Abstract
In this paper, we study the periodic averaging principle for neutral stochastic delay differential equations with impulses under non-Lipschitz condition. By using the linear operator theory, we deal with the difficulty brought by delay term of the neutral system and obtain the conclusion that the solutions of neutral stochastic delay differential equations with impulses converge to the solutions of the corresponding averaged stochastic delay differential equations without impulses in the sense of mean square and in probability. At last, an example is presented to show the validity of the proposed theories.
1. Introduction
Delay, impulse, and noise are natural phenomena in most practical issues and those phenomena are generally modeled by stochastic delay differential equations with impulses, and noise can be described by Brownian motion. Recent theoretical and computational advancements indicate that stochastic delay differential equations with impulses tend to generate rich and complex dynamics. However, because of the complexity of the system, it is difficult to obtain the exact solution of the vast majority of stochastic delay differential equations with impulses. In this background, it is very important to look for an approximate system which is more amenable for analysis and simulation, and it governs the evolution of the original system over a long time scale.
The averaging principle is an effective method to understand the main part of the behavior of dynamical systems. It allows to avoid the detailed analysis of complex original systems and consider the simplified equations. The first analysis for averaging principle for stochastic differential equations was deeply addressed by Khasminskij [1]. And then the averaging principle has been applied for various types of stochastic differential equations. Generally speaking, the results of averaging principle in standard form mainly fall into two categories. The first one is considered in slow-fast systems or two-time-scale systems. It approximates coupled slow component equation in two-time-scale systems by a noncoupled equation often called an averaged equation. Many important results for averaging principle for two-time-scale stochastic differential equations have been carried out, see, for example, [2–6]. The second is approximating a nonautonomous stochastic differential equation by an autonomous stochastic differential equation. The results obtained by this method can be referred to the papers [7–9] and the references therein. However, there are few results on average principle for neutral stochastic delay differential equations. Recently, averaging principle for stochastic delay differential equations of the neutral type driven by G-Brownian motion was studied [10], in which the impulses are not considered in the system.
Motivated by the previous discussion, in this paper, we study the average principle for the neutral stochastic delay differential equations with impulses. We overcome the difficulties caused by the delay term which is included under the differentiation at the left-hand side, and the impulse appears in neutral stochastic differential equations.
The structure of the paper is the following. In Section 2, we introduce some basic concepts, notations, and necessary hypotheses. In Section 3, under several sufficient conditions, we obtain the main results that the solutions of neutral stochastic delay differential equations with impulses converge to the solutions of the corresponding averaged stochastic delay differential equations without impulses in the sense of mean square and in probability. In Section 4, we offer an example to illustrate the effectiveness of the obtained results.
2. Model Description and Preliminaries
In this section, we will introduce the basic concepts, the model, and some preliminary lemmas.
Let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets) and is -dimensional Brownian motion defined on the space. denotes the family of all concave continuous nondecreasing functions such that . Let denote the family of all continuous functions with the norm and denote any norm in . Let denote the family of all -measurable, -valued random variables such that .
In this paper, we will discuss the following neutral stochastic delay differential equations with impulses:where is a -periodic function, . The mappings and are Borel measurable and -periodic in the first argument. , and represent the right and the left limits of . We assume that there exists a positive constant such that , and . The initial condition is defined by
The following lemmas are important to obtain our results.
Lemma 1 (see [11]). Suppose is a bounded linear operator on Banach space ; if , then has bounded inverse operator and .
Lemma 2 (see [11]). Suppose is a bounded linear operator on Banach space and has an inverse bounded operator, for arbitrary , if , then has a bounded inverse and .
Letand consider the Banach space
Let be defined by
Lemma 3 (see [12]). Let . If , then has a bounded inverse on , and for all ,and .
Since and are both linear operators, we can change system (1) by using the inverse transformation of into the following form:
Hence, is an -periodic solution of system (2) if and only if is an -periodic solution of system (1).
To study the averaging principle of system (1), we impose the following hypotheses on the coefficients.
For all and , there exist such thatwhere .
Furthermore, by the definition of , there must exist positive constants and such that
For every , there exists a positive constant such that .
Consider the standard form of system (2)
Accordingly, the standard form of system (1) iswhere the functions have the same conditions as in , and is a positive small parameter with is a fixed number.
Let , , and be measurable functions and satisfy the conditions in and the following definitions:
We also assume that the following hypothesis is satisfied.
There exists a constant such that , , , and for every .
Now, we consider the following averaged stochastic delay differential equations which correspond to the original standard form (10):
Let , for system (11), we consider averaged stochastic delay differential equations:
Obviously, under hypotheses , one can follow [13, 14] to prove the existence and uniqueness of the periodic probability solutions on of the standard form (10) and (11) and the averaged form (13) and (14), respectively. Now, we offer the proof for the relationship between and .
3. Main Results and Proofs
In this section, we will use the periodic averaging principle to investigate the neutral stochastic delay differential equations with impulses.
In the rest of the paper, , and are all constants. Our main results are the following.
Theorem 1. Suppose hypotheses are satisfied and systems (10) and (13)–(11) and (14) have solutions , , , and , respectively, where is a positive small parameter with a constant, then there exist constants , , and such that, for any ,and then, we obtain
Proof. By using the elementary inequality, for any , we haveFor the first term , thanks again to the elementary inequality yields:According Hlder’s inequality and hypothesis , we arrive atLet be the largest integer such that . Then, for every , we obtainBy the Hlder’s inequality, Burkholder–Davis–Gundy’s inequality, and hypotheses and , it follows thatThe definition of implies thatThus, we haveFor the second term , apply Burkholder–Davis–Gundy’s inequality to deduceFurthermore, on account of hypothesis , we haveHence, we obtainFor the third term , utilizing hypothesis , we deduceCombining (23), (23), and (27), we conclude thatand in addition, using Gronwall’s inequality, we haveTherefore, selecting such that, for every , setting , we can choose such that, for each and ,and then, we obtain
Corollary 1. Suppose hypotheses are satisfied. Then, for any parameter , we have
Proof. In fact, under the consequence of Theorem 1, by using Chebyshev–Markov inequality, for any given number , we can deriveThe conclusion follows by letting .
The proof is complete.
4. Illustrative Example
Example 1. Consider the following neutral stochastic delay differential equations with impulses:where .
Let and consider the following stochastic delay differential equations with impulses:Then,and define a new averaged stochastic delay differential equations:namely,Let , we haveWhen , the solution of system (3) can be deduced asWhen , the solution of system (3) is deduced asRepeat the steps above on , etc., and we can get the solution on the whole interval . In addition, it is easy to verify that the hypotheses of Theorem 1 and Corollary 1 holds; hence, the solution of the averaged system (3) converges to that of the standard system (1) in the sense of mean square and in probability.
5. Conclusion
In this paper, we generalize the periodic averaging principle for neutral stochastic delay differential equations with impulses. The main difficulties in the application of periodic averaging principle for neutral system are caused by the delay term which is included under the differentiation at the left-hand side of the system. Using the linear operator theory, Theorem 1 and Corollary 1 show that the solutions of neutral stochastic delay differential equations with impulses converge to the solutions of the corresponding averaged stochastic delay differential equations without impulses in the sense of mean square and in probability. We remark that when the noises are Lvy processes other than Brownian motion considered in this paper, and the approach presented in the paper remains valid.
Data Availability
No data were used to support the findings of the 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 work was supported by the National Natural Science Foundation of China (NSFC) under Grant nos. 11771115, 11271106, and 11801128 and Natural Science Foundation of Hebei Province under Grant no. A2018201109.