Abstract
This paper is devoted to investigating a class of stochastic neutral-type neural networks with delays. By using the fixed point theorem and the properties of neutral-type operator, we obtain the existence conditions for periodic solutions of stochastic neutral-type neural networks. Furthermore, we obtain the conditions for the exponential stability of periodic solutions using Gronwall-Bellman inequality and stochastic analysis technique. Finally, a numerical example is given to show the effectiveness and merits of the present results. Our results can be used to obtain the existence and exponential stability of periodic solution to the corresponding deterministic systems.
1 Introduction
During the past years, the theory of stochastic differential equations has been extensively studied, see, e.g., [1–5] and references therein. However, periodic solution problems of stochastic differential equations have been studied by few authors. To be best our knowledge, we only find that very few results for periodic solution of stochastic differential equations and stochastic differential systems have been obtained. In [6], Kolmanovskii and Myshkis introduced basic theory of T − periodic stochastic process and T − periodic solution for stochastic retard differential equation which greatly promoted the study of periodic solutions of stochastic differential equations. It and Nisio [7] studied stationary solutions of a stochastic differential equation. Has’minskii [8] studied the existence of periodic solution of differential equations with random right sides. In [9], the authors considered existence problems for periodic Markov process and stochastic functional differential equations by using the properties of periodic Markov processes. After that, Li and Xu [10] also obtained the existence of periodic solution for a stochastic functional differential equation with unbounded delays. In [11], Zhang and Gopalsamy considered two classes of n − species stochastic population models with periodic coefficients and obtained some sufficient conditions for the existence of a stochastically asymptotically stable in the large periodic solution process. In [12], the authors studied mean-square almost periodic solution for impulsive stochastic by applying Cauchy matrix. Jiang, etc. [13, 14] dealt with periodic solution of nonautonomous logistic equation with random perturbation.
In the real world, a specific neural network is always affected by various uncertain factors, stochastic perturbations are almost inevitable [15]. Therefore, it is necessary to investigate effects of stochastic perturbations on the dynamic properties of neural networks. Recent years, stochastic neural networks has been extensively studied. Stability analysis of various stochastic neural networks see, e.g., [16–19]. In [16], the authors considered the distributed synchronization of coupled neural networks. Zhu and Cao [17, 18] investigated exponential stability of stochastic neural networks with both Markovian jump parameters and mixed time delays. Stability analysis of switched stochastic neural networks with time-varying delays has been studied in [19]. The stability and stabilization for a class of stochastic systems with impulsive effects, see [20]; the pth moment (p ≥ 2) and the almost-sure stability of stochastic Cohen-Grossberg neural networks, see [21]; stochastic neural networks with local impulsive effects, see [22]. For deterministic neural network, see, e.g., [23, 24]. However, there are not many achievements in the study of periodic solutions of stochastic neural networks. Using fixed points principle and Gronwall-Bellman inequality, the authors [25] concerned with the periodic solutions for a class of stochastic Cohen-Grossberg neural networks with time-varying delays. Yang and Li [26] considered existence and exponential stability of periodic solution for stochastic Hopfield neural networks on time scales. Wang and Wu [27] studied mean square exponential stability and periodic solutions of stochastic interval neural networks with mixed time delays. For recent advances in periodic solution of stochastic differential equation and neural networks, see [28–34].
Our main purpose of this paper is to study periodic solution of a stochastic neutral-type neural networks by using the contraction mapping theorem and Gronwall-Bellman inequality. For First, an effective existence and uniqueness theorem of periodic solution for considered system is established. Then, some sufficient conditions for the exponential stability of periodic solution are given. Because the system we study contains neutral terms and random perturbations, it is difficult to obtain the existence conditions of periodic solutions. To overcome the above difficulties, as one will see, several novel mathematic analysis methods are applied. These existence and stability theorems are rather general and therefore have great power in applications.
The distinctive contributions of this paper are outlined as follows:
(1) It is noted that most existing results on stochastic neural networks are mainly pertaining to the stability of considered systems, see e.g., [9–11] and related references. In this paper, we obtained existence results of periodic solution by using the contraction mapping theorem. Hence, the research content of this article expands the scope of research on nonlinear neutral stochastic differential systems.
(2) We develop some techniques of stochastic analysis for studying stochastic neutral-type neural networks with delays, our methods for the proof of main results can more easily be understood. Particularly, we use contraction mapping theorem and Gronwall-Bellman inequality for obtaining stability results which is different from ones in [14–16, 35–37].
(3) Our main results are also valid for the case of the corresponding deterministic systems.
The following sections are organized as follows: In Section 2, we introduce some useful Lemmas and Definitions. In Section 3, some sufficient conditions are established for existence and uniqueness of periodic solution of the considered system. Section 4 gives some sufficient conditions for guaranteeing the exponential stability of periodic solution. In Section 5, an example is given to show the feasibility of our results. Finally, some conclusions are given for this paper.
2 Preliminaries
In the present paper, we consider a stochastic neutral-type neural networks with delays of the form
where n is the number of units in the considered system, γ > 0 is a delay, xi(t) is the state of the ith neuron at time t, fj(⋅) and gj(⋅) are the activation functions of the jth unit, ai(t) ≥ 0 denotes the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the networks and external inputs, aij(t) and dij(t) denote the strength of the jth neuron on ith unit at time t and t − τij(t), respectively, Ii(t) denotes the ith component of an external input source, is the standard Wiener process defined on complete probability space σij is Borel measurable function. We assume that ci(t), τij(t), ai(t), aij(t), dij(t) and Ij(t) are defined on , are T − periodic and continuous functions. The initial condition of Eq. 2.1 is
where .
Let B be a Banach space with the norm ‖ ⋅‖ and be the space of all B-value random variable X such that . Let f(x) be a continuous T − peridic function on . Denote
Throughout this paper, we assume that.
(H1) are Lipschitz-continuous with Lipschitz constants and lij, respectively, i, j = 1, 2, … , n.
() are Lipschitz-continuous with Lipschitz constants and , respectively, j = 1, 2, … , n.
Definition 2.1[8] A stochastic process xt(s) is said to be periodic with period T if its finite dimensional distributions are periodic with period T, that is, for any positive integer n and any moments of time t1, t2, … , tn the joint distribution of the random variables are independent of .
Remark 2.1[8] Ifx(t) is anT − periodic stochastic process, then its mathematical expectation and variance areT − periodic.
Lemma 2.1[5] (The It isometry) If f(t, ω) is is bounded and elementary, then
Lemma 2.2[38] For each and p > 0,where ∨ denotes the Min operator, that is,
Lemma 2.3[39] Let
|
c(
t)| ≠ 1
, thenoperator
Ahas continuous inverseA−1onCT, satisfying.
1)
2)
3)
where .
Remark 2.2From Lemma 2.3, we havewhere .
Definition 2.2[25] The periodic solutionx(t, t0, ϕ) with initial valueϕof system (2.1) is said to be globally exponentially stable, if there are constantsλ > 0 andM > 1 such that for any solutiony(t, t0, ϕ1) with initial valueϕ1of system (2.1) satisfies
Let (Aixi) (t) = xi(t) − ci(t)xi(t − γ) = yi(t). From Lemma 2.3, then , and system (2.1) can be rewritten bywith initial condition
Remark 2.3system (2.1) is equivalent to system (2.3). Thus, system (2.1) has a globally exponentially stable periodic solution, if and only if, system (2.3) has a globally exponentially stable periodic solution. Since system (2.3) has not neutral-type term, we can easily obtain existence and stability results for system (2.3).
Remark 2.4System (2.1) is a neutral-type stochastic system which shows the neutral properties byD − operatorxi(t) − ci(t)xi(t − γ). For the details aboutD − operator, see [40]. Some results of stochastic system withD − operator have been obtained, see [41–43] and related references. However, there exist few results for the periodic solution of stochastic system withD − operator. This paper is devoted to investigating the above problem and obtaining the new results.
3 Existence of periodic solution
In this section, we will show the existence of periodic solutions for system (2.1). Now, consider the linear section for system (2.3)By basic theory for ordinary differential equation, system (3.1) has a solutionwhere . It is easy to see that
Suppose that |ci(t)| ≠ 1, i = 1, 2, … , nand assumption (H1) holds. Then system (2.1) has uniqueT-periodic solution, provided thatwhere
Proof. Letwith the norm , where . Then is a Banach space. Define a map Γ on bywhereObviously, (Γϕ)i(t + T) = (Γϕ)i(t). Hence, Γ maps to . Next, we show that Γ is a contraction mapping. For any , we haveFor i = 1, 2, … , n, letandTaking expectations for the above F1i and F2i, by Lemma 2.2, we haveEvaluating the first term of the right-hand side of (3.5), in view of Lemma 2.3, (3.2) and assumption (H1), we haveAs to the second term of the right-hand side of (3.5), in view of Lemma 2.1, Lemma 2.3, (3.2) and assumption (H1), we also haveFrom (3.5-3.7), we haveThus,By (3.1) Γ is a contraction mapping on system (2.3) has a unique periodic solution yi(t), i.e., system (2.1) has a unique periodic solution .
Remark 3.1Consider the corresponding deterministic system of system (2.1)wherei = 1, 2, … , n.
Corollary 3.1Suppose that |ci(t)| ≠ 1, i = 1, 2, … , n and assumption () holds. Then system (3.8) has unique T-periodic solution, provided thatwhere
Remark 3.2To the best of our knowledge, few authors deal with the existence and exponential stability of periodic solutions to stochastic neutral-type neural networks by using contraction mapping theorem and Gronwall-Bellman inequality. Most articles only studied the stability of stochastic neural networks, and the results on the existence of solutions are not many, see e.g., [17–19, 21, 22]. Therefore, the results of this article enrich and develop the research content and methods of stochastic neural networks. It should be pointed out that the properties of regarding neutral-type operators in Lemma 2.3 have important applications for obtaining the main results of this paper. I believe that the above properties of neutral-type operators will have wide applications in studying other types of neutral-type systems.
Remark 3.3In [44], the authors studied periodic solution problem of a class of stochastic nonlinear system with delays; in this paper, we investigated periodic solution problem of a class of stochastic neutral-type neural networks with delays. The above two systems are obviously different. Furthermore The main research methods in [44] are stochastic analysis technique and Lyaplov functional method, see Theorem 2.2, Lemma 2.3 and Lemma 2.4 in [44]; the main research methods in this paper are contraction mapping theorem and Gronwall-Bellman inequality which are different from the corresponding ones in [44].
4 Globally exponential stability of periodic solution
In this section, we firstly show the exponential stability of periodic solutions for system (2.3) with initial condition (2.4). Then, we further obtain the exponential stability of periodic solutions for system (2.1) with initial condition (2.2).
Suppose that all conditions of Theorem 3.1 hold. Then, the periodic solution of system (2.1) is globally exponentially stable, provided that
Proof. From Theorem 3.1, system (2.3) has a periodic solution yi(t) with initial condition ψi(s), where i = 1, 2, … , n, s ∈ (−∞, t0]. Assume that is an arbitrary solution of system (2.3) with the initial condition , where i = 1, 2, … , n, s ∈ (−∞, t0]. From basic theory of ordinary differential equation, system (2.3) has a solutionwhere i = 1, 2, … , n, t ≥ t0. Let . By (4.2), we haveLetTaking expectations for the above H1i − H3i, by Lemma 2.2 and (4.3), we haveEvaluating the first term of the right-hand side of (4.4), by (3.2) we haveEvaluating the second term of the right-hand side of (4.4), in view of Lemma 2.3, (3.2) and assumption (H1), we haveAs to the third term of the right-hand side of (4.4), in view of Lemma 2.1, Lemma 2.3, (3.2) and assumption (H1), we also haveFrom (4.4-4.7), we haveUsing Gronwall-Bellman inequality and (4.8), we havewhere λ is defined by (4.1). Furthermore, form Lemma 2.3 and (4.9), we haveHence, the periodic solution of system (2.3) is globally exponentially stable, i.e., the periodic solution of system (2.1) is globally exponentially stable.
Corollary 4.1Suppose that all conditions of Theorem 3.1 hold. Then, the periodic solution of system (2.1) is globally exponentially stable, provided thatwhere is defined by corollary 3.1.
Remark 4.1From the above results, it is easy to see that the random terms have no effect on the periodicity of the considered system. That is, both stochastic neutral-type neural networks and its corresponding deterministic systems have the similar periodicity.
Remark 4.2In recent years, fractional-order system have been extensively studied, see [45, 46] and related references. However, the periodic solution problems for fractional-order system or stochastic fractional-order system are rarely studied. In future research, we will focus on the aforementioned issues.
5 A numerical example
In this section, we present an example to illustrate the feasibility of our results obtained in previous sections. For i = 2, consider the following stochastic neutral-type neural networks:whereAfter a simple calculation, we haveThus,
It follows that all conditions of Theorem 4.1 hold. Hence, system (5.1) has a periodic solution, which is globally exponentially stable. The numerical solutions with different initial values are shown in Figure 1 and Figure 2.
FIGURE 1
The states’ evolution of x1(t) for Eq. 5.1 with different initial values.
FIGURE 2
The states’ evolution of x2(t) for Eq. 5.1 with different initial values.
6 Conclusion and discussions
In this paper, we have obtained some new sufficient conditions for the existence, uniqueness and exponential stability of periodic solution for a stochastic neutral-type neural networks with delays. The existence results have been obtained by the contraction mapping theorem which extend the previous corresponding results. The stability results have been obtained by stochastic analysis and Gronwall-Bellman inequality. It should be pointed out that the properties of neutral type operators have important applications in this study. We believe that the above properties can also be used to study other types of neutral-type neural networks.
In the future, we will explore existence and stability of periodic solution for neutral-type stochastic differential system with impulse, markov jumps, Lvy jumps and so on. Also, we will study periodic solution problems for neutral-type stochastic differential system on time scales.
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding authors.
Author contributions
FZ: Methodology, Funding acquisition, Writing–review and editing. XL: Methodology, Funding acquisition, Investigation, Writing–review and editing. BD: Writing–original draft, Methodology.
Funding
The author(s) declare financial support was received for the research, authorship, and/or publication of this article. This work is supported by the National Natural Science Foundation of China (No. 11971197) and Doctor Training Program of Jiyang College, Zhejiang Agriculture and Forestry University (RC2022D03).
Acknowledgments
The authors would like to thanks the editor and the referees for their valuable comments and suggestions, that improve the quality of our paper.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
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Summary
Keywords
periodic solution, stochastic, neutral-type neural networks, existence, exponential stability
Citation
Zheng F, Li X and Du B (2024) Periodic solution problems of neutral-type stochastic neural networks with time-varying delays. Front. Phys. 12:1338799. doi: 10.3389/fphy.2024.1338799
Received
15 November 2023
Accepted
20 March 2024
Published
05 April 2024
Volume
12 - 2024
Edited by
Grienggrai Rajchakit, Maejo University, Thailand
Reviewed by
Balasubramaniam. P, The Gandhigram Rural Institute, India
Peiluan Li, Henan University of Science and Technology, China
Updates
Copyright
© 2024 Zheng, Li and Du.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Xiaoliang Li, lixiaoliang@zafu.edu.cn; Bo Du, dubo7307@163.com
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.