Asymptotic Stability of Neutral Set-Valued Functional Differential Equation by Fixed Point Method

Bao, Junyan; Wang, Peiguang

doi:https://doi.org/10.1155/2020/6569308

Discrete Dynamics in Nature and Society

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

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Stability and Bifurcation Analysis of Discrete Dynamical Systems 2020

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Research Article | Open Access

Volume 2020 | Article ID 6569308 | https://doi.org/10.1155/2020/6569308

Asymptotic Stability of Neutral Set-Valued Functional Differential Equation by Fixed Point Method

Junyan Bao¹and Peiguang Wang¹

Guest Editor: Abdul Qadeer Khan

Received28 Feb 2020

Accepted04 Apr 2020

Published27 Apr 2020

Abstract

This paper studies a class of nonlinear neutral set-valued functional differential equations. The globally asymptotic stability theorem with necessary and sufficient conditions is obtained via the fixed point method. Meanwhile, we give an example to illustrate the obtained result.

1. Introduction

It is well known that Lyapunov’s direct method is an important technique to consider the stability of various differential equations. However, this method is not always valid for stability analysis in the functional differential equation when the delay is unbounded or when the equation has unbounded terms [1–3]. Burton and many researchers found a way to these difficulties by using various fixed point theorems; we can refer to the literature studies [4–14].

Recently, the study of qualitative analysis of the set-valued differential equation has attracted much attention. The stability results of various set-valued differential equations were obtained by applying Lyapunov’s direct method. The results can be found in the monograph [15], the papers for the set-valued differential equation [16–24], set-valued functional differential equation [25–29], and other equations [6, 9, 30–32]. However, according to what we know so far, there are few stability results for the set-valued differential equation via the fixed point method. Inspired by the application of the fixed point method mentioned above, in this paper, we study a class of nonlinear neutral set-valued functional differential equations:where , , and ; , , , and . denote the collection of all nonempty, compact convex subsets of Banach space ; denotes the null set-valued function , and for , .

The aim of this paper is to obtain an asymptotic stability theorem with a necessary and sufficient condition via the fixed point method. In addition, an application of the main result is presented.

2. Preliminaries

To get the desired result, we first give some notations, definitions, and propositions briefly; for the details, see the literature [15].

Let denote the collection of all nonempty, compact convex subsets of Banach space , given , defining the Hausdorff metric between and as follows:where and . The Hausdorff metric satisfies the properties as follows:for all and .

In the sense of the above metric , the set is a complete metric space.

Definition 1 (see [15]). The set-valued function is Hukuhara differentiable at if the limitsexist in and equal to . is called the Hukuhara derivative of at .
By embedding as a complete cone in a corresponding Banach space and taking into account the differentiation of the Bochner integral, we can find that ifthen exists and a.e. on holds, where is integrable in the sense of Bochner.

Proposition 1 (see [15]). Let be Hukuhara differentiable and have continuous Hukuhara derivative on ; then, we have

Proposition 2 (see [15]). If is integrable, then

For each , is said to be a solution of (1) through if satisfies (1) on and on , and we denote the solution by .

Let and with normrespectively, and . By the properties of Hausdorff metric and the definition of , we can get the following:(i), and if and only if ,(ii),(iii),where and .

Definition 2. The trivial solution of (1) is said to be(i)Stable in if for any and , there exists such that , and implies that and .(ii)Globally asymptotically stable in if it is stable, and for any , implies thatFor , and , we can define the following addition and scalar multiplication as follows: , . Then, with the algebraic operations of addition and nonnegative scalar multiplication, becomes a semilinear metric space.
In order to apply the fixed point method, we first need to prove the following lemma.

Lemma 1. Let . Then, is a semilinear complete metric space.

Proof. Firstly, from the previous discussion, we know that is a semilinear metric space. Next, we prove the space is complete. Assume that is a Cauchy sequence; then, for any , there exists positive integer such that, for all , we have , . Hence,Then, and are Cauchy sequences in , respectively. Since is a complete metric space, there exist set-valued functions and such thatFirst, we prove that is a continuous function on . From (11), there exists a positive integer such that for all and . In addition, since is a continuous set-valued function, there exists such that implies that for . Then, for any with and , we haveUsing the same way, we can also prove that is a continuous function.
Next, we prove that . In fact, for any and , we haveBy (11) and (13), holds when . Using Proposition 1, we get . Meanwhile, we can prove that . Therefore, . This completes the proof of Lemma 1.

3. Main Result

In this section, we establish the necessary and sufficient conditions for the global asymptotic stability of trivial solution of equation (1) by using the fixed point method.

Theorem 1. Assume that the following conditions hold: : the function is bounded and . : there exist functions such that : there exists a constant such that

Then, the trivial solution of equation (1) is globally asymptotically stable in if and only if

Proof. We will prove this conclusion in two steps.

3.1. Proof of Sufficiency

Let . Then, is a nonempty, closed subset of . At the same time, we can get equation (1) with initial condition which is equivalent to the following integral equation:

We define the mapping as follows:

Firstly, we prove that is a mapping of . In fact, we just need to prove that and as . Since , for any , there exists such that implies

Thus, from (18), (19), , and (A₃), we have

Then, by condition (16), there exists such that, for implies

Thus, we have for , and we can obtain as . In addition, we can also get

Furthermore, from (22), we can get

From () and , we have for . Therefore, for , that is, .

Secondly, we prove that the mapping has a unique fixed point.

Let , and by (18), (), and (), we have

Similarly, from (18), (24), (), and (), we can also get

By (24) and (25), we know that is a contraction mapping. Therefore, using the principle of contraction mapping, has a unique fixed point which is a unique solution of equation (1) and satisfies

Finally, we prove the global asymptotic stability of equation (1). To do this, we first prove that the trivial solution of (1) is stable. Let

For any , we choose satisfying . If is a solution of equation (1) with , we can claim that on . In fact, if this is not true and we notice that on , then there exists such that

If , then it follows from () and () thatwhich contradicts the definition of .

If , then it follows from () and () thatwhich also contradicts the definition . Hence, for all , and the trivial solution of (1) is stable. This, together with (26), claims that the trivial solution of (1) is globally asymptotically stable.

3.2. Proof of Necessity

Assume that (16) is not valid. Then, by condition (), there exists a sequence such that for some , where as . Furthermore, we also choose a positive constant satisfying

By (), we havethat is, the sequence of is bounded. Thus, there exists a convergent subsequence. For convenience, we may assume that . Let

Then, there exists a sufficient large positive integer such that, for all ,where .

For any , we consider the solution of (1) with and , . It follows from (17), (22), (31), (), and () that

Hence, we have , that is, , where for . Furthermore, it follows from (17), (34), and () that, for ,which contradicts with (26). Hence, condition (16) is necessary for globally asymptotic stability of the trivial solution of (1). This completes the proof of Theorem 1.

To illustrate our result, we give an example.

Example 1. In equation (1), let , , andBy a straightforward computation, we can obtainIn addition,Let ; then, condition () of Theorem 1 holds. By direct calculations, we haveLet , and by (38)–(41), we find that conditions ()–() and (16) hold. Thus, (1) is globally asymptotically stable.

4. Conclusion

Stability is one of the main problems encountered in applications and has recently attracted considerable attention. The fixed point method is an effective method to discuss the stability for the differential equation with unbounded delay or the differential equation with unbounded terms. In this paper, we investigate a class of neutral set-valued functional differential equations and obtain a criterion for the globally asymptotic stability theorem with necessary and sufficient conditions by the fixed point method. Finally, we verify the validity of the result by an example.

Data Availability

Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors completed the paper together. All authors read and approved the final manuscript.

Acknowledgments

This paper was supported by the National Natural Science Foundation of China (11771115 and 11271106).

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Copyright © 2020 Junyan Bao and Peiguang Wang. 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.

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