Periodic Averaging Principle for Neutral Stochastic Delay Differential Equations with Impulses

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.


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. e 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. e 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. e 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][3][4][5][6]. e second is approximating a nonautonomous stochastic differential equation by an autonomous stochastic differential equation. e results obtained by this method can be referred to the papers [7][8][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. e 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.

Model Description and Preliminaries
In this section, we will introduce the basic concepts, the model, and some preliminary lemmas.
Let (Ω, F, F t t ≥ 0 , P) be a complete probability space with a filtration F t t ≥ 0 satisfying the usual conditions (i.e., it is right continuous and F 0 contains all P-null sets) and B(t) is m-dimensional Brownian motion defined on the space. K denotes the family of all concave continuous nondecreasing functions α: R + ⟶ R + such that α(0) � 0, 0 + (ds/α(s)) � ∞. Let C((− ∞, 0]; R d ) denote the family of all continuous functions ξ: (− ∞, 0] ⟶ R d with the norm ‖ξ‖ � sup − ∞<θ≤0 |ξ(θ)| and |·| denote any norm in In this paper, we will discuss the following neutral stochastic delay differential equations with impulses: represent the right and the left limits of e initial condition x 0 is defined by e following lemmas are important to obtain our results.

Lemma 1 (see [11]). Suppose H is a bounded linear operator on Banach space
Lemma 2 (see [11]). Suppose H is a bounded linear operator on Banach space X and has an inverse bounded operator, for arbitrary ΔH: and consider the Banach space Lemma 3 (see [12]). and Since Φ and Φ − 1 are both linear operators, we can change system (1) by using the inverse transformation of Φ into the following form: Hence, u(t) is an T-periodic solution of system (2) if and only if (Φ − 1 u)(t) is an T-periodic solution of system (1).
To study the averaging principle of system (1), we impose the following hypotheses on the coefficients. 2 Complexity Furthermore, by the definition of α, there must exist positive constants k 1 and k 2 such that Consider the standard form of system (2) du Accordingly, the standard form of system (1) is where the functions f, g, I j , j ∈ 1, 2, . . . , l { } have the same conditions as in (H 1 ) and (H 2 ), and ε ∈ [0, ε 0 ] is a positive small parameter with ε 0 is a fixed number. Let We also assume that the following hypothesis is satisfied.

Main Results and Proofs
In this section, we will use the periodic averaging principle to investigate the neutral stochastic delay differential equations with impulses.

Theorem 1. Suppose hypotheses (H 1 ) − (H 3 ) are satisfied and systems
and then, we obtain Proof. By using the elementary inequality, for any t ∈ [0, U], we have For the first term J 1 , thanks again to the elementary inequality yields: According H€ older's inequality and hypothesis (H 1 ), we arrive at Let n be the largest integer such that nT ≤ t. en, for every i � 1, . . . , n { }, we obtain 4 Complexity (20) By the H€ older's inequality, Burkholder-Davis-Gundy's inequality, and hypotheses (H 1 ) and (H 3 ), it follows that e definition of f implies that For the second term J 2 , apply Burkholder-Davis-Gundy's inequality to deduce Furthermore, on account of hypothesis (H 1 ), we have For the third term J 3 , utilizing hypothesis (H 2 ), we deduce ≤ 6ε 2 l 2 (n + 1)N 1 + 6ε 2 l 2 (n + 1) 2 N 1 ≔ εP 3 .