575minus11 1205754Hence from inequal-ities (29) and (50) hypotheses of Lemma 10 hold Moreoverfrom inequalities (45) (46) (48) and (49) assumption (ii)of Corollary 11 holds Thus by Corollary 11 all solutionsof the stochastic differential equation (9) are stochasticallybounded This completes the proof of Theorem 15

Next we shall discuss the stability of the trivial solutionof the stochastic differential equation (8) Suppose that119901(119905 119909 1199091015840) = 0 (8) specializes to

11990910158401015840 (119905) + 119892 (119909 (119905) 1199091015840 (119905)) 1199091015840 (119905) + 119891 (119909 (119905))+ 120590119909 (119905) 1205961015840 (119905) = 0 (51)

Equation (51) has the following equivalent system

1199091015840 (119905) = 119910 (119905) 1199101015840 (119905) = minus119891 (119909) minus 119892 (119909 (119905) 119910 (119905)) 119910 (119905) minus 120590119909 (119905) 1205961015840 (119905) (52)

where the functions 119891 119892 and 120596 are defined in Section 1

Theorem 16 If assumptions (i) and (ii) of Theorem 12 holdthen the trivial solution of the stochastic differential equation(52) is stochastically stable

Proof Let (119909(119905) 119910(119905)) be any solution of the stochastic differ-ential equation (52) From equation (28) and estimate (29)assumptions (i) and (ii) of Lemma 5 hold so that the function119881(119905 119883) is positive definite Furthermore using Itorsquos formulaalong the solution path of (52) we obtain

119871119881 (119905 119883) le minus1205752 (1199092 (119905) + 1199102 (119905)) le 0 (53)

for all 119905 ge 0 119909 and 119910 where 1205752 is defined in (40)Inequality (53) satisfies hypothesis (iii) of Lemma 5 henceby Lemma 5 the trivial solution of the stochastic differentialequation (52) is stochastically stableThis completes the proofof Theorem 16

Theorem 17 If assumptions (i) and (ii) of Theorem 12 holdthen the trivial solution of the stochastic differential equation(52) is not only uniformly stochastically asymptotically stablebut also uniformly stochastically asymptotically stable in thelarge

Proof Let (119909(119905) 119910(119905)) be any solution of the stochastic differ-ential equation (52) In view of (28) and estimate (29) thefunction 119881(119905 119883) is positive definite Furthermore estimate(32) and inequality (33) show that the function 119881(119905 119883) isradially unbounded and decrescent respectively It followsfrom (28) estimate (32) inequality (35) and the first inequal-ity in (53) that all assumptions of Lemma 6 hold Thus byLemma 6 the trivial solution of the stochastic differentialequation (52) is uniformly stochastically asymptotically stablein the large If estimate (32) is omitted then the trivial solutionof the stochastic differential equation (52) is uniformlystochastically asymptotically stable This completes the proofof Theorem 17

Next if the function 119901(119905 119909 1199091015840) is replaced by 119901(119905) isin119862(R+R+) we have the following special case11990910158401015840 (119905) + 119892 (119909 (119905) 1199091015840 (119905)) 1199091015840 (119905) + 119891 (119909 (119905))

+ 120590119909 (119905) 1205961015840 (119905) = 119901 (119905) (54)

of (8) Equation (54) has the following equivalent system

1199091015840 (119905) = 119910 (119905) 1199101015840 (119905) = 119901 (119905) minus 119891 (119909) minus 119892 (119909 (119905) 119910 (119905)) 119910 (119905)

minus 120590119909 (119905) 1205961015840 (119905) (55)

with the following result

Corollary 18 If assumptions (i) and (ii) of Theorem 12 holdand hypothesis (iii) is replaced by the boundedness of thefunction 119901(119905) then the solutions (119909(119905) 119910(119905)) of the stochasticdifferential equation (55) are not only stochastically boundedbut also uniformly stochastically bounded

Proof Theproof of Corollary 18 is similar to the proof ofThe-orems 12 and 15This completes the proof of Corollary 18

4 Examples

In this section we shall present two examples to illustrate theapplications of the results we obtained in the previous section

Example 1 Consider the second-order nonlinear nonau-tonomous stochastic differential equation

11990910158401015840 + (3 + 10038161003816100381610038161003816cos (1199091199091015840)10038161003816100381610038161003816) 1199091015840 + 119909 + sin119909 + 011199091205961015840 (119905)= (1 + 2119905 + 10038161003816100381610038161003816119909119909101584010038161003816100381610038161003816)minus1

(56)

International Journal of Analysis 7

Equation (56) is equivalent to system

1199091015840 = 1199101199101015840 = (1 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816)minus1 minus (119909 + sin119909)

minus [3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816] 119910 minus 011199091205961015840 (119905) (57)

Now from systems (9) and (57) we have the followingrelations

(i) The function

119892 (119909 119910) fl 3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816 (58)

Noting that 1003816100381610038161003816cos (119909119910)1003816100381610038161003816 ge 0 (59)

for all 119909 and 119910 it follows that119892 (119909 119910) = 3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816 ge 119886 = 3 (60)

for all 119909 and 119910The behaviour of the function 119892(119909 119910)is shown below in Figure 1

(ii) The function

119891 (119909) fl 119909 + sin119909 (61)

Since

minus02 le 119865 (119909) = sin119909119909 le 1 (62)

for all 119909 = 0 then we have

1 = 119887 le 119891 (119909)119909 = 1 + sin119909119909 le 119861 = 2 (63)

for all 119909 = 0 and since 120590 fl 01 it follows that1205902 lt 2119886119887(119887 + 1)minus1 implies that 0 lt 299 The function119891(119909)119909 and its bounds are shown in Figure 2(iii) The function

119901 (119905 119909 119910) fl 11 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816 (64)

Clearly

1003816100381610038161003816119901 (119905 119909 119910)1003816100381610038161003816 = 11 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816 le 1 = 1198720 (65)

for all 119905 ge 0 119909 and 119910Now from items (i) (ii) above and (24) the continuouslydifferentiable function 119881(119905 119883) used for system (57) is

2119881 (119905 119883) = 31199092 + 1199102 + (3119909 + 119910)2 (66)

Different views of the function119881(119905 119883) are shown in Figure 3From (66) it is not difficult to show that

(1199092 + 1199102) le 119881 (119905 119883) le 3 (1199092 + 1199102) (67)

Figure 1 Behaviour of the function 119892(119909 119910)

minus6120587 minus4120587 minus2120587 612058741205872120587

ge minus0225

25

2

15

1

05

minus05

minus1

F(x) f(x)x

f(x)x = 1 + sin(x)x

F(x) = sin(x)xx

b = 076

F(x)

Figure 2 Bounds on the function 119891(119909)119909

for all 119905 ge 0 119909 and 119910 From (35) and (67) we have 1205750 = 11205751 = 3 119901 = 2 and 119902 = 2 and thus inequalities (67) satisfycondition (i) of Lemma 9 Also from the first inequality in(67) we have

119881 (119905 119883) 997888rarr +infin as 1199092 + 1199102 997888rarr infin (68)

Estimate (68) verifies (32) (ie the function 119881(119905 119883) definedby (66) is radially unbounded) Next applying Itorsquos formulain (66) using system (57) we find that

119871119881 (119905 119883) = 12119909119910 + 31199102 minus 119909 (3119909 + 2119910) (1 + sin119909119909 )minus 119910 (3119909 + 2119910) (3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816) + 11001199092minus 11990910 (3119909 + 2119910)+ (3119909 + 2119910) (1 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816)minus1

(69)

Using the estimates in items (i) to (iii) of Example 1 and theinequality 211990911199092 le 11990921 + 11990922 in (69) we obtain

119871119881 (119905 119883) le minus29 (1199092 + 1199102) + 3 (|119909| + 10038161003816100381610038161199101003816100381610038161003816) (70)

for all 119905 ge 0 119909 and 119910 Inequality (70) satisfies inequality (40)where 1205752 = 29 and 1205753 = 3 Since

(|119909| minus 105)2 + (10038161003816100381610038161199101003816100381610038161003816 minus 105)2 ge 0 (71)

for all 119909 and 119910 it follows from inequality (70) that

119871119881 (119905 119883) le minus145 (1199092 + 1199102) + 32 (72)

8 International Journal of Analysis

Figure 3 The behaviour of the function 119881(119905 119883)

for all 119905 ge 0 119909 and 119910 Inequality (72) satisfies assumption(ii) of Lemma 9 and estimate (44) with 120572(119905) = 1205754 = 145 and120573(119905) = 1205755 = 32 Since 119903 = 119901 = 119902 = 2 it follows that 120574 = 0 sothat assumption (iii) of Lemma 9 holds In addition

int1199051199050[(120574120572 (119906) + 120573 (119906)) 119890minusint119905119906120572 (119904) 119889119904] 119889119906 le 16 (73)

for all 119905 ge 1199050 ge 0 Estimate (73) satisfies (23) and (45) with119872 = 26 Furthermore

119881119909119894 (119905 119883)119866119894119896 (119905 119883) = minus 110 (31199092 + 2119909119910) (74)

and10038161003816100381610038161003816119881119909119894 (119905 119883)119866119894119896 (119905 119883)10038161003816100381610038161003816 le 25 (1199092 + 1199102) (75)

for all 119905 ge 0 119909 and119910 Inequality (75) satisfies inequalities (20)and (21) with

120582 (119905) = 25 (1199092 + 1199102) (76)

Hence by Corollary 11 (i) all solutions of stochastic differen-tial equation (57) are uniformly stochastically bounded

Example 2 If 119901(119905 119909 1199091015840) = 119901(119905 119909 119910) = 0 in (56) and system(57) we have the following stochastic differential equation

11990910158401015840 + (3 + 10038161003816100381610038161003816cos (1199091199091015840)10038161003816100381610038161003816) 1199091015840 + 119909 + sin119909 + 011199091205961015840 (119905)= 0 (77)

Equation (77) is equivalent to system

1199091015840 = 1199101199101015840 = minus (119909 + sin119909) minus [3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816] 119910 minus 011199091205961015840 (119905) (78)

Now from systems (52) and (78) items (i) and (ii) of Example 1hold Also equations (66) (67) and estimate (68) hold thatis

2119881 (119905 119883) = 31199092 + 1199102 + (3119909 + 119910)2 119881 (119905 0) = 0 forall119905 ge 0

(1199092 + 1199102) le 119881 (119905 119883) le 3 (1199092 + 1199102) forall119905 ge 0 119909 119910119881 (119905 119883) 997888rarr +infin as 1199092 + 1199102 997888rarr infin

(79)

112

1416

182

0500

10001500

0

02

04

06

08

1

t

x(t) y(t)

Figure 4 Graph of solutions of (56) in 3D

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(b)

Figure 5

Furthermore application of Itorsquos formula in (66) and usingsystem (78) yield

119871119881 (119905 119883) le minus29 (1199092 + 1199102) (80)

for all 119905 ge 0 119909 119910 and thus

119871119881 (119905 119883) le 0 (81)

International Journal of Analysis 9

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

000200400600801012

t

minus004

minus002

x(t)y(t)

x(t)

y(t)

times1011

(b)

Figure 6

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

000200400600801012

t

minus002

minus008

minus006

minus004

times1011

x(t)y(t)

x(t)

y(t)

(b)

Figure 7

for all 119905 ge 0 119909 and 119910 Moreover from (79) and (80)all assumptions of Theorem 17 and Lemma 6 are satisfiedThus by Lemma 6 the trivial solution of system (78) is notonly uniformly stochastically asymptotically stable but alsouniformly stochastically asymptotically stable in the largeFinally from (79) and (81) the function 119881(119905 119883) is positivedefinite and

119871119881 (119905 119883) le 0 forall (119905 119883) isin R+ timesR

2 (82)

Hence assumptions of Theorem 17 and Lemma 5 hold byTheorem 17 and Lemma 5 the trivial solution of system (78)is stochastically stable

Simulation of Solutions In what follows we shall nowsimulate the solutions of (56) (resp system (57)) and (78)(resp system (79)) Our approach depends on the Euler-Maruyama method which enables us to get approximatenumerical solution for the considered systems It will be seenfrom our figures that the simulated solutions are bounded

which justifies our given results For instance when 120590 = 01the numerical solutions of (56) in three-dimensional spaceare shown in Figure 4 If we vary the value of the noise inthe numerical solution (119909(119905) 119910(119905)) of system (57) as 120590 = 01and 120590 = 10 we have Figures 5(a) and 5(b) respectively Itcan be seen that when the noise is increased the stochasticitybecomes more pronounced The behaviour of the numericalsolution (119909(119905) 119910(119905)) of system (57) when 120590 = 05 and 120590 = 20is shown in Figures 6(a) and 6(b) respectivelyThe behaviourof the numerical solution (119909(119905) 119910(119905)) of system (57) for 120590 = 0and 120590 = 50 is shown in Figures 7(a) and 7(b) respectivelyFor the case of (78) Figure 8 shows the closeness of thesolution (119909(119905)) and the perturbed solution (119909120598(119905)) for a verylarge 119905 which implies asymptotic stability in the large for theconsidered SDE

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

10 International Journal of Analysis

0 1 2 3 4 5 6 7 8 9 100

0005

001

0015

002

0025

t times1011

x(t)x120576(t)

x(t)x120576(t)

Figure 8 Graph of solutions of (78)

References

[1] L Arnold Stochastic Differential Equations Theory and Appli-cations John Wiley amp Sons 1974

[2] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations vol 178 of Mathematics inScience and Engineering Academic Press Inc Orlando FlaUSA 1985

[3] T A Burton Volterra Integral and Differential EquationsAcademic Press New York NY USA 1983

[4] J K HaleTheory of Functional Differential Equations SpringerNew York NY USA 1977

[5] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer 2000

[6] L Shaikihet Lyapunov Functionals and Stability of StochasticFunctional Differential Equations Springer International 2013

[7] T Yoshizawa StabilityTheory and Existence of Periodic Solutionsand almost Periodic Solutions Spriger New York NY USA1975

[8] T Yoshizawa StabilityTheory by Liapunovrsquos SecondMethodTheMathematical Society of Japan 1966

[9] A M A Abou-El-Ela A I Sadek and A M Mahmoud ldquoOnthe stability of solutions for certain second-order stochasticdelay differential equationsrdquo Differential Equations and ControlProcesses no 2 pp 1ndash13 2015

[10] A M Abou-El-Ela A I Sadek A M Mahmoud and R OTaie ldquoOn the stochastic stability and boundedness of solutionsfor stochastic delay differential equation of the second orderrdquoChinese Journal of Mathematics vol 2015 Article ID 358936 8pages 2015

[11] A T Ademola Boundedness and Stability of Solutions to CertainSecond Order Differential Equations Differential Equations andControl Processes 2015

[12] A T Ademola B S Ogundare M O Ogundiran and O AAdesina ldquoPeriodicity stability and boundedness of solutions tocertain second order delay differential equationsrdquo InternationalJournal of Differential Equations vol 2016 Article ID 284370910 pages 2016

[13] J G Alaba and B S Ogundare ldquoOn stability and bound-edness properties of solutions of certain second order non-autonomous nonlinear ordinary differential equationrdquo Kragu-jevac Journal of Mathematics vol 39 no 2 pp 255ndash266 2015

[14] T A Burton and L Hatvani ldquoAsymptotic stability of secondorder ordinary functional and partial differential equationsrdquoJournal of Mathematical Analysis and Applications vol 176 no1 pp 261ndash281 1993

[15] B Cahlon andD Schmidt ldquoStability criteria for certain second-order delay differential equations with mixed coefficientsrdquoJournal of Computational and AppliedMathematics vol 170 no1 pp 79ndash102 2004

[16] T Caraballo M A Diop and A S Ndoye ldquoFixed points andexponential stability for stochastic partial integro-differentialequations with delaysrdquo Advances in Dynamical Systems andApplications vol 9 no 2 pp 133ndash147 2014

[17] A Domoshnitsky ldquoNonoscillation maximum principles andexponential stability of secondorder delay differential equationswithout damping termrdquo Domoshnitsky Journal of Inequalitiesand Applications vol 2014 article 361 2014

[18] I I Gikhman and A V Skorokhod Stochastische Differential-gleichungen Akademie Berlin Germany 1971 (Russian)

[19] I I Gikhman On the Stability of the Solutions of StochasticDifferential Equations Predelrsquonyye Teoremy i StatisticheskiyeVyvody Tashkent Uzbekistan 1966

[20] G A Grigoryan ldquoBoundedness and stability criteria for linearordinary differential equations of the second orderrdquo RussianMathematics vol 57 no 12 pp 8ndash15 2013

[21] A F Ivanov Y I Kazmerchuk and A V Swishchuk ldquoThe-ory stochastic stability and applications of stochastic delaydifferential equations a survey of recent resultsrdquo in DifferentialEquations and Dynamical Systems vol 11 no 1 2003

[22] F Jedrzejewski and D Brochard ldquoLyapounv exponents andstability stochastic dynamical structuresrdquo 2000

[23] Z Jin and L Zengrong ldquoOn the global asymptotic behaviorof solutions to a non autonomous generalized Lienard systemrdquoJournal of Mathematical Research and Exposition vol 21 no 3pp 410ndash414 2001

[24] E Kolarova ldquoAn application of stochastic integral equations toelectrical networksrdquo Acta Electrotechnica et Informatica vol 8no 3 pp 14ndash17 2008

[25] V B Kolmanovskii and L E Shaikhet ldquoA method for con-structing Lyapunov functionals for stochastic systems with aftereffectrdquo Differentsialrsquonye Uravneniya vol 29 no 11 pp 1909ndash2022 1993

[26] V Kolmanovskii and L Shaikhet ldquoConstruction of Lyapunovfunctionals for stochastic hereditary systems a survey of somerecent resultsrdquo Mathematical and Computer Modelling vol 36no 6 pp 691ndash716 2002

[27] A J Kroopnick ldquoBounded solutions to 11990910158401015840 + 119902(119905)119887(119909) = 119891(119905)rdquoInternational Journal of Mathematical Education in Science andTechnology vol 41 no 6 pp 829ndash836 2010

[28] R Liu and Y Raffoul ldquoBoundedness and exponential stabilityof highly nonlinear stochastic differential equationsrdquo ElectronicJournal of Differential Equations vol 2009 no 143 pp 1ndash102009

[29] X Mao ldquoSome contributions to stochastic asymptotic stabilityand boundedness via multiple Lyapunov functionsrdquo Journal ofMathematical Analysis and Applications vol 260 no 2 pp 325ndash340 2001

[30] B S Ogundare A T Ademola M O Ogundiran and O AAdesina ldquoOn the qualitative behaviour of solutions to certainsecond order nonlinear differential equation with delayrdquoAnnalidellrsquoUniversitarsquo di Ferrara 2016

International Journal of Analysis 11

[31] B S Ogundare and A U Afuwape ldquoBoundedness and stabilityproperties of solutions of generalized Lienard equationrdquo KochiJournal of Mathematics vol 9 pp 97ndash108 2014

[32] B S Ogundare and G E Okecha ldquoBoundedness periodicityand stability of solutions to x(t) + g(x) + b(t)h(x) = p(t x x)rdquoMathematical Sciences Research Journal vol 11 no 5 pp 432ndash443 2007

[33] Y N Raffoul ldquoBoundedness and exponential asymptotic sta-bility in dynamical systems with applications to nonlineardifferential equations with unbounded termsrdquo Advances inDynamical Systems and Applications vol 2 no 1 pp 107ndash1212007

[34] R Rezaeyan and R Farnoosh ldquoStochastic differential equationsand application of the Kalman-Bucy filter in the modeling ofRC circuitrdquo Applied Mathematical Sciences vol 4 no 21-24 pp1119ndash1127 2010

[35] C Tunc ldquoA note on the stability and boundedness of non-autonomous differential equations of second order with avariable deviating argumentrdquo Afrika Matematika vol 25 no 2pp 417ndash425 2014

[36] C Tunc ldquoA note on the bounded solutions to 11990910158401015840 + 119888(119905 119909 1199091015840 ) +119902(119905)119887(119909) = 119891(119905)rdquo Applied Mathematics amp Information Sciencesvol 8 no 1 pp 393ndash399 2014

[37] C Tunc ldquoBoundedness analysis for certain two-dimensionaldifferential systems via a Lyapunov approachrdquo Bulletin Mathe-matique de la Societe des Sciences Mathematiques de Roumanievol 53 no 1 pp 61ndash68 2010

[38] 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

[39] C Tunc ldquoStability and boundedness in multi delay vectorLienard equationrdquo Filomat vol 27 no 3 pp 435ndash445 2013

[40] C Tunc ldquoStability and boundedness of solutions of non-autonomous differential equations of second orderrdquo Journalof Computational Analysis and Applications vol 13 no 6 pp1067ndash1074 2011

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

[42] 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

[43] C Tunc and T Ayhan ldquoGlobal existence and boundedness ofsolutions of a certain nonlinear integro-differential equationof second order with multiple deviating argumentsrdquo Journal ofInequalities and Applications vol 2016 article no 46 2016

[44] F Wang and H Zhu ldquoExistence uniqueness and stabilityof periodic solutions of a duffing equation under periodicand anti-periodic eigenvalues conditionsrdquo Taiwanese Journal ofMathematics vol 19 no 5 pp 1457ndash1468 2015

[45] Z Xianfeng and J Wei ldquoStability and boundedness of aretarded Lienard-type equationrdquo Chinese Quarterly Journal ofMathematics vol 18 no 1 pp 7ndash12 2003

[46] A F Yenicerioglu ldquoThe behavior of solutions of second orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 332 no 2 pp 1278ndash1290 2007

[47] A F Yenicerioglu ldquoStability properties of second order delayintegro-differential equationsrdquo Computers and Mathematicswith Applications vol 56 no 12 pp 3109ndash3117 2008

[48] T Yoshizawa ldquoLiapunovrsquos function and boundedness of solu-tionsrdquo Funkcialaj Ekvacioj vol 2 pp 71ndash103 1958

[49] W Zhu J Huang X Ruan and Z Zhao ldquoExponential stabilityof stochastic differential equation with mixed delayrdquo Journal ofAppliedMathematics vol 2014 Article ID 187037 11 pages 2014

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

6 International Journal of Analysis

Also

int119879119905012057526 (1199092 (119905) + 1199102 (119905))2 119889119905 lt infin (48)

for any fixed 0 le 1199050 le 119879 lt infin Thus from inequalities (46)and (48) estimates (20) and (21) hold respectively Finallyfrom inequalities (33) and (45) we have

1198641198830 1003817100381710038171003817119883 (119905 1198830)1003817100381710038171003817 le (120575111988320 + 21198720120575minus22 12057523)12 (49)

for all 119905 ge 1199050 ge 0 where1198830 fl (11990920 +11991020) and119862 fl 1205751Thus thesolutions (119909(119905) 119910(119905)) of the stochastic differential equation (9)are uniformly stochastically bounded

Theorem 15 If assumptions of Theorem 12 hold then thesolution (119909(119905) 119910(119905)) of the stochastic differential equation (9)is stochastically bounded

Proof Suppose that (119909(119905) 119910(119905)) is any solution of the stochas-tic differential equation (9) From inequalities (33) and (44)there exists a positive constant 1205757 such that

119871119881 (119905 119883) le minus1205757119881 (119905 119883) + 1205755 (50)

for all 119905 ge 0 119909 and119910 where1205757 fl 120575minus11 1205754Hence from inequal-ities (29) and (50) hypotheses of Lemma 10 hold Moreoverfrom inequalities (45) (46) (48) and (49) assumption (ii)of Corollary 11 holds Thus by Corollary 11 all solutionsof the stochastic differential equation (9) are stochasticallybounded This completes the proof of Theorem 15

Next we shall discuss the stability of the trivial solutionof the stochastic differential equation (8) Suppose that119901(119905 119909 1199091015840) = 0 (8) specializes to

11990910158401015840 (119905) + 119892 (119909 (119905) 1199091015840 (119905)) 1199091015840 (119905) + 119891 (119909 (119905))+ 120590119909 (119905) 1205961015840 (119905) = 0 (51)

Equation (51) has the following equivalent system

1199091015840 (119905) = 119910 (119905) 1199101015840 (119905) = minus119891 (119909) minus 119892 (119909 (119905) 119910 (119905)) 119910 (119905) minus 120590119909 (119905) 1205961015840 (119905) (52)

where the functions 119891 119892 and 120596 are defined in Section 1

Theorem 16 If assumptions (i) and (ii) of Theorem 12 holdthen the trivial solution of the stochastic differential equation(52) is stochastically stable

Proof Let (119909(119905) 119910(119905)) be any solution of the stochastic differ-ential equation (52) From equation (28) and estimate (29)assumptions (i) and (ii) of Lemma 5 hold so that the function119881(119905 119883) is positive definite Furthermore using Itorsquos formulaalong the solution path of (52) we obtain

119871119881 (119905 119883) le minus1205752 (1199092 (119905) + 1199102 (119905)) le 0 (53)

for all 119905 ge 0 119909 and 119910 where 1205752 is defined in (40)Inequality (53) satisfies hypothesis (iii) of Lemma 5 henceby Lemma 5 the trivial solution of the stochastic differentialequation (52) is stochastically stableThis completes the proofof Theorem 16

Theorem 17 If assumptions (i) and (ii) of Theorem 12 holdthen the trivial solution of the stochastic differential equation(52) is not only uniformly stochastically asymptotically stablebut also uniformly stochastically asymptotically stable in thelarge

Proof Let (119909(119905) 119910(119905)) be any solution of the stochastic differ-ential equation (52) In view of (28) and estimate (29) thefunction 119881(119905 119883) is positive definite Furthermore estimate(32) and inequality (33) show that the function 119881(119905 119883) isradially unbounded and decrescent respectively It followsfrom (28) estimate (32) inequality (35) and the first inequal-ity in (53) that all assumptions of Lemma 6 hold Thus byLemma 6 the trivial solution of the stochastic differentialequation (52) is uniformly stochastically asymptotically stablein the large If estimate (32) is omitted then the trivial solutionof the stochastic differential equation (52) is uniformlystochastically asymptotically stable This completes the proofof Theorem 17

Next if the function 119901(119905 119909 1199091015840) is replaced by 119901(119905) isin119862(R+R+) we have the following special case11990910158401015840 (119905) + 119892 (119909 (119905) 1199091015840 (119905)) 1199091015840 (119905) + 119891 (119909 (119905))

+ 120590119909 (119905) 1205961015840 (119905) = 119901 (119905) (54)

of (8) Equation (54) has the following equivalent system

1199091015840 (119905) = 119910 (119905) 1199101015840 (119905) = 119901 (119905) minus 119891 (119909) minus 119892 (119909 (119905) 119910 (119905)) 119910 (119905)

minus 120590119909 (119905) 1205961015840 (119905) (55)

with the following result

Corollary 18 If assumptions (i) and (ii) of Theorem 12 holdand hypothesis (iii) is replaced by the boundedness of thefunction 119901(119905) then the solutions (119909(119905) 119910(119905)) of the stochasticdifferential equation (55) are not only stochastically boundedbut also uniformly stochastically bounded

Proof Theproof of Corollary 18 is similar to the proof ofThe-orems 12 and 15This completes the proof of Corollary 18

4 Examples

In this section we shall present two examples to illustrate theapplications of the results we obtained in the previous section

Example 1 Consider the second-order nonlinear nonau-tonomous stochastic differential equation

11990910158401015840 + (3 + 10038161003816100381610038161003816cos (1199091199091015840)10038161003816100381610038161003816) 1199091015840 + 119909 + sin119909 + 011199091205961015840 (119905)= (1 + 2119905 + 10038161003816100381610038161003816119909119909101584010038161003816100381610038161003816)minus1

(56)

International Journal of Analysis 7

Equation (56) is equivalent to system

1199091015840 = 1199101199101015840 = (1 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816)minus1 minus (119909 + sin119909)

minus [3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816] 119910 minus 011199091205961015840 (119905) (57)

Now from systems (9) and (57) we have the followingrelations

(i) The function

119892 (119909 119910) fl 3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816 (58)

Noting that 1003816100381610038161003816cos (119909119910)1003816100381610038161003816 ge 0 (59)

for all 119909 and 119910 it follows that119892 (119909 119910) = 3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816 ge 119886 = 3 (60)

for all 119909 and 119910The behaviour of the function 119892(119909 119910)is shown below in Figure 1

(ii) The function

119891 (119909) fl 119909 + sin119909 (61)

Since

minus02 le 119865 (119909) = sin119909119909 le 1 (62)

for all 119909 = 0 then we have

1 = 119887 le 119891 (119909)119909 = 1 + sin119909119909 le 119861 = 2 (63)

for all 119909 = 0 and since 120590 fl 01 it follows that1205902 lt 2119886119887(119887 + 1)minus1 implies that 0 lt 299 The function119891(119909)119909 and its bounds are shown in Figure 2(iii) The function

119901 (119905 119909 119910) fl 11 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816 (64)

Clearly

1003816100381610038161003816119901 (119905 119909 119910)1003816100381610038161003816 = 11 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816 le 1 = 1198720 (65)

for all 119905 ge 0 119909 and 119910Now from items (i) (ii) above and (24) the continuouslydifferentiable function 119881(119905 119883) used for system (57) is

2119881 (119905 119883) = 31199092 + 1199102 + (3119909 + 119910)2 (66)

Different views of the function119881(119905 119883) are shown in Figure 3From (66) it is not difficult to show that

(1199092 + 1199102) le 119881 (119905 119883) le 3 (1199092 + 1199102) (67)

Figure 1 Behaviour of the function 119892(119909 119910)

minus6120587 minus4120587 minus2120587 612058741205872120587

ge minus0225

25

2

15

1

05

minus05

minus1

F(x) f(x)x

f(x)x = 1 + sin(x)x

F(x) = sin(x)xx

b = 076

F(x)

Figure 2 Bounds on the function 119891(119909)119909

for all 119905 ge 0 119909 and 119910 From (35) and (67) we have 1205750 = 11205751 = 3 119901 = 2 and 119902 = 2 and thus inequalities (67) satisfycondition (i) of Lemma 9 Also from the first inequality in(67) we have

119881 (119905 119883) 997888rarr +infin as 1199092 + 1199102 997888rarr infin (68)

Estimate (68) verifies (32) (ie the function 119881(119905 119883) definedby (66) is radially unbounded) Next applying Itorsquos formulain (66) using system (57) we find that

119871119881 (119905 119883) = 12119909119910 + 31199102 minus 119909 (3119909 + 2119910) (1 + sin119909119909 )minus 119910 (3119909 + 2119910) (3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816) + 11001199092minus 11990910 (3119909 + 2119910)+ (3119909 + 2119910) (1 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816)minus1

(69)

Using the estimates in items (i) to (iii) of Example 1 and theinequality 211990911199092 le 11990921 + 11990922 in (69) we obtain

119871119881 (119905 119883) le minus29 (1199092 + 1199102) + 3 (|119909| + 10038161003816100381610038161199101003816100381610038161003816) (70)

for all 119905 ge 0 119909 and 119910 Inequality (70) satisfies inequality (40)where 1205752 = 29 and 1205753 = 3 Since

(|119909| minus 105)2 + (10038161003816100381610038161199101003816100381610038161003816 minus 105)2 ge 0 (71)

for all 119909 and 119910 it follows from inequality (70) that

119871119881 (119905 119883) le minus145 (1199092 + 1199102) + 32 (72)

8 International Journal of Analysis

Figure 3 The behaviour of the function 119881(119905 119883)

for all 119905 ge 0 119909 and 119910 Inequality (72) satisfies assumption(ii) of Lemma 9 and estimate (44) with 120572(119905) = 1205754 = 145 and120573(119905) = 1205755 = 32 Since 119903 = 119901 = 119902 = 2 it follows that 120574 = 0 sothat assumption (iii) of Lemma 9 holds In addition

int1199051199050[(120574120572 (119906) + 120573 (119906)) 119890minusint119905119906120572 (119904) 119889119904] 119889119906 le 16 (73)

for all 119905 ge 1199050 ge 0 Estimate (73) satisfies (23) and (45) with119872 = 26 Furthermore

119881119909119894 (119905 119883)119866119894119896 (119905 119883) = minus 110 (31199092 + 2119909119910) (74)

and10038161003816100381610038161003816119881119909119894 (119905 119883)119866119894119896 (119905 119883)10038161003816100381610038161003816 le 25 (1199092 + 1199102) (75)

for all 119905 ge 0 119909 and119910 Inequality (75) satisfies inequalities (20)and (21) with

120582 (119905) = 25 (1199092 + 1199102) (76)

Hence by Corollary 11 (i) all solutions of stochastic differen-tial equation (57) are uniformly stochastically bounded

Example 2 If 119901(119905 119909 1199091015840) = 119901(119905 119909 119910) = 0 in (56) and system(57) we have the following stochastic differential equation

11990910158401015840 + (3 + 10038161003816100381610038161003816cos (1199091199091015840)10038161003816100381610038161003816) 1199091015840 + 119909 + sin119909 + 011199091205961015840 (119905)= 0 (77)

Equation (77) is equivalent to system

1199091015840 = 1199101199101015840 = minus (119909 + sin119909) minus [3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816] 119910 minus 011199091205961015840 (119905) (78)

Now from systems (52) and (78) items (i) and (ii) of Example 1hold Also equations (66) (67) and estimate (68) hold thatis

2119881 (119905 119883) = 31199092 + 1199102 + (3119909 + 119910)2 119881 (119905 0) = 0 forall119905 ge 0

(1199092 + 1199102) le 119881 (119905 119883) le 3 (1199092 + 1199102) forall119905 ge 0 119909 119910119881 (119905 119883) 997888rarr +infin as 1199092 + 1199102 997888rarr infin

(79)

112

1416

182

0500

10001500

0

02

04

06

08

1

t

x(t) y(t)

Figure 4 Graph of solutions of (56) in 3D

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(b)

Figure 5

Furthermore application of Itorsquos formula in (66) and usingsystem (78) yield

119871119881 (119905 119883) le minus29 (1199092 + 1199102) (80)

for all 119905 ge 0 119909 119910 and thus

119871119881 (119905 119883) le 0 (81)

International Journal of Analysis 9

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

000200400600801012

t

minus004

minus002

x(t)y(t)

x(t)

y(t)

times1011

(b)

Figure 6

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

000200400600801012

t

minus002

minus008

minus006

minus004

times1011

x(t)y(t)

x(t)

y(t)

(b)

Figure 7

for all 119905 ge 0 119909 and 119910 Moreover from (79) and (80)all assumptions of Theorem 17 and Lemma 6 are satisfiedThus by Lemma 6 the trivial solution of system (78) is notonly uniformly stochastically asymptotically stable but alsouniformly stochastically asymptotically stable in the largeFinally from (79) and (81) the function 119881(119905 119883) is positivedefinite and

119871119881 (119905 119883) le 0 forall (119905 119883) isin R+ timesR

2 (82)

Hence assumptions of Theorem 17 and Lemma 5 hold byTheorem 17 and Lemma 5 the trivial solution of system (78)is stochastically stable

Simulation of Solutions In what follows we shall nowsimulate the solutions of (56) (resp system (57)) and (78)(resp system (79)) Our approach depends on the Euler-Maruyama method which enables us to get approximatenumerical solution for the considered systems It will be seenfrom our figures that the simulated solutions are bounded

which justifies our given results For instance when 120590 = 01the numerical solutions of (56) in three-dimensional spaceare shown in Figure 4 If we vary the value of the noise inthe numerical solution (119909(119905) 119910(119905)) of system (57) as 120590 = 01and 120590 = 10 we have Figures 5(a) and 5(b) respectively Itcan be seen that when the noise is increased the stochasticitybecomes more pronounced The behaviour of the numericalsolution (119909(119905) 119910(119905)) of system (57) when 120590 = 05 and 120590 = 20is shown in Figures 6(a) and 6(b) respectivelyThe behaviourof the numerical solution (119909(119905) 119910(119905)) of system (57) for 120590 = 0and 120590 = 50 is shown in Figures 7(a) and 7(b) respectivelyFor the case of (78) Figure 8 shows the closeness of thesolution (119909(119905)) and the perturbed solution (119909120598(119905)) for a verylarge 119905 which implies asymptotic stability in the large for theconsidered SDE

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

10 International Journal of Analysis

0 1 2 3 4 5 6 7 8 9 100

0005

001

0015

002

0025

t times1011

x(t)x120576(t)

x(t)x120576(t)

Figure 8 Graph of solutions of (78)

References

[1] L Arnold Stochastic Differential Equations Theory and Appli-cations John Wiley amp Sons 1974

[2] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations vol 178 of Mathematics inScience and Engineering Academic Press Inc Orlando FlaUSA 1985

[3] T A Burton Volterra Integral and Differential EquationsAcademic Press New York NY USA 1983

[4] J K HaleTheory of Functional Differential Equations SpringerNew York NY USA 1977

[5] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer 2000

[6] L Shaikihet Lyapunov Functionals and Stability of StochasticFunctional Differential Equations Springer International 2013

[7] T Yoshizawa StabilityTheory and Existence of Periodic Solutionsand almost Periodic Solutions Spriger New York NY USA1975

[8] T Yoshizawa StabilityTheory by Liapunovrsquos SecondMethodTheMathematical Society of Japan 1966

[9] A M A Abou-El-Ela A I Sadek and A M Mahmoud ldquoOnthe stability of solutions for certain second-order stochasticdelay differential equationsrdquo Differential Equations and ControlProcesses no 2 pp 1ndash13 2015

[10] A M Abou-El-Ela A I Sadek A M Mahmoud and R OTaie ldquoOn the stochastic stability and boundedness of solutionsfor stochastic delay differential equation of the second orderrdquoChinese Journal of Mathematics vol 2015 Article ID 358936 8pages 2015

[11] A T Ademola Boundedness and Stability of Solutions to CertainSecond Order Differential Equations Differential Equations andControl Processes 2015

[12] A T Ademola B S Ogundare M O Ogundiran and O AAdesina ldquoPeriodicity stability and boundedness of solutions tocertain second order delay differential equationsrdquo InternationalJournal of Differential Equations vol 2016 Article ID 284370910 pages 2016

[13] J G Alaba and B S Ogundare ldquoOn stability and bound-edness properties of solutions of certain second order non-autonomous nonlinear ordinary differential equationrdquo Kragu-jevac Journal of Mathematics vol 39 no 2 pp 255ndash266 2015

[14] T A Burton and L Hatvani ldquoAsymptotic stability of secondorder ordinary functional and partial differential equationsrdquoJournal of Mathematical Analysis and Applications vol 176 no1 pp 261ndash281 1993

[15] B Cahlon andD Schmidt ldquoStability criteria for certain second-order delay differential equations with mixed coefficientsrdquoJournal of Computational and AppliedMathematics vol 170 no1 pp 79ndash102 2004

[16] T Caraballo M A Diop and A S Ndoye ldquoFixed points andexponential stability for stochastic partial integro-differentialequations with delaysrdquo Advances in Dynamical Systems andApplications vol 9 no 2 pp 133ndash147 2014

[17] A Domoshnitsky ldquoNonoscillation maximum principles andexponential stability of secondorder delay differential equationswithout damping termrdquo Domoshnitsky Journal of Inequalitiesand Applications vol 2014 article 361 2014

[18] I I Gikhman and A V Skorokhod Stochastische Differential-gleichungen Akademie Berlin Germany 1971 (Russian)

[19] I I Gikhman On the Stability of the Solutions of StochasticDifferential Equations Predelrsquonyye Teoremy i StatisticheskiyeVyvody Tashkent Uzbekistan 1966

[20] G A Grigoryan ldquoBoundedness and stability criteria for linearordinary differential equations of the second orderrdquo RussianMathematics vol 57 no 12 pp 8ndash15 2013

[21] A F Ivanov Y I Kazmerchuk and A V Swishchuk ldquoThe-ory stochastic stability and applications of stochastic delaydifferential equations a survey of recent resultsrdquo in DifferentialEquations and Dynamical Systems vol 11 no 1 2003

[22] F Jedrzejewski and D Brochard ldquoLyapounv exponents andstability stochastic dynamical structuresrdquo 2000

[23] Z Jin and L Zengrong ldquoOn the global asymptotic behaviorof solutions to a non autonomous generalized Lienard systemrdquoJournal of Mathematical Research and Exposition vol 21 no 3pp 410ndash414 2001

[24] E Kolarova ldquoAn application of stochastic integral equations toelectrical networksrdquo Acta Electrotechnica et Informatica vol 8no 3 pp 14ndash17 2008

[25] V B Kolmanovskii and L E Shaikhet ldquoA method for con-structing Lyapunov functionals for stochastic systems with aftereffectrdquo Differentsialrsquonye Uravneniya vol 29 no 11 pp 1909ndash2022 1993

[26] V Kolmanovskii and L Shaikhet ldquoConstruction of Lyapunovfunctionals for stochastic hereditary systems a survey of somerecent resultsrdquo Mathematical and Computer Modelling vol 36no 6 pp 691ndash716 2002

[27] A J Kroopnick ldquoBounded solutions to 11990910158401015840 + 119902(119905)119887(119909) = 119891(119905)rdquoInternational Journal of Mathematical Education in Science andTechnology vol 41 no 6 pp 829ndash836 2010

[28] R Liu and Y Raffoul ldquoBoundedness and exponential stabilityof highly nonlinear stochastic differential equationsrdquo ElectronicJournal of Differential Equations vol 2009 no 143 pp 1ndash102009

[29] X Mao ldquoSome contributions to stochastic asymptotic stabilityand boundedness via multiple Lyapunov functionsrdquo Journal ofMathematical Analysis and Applications vol 260 no 2 pp 325ndash340 2001

[30] B S Ogundare A T Ademola M O Ogundiran and O AAdesina ldquoOn the qualitative behaviour of solutions to certainsecond order nonlinear differential equation with delayrdquoAnnalidellrsquoUniversitarsquo di Ferrara 2016

International Journal of Analysis 11

[31] B S Ogundare and A U Afuwape ldquoBoundedness and stabilityproperties of solutions of generalized Lienard equationrdquo KochiJournal of Mathematics vol 9 pp 97ndash108 2014

[32] B S Ogundare and G E Okecha ldquoBoundedness periodicityand stability of solutions to x(t) + g(x) + b(t)h(x) = p(t x x)rdquoMathematical Sciences Research Journal vol 11 no 5 pp 432ndash443 2007

[33] Y N Raffoul ldquoBoundedness and exponential asymptotic sta-bility in dynamical systems with applications to nonlineardifferential equations with unbounded termsrdquo Advances inDynamical Systems and Applications vol 2 no 1 pp 107ndash1212007

[34] R Rezaeyan and R Farnoosh ldquoStochastic differential equationsand application of the Kalman-Bucy filter in the modeling ofRC circuitrdquo Applied Mathematical Sciences vol 4 no 21-24 pp1119ndash1127 2010

[35] C Tunc ldquoA note on the stability and boundedness of non-autonomous differential equations of second order with avariable deviating argumentrdquo Afrika Matematika vol 25 no 2pp 417ndash425 2014

[36] C Tunc ldquoA note on the bounded solutions to 11990910158401015840 + 119888(119905 119909 1199091015840 ) +119902(119905)119887(119909) = 119891(119905)rdquo Applied Mathematics amp Information Sciencesvol 8 no 1 pp 393ndash399 2014

[37] C Tunc ldquoBoundedness analysis for certain two-dimensionaldifferential systems via a Lyapunov approachrdquo Bulletin Mathe-matique de la Societe des Sciences Mathematiques de Roumanievol 53 no 1 pp 61ndash68 2010

[38] 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

[39] C Tunc ldquoStability and boundedness in multi delay vectorLienard equationrdquo Filomat vol 27 no 3 pp 435ndash445 2013

[40] C Tunc ldquoStability and boundedness of solutions of non-autonomous differential equations of second orderrdquo Journalof Computational Analysis and Applications vol 13 no 6 pp1067ndash1074 2011

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

[42] 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

[43] C Tunc and T Ayhan ldquoGlobal existence and boundedness ofsolutions of a certain nonlinear integro-differential equationof second order with multiple deviating argumentsrdquo Journal ofInequalities and Applications vol 2016 article no 46 2016

[44] F Wang and H Zhu ldquoExistence uniqueness and stabilityof periodic solutions of a duffing equation under periodicand anti-periodic eigenvalues conditionsrdquo Taiwanese Journal ofMathematics vol 19 no 5 pp 1457ndash1468 2015

[45] Z Xianfeng and J Wei ldquoStability and boundedness of aretarded Lienard-type equationrdquo Chinese Quarterly Journal ofMathematics vol 18 no 1 pp 7ndash12 2003

[46] A F Yenicerioglu ldquoThe behavior of solutions of second orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 332 no 2 pp 1278ndash1290 2007

[47] A F Yenicerioglu ldquoStability properties of second order delayintegro-differential equationsrdquo Computers and Mathematicswith Applications vol 56 no 12 pp 3109ndash3117 2008

[48] T Yoshizawa ldquoLiapunovrsquos function and boundedness of solu-tionsrdquo Funkcialaj Ekvacioj vol 2 pp 71ndash103 1958

[49] W Zhu J Huang X Ruan and Z Zhao ldquoExponential stabilityof stochastic differential equation with mixed delayrdquo Journal ofAppliedMathematics vol 2014 Article ID 187037 11 pages 2014

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

International Journal of Analysis 7

Equation (56) is equivalent to system

1199091015840 = 1199101199101015840 = (1 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816)minus1 minus (119909 + sin119909)

minus [3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816] 119910 minus 011199091205961015840 (119905) (57)

Now from systems (9) and (57) we have the followingrelations

(i) The function

119892 (119909 119910) fl 3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816 (58)

Noting that 1003816100381610038161003816cos (119909119910)1003816100381610038161003816 ge 0 (59)

for all 119909 and 119910 it follows that119892 (119909 119910) = 3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816 ge 119886 = 3 (60)

for all 119909 and 119910The behaviour of the function 119892(119909 119910)is shown below in Figure 1

(ii) The function

119891 (119909) fl 119909 + sin119909 (61)

Since

minus02 le 119865 (119909) = sin119909119909 le 1 (62)

for all 119909 = 0 then we have

1 = 119887 le 119891 (119909)119909 = 1 + sin119909119909 le 119861 = 2 (63)

for all 119909 = 0 and since 120590 fl 01 it follows that1205902 lt 2119886119887(119887 + 1)minus1 implies that 0 lt 299 The function119891(119909)119909 and its bounds are shown in Figure 2(iii) The function

119901 (119905 119909 119910) fl 11 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816 (64)

Clearly

1003816100381610038161003816119901 (119905 119909 119910)1003816100381610038161003816 = 11 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816 le 1 = 1198720 (65)

for all 119905 ge 0 119909 and 119910Now from items (i) (ii) above and (24) the continuouslydifferentiable function 119881(119905 119883) used for system (57) is

2119881 (119905 119883) = 31199092 + 1199102 + (3119909 + 119910)2 (66)

Different views of the function119881(119905 119883) are shown in Figure 3From (66) it is not difficult to show that

(1199092 + 1199102) le 119881 (119905 119883) le 3 (1199092 + 1199102) (67)

Figure 1 Behaviour of the function 119892(119909 119910)

minus6120587 minus4120587 minus2120587 612058741205872120587

ge minus0225

25

2

15

1

05

minus05

minus1

F(x) f(x)x

f(x)x = 1 + sin(x)x

F(x) = sin(x)xx

b = 076

F(x)

Figure 2 Bounds on the function 119891(119909)119909

for all 119905 ge 0 119909 and 119910 From (35) and (67) we have 1205750 = 11205751 = 3 119901 = 2 and 119902 = 2 and thus inequalities (67) satisfycondition (i) of Lemma 9 Also from the first inequality in(67) we have

119881 (119905 119883) 997888rarr +infin as 1199092 + 1199102 997888rarr infin (68)

Estimate (68) verifies (32) (ie the function 119881(119905 119883) definedby (66) is radially unbounded) Next applying Itorsquos formulain (66) using system (57) we find that

119871119881 (119905 119883) = 12119909119910 + 31199102 minus 119909 (3119909 + 2119910) (1 + sin119909119909 )minus 119910 (3119909 + 2119910) (3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816) + 11001199092minus 11990910 (3119909 + 2119910)+ (3119909 + 2119910) (1 + 2119905 + 10038161003816100381610038161199091199101003816100381610038161003816)minus1

(69)

Using the estimates in items (i) to (iii) of Example 1 and theinequality 211990911199092 le 11990921 + 11990922 in (69) we obtain

119871119881 (119905 119883) le minus29 (1199092 + 1199102) + 3 (|119909| + 10038161003816100381610038161199101003816100381610038161003816) (70)

for all 119905 ge 0 119909 and 119910 Inequality (70) satisfies inequality (40)where 1205752 = 29 and 1205753 = 3 Since

(|119909| minus 105)2 + (10038161003816100381610038161199101003816100381610038161003816 minus 105)2 ge 0 (71)

for all 119909 and 119910 it follows from inequality (70) that

119871119881 (119905 119883) le minus145 (1199092 + 1199102) + 32 (72)

8 International Journal of Analysis

Figure 3 The behaviour of the function 119881(119905 119883)

for all 119905 ge 0 119909 and 119910 Inequality (72) satisfies assumption(ii) of Lemma 9 and estimate (44) with 120572(119905) = 1205754 = 145 and120573(119905) = 1205755 = 32 Since 119903 = 119901 = 119902 = 2 it follows that 120574 = 0 sothat assumption (iii) of Lemma 9 holds In addition

int1199051199050[(120574120572 (119906) + 120573 (119906)) 119890minusint119905119906120572 (119904) 119889119904] 119889119906 le 16 (73)

for all 119905 ge 1199050 ge 0 Estimate (73) satisfies (23) and (45) with119872 = 26 Furthermore

119881119909119894 (119905 119883)119866119894119896 (119905 119883) = minus 110 (31199092 + 2119909119910) (74)

and10038161003816100381610038161003816119881119909119894 (119905 119883)119866119894119896 (119905 119883)10038161003816100381610038161003816 le 25 (1199092 + 1199102) (75)

for all 119905 ge 0 119909 and119910 Inequality (75) satisfies inequalities (20)and (21) with

120582 (119905) = 25 (1199092 + 1199102) (76)

Hence by Corollary 11 (i) all solutions of stochastic differen-tial equation (57) are uniformly stochastically bounded

Example 2 If 119901(119905 119909 1199091015840) = 119901(119905 119909 119910) = 0 in (56) and system(57) we have the following stochastic differential equation

11990910158401015840 + (3 + 10038161003816100381610038161003816cos (1199091199091015840)10038161003816100381610038161003816) 1199091015840 + 119909 + sin119909 + 011199091205961015840 (119905)= 0 (77)

Equation (77) is equivalent to system

1199091015840 = 1199101199101015840 = minus (119909 + sin119909) minus [3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816] 119910 minus 011199091205961015840 (119905) (78)

Now from systems (52) and (78) items (i) and (ii) of Example 1hold Also equations (66) (67) and estimate (68) hold thatis

2119881 (119905 119883) = 31199092 + 1199102 + (3119909 + 119910)2 119881 (119905 0) = 0 forall119905 ge 0

(1199092 + 1199102) le 119881 (119905 119883) le 3 (1199092 + 1199102) forall119905 ge 0 119909 119910119881 (119905 119883) 997888rarr +infin as 1199092 + 1199102 997888rarr infin

(79)

112

1416

182

0500

10001500

0

02

04

06

08

1

t

x(t) y(t)

Figure 4 Graph of solutions of (56) in 3D

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(b)

Figure 5

Furthermore application of Itorsquos formula in (66) and usingsystem (78) yield

119871119881 (119905 119883) le minus29 (1199092 + 1199102) (80)

for all 119905 ge 0 119909 119910 and thus

119871119881 (119905 119883) le 0 (81)

International Journal of Analysis 9

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

000200400600801012

t

minus004

minus002

x(t)y(t)

x(t)

y(t)

times1011

(b)

Figure 6

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

000200400600801012

t

minus002

minus008

minus006

minus004

times1011

x(t)y(t)

x(t)

y(t)

(b)

Figure 7

for all 119905 ge 0 119909 and 119910 Moreover from (79) and (80)all assumptions of Theorem 17 and Lemma 6 are satisfiedThus by Lemma 6 the trivial solution of system (78) is notonly uniformly stochastically asymptotically stable but alsouniformly stochastically asymptotically stable in the largeFinally from (79) and (81) the function 119881(119905 119883) is positivedefinite and

119871119881 (119905 119883) le 0 forall (119905 119883) isin R+ timesR

2 (82)

Hence assumptions of Theorem 17 and Lemma 5 hold byTheorem 17 and Lemma 5 the trivial solution of system (78)is stochastically stable

Simulation of Solutions In what follows we shall nowsimulate the solutions of (56) (resp system (57)) and (78)(resp system (79)) Our approach depends on the Euler-Maruyama method which enables us to get approximatenumerical solution for the considered systems It will be seenfrom our figures that the simulated solutions are bounded

which justifies our given results For instance when 120590 = 01the numerical solutions of (56) in three-dimensional spaceare shown in Figure 4 If we vary the value of the noise inthe numerical solution (119909(119905) 119910(119905)) of system (57) as 120590 = 01and 120590 = 10 we have Figures 5(a) and 5(b) respectively Itcan be seen that when the noise is increased the stochasticitybecomes more pronounced The behaviour of the numericalsolution (119909(119905) 119910(119905)) of system (57) when 120590 = 05 and 120590 = 20is shown in Figures 6(a) and 6(b) respectivelyThe behaviourof the numerical solution (119909(119905) 119910(119905)) of system (57) for 120590 = 0and 120590 = 50 is shown in Figures 7(a) and 7(b) respectivelyFor the case of (78) Figure 8 shows the closeness of thesolution (119909(119905)) and the perturbed solution (119909120598(119905)) for a verylarge 119905 which implies asymptotic stability in the large for theconsidered SDE

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

10 International Journal of Analysis

0 1 2 3 4 5 6 7 8 9 100

0005

001

0015

002

0025

t times1011

x(t)x120576(t)

x(t)x120576(t)

Figure 8 Graph of solutions of (78)

References

[1] L Arnold Stochastic Differential Equations Theory and Appli-cations John Wiley amp Sons 1974

[2] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations vol 178 of Mathematics inScience and Engineering Academic Press Inc Orlando FlaUSA 1985

[3] T A Burton Volterra Integral and Differential EquationsAcademic Press New York NY USA 1983

[4] J K HaleTheory of Functional Differential Equations SpringerNew York NY USA 1977

[5] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer 2000

[6] L Shaikihet Lyapunov Functionals and Stability of StochasticFunctional Differential Equations Springer International 2013

[7] T Yoshizawa StabilityTheory and Existence of Periodic Solutionsand almost Periodic Solutions Spriger New York NY USA1975

[8] T Yoshizawa StabilityTheory by Liapunovrsquos SecondMethodTheMathematical Society of Japan 1966

[9] A M A Abou-El-Ela A I Sadek and A M Mahmoud ldquoOnthe stability of solutions for certain second-order stochasticdelay differential equationsrdquo Differential Equations and ControlProcesses no 2 pp 1ndash13 2015

[10] A M Abou-El-Ela A I Sadek A M Mahmoud and R OTaie ldquoOn the stochastic stability and boundedness of solutionsfor stochastic delay differential equation of the second orderrdquoChinese Journal of Mathematics vol 2015 Article ID 358936 8pages 2015

[11] A T Ademola Boundedness and Stability of Solutions to CertainSecond Order Differential Equations Differential Equations andControl Processes 2015

[12] A T Ademola B S Ogundare M O Ogundiran and O AAdesina ldquoPeriodicity stability and boundedness of solutions tocertain second order delay differential equationsrdquo InternationalJournal of Differential Equations vol 2016 Article ID 284370910 pages 2016

[13] J G Alaba and B S Ogundare ldquoOn stability and bound-edness properties of solutions of certain second order non-autonomous nonlinear ordinary differential equationrdquo Kragu-jevac Journal of Mathematics vol 39 no 2 pp 255ndash266 2015

[14] T A Burton and L Hatvani ldquoAsymptotic stability of secondorder ordinary functional and partial differential equationsrdquoJournal of Mathematical Analysis and Applications vol 176 no1 pp 261ndash281 1993

[15] B Cahlon andD Schmidt ldquoStability criteria for certain second-order delay differential equations with mixed coefficientsrdquoJournal of Computational and AppliedMathematics vol 170 no1 pp 79ndash102 2004

[16] T Caraballo M A Diop and A S Ndoye ldquoFixed points andexponential stability for stochastic partial integro-differentialequations with delaysrdquo Advances in Dynamical Systems andApplications vol 9 no 2 pp 133ndash147 2014

[17] A Domoshnitsky ldquoNonoscillation maximum principles andexponential stability of secondorder delay differential equationswithout damping termrdquo Domoshnitsky Journal of Inequalitiesand Applications vol 2014 article 361 2014

[18] I I Gikhman and A V Skorokhod Stochastische Differential-gleichungen Akademie Berlin Germany 1971 (Russian)

[19] I I Gikhman On the Stability of the Solutions of StochasticDifferential Equations Predelrsquonyye Teoremy i StatisticheskiyeVyvody Tashkent Uzbekistan 1966

[20] G A Grigoryan ldquoBoundedness and stability criteria for linearordinary differential equations of the second orderrdquo RussianMathematics vol 57 no 12 pp 8ndash15 2013

[21] A F Ivanov Y I Kazmerchuk and A V Swishchuk ldquoThe-ory stochastic stability and applications of stochastic delaydifferential equations a survey of recent resultsrdquo in DifferentialEquations and Dynamical Systems vol 11 no 1 2003

[22] F Jedrzejewski and D Brochard ldquoLyapounv exponents andstability stochastic dynamical structuresrdquo 2000

[23] Z Jin and L Zengrong ldquoOn the global asymptotic behaviorof solutions to a non autonomous generalized Lienard systemrdquoJournal of Mathematical Research and Exposition vol 21 no 3pp 410ndash414 2001

[24] E Kolarova ldquoAn application of stochastic integral equations toelectrical networksrdquo Acta Electrotechnica et Informatica vol 8no 3 pp 14ndash17 2008

[25] V B Kolmanovskii and L E Shaikhet ldquoA method for con-structing Lyapunov functionals for stochastic systems with aftereffectrdquo Differentsialrsquonye Uravneniya vol 29 no 11 pp 1909ndash2022 1993

[26] V Kolmanovskii and L Shaikhet ldquoConstruction of Lyapunovfunctionals for stochastic hereditary systems a survey of somerecent resultsrdquo Mathematical and Computer Modelling vol 36no 6 pp 691ndash716 2002

[27] A J Kroopnick ldquoBounded solutions to 11990910158401015840 + 119902(119905)119887(119909) = 119891(119905)rdquoInternational Journal of Mathematical Education in Science andTechnology vol 41 no 6 pp 829ndash836 2010

[28] R Liu and Y Raffoul ldquoBoundedness and exponential stabilityof highly nonlinear stochastic differential equationsrdquo ElectronicJournal of Differential Equations vol 2009 no 143 pp 1ndash102009

[29] X Mao ldquoSome contributions to stochastic asymptotic stabilityand boundedness via multiple Lyapunov functionsrdquo Journal ofMathematical Analysis and Applications vol 260 no 2 pp 325ndash340 2001

[30] B S Ogundare A T Ademola M O Ogundiran and O AAdesina ldquoOn the qualitative behaviour of solutions to certainsecond order nonlinear differential equation with delayrdquoAnnalidellrsquoUniversitarsquo di Ferrara 2016

International Journal of Analysis 11

[31] B S Ogundare and A U Afuwape ldquoBoundedness and stabilityproperties of solutions of generalized Lienard equationrdquo KochiJournal of Mathematics vol 9 pp 97ndash108 2014

[32] B S Ogundare and G E Okecha ldquoBoundedness periodicityand stability of solutions to x(t) + g(x) + b(t)h(x) = p(t x x)rdquoMathematical Sciences Research Journal vol 11 no 5 pp 432ndash443 2007

[33] Y N Raffoul ldquoBoundedness and exponential asymptotic sta-bility in dynamical systems with applications to nonlineardifferential equations with unbounded termsrdquo Advances inDynamical Systems and Applications vol 2 no 1 pp 107ndash1212007

[34] R Rezaeyan and R Farnoosh ldquoStochastic differential equationsand application of the Kalman-Bucy filter in the modeling ofRC circuitrdquo Applied Mathematical Sciences vol 4 no 21-24 pp1119ndash1127 2010

[35] C Tunc ldquoA note on the stability and boundedness of non-autonomous differential equations of second order with avariable deviating argumentrdquo Afrika Matematika vol 25 no 2pp 417ndash425 2014

[36] C Tunc ldquoA note on the bounded solutions to 11990910158401015840 + 119888(119905 119909 1199091015840 ) +119902(119905)119887(119909) = 119891(119905)rdquo Applied Mathematics amp Information Sciencesvol 8 no 1 pp 393ndash399 2014

[37] C Tunc ldquoBoundedness analysis for certain two-dimensionaldifferential systems via a Lyapunov approachrdquo Bulletin Mathe-matique de la Societe des Sciences Mathematiques de Roumanievol 53 no 1 pp 61ndash68 2010

[38] 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

[39] C Tunc ldquoStability and boundedness in multi delay vectorLienard equationrdquo Filomat vol 27 no 3 pp 435ndash445 2013

[40] C Tunc ldquoStability and boundedness of solutions of non-autonomous differential equations of second orderrdquo Journalof Computational Analysis and Applications vol 13 no 6 pp1067ndash1074 2011

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

[42] 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

[43] C Tunc and T Ayhan ldquoGlobal existence and boundedness ofsolutions of a certain nonlinear integro-differential equationof second order with multiple deviating argumentsrdquo Journal ofInequalities and Applications vol 2016 article no 46 2016

[44] F Wang and H Zhu ldquoExistence uniqueness and stabilityof periodic solutions of a duffing equation under periodicand anti-periodic eigenvalues conditionsrdquo Taiwanese Journal ofMathematics vol 19 no 5 pp 1457ndash1468 2015

[45] Z Xianfeng and J Wei ldquoStability and boundedness of aretarded Lienard-type equationrdquo Chinese Quarterly Journal ofMathematics vol 18 no 1 pp 7ndash12 2003

[46] A F Yenicerioglu ldquoThe behavior of solutions of second orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 332 no 2 pp 1278ndash1290 2007

[47] A F Yenicerioglu ldquoStability properties of second order delayintegro-differential equationsrdquo Computers and Mathematicswith Applications vol 56 no 12 pp 3109ndash3117 2008

[48] T Yoshizawa ldquoLiapunovrsquos function and boundedness of solu-tionsrdquo Funkcialaj Ekvacioj vol 2 pp 71ndash103 1958

[49] W Zhu J Huang X Ruan and Z Zhao ldquoExponential stabilityof stochastic differential equation with mixed delayrdquo Journal ofAppliedMathematics vol 2014 Article ID 187037 11 pages 2014

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

8 International Journal of Analysis

Figure 3 The behaviour of the function 119881(119905 119883)

for all 119905 ge 0 119909 and 119910 Inequality (72) satisfies assumption(ii) of Lemma 9 and estimate (44) with 120572(119905) = 1205754 = 145 and120573(119905) = 1205755 = 32 Since 119903 = 119901 = 119902 = 2 it follows that 120574 = 0 sothat assumption (iii) of Lemma 9 holds In addition

int1199051199050[(120574120572 (119906) + 120573 (119906)) 119890minusint119905119906120572 (119904) 119889119904] 119889119906 le 16 (73)

for all 119905 ge 1199050 ge 0 Estimate (73) satisfies (23) and (45) with119872 = 26 Furthermore

119881119909119894 (119905 119883)119866119894119896 (119905 119883) = minus 110 (31199092 + 2119909119910) (74)

and10038161003816100381610038161003816119881119909119894 (119905 119883)119866119894119896 (119905 119883)10038161003816100381610038161003816 le 25 (1199092 + 1199102) (75)

for all 119905 ge 0 119909 and119910 Inequality (75) satisfies inequalities (20)and (21) with

120582 (119905) = 25 (1199092 + 1199102) (76)

Hence by Corollary 11 (i) all solutions of stochastic differen-tial equation (57) are uniformly stochastically bounded

Example 2 If 119901(119905 119909 1199091015840) = 119901(119905 119909 119910) = 0 in (56) and system(57) we have the following stochastic differential equation

11990910158401015840 + (3 + 10038161003816100381610038161003816cos (1199091199091015840)10038161003816100381610038161003816) 1199091015840 + 119909 + sin119909 + 011199091205961015840 (119905)= 0 (77)

Equation (77) is equivalent to system

1199091015840 = 1199101199101015840 = minus (119909 + sin119909) minus [3 + 1003816100381610038161003816cos (119909119910)1003816100381610038161003816] 119910 minus 011199091205961015840 (119905) (78)

Now from systems (52) and (78) items (i) and (ii) of Example 1hold Also equations (66) (67) and estimate (68) hold thatis

2119881 (119905 119883) = 31199092 + 1199102 + (3119909 + 119910)2 119881 (119905 0) = 0 forall119905 ge 0

(1199092 + 1199102) le 119881 (119905 119883) le 3 (1199092 + 1199102) forall119905 ge 0 119909 119910119881 (119905 119883) 997888rarr +infin as 1199092 + 1199102 997888rarr infin

(79)

112

1416

182

0500

10001500

0

02

04

06

08

1

t

x(t) y(t)

Figure 4 Graph of solutions of (56) in 3D

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(b)

Figure 5

Furthermore application of Itorsquos formula in (66) and usingsystem (78) yield

119871119881 (119905 119883) le minus29 (1199092 + 1199102) (80)

for all 119905 ge 0 119909 119910 and thus

119871119881 (119905 119883) le 0 (81)

International Journal of Analysis 9

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

000200400600801012

t

minus004

minus002

x(t)y(t)

x(t)

y(t)

times1011

(b)

Figure 6

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

000200400600801012

t

minus002

minus008

minus006

minus004

times1011

x(t)y(t)

x(t)

y(t)

(b)

Figure 7

for all 119905 ge 0 119909 and 119910 Moreover from (79) and (80)all assumptions of Theorem 17 and Lemma 6 are satisfiedThus by Lemma 6 the trivial solution of system (78) is notonly uniformly stochastically asymptotically stable but alsouniformly stochastically asymptotically stable in the largeFinally from (79) and (81) the function 119881(119905 119883) is positivedefinite and

119871119881 (119905 119883) le 0 forall (119905 119883) isin R+ timesR

2 (82)

Hence assumptions of Theorem 17 and Lemma 5 hold byTheorem 17 and Lemma 5 the trivial solution of system (78)is stochastically stable

Simulation of Solutions In what follows we shall nowsimulate the solutions of (56) (resp system (57)) and (78)(resp system (79)) Our approach depends on the Euler-Maruyama method which enables us to get approximatenumerical solution for the considered systems It will be seenfrom our figures that the simulated solutions are bounded

which justifies our given results For instance when 120590 = 01the numerical solutions of (56) in three-dimensional spaceare shown in Figure 4 If we vary the value of the noise inthe numerical solution (119909(119905) 119910(119905)) of system (57) as 120590 = 01and 120590 = 10 we have Figures 5(a) and 5(b) respectively Itcan be seen that when the noise is increased the stochasticitybecomes more pronounced The behaviour of the numericalsolution (119909(119905) 119910(119905)) of system (57) when 120590 = 05 and 120590 = 20is shown in Figures 6(a) and 6(b) respectivelyThe behaviourof the numerical solution (119909(119905) 119910(119905)) of system (57) for 120590 = 0and 120590 = 50 is shown in Figures 7(a) and 7(b) respectivelyFor the case of (78) Figure 8 shows the closeness of thesolution (119909(119905)) and the perturbed solution (119909120598(119905)) for a verylarge 119905 which implies asymptotic stability in the large for theconsidered SDE

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

10 International Journal of Analysis

0 1 2 3 4 5 6 7 8 9 100

0005

001

0015

002

0025

t times1011

x(t)x120576(t)

x(t)x120576(t)

Figure 8 Graph of solutions of (78)

References

[1] L Arnold Stochastic Differential Equations Theory and Appli-cations John Wiley amp Sons 1974

[2] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations vol 178 of Mathematics inScience and Engineering Academic Press Inc Orlando FlaUSA 1985

[3] T A Burton Volterra Integral and Differential EquationsAcademic Press New York NY USA 1983

[4] J K HaleTheory of Functional Differential Equations SpringerNew York NY USA 1977

[5] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer 2000

[6] L Shaikihet Lyapunov Functionals and Stability of StochasticFunctional Differential Equations Springer International 2013

[7] T Yoshizawa StabilityTheory and Existence of Periodic Solutionsand almost Periodic Solutions Spriger New York NY USA1975

[8] T Yoshizawa StabilityTheory by Liapunovrsquos SecondMethodTheMathematical Society of Japan 1966

[9] A M A Abou-El-Ela A I Sadek and A M Mahmoud ldquoOnthe stability of solutions for certain second-order stochasticdelay differential equationsrdquo Differential Equations and ControlProcesses no 2 pp 1ndash13 2015

[10] A M Abou-El-Ela A I Sadek A M Mahmoud and R OTaie ldquoOn the stochastic stability and boundedness of solutionsfor stochastic delay differential equation of the second orderrdquoChinese Journal of Mathematics vol 2015 Article ID 358936 8pages 2015

[11] A T Ademola Boundedness and Stability of Solutions to CertainSecond Order Differential Equations Differential Equations andControl Processes 2015

[12] A T Ademola B S Ogundare M O Ogundiran and O AAdesina ldquoPeriodicity stability and boundedness of solutions tocertain second order delay differential equationsrdquo InternationalJournal of Differential Equations vol 2016 Article ID 284370910 pages 2016

[13] J G Alaba and B S Ogundare ldquoOn stability and bound-edness properties of solutions of certain second order non-autonomous nonlinear ordinary differential equationrdquo Kragu-jevac Journal of Mathematics vol 39 no 2 pp 255ndash266 2015

[14] T A Burton and L Hatvani ldquoAsymptotic stability of secondorder ordinary functional and partial differential equationsrdquoJournal of Mathematical Analysis and Applications vol 176 no1 pp 261ndash281 1993

[15] B Cahlon andD Schmidt ldquoStability criteria for certain second-order delay differential equations with mixed coefficientsrdquoJournal of Computational and AppliedMathematics vol 170 no1 pp 79ndash102 2004

[16] T Caraballo M A Diop and A S Ndoye ldquoFixed points andexponential stability for stochastic partial integro-differentialequations with delaysrdquo Advances in Dynamical Systems andApplications vol 9 no 2 pp 133ndash147 2014

[17] A Domoshnitsky ldquoNonoscillation maximum principles andexponential stability of secondorder delay differential equationswithout damping termrdquo Domoshnitsky Journal of Inequalitiesand Applications vol 2014 article 361 2014

[18] I I Gikhman and A V Skorokhod Stochastische Differential-gleichungen Akademie Berlin Germany 1971 (Russian)

[19] I I Gikhman On the Stability of the Solutions of StochasticDifferential Equations Predelrsquonyye Teoremy i StatisticheskiyeVyvody Tashkent Uzbekistan 1966

[20] G A Grigoryan ldquoBoundedness and stability criteria for linearordinary differential equations of the second orderrdquo RussianMathematics vol 57 no 12 pp 8ndash15 2013

[21] A F Ivanov Y I Kazmerchuk and A V Swishchuk ldquoThe-ory stochastic stability and applications of stochastic delaydifferential equations a survey of recent resultsrdquo in DifferentialEquations and Dynamical Systems vol 11 no 1 2003

[22] F Jedrzejewski and D Brochard ldquoLyapounv exponents andstability stochastic dynamical structuresrdquo 2000

[23] Z Jin and L Zengrong ldquoOn the global asymptotic behaviorof solutions to a non autonomous generalized Lienard systemrdquoJournal of Mathematical Research and Exposition vol 21 no 3pp 410ndash414 2001

[24] E Kolarova ldquoAn application of stochastic integral equations toelectrical networksrdquo Acta Electrotechnica et Informatica vol 8no 3 pp 14ndash17 2008

[25] V B Kolmanovskii and L E Shaikhet ldquoA method for con-structing Lyapunov functionals for stochastic systems with aftereffectrdquo Differentsialrsquonye Uravneniya vol 29 no 11 pp 1909ndash2022 1993

[26] V Kolmanovskii and L Shaikhet ldquoConstruction of Lyapunovfunctionals for stochastic hereditary systems a survey of somerecent resultsrdquo Mathematical and Computer Modelling vol 36no 6 pp 691ndash716 2002

[27] A J Kroopnick ldquoBounded solutions to 11990910158401015840 + 119902(119905)119887(119909) = 119891(119905)rdquoInternational Journal of Mathematical Education in Science andTechnology vol 41 no 6 pp 829ndash836 2010

[28] R Liu and Y Raffoul ldquoBoundedness and exponential stabilityof highly nonlinear stochastic differential equationsrdquo ElectronicJournal of Differential Equations vol 2009 no 143 pp 1ndash102009

[29] X Mao ldquoSome contributions to stochastic asymptotic stabilityand boundedness via multiple Lyapunov functionsrdquo Journal ofMathematical Analysis and Applications vol 260 no 2 pp 325ndash340 2001

[30] B S Ogundare A T Ademola M O Ogundiran and O AAdesina ldquoOn the qualitative behaviour of solutions to certainsecond order nonlinear differential equation with delayrdquoAnnalidellrsquoUniversitarsquo di Ferrara 2016

International Journal of Analysis 11

[31] B S Ogundare and A U Afuwape ldquoBoundedness and stabilityproperties of solutions of generalized Lienard equationrdquo KochiJournal of Mathematics vol 9 pp 97ndash108 2014

[32] B S Ogundare and G E Okecha ldquoBoundedness periodicityand stability of solutions to x(t) + g(x) + b(t)h(x) = p(t x x)rdquoMathematical Sciences Research Journal vol 11 no 5 pp 432ndash443 2007

[33] Y N Raffoul ldquoBoundedness and exponential asymptotic sta-bility in dynamical systems with applications to nonlineardifferential equations with unbounded termsrdquo Advances inDynamical Systems and Applications vol 2 no 1 pp 107ndash1212007

[34] R Rezaeyan and R Farnoosh ldquoStochastic differential equationsand application of the Kalman-Bucy filter in the modeling ofRC circuitrdquo Applied Mathematical Sciences vol 4 no 21-24 pp1119ndash1127 2010

[35] C Tunc ldquoA note on the stability and boundedness of non-autonomous differential equations of second order with avariable deviating argumentrdquo Afrika Matematika vol 25 no 2pp 417ndash425 2014

[36] C Tunc ldquoA note on the bounded solutions to 11990910158401015840 + 119888(119905 119909 1199091015840 ) +119902(119905)119887(119909) = 119891(119905)rdquo Applied Mathematics amp Information Sciencesvol 8 no 1 pp 393ndash399 2014

[37] C Tunc ldquoBoundedness analysis for certain two-dimensionaldifferential systems via a Lyapunov approachrdquo Bulletin Mathe-matique de la Societe des Sciences Mathematiques de Roumanievol 53 no 1 pp 61ndash68 2010

[38] 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

[39] C Tunc ldquoStability and boundedness in multi delay vectorLienard equationrdquo Filomat vol 27 no 3 pp 435ndash445 2013

[40] C Tunc ldquoStability and boundedness of solutions of non-autonomous differential equations of second orderrdquo Journalof Computational Analysis and Applications vol 13 no 6 pp1067ndash1074 2011

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

[42] 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

[43] C Tunc and T Ayhan ldquoGlobal existence and boundedness ofsolutions of a certain nonlinear integro-differential equationof second order with multiple deviating argumentsrdquo Journal ofInequalities and Applications vol 2016 article no 46 2016

[44] F Wang and H Zhu ldquoExistence uniqueness and stabilityof periodic solutions of a duffing equation under periodicand anti-periodic eigenvalues conditionsrdquo Taiwanese Journal ofMathematics vol 19 no 5 pp 1457ndash1468 2015

[45] Z Xianfeng and J Wei ldquoStability and boundedness of aretarded Lienard-type equationrdquo Chinese Quarterly Journal ofMathematics vol 18 no 1 pp 7ndash12 2003

[46] A F Yenicerioglu ldquoThe behavior of solutions of second orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 332 no 2 pp 1278ndash1290 2007

[47] A F Yenicerioglu ldquoStability properties of second order delayintegro-differential equationsrdquo Computers and Mathematicswith Applications vol 56 no 12 pp 3109ndash3117 2008

[48] T Yoshizawa ldquoLiapunovrsquos function and boundedness of solu-tionsrdquo Funkcialaj Ekvacioj vol 2 pp 71ndash103 1958

[49] W Zhu J Huang X Ruan and Z Zhao ldquoExponential stabilityof stochastic differential equation with mixed delayrdquo Journal ofAppliedMathematics vol 2014 Article ID 187037 11 pages 2014

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

International Journal of Analysis 9

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

000200400600801012

t

minus004

minus002

x(t)y(t)

x(t)

y(t)

times1011

(b)

Figure 6

0 1 2 3 4 5 6 7 8 9 10

0

002

004

006

008

01

012

t

minus002

times1011

x(t)y(t)

x(t)

y(t)

(a)

0 1 2 3 4 5 6 7 8 9 10

000200400600801012

t

minus002

minus008

minus006

minus004

times1011

x(t)y(t)

x(t)

y(t)

(b)

Figure 7

for all 119905 ge 0 119909 and 119910 Moreover from (79) and (80)all assumptions of Theorem 17 and Lemma 6 are satisfiedThus by Lemma 6 the trivial solution of system (78) is notonly uniformly stochastically asymptotically stable but alsouniformly stochastically asymptotically stable in the largeFinally from (79) and (81) the function 119881(119905 119883) is positivedefinite and

119871119881 (119905 119883) le 0 forall (119905 119883) isin R+ timesR

2 (82)

Hence assumptions of Theorem 17 and Lemma 5 hold byTheorem 17 and Lemma 5 the trivial solution of system (78)is stochastically stable

Simulation of Solutions In what follows we shall nowsimulate the solutions of (56) (resp system (57)) and (78)(resp system (79)) Our approach depends on the Euler-Maruyama method which enables us to get approximatenumerical solution for the considered systems It will be seenfrom our figures that the simulated solutions are bounded

which justifies our given results For instance when 120590 = 01the numerical solutions of (56) in three-dimensional spaceare shown in Figure 4 If we vary the value of the noise inthe numerical solution (119909(119905) 119910(119905)) of system (57) as 120590 = 01and 120590 = 10 we have Figures 5(a) and 5(b) respectively Itcan be seen that when the noise is increased the stochasticitybecomes more pronounced The behaviour of the numericalsolution (119909(119905) 119910(119905)) of system (57) when 120590 = 05 and 120590 = 20is shown in Figures 6(a) and 6(b) respectivelyThe behaviourof the numerical solution (119909(119905) 119910(119905)) of system (57) for 120590 = 0and 120590 = 50 is shown in Figures 7(a) and 7(b) respectivelyFor the case of (78) Figure 8 shows the closeness of thesolution (119909(119905)) and the perturbed solution (119909120598(119905)) for a verylarge 119905 which implies asymptotic stability in the large for theconsidered SDE

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

10 International Journal of Analysis

0 1 2 3 4 5 6 7 8 9 100

0005

001

0015

002

0025

t times1011

x(t)x120576(t)

x(t)x120576(t)

Figure 8 Graph of solutions of (78)

References

[1] L Arnold Stochastic Differential Equations Theory and Appli-cations John Wiley amp Sons 1974

[2] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations vol 178 of Mathematics inScience and Engineering Academic Press Inc Orlando FlaUSA 1985

[3] T A Burton Volterra Integral and Differential EquationsAcademic Press New York NY USA 1983

[4] J K HaleTheory of Functional Differential Equations SpringerNew York NY USA 1977

[5] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer 2000

[6] L Shaikihet Lyapunov Functionals and Stability of StochasticFunctional Differential Equations Springer International 2013

[7] T Yoshizawa StabilityTheory and Existence of Periodic Solutionsand almost Periodic Solutions Spriger New York NY USA1975

[8] T Yoshizawa StabilityTheory by Liapunovrsquos SecondMethodTheMathematical Society of Japan 1966

[9] A M A Abou-El-Ela A I Sadek and A M Mahmoud ldquoOnthe stability of solutions for certain second-order stochasticdelay differential equationsrdquo Differential Equations and ControlProcesses no 2 pp 1ndash13 2015

[10] A M Abou-El-Ela A I Sadek A M Mahmoud and R OTaie ldquoOn the stochastic stability and boundedness of solutionsfor stochastic delay differential equation of the second orderrdquoChinese Journal of Mathematics vol 2015 Article ID 358936 8pages 2015

[11] A T Ademola Boundedness and Stability of Solutions to CertainSecond Order Differential Equations Differential Equations andControl Processes 2015

[12] A T Ademola B S Ogundare M O Ogundiran and O AAdesina ldquoPeriodicity stability and boundedness of solutions tocertain second order delay differential equationsrdquo InternationalJournal of Differential Equations vol 2016 Article ID 284370910 pages 2016

[13] J G Alaba and B S Ogundare ldquoOn stability and bound-edness properties of solutions of certain second order non-autonomous nonlinear ordinary differential equationrdquo Kragu-jevac Journal of Mathematics vol 39 no 2 pp 255ndash266 2015

[14] T A Burton and L Hatvani ldquoAsymptotic stability of secondorder ordinary functional and partial differential equationsrdquoJournal of Mathematical Analysis and Applications vol 176 no1 pp 261ndash281 1993

[15] B Cahlon andD Schmidt ldquoStability criteria for certain second-order delay differential equations with mixed coefficientsrdquoJournal of Computational and AppliedMathematics vol 170 no1 pp 79ndash102 2004

[16] T Caraballo M A Diop and A S Ndoye ldquoFixed points andexponential stability for stochastic partial integro-differentialequations with delaysrdquo Advances in Dynamical Systems andApplications vol 9 no 2 pp 133ndash147 2014

[17] A Domoshnitsky ldquoNonoscillation maximum principles andexponential stability of secondorder delay differential equationswithout damping termrdquo Domoshnitsky Journal of Inequalitiesand Applications vol 2014 article 361 2014

[18] I I Gikhman and A V Skorokhod Stochastische Differential-gleichungen Akademie Berlin Germany 1971 (Russian)

[19] I I Gikhman On the Stability of the Solutions of StochasticDifferential Equations Predelrsquonyye Teoremy i StatisticheskiyeVyvody Tashkent Uzbekistan 1966

[20] G A Grigoryan ldquoBoundedness and stability criteria for linearordinary differential equations of the second orderrdquo RussianMathematics vol 57 no 12 pp 8ndash15 2013

[21] A F Ivanov Y I Kazmerchuk and A V Swishchuk ldquoThe-ory stochastic stability and applications of stochastic delaydifferential equations a survey of recent resultsrdquo in DifferentialEquations and Dynamical Systems vol 11 no 1 2003

[22] F Jedrzejewski and D Brochard ldquoLyapounv exponents andstability stochastic dynamical structuresrdquo 2000

[23] Z Jin and L Zengrong ldquoOn the global asymptotic behaviorof solutions to a non autonomous generalized Lienard systemrdquoJournal of Mathematical Research and Exposition vol 21 no 3pp 410ndash414 2001

[24] E Kolarova ldquoAn application of stochastic integral equations toelectrical networksrdquo Acta Electrotechnica et Informatica vol 8no 3 pp 14ndash17 2008

[25] V B Kolmanovskii and L E Shaikhet ldquoA method for con-structing Lyapunov functionals for stochastic systems with aftereffectrdquo Differentsialrsquonye Uravneniya vol 29 no 11 pp 1909ndash2022 1993

[26] V Kolmanovskii and L Shaikhet ldquoConstruction of Lyapunovfunctionals for stochastic hereditary systems a survey of somerecent resultsrdquo Mathematical and Computer Modelling vol 36no 6 pp 691ndash716 2002

[27] A J Kroopnick ldquoBounded solutions to 11990910158401015840 + 119902(119905)119887(119909) = 119891(119905)rdquoInternational Journal of Mathematical Education in Science andTechnology vol 41 no 6 pp 829ndash836 2010

[28] R Liu and Y Raffoul ldquoBoundedness and exponential stabilityof highly nonlinear stochastic differential equationsrdquo ElectronicJournal of Differential Equations vol 2009 no 143 pp 1ndash102009

[29] X Mao ldquoSome contributions to stochastic asymptotic stabilityand boundedness via multiple Lyapunov functionsrdquo Journal ofMathematical Analysis and Applications vol 260 no 2 pp 325ndash340 2001

[30] B S Ogundare A T Ademola M O Ogundiran and O AAdesina ldquoOn the qualitative behaviour of solutions to certainsecond order nonlinear differential equation with delayrdquoAnnalidellrsquoUniversitarsquo di Ferrara 2016

International Journal of Analysis 11

[31] B S Ogundare and A U Afuwape ldquoBoundedness and stabilityproperties of solutions of generalized Lienard equationrdquo KochiJournal of Mathematics vol 9 pp 97ndash108 2014

[32] B S Ogundare and G E Okecha ldquoBoundedness periodicityand stability of solutions to x(t) + g(x) + b(t)h(x) = p(t x x)rdquoMathematical Sciences Research Journal vol 11 no 5 pp 432ndash443 2007

[33] Y N Raffoul ldquoBoundedness and exponential asymptotic sta-bility in dynamical systems with applications to nonlineardifferential equations with unbounded termsrdquo Advances inDynamical Systems and Applications vol 2 no 1 pp 107ndash1212007

[34] R Rezaeyan and R Farnoosh ldquoStochastic differential equationsand application of the Kalman-Bucy filter in the modeling ofRC circuitrdquo Applied Mathematical Sciences vol 4 no 21-24 pp1119ndash1127 2010

[35] C Tunc ldquoA note on the stability and boundedness of non-autonomous differential equations of second order with avariable deviating argumentrdquo Afrika Matematika vol 25 no 2pp 417ndash425 2014

[36] C Tunc ldquoA note on the bounded solutions to 11990910158401015840 + 119888(119905 119909 1199091015840 ) +119902(119905)119887(119909) = 119891(119905)rdquo Applied Mathematics amp Information Sciencesvol 8 no 1 pp 393ndash399 2014

[37] C Tunc ldquoBoundedness analysis for certain two-dimensionaldifferential systems via a Lyapunov approachrdquo Bulletin Mathe-matique de la Societe des Sciences Mathematiques de Roumanievol 53 no 1 pp 61ndash68 2010

[38] 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

[39] C Tunc ldquoStability and boundedness in multi delay vectorLienard equationrdquo Filomat vol 27 no 3 pp 435ndash445 2013

[40] C Tunc ldquoStability and boundedness of solutions of non-autonomous differential equations of second orderrdquo Journalof Computational Analysis and Applications vol 13 no 6 pp1067ndash1074 2011

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

[42] 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

[43] C Tunc and T Ayhan ldquoGlobal existence and boundedness ofsolutions of a certain nonlinear integro-differential equationof second order with multiple deviating argumentsrdquo Journal ofInequalities and Applications vol 2016 article no 46 2016

[44] F Wang and H Zhu ldquoExistence uniqueness and stabilityof periodic solutions of a duffing equation under periodicand anti-periodic eigenvalues conditionsrdquo Taiwanese Journal ofMathematics vol 19 no 5 pp 1457ndash1468 2015

[45] Z Xianfeng and J Wei ldquoStability and boundedness of aretarded Lienard-type equationrdquo Chinese Quarterly Journal ofMathematics vol 18 no 1 pp 7ndash12 2003

[46] A F Yenicerioglu ldquoThe behavior of solutions of second orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 332 no 2 pp 1278ndash1290 2007

[47] A F Yenicerioglu ldquoStability properties of second order delayintegro-differential equationsrdquo Computers and Mathematicswith Applications vol 56 no 12 pp 3109ndash3117 2008

[48] T Yoshizawa ldquoLiapunovrsquos function and boundedness of solu-tionsrdquo Funkcialaj Ekvacioj vol 2 pp 71ndash103 1958

[49] W Zhu J Huang X Ruan and Z Zhao ldquoExponential stabilityof stochastic differential equation with mixed delayrdquo Journal ofAppliedMathematics vol 2014 Article ID 187037 11 pages 2014

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

10 International Journal of Analysis

0 1 2 3 4 5 6 7 8 9 100

0005

001

0015

002

0025

t times1011

x(t)x120576(t)

x(t)x120576(t)

Figure 8 Graph of solutions of (78)

References

[1] L Arnold Stochastic Differential Equations Theory and Appli-cations John Wiley amp Sons 1974

[2] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations vol 178 of Mathematics inScience and Engineering Academic Press Inc Orlando FlaUSA 1985

[3] T A Burton Volterra Integral and Differential EquationsAcademic Press New York NY USA 1983

[4] J K HaleTheory of Functional Differential Equations SpringerNew York NY USA 1977

[5] B Oksendal Stochastic Differential Equations An Introductionwith Applications Springer 2000

[6] L Shaikihet Lyapunov Functionals and Stability of StochasticFunctional Differential Equations Springer International 2013

[7] T Yoshizawa StabilityTheory and Existence of Periodic Solutionsand almost Periodic Solutions Spriger New York NY USA1975

[8] T Yoshizawa StabilityTheory by Liapunovrsquos SecondMethodTheMathematical Society of Japan 1966

[9] A M A Abou-El-Ela A I Sadek and A M Mahmoud ldquoOnthe stability of solutions for certain second-order stochasticdelay differential equationsrdquo Differential Equations and ControlProcesses no 2 pp 1ndash13 2015

[10] A M Abou-El-Ela A I Sadek A M Mahmoud and R OTaie ldquoOn the stochastic stability and boundedness of solutionsfor stochastic delay differential equation of the second orderrdquoChinese Journal of Mathematics vol 2015 Article ID 358936 8pages 2015

[11] A T Ademola Boundedness and Stability of Solutions to CertainSecond Order Differential Equations Differential Equations andControl Processes 2015

[12] A T Ademola B S Ogundare M O Ogundiran and O AAdesina ldquoPeriodicity stability and boundedness of solutions tocertain second order delay differential equationsrdquo InternationalJournal of Differential Equations vol 2016 Article ID 284370910 pages 2016

[13] J G Alaba and B S Ogundare ldquoOn stability and bound-edness properties of solutions of certain second order non-autonomous nonlinear ordinary differential equationrdquo Kragu-jevac Journal of Mathematics vol 39 no 2 pp 255ndash266 2015

[14] T A Burton and L Hatvani ldquoAsymptotic stability of secondorder ordinary functional and partial differential equationsrdquoJournal of Mathematical Analysis and Applications vol 176 no1 pp 261ndash281 1993

[15] B Cahlon andD Schmidt ldquoStability criteria for certain second-order delay differential equations with mixed coefficientsrdquoJournal of Computational and AppliedMathematics vol 170 no1 pp 79ndash102 2004

[16] T Caraballo M A Diop and A S Ndoye ldquoFixed points andexponential stability for stochastic partial integro-differentialequations with delaysrdquo Advances in Dynamical Systems andApplications vol 9 no 2 pp 133ndash147 2014

[17] A Domoshnitsky ldquoNonoscillation maximum principles andexponential stability of secondorder delay differential equationswithout damping termrdquo Domoshnitsky Journal of Inequalitiesand Applications vol 2014 article 361 2014

[18] I I Gikhman and A V Skorokhod Stochastische Differential-gleichungen Akademie Berlin Germany 1971 (Russian)

[19] I I Gikhman On the Stability of the Solutions of StochasticDifferential Equations Predelrsquonyye Teoremy i StatisticheskiyeVyvody Tashkent Uzbekistan 1966

[20] G A Grigoryan ldquoBoundedness and stability criteria for linearordinary differential equations of the second orderrdquo RussianMathematics vol 57 no 12 pp 8ndash15 2013

[21] A F Ivanov Y I Kazmerchuk and A V Swishchuk ldquoThe-ory stochastic stability and applications of stochastic delaydifferential equations a survey of recent resultsrdquo in DifferentialEquations and Dynamical Systems vol 11 no 1 2003

[22] F Jedrzejewski and D Brochard ldquoLyapounv exponents andstability stochastic dynamical structuresrdquo 2000

[23] Z Jin and L Zengrong ldquoOn the global asymptotic behaviorof solutions to a non autonomous generalized Lienard systemrdquoJournal of Mathematical Research and Exposition vol 21 no 3pp 410ndash414 2001

[24] E Kolarova ldquoAn application of stochastic integral equations toelectrical networksrdquo Acta Electrotechnica et Informatica vol 8no 3 pp 14ndash17 2008

[25] V B Kolmanovskii and L E Shaikhet ldquoA method for con-structing Lyapunov functionals for stochastic systems with aftereffectrdquo Differentsialrsquonye Uravneniya vol 29 no 11 pp 1909ndash2022 1993

[26] V Kolmanovskii and L Shaikhet ldquoConstruction of Lyapunovfunctionals for stochastic hereditary systems a survey of somerecent resultsrdquo Mathematical and Computer Modelling vol 36no 6 pp 691ndash716 2002

[27] A J Kroopnick ldquoBounded solutions to 11990910158401015840 + 119902(119905)119887(119909) = 119891(119905)rdquoInternational Journal of Mathematical Education in Science andTechnology vol 41 no 6 pp 829ndash836 2010

[28] R Liu and Y Raffoul ldquoBoundedness and exponential stabilityof highly nonlinear stochastic differential equationsrdquo ElectronicJournal of Differential Equations vol 2009 no 143 pp 1ndash102009

[29] X Mao ldquoSome contributions to stochastic asymptotic stabilityand boundedness via multiple Lyapunov functionsrdquo Journal ofMathematical Analysis and Applications vol 260 no 2 pp 325ndash340 2001

[30] B S Ogundare A T Ademola M O Ogundiran and O AAdesina ldquoOn the qualitative behaviour of solutions to certainsecond order nonlinear differential equation with delayrdquoAnnalidellrsquoUniversitarsquo di Ferrara 2016

International Journal of Analysis 11

[31] B S Ogundare and A U Afuwape ldquoBoundedness and stabilityproperties of solutions of generalized Lienard equationrdquo KochiJournal of Mathematics vol 9 pp 97ndash108 2014

[32] B S Ogundare and G E Okecha ldquoBoundedness periodicityand stability of solutions to x(t) + g(x) + b(t)h(x) = p(t x x)rdquoMathematical Sciences Research Journal vol 11 no 5 pp 432ndash443 2007

[33] Y N Raffoul ldquoBoundedness and exponential asymptotic sta-bility in dynamical systems with applications to nonlineardifferential equations with unbounded termsrdquo Advances inDynamical Systems and Applications vol 2 no 1 pp 107ndash1212007

[34] R Rezaeyan and R Farnoosh ldquoStochastic differential equationsand application of the Kalman-Bucy filter in the modeling ofRC circuitrdquo Applied Mathematical Sciences vol 4 no 21-24 pp1119ndash1127 2010

[35] C Tunc ldquoA note on the stability and boundedness of non-autonomous differential equations of second order with avariable deviating argumentrdquo Afrika Matematika vol 25 no 2pp 417ndash425 2014

[36] C Tunc ldquoA note on the bounded solutions to 11990910158401015840 + 119888(119905 119909 1199091015840 ) +119902(119905)119887(119909) = 119891(119905)rdquo Applied Mathematics amp Information Sciencesvol 8 no 1 pp 393ndash399 2014

[37] C Tunc ldquoBoundedness analysis for certain two-dimensionaldifferential systems via a Lyapunov approachrdquo Bulletin Mathe-matique de la Societe des Sciences Mathematiques de Roumanievol 53 no 1 pp 61ndash68 2010

[38] 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

[39] C Tunc ldquoStability and boundedness in multi delay vectorLienard equationrdquo Filomat vol 27 no 3 pp 435ndash445 2013

[40] C Tunc ldquoStability and boundedness of solutions of non-autonomous differential equations of second orderrdquo Journalof Computational Analysis and Applications vol 13 no 6 pp1067ndash1074 2011

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

[42] 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

[43] C Tunc and T Ayhan ldquoGlobal existence and boundedness ofsolutions of a certain nonlinear integro-differential equationof second order with multiple deviating argumentsrdquo Journal ofInequalities and Applications vol 2016 article no 46 2016

[44] F Wang and H Zhu ldquoExistence uniqueness and stabilityof periodic solutions of a duffing equation under periodicand anti-periodic eigenvalues conditionsrdquo Taiwanese Journal ofMathematics vol 19 no 5 pp 1457ndash1468 2015

[45] Z Xianfeng and J Wei ldquoStability and boundedness of aretarded Lienard-type equationrdquo Chinese Quarterly Journal ofMathematics vol 18 no 1 pp 7ndash12 2003

[46] A F Yenicerioglu ldquoThe behavior of solutions of second orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 332 no 2 pp 1278ndash1290 2007

[47] A F Yenicerioglu ldquoStability properties of second order delayintegro-differential equationsrdquo Computers and Mathematicswith Applications vol 56 no 12 pp 3109ndash3117 2008

[48] T Yoshizawa ldquoLiapunovrsquos function and boundedness of solu-tionsrdquo Funkcialaj Ekvacioj vol 2 pp 71ndash103 1958

[49] W Zhu J Huang X Ruan and Z Zhao ldquoExponential stabilityof stochastic differential equation with mixed delayrdquo Journal ofAppliedMathematics vol 2014 Article ID 187037 11 pages 2014

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

International Journal of Analysis 11

[31] B S Ogundare and A U Afuwape ldquoBoundedness and stabilityproperties of solutions of generalized Lienard equationrdquo KochiJournal of Mathematics vol 9 pp 97ndash108 2014

[32] B S Ogundare and G E Okecha ldquoBoundedness periodicityand stability of solutions to x(t) + g(x) + b(t)h(x) = p(t x x)rdquoMathematical Sciences Research Journal vol 11 no 5 pp 432ndash443 2007

[33] Y N Raffoul ldquoBoundedness and exponential asymptotic sta-bility in dynamical systems with applications to nonlineardifferential equations with unbounded termsrdquo Advances inDynamical Systems and Applications vol 2 no 1 pp 107ndash1212007

[34] R Rezaeyan and R Farnoosh ldquoStochastic differential equationsand application of the Kalman-Bucy filter in the modeling ofRC circuitrdquo Applied Mathematical Sciences vol 4 no 21-24 pp1119ndash1127 2010

[35] C Tunc ldquoA note on the stability and boundedness of non-autonomous differential equations of second order with avariable deviating argumentrdquo Afrika Matematika vol 25 no 2pp 417ndash425 2014

[36] C Tunc ldquoA note on the bounded solutions to 11990910158401015840 + 119888(119905 119909 1199091015840 ) +119902(119905)119887(119909) = 119891(119905)rdquo Applied Mathematics amp Information Sciencesvol 8 no 1 pp 393ndash399 2014

[37] C Tunc ldquoBoundedness analysis for certain two-dimensionaldifferential systems via a Lyapunov approachrdquo Bulletin Mathe-matique de la Societe des Sciences Mathematiques de Roumanievol 53 no 1 pp 61ndash68 2010

[38] 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

[39] C Tunc ldquoStability and boundedness in multi delay vectorLienard equationrdquo Filomat vol 27 no 3 pp 435ndash445 2013

[40] C Tunc ldquoStability and boundedness of solutions of non-autonomous differential equations of second orderrdquo Journalof Computational Analysis and Applications vol 13 no 6 pp1067ndash1074 2011

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

[42] 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

[43] C Tunc and T Ayhan ldquoGlobal existence and boundedness ofsolutions of a certain nonlinear integro-differential equationof second order with multiple deviating argumentsrdquo Journal ofInequalities and Applications vol 2016 article no 46 2016

[44] F Wang and H Zhu ldquoExistence uniqueness and stabilityof periodic solutions of a duffing equation under periodicand anti-periodic eigenvalues conditionsrdquo Taiwanese Journal ofMathematics vol 19 no 5 pp 1457ndash1468 2015

[45] Z Xianfeng and J Wei ldquoStability and boundedness of aretarded Lienard-type equationrdquo Chinese Quarterly Journal ofMathematics vol 18 no 1 pp 7ndash12 2003

[46] A F Yenicerioglu ldquoThe behavior of solutions of second orderdelay differential equationsrdquo Journal of Mathematical Analysisand Applications vol 332 no 2 pp 1278ndash1290 2007

[47] A F Yenicerioglu ldquoStability properties of second order delayintegro-differential equationsrdquo Computers and Mathematicswith Applications vol 56 no 12 pp 3109ndash3117 2008

[48] T Yoshizawa ldquoLiapunovrsquos function and boundedness of solu-tionsrdquo Funkcialaj Ekvacioj vol 2 pp 71ndash103 1958

[49] W Zhu J Huang X Ruan and Z Zhao ldquoExponential stabilityof stochastic differential equation with mixed delayrdquo Journal ofAppliedMathematics vol 2014 Article ID 187037 11 pages 2014

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