International Journal of Analysis
Volume 2016 (2016), Article ID 2012315, 11 pages
http://dx.doi.org/10.1155/2016/2012315

Research Article

Stability and Boundedness of Solutions to a Certain Second-Order Nonautonomous Stochastic Differential Equation

A. T. Ademola,¹ S. Moyo,² B. S. Ogundare,¹ M. O. Ogundiran,¹ and O. A. Adesina¹

¹Research Group in Differential Equations and Applications (RGDEA), Department of Mathematics, Obafemi Awolowo University, Ile-Ife 220005, Nigeria
²Institute for Systems Science & Research and Postgraduate Support Directorate, Durban University of Technology, Durban 4000, South Africa

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

Academic Editor: Ying Hu

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

Abstract.

This paper focuses on stability and boundedness of certain nonlinear nonautonomous second-order stochastic differential equations. Lyapunov’s second method is employed by constructing a suitable complete Lyapunov function and is used to obtain criteria, on the nonlinear functions, that guarantee stability and boundedness of solutions. Our results are new; in fact, according to our observations from the relevant literature, this is the first attempt on stability and boundedness of solutions of second-order nonlinear nonautonomous stochastic differential equations. Finally, examples together with their numerical simulations are given to authenticate and affirm the correctness of the obtained results.

1. Introduction

Differential equations of second-order have generated a great deal of applications in various fields of science and technology such as biology, chemistry, physics, mechanics, control technology, communication network, automatic regulation, economy, and ecology to mention few. In addition, the study of problems that involve the behaviour of solutions of ordinary differential equations (ODE), delay or functional differential equations (DDE), and stochastic differential equations (SDE) has been dealt with by many outstanding authors; see, for instance, Arnold [1], Burton [2, 3], Hale [4], Oksendal [5], Shaikihet [6], and Yoshizawa [7, 8], which contain the background to the study and the expository papers of Abou-El-Ela et al. [9, 10], Ademola et al. [11, 12], Alaba and Ogundare [13], Burton and Hatvani [14], Cahlon and Schmidt [15], Caraballo et al. [16], Domoshnitsky [17], Gikhman and Skorokhod [18, 19], Grigoryan [20], Ivanov et al. [21], Jedrzejewski and Brochard [22], Jin and Zengrong [23], Kolarova [24], Kolmanovskii and Shaikhet [25, 26], Kroopnick [27], Liu and Raffoul [28], Mao [29], Ogundare et al. [30–32], Raffoul [33], Rezaeyan and Farnoosh [34], Tunç [35–43], Wang and Zhu [44], Xianfeng and Wei [45], Yeniçerioğlu [46, 47], Yoshizawa [48], Zhu et al. [49], and the references cited therein.

The authors in [18, 19] investigated the second-order linear scalar equations of the formwhere is a general disturbance process (the derivative of a martingale). In [11, 12] the authors discussed stability, boundedness, and periodic solutions to the following second-order ordinary and delay differential equations: respectively, where , , , , , and are continuous functions in their respective arguments. In their contributions, the authors in [9, 10] investigated asymptotic stability and boundedness of solutions of the following second-order stochastic delay differential equations:respectively, where , , and are positive constants; , are delay constants; , , and are continuous functions in their respective arguments and is an -dimensional standard Brownian motion defined on the probability space (also called Wiener process). Recently, in 2016 the authors in [43] discussed global existence and boundedness of solutions of a certain nonlinear integrodifferential equation of second-order with multiple deviating argumentswhere are positive constants, , , and are defined on , and , and are continuous functions defined in their respective arguments.

Although second-order stochastic delay differential equations have started receiving attention of authors, according to our observation from relevant literature, there is no previous literature available on the stability and boundedness of solutions of second-order nonlinear nonautonomous stochastic differential equation. The aim of this paper is to bridge this gap. Consider the following second-order nonlinear nonautonomous stochastic differential equation:where is a positive constant, the functions , , and are continuous in their respective arguments on , and , respectively, with , , and (a standard Wiener process, representing the noise) is defined on . Furthermore, it is assumed that the continuity of the functions , , and is sufficient for the existence of solutions and the local Lipschitz condition for (8) to have a unique continuous solution denoted by The primes denote differentiation with respect to the independent variable If , then (8) is equivalent to the system:where the derivative of the function (i.e., ) exists and is continuous for all Despite the applicability of these classes of equations, there is no previous result on nonautonomous second-order nonlinear stochastic differential equation (8). The motivation for this investigation comes from the works in [9–12, 18, 19]. If in (8), then we have a general second-order nonlinear ordinary differential equation which has been discussed extensively in relevant literature. The remaining parts of this paper are organized as follows. In Section 2, we give the preliminary results on stochastic differential equations. Main results and their proofs are presented in Section 3 while examples and simulation of solutions are given in Section 4 to validate our results.

2. Preliminary Results

Let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets). Let be an -dimensional Brownian motion defined on the probability space. Let denotes the Euclidean norm in . If is a vector or matrix, its transpose is denoted by . If is a matrix, its trace norm is denoted by For more exposition in this regard, see Mao [29] and Arnold [1]. Now let us consider a nonautonomous -dimensional stochastic differential equationon with initial value Here and are measurable functions. Suppose that both and are sufficiently smooth for (11) to have a unique continuous solution on which is denoted by , if . Assume further that for all Then, the stochastic differential equation (11) admits zero solution

Definition 1 (see [1]). The zero solution of the stochastic differential equation (11) is said to be stochastically stable or stable in probability, if for every pair of and , there exists a such that Otherwise, it is said to be stochastically unstable.

Definition 2 (see [1]). The zero solution of the stochastic differential equation (11) is said to be stochastically asymptotically stable if it is stochastically stable and in addition if for every and , there exists a such that

Definition 3. A solution of the stochastic differential equation (11) is said to be stochastically bounded or bounded in probability, if it satisfieswhere denotes the expectation operator with respect to the probability law associated with and is a constant depending on and

Definition 4. The solutions of the stochastic differential equation (11) are said to be uniformly stochastically bounded if in inequality (15) is independent of .

For , let and let denote the family of all nonnegative functions (Lyapunov function) defined on which are twice continuously differentiable in and once in By Itô’s formula we have whereFurthermore, In this study we will use the diffusion operator defined in (17) to replace We now present the basic results that will be used in the proofs of the main results.

Lemma 5 (see [1]). Assume that there exist and such that (i);(ii);(iii) for all Then the zero solution of stochastic differential equation (11) is stochastically stable.

Lemma 6 (see [1]). Suppose that there exist and such that (i);(ii), as ;(iii) for all Then the zero solution of stochastic differential equation (11) is uniformly stochastically asymptotically stable in the large.

Assumption 7 (see [28, 33]). Let , and suppose that for any solutions of stochastic differential equation (11) and for any fixed , we have

Assumption 8 (see [28, 33]). A special case of the general condition (19) is the following condition. Assume that there exits a function such thatand for any fixed ,

Lemma 9 (see [28, 33]). Assume there exists a Lyapunov function , satisfying Assumption 7, such that, for all , (i),(ii),(iii),where , , , and are positive constants, , and is a nonnegative constant. Then all solutions of the stochastic differential equation (11) satisfyfor all

Lemma 10 (see [28, 33]). Assume there exists a Lyapunov function , satisfying Assumption 7, such that, for all , (i),(ii),(iii),where , , are positive constants, , and is a nonnegative constant. Then all solutions of the stochastic differential equation (11) satisfy (22) for all

Corollary 11 (see [28, 33]). (i) Assume that hypotheses (i) to (iii) of Lemma 9 hold. In addition,for some positive constant ; then all solutions of stochastic differential equation (11) are uniformly stochastically bounded.
(ii) Assume that hypotheses (i) to (iii) of Lemma 10 hold. If condition (23) is satisfied, then all solutions of the stochastic differential equation (11) are stochastically bounded.

3. Main Results

Let be any solution of the stochastic differential equation (9); the main tool employed in the proofs of our results is the continuously differentiable function defined aswhere and are positive constants and the function is as defined in Section 1.

Theorem 12. Suppose that , , , and are positive constants such that (i) for all and ,(ii) for all and ,(iii) for all and Then solution of the stochastic differential equation (9) is uniformly stochastically bounded.

Remark 13. We note the following: (i)Whenever the functions and , then the stochastic differential equation (8) becomes a second-order linear ordinary differential equation and conditions (i) to (iii) of Theorem 12 reduce to Routh Hurwitz criteria and for the asymptotic stability of the second-order linear differential equation (25).(ii)The term in the stochastic differential equation (8) is an extension of the ordinary case discussed recently by authors in [11, 18, 23, 31, 32, 35–37, 40].

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

Lemma 14. Under the hypotheses of Theorem 12, there exist positive constants and such thatfor all , , and In addition, there exist positive constants and such thatfor all , and

Proof. Let be any solution of the stochastic differential equation (9); since , it follows from (24) thatfor all Moreover, from (24) and the fact that for all , there exists a positive constant such thatfor all , , and , whereIt is clear from inequality (29) that Furthermore, since for all , it follows from (24) that there exists a positive constant such thatfor all , , and , where From inequalities (29) and (33), we havefor all , , and It is not difficult to see that estimates (35) satisfy inequalities (26) of Lemma 14 with and equivalent to and , respectively.
Moreover, applying Itô’s formula in (24) using system (9), we find thatwhere It is clear from the inequalitiesthatfor all and . Using inequality (39) and hypotheses (i) and (ii) of Theorem 12 in (36), there exist positive constants and such thatfor all , , and , where Inequality (40) satisfies inequality (27) with and equivalent to and , respectively. This completes the proof of Lemma 14.

Proof of Theorem 12. Let be any solution of system (9). From inequality (40) and assumption (iii) of Theorem 12, we have for , , and Since , and are positives and for all and , there exist positive constants and such thatfor all , , where and Hence, condition (ii) of Lemma 9 is satisfied with , and Also from inequality (35), hypotheses (i) and (iii) of Lemma 9 hold with so that
Furthermore, from inequality (23) we havefor all Inequality (45) satisfies estimate (23) with Moreover, from (9) and (24) there exists a positive constant such thatwhere Also,for any fixed Thus, from inequalities (46) and (48) estimates (20) and (21) hold, respectively. Finally, from inequalities (33) and (45), we havefor all , where and Thus, the solutions of the stochastic differential equation (9) are uniformly stochastically bounded.

Theorem 15. If assumptions of Theorem 12 hold, then the solution of the stochastic differential equation (9) is stochastically bounded.

Proof. Suppose that is any solution of the stochastic differential equation (9). From inequalities (33) and (44) there exists a positive constant such thatfor all , and , where Hence, from inequalities (29) and (50) hypotheses of Lemma 10 hold. Moreover, from inequalities (45), (46), (48), and (49) assumption (ii) of Corollary 11 holds. Thus, by Corollary 11, all solutions of the stochastic differential equation (9) are stochastically bounded. This completes the proof of Theorem 15.

Next, we shall discuss the stability of the trivial solution of the stochastic differential equation (8). Suppose that , (8) specializes toEquation (51) has the following equivalent system:where the functions , and are defined in Section 1.

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

Proof. Let be any solution of the stochastic differential equation (52). From equation (28) and estimate (29) assumptions (i) and (ii) of Lemma 5 hold so that the function is positive definite. Furthermore, using Itô’s formula along the solution path of (52), we obtainfor all , , and , where is defined in (40). Inequality (53) satisfies hypothesis (iii) of Lemma 5; hence, by Lemma 5 the trivial solution of the stochastic differential equation (52) is stochastically stable. This completes the proof of Theorem 16.

Theorem 17. If assumptions (i) and (ii) of Theorem 12 hold, then the trivial solution of the stochastic differential equation (52) is not only uniformly stochastically asymptotically stable, but also uniformly stochastically asymptotically stable in the large.

Proof. Let be any solution of the stochastic differential equation (52). In view of (28) and estimate (29), the function is positive definite. Furthermore, estimate (32) and inequality (33) show that the function is radially unbounded and decrescent, respectively. It follows from (28), estimate (32), inequality (35), and the first inequality in (53) that all assumptions of Lemma 6 hold. Thus, by Lemma 6 the trivial solution of the stochastic differential equation (52) is uniformly stochastically asymptotically stable in the large. If estimate (32) is omitted then the trivial solution of the stochastic differential equation (52) is uniformly stochastically asymptotically stable. This completes the proof of Theorem 17.

Next, if the function is replaced by , we have the following special case:of (8). Equation (54) has the following equivalent system:with the following result.

Corollary 18. If assumptions (i) and (ii) of Theorem 12 hold and hypothesis (iii) is replaced by the boundedness of the function , then the solutions of the stochastic differential equation (55) are not only stochastically bounded but also uniformly stochastically bounded.

Proof. The proof of Corollary 18 is similar to the proof of Theorems 12 and 15. This completes the proof of Corollary 18.

4. Examples

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

Example 1. Consider the second-order nonlinear nonautonomous stochastic differential equationEquation (56) is equivalent to systemNow from systems (9) and (57) we have the following relations: (i)The function  Noting that  for all and , it follows that  for all and The behaviour of the function is shown below in Figure 1.(ii)The function  Since  for all , then we have  for all and since it follows that implies that . The function and its bounds are shown in Figure 2.(iii)The function  Clearly,  for all , , and Now from items (i), (ii) above and (24), the continuously differentiable function used for system (57) isDifferent views of the function are shown in Figure 3. From (66), it is not difficult to show thatfor all , and From (35) and (67) we have , , , and , and thus, inequalities (67) satisfy condition (i) of Lemma 9. Also, from the first inequality in (67), we haveEstimate (68) verifies (32) (i.e., the function defined by (66) is radially unbounded). Next, applying Itô’s formula in (66) using system (57), we find thatUsing the estimates in items (i) to (iii) of Example 1 and the inequality in (69), we obtainfor all , and Inequality (70) satisfies inequality (40) where and . Since for all and , it follows from inequality (70) thatfor all , , and Inequality (72) satisfies assumption (ii) of Lemma 9 and estimate (44) with and Since , it follows that , so that assumption (iii) of Lemma 9 holds. In addition,for all Estimate (73) satisfies (23) and (45), with . Furthermore,and for all , , and Inequality (75) satisfies inequalities (20) and (21) with Hence, by Corollary 11 (i), all solutions of stochastic differential equation (57) are uniformly stochastically bounded.

Figure 1: Behaviour of the function .

Figure 2: Bounds on the function .

Figure 3: The behaviour of the function

Example 2. If in (56) and system (57), we have the following stochastic differential equation:Equation (77) is equivalent to systemNow from systems (52) and (78) items (i) and (ii) of Example 1 hold. Also, equations (66), (67) and estimate (68) hold: that is,Furthermore, application of Itô’s formula in (66) and using system (78) yieldfor all , and thusfor all , , and Moreover, from (79) and (80) all assumptions of Theorem 17 and Lemma 6 are satisfied. Thus, by Lemma 6 the trivial solution of system (78) is not only uniformly stochastically asymptotically stable but also uniformly stochastically asymptotically stable in the large. Finally, from (79) and (81) the function is positive definite and Hence, assumptions of Theorem 17 and Lemma 5 hold; by Theorem 17 and Lemma 5 the trivial solution of system (78) is stochastically stable.

Simulation of Solutions. In what follows, we shall now simulate 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 approximate numerical solution for the considered systems. It will be seen from our figures that the simulated solutions are bounded which justifies our given results. For instance, when , the numerical solutions of (56) in three-dimensional space are shown in Figure 4. If we vary the value of the noise in the numerical solution of system (57), as and , we have Figures 5(a) and 5(b), respectively. It can be seen that, when the noise is increased, the stochasticity becomes more pronounced. The behaviour of the numerical solution of system (57) when and is shown in Figures 6(a) and 6(b), respectively. The behaviour of the numerical solution of system (57) for and is shown in Figures 7(a) and 7(b), respectively. For the case of (78), Figure 8 shows the closeness of the solution () and the perturbed solution () for a very large which implies asymptotic stability in the large for the considered SDE.

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

(a)

(b)

Figure 5

(a)

(b)

Figure 6

(a)

(b)

Figure 7

Figure 8: Graph of solutions of (78).

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

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