Existence and global attractivity of periodic solution for impulsive stochastic Volterra-Levin

In this paper, we consider a class of impulsive stochastic Volterra-Levin equations. By establishing a new integral inequality, some sufficient conditions for the existence and global attractivity of periodic solution for impulsive stochastic Volterra-Levin equations are given. Our results imply that under the appropriate linear periodic impulsive perturbations, the impulsive stochastic Volterra-Levin equations preserve the original periodic property of the nonimpulsive stochastic Volterra-Levin equations. An example is provided to show the effectiveness of the theoretical results.


Introduction
Since Itô introduced his stochastic calculus about 50 years ago, the theory of stochastic differential equations has been developed very quickly [1][2][3].It is now being recognized to be not only richer than the corresponding theory of differential equations without stochastic perturbation but also represent a more natural framework for mathematical modeling of many real-world phenomena.Now there also exists a well-developed qualitative theory of stochastic differential equations [4][5][6].However, not so much has been developed in the direction of the periodically stochastic differential equations.Till now only a few papers have been published on this topic [7][8][9][10].In [10], Xu et al. showed that stochastic differential equations with delay has a periodic solution if its solutions are uniformly bounded and point dissipativity.
Meanwhile, the theory of impulsive differential equations has attracted the interest of many researchers in the past twenty years [11][12][13][14][15] since they provide a natural description of several real processes subject to certain perturbations whose duration is negligible in comparison with the duration of the process.Such processes are often investigated in various fields of science and technology such as physics, population dynamics, ecology, biological systems, optimal control, etc.For details, see [11,13] and references therein.In [16], the stability of nonlinear stochastic differential delay systems with impulsive are studied by constructing an impulse control for a nonlinear stochastic differential delay system.Recently, the corresponding theory for the existence of periodic solution for impulsive functional differential equations has been studied by several authors [17][18][19][20].
To the best of our knowledge, there are no results on the existence of periodic solution for impulsive stochastic differential equation, which is very important in both theories and applications and also is a very challenging problem.Motivated by the above discussions, in this paper, we will focus on the existence and global attractivity of periodic solution for impulsive stochastic Volterra-Levin equations [21,22].First we will establish the equivalence between the solution of impulsive stochastic Volterra-Levin equations and that of a corresponding nonimpulsive stochastic Volterra-Levin equations by the method given in [16].Then, by establishing a new integral inequality, some sufficient conditions for the existence and global attractivity of periodic solution for impulsive stochastic Volterra-Levin equations are given.Our results imply that under the appropriate linear periodic impulsive perturbations, the impulsive stochastic Volterra-Levin equations preserve the original periodic property of the nonimpulsive stochastic Volterra-Levin equations.An example is provided to show the effectiveness of the theoretical results.

Model description and preliminaries
For convenience, we introduce several notations and recall some basic definitions.
C(X, Y ) denotes the space of continuous mappings from the topological space X to the topological space is continuous for all but at most countable points s ∈ J and at these points s ∈ J, ψ(s + ) and ψ(s where J ⊂ R is an interval, H is a complete metric space, ψ(s + ) and ψ(s − ) denote the right-hand and left-hand limit of the function ψ(s), respectively.Especially, let P C 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 where f (t) is a continuous ω-periodic function, where ω > 0.
Throughout this paper, we make the following assumptions: (H 4 ) g(t, x(t)) and σ(t) are periodic continuous functions with periodic ω for t ≥ t 0 .
Remark 2.3.It follows from (H 4 ) and (H 5 ) that function h(t) is nonnegative integral function and satisfies that sup satisfies the differential equation of (1) for almost everywhere in [t 0 , ∞)\t k and the impulsive condtions for every Under Condition (H 5 ), Eq. ( 1) can be rewritten as follows: Under the assumptions (H 1 ) − (H 5 ), we consider the following system: with initial condition By a solution y(t) of ( 4) with initial condition (5), we mean an absolutely continuous function y(t) defined on [t 0 , ∞) satisfying (4) a.e. for t ≥ t 0 and y(t The following lemma will be useful to prove our results.The proof is similar to that of Lemma 3.1 in [16].4) and ( 5), then ) is a solution of ( 4) and ( 5) on [t 0 −τ, ∞).
Proof.(i) Suppose that y(t) is a solution of ( 4) on [t 0 , ∞), then we have for any which implies that x(t) satisfies the first equation of (3) for almost everywhere in [t 0 , ∞)\t k .
On the other hand, for every and this means that, for every Therefore, we arrive at a conclusion that x(t) is the solution of (3) corresponding to initial condition (2).
In fact, if y(t) is the solution of (4) with initial condition (5), then (ii) Since x(t) is a solution of (3) and (2), so x(t) is absolutely continuous on each interval (t and which implies that y(t) is continuous and easy to prove absolutely continuous on [t 0 , ∞).Now, similar proof in the case (i), we can easily check that y (t) = is the solution of ( 4) on [t 0 − τ, ∞) corresponding to the initial condition (5).
From the above analysis, we know that the conclusion of Lemma 2.1 is true.The proof is complete.
Definition 2.2.A stochastic process x t (s) is said to be periodic with period ω if its finite dimensional distributions are periodic with periodic ω, i.e., for any positive integer m and any moments of time t 1 , . . ., t m , the joint distributions of the random variables x t 1+kω (s), . . ., x t m+kω (s) are independent of k, Remark 2.4.By the definition of periodicity, if x t (s) is an ω-periodic stochastic process, then its mathematic expectation and variance are ω-periodic [8, p49].
Definition 2.4.The periodic solution x(t, t 0 , ϕ) with the initial condition 3) is called globally attractive if for any solution x(t, t 0 , ϕ 1 ) with the initial condition Remark 2.5.Similarly as Definition 2.2-2.4,the periodicity, attracting set and global attractivity of the solution of (4) can be defined.
Remark 2.6.From Lemma 2.1, we can easily obtain that if the periodic solution of (4) is globally attractive, then the periodic solution of (3) is also globally attractive.
Definition 2.5.The solutions y t (t 0 , ϕ) of ( 4) are said to be (i) p-uniformly bounded, if for each α > 0, t 0 ∈ R, there exists a positive constant θ = θ (α) which is We recall the following result [10, Theorem 3.5] which lays the foundation for the existence of periodic solution to Eq. (4).
If η 4 = 0, we can easily get the following corollary: Corollary 2.1.Assume that all conditions of Lemma 2.3 hold and lim t→∞ t t0 h (s)ds = ∞.Then all solutions of the inequality (6) convergence to zero.
To obtain the existence and global attractivity of periodic solution of Eq. ( 1), we introduce the following assumption.
(H 6 ) There exist positive constants p > 2 and I such that and Theorem 3.1.Suppose that (H 1 ) − (H 6 ) hold, then the system (1) must have a periodic solution, which is globally attractive and in the attracting set Proof.By the method of variation parameter, we have from ( 4) that for t ≥ t 0 , By using the inequality (a for any positive real numbers a, b, c and d, taking expectations, we find for all t ≥ t 0 , We first evaluate the first term of the right-hand side as follows: As to the second term, by (H 5 ) and (H 3 ), we have EJQTDE, 2012 No. 46, p. 7 As to the third term, by Hölder inequality, (H 5 ) and (H 3 ), we have As far as the last term is concerned, using an estimate on the Itô integral established in [24, Proposition 1.9], Hölder inequality, (H 5 ) and (H 3 ), we obtain: where c p = (p (p − 1) /2) p/2 .
It follows from ( 13)-( 17 From Lemma 2.3 and Condition (H 6 ), the solutions of (4) are p-uniformly bounded and is an attracting set of (4) (i.e., the family of all solutions of (4) is p-point dissipative).From Lemma 2.2, then system (4) must exist an ω-periodic solution.It follows from Lemma 2.1, (H 3 ) and the equivalence between (1) and (3) that the system (1) must have an ω-periodic solution.
In view of (ii) of Lemma 2.1 and (H 3 ), it's easy to see that i,e, EJQTDE, 2012 No. 46, p. 8 is an attracting set of (1) Denote y * (t) be the ω-periodic solution and y(t) be an arbitrary solution of Eq. ( 4).
We rewrite the Eq. ( 4) by Proceeding as the proof of the existence of periodic solution, we have From Corollary 2.1 and Condition (H 6 ), we get that the periodic solution is globally attractive .And the proof is completed.
Corollary 3.1.Suppose that (H 4 ), (H 5 ) and (H 6 ) with m = M = 1 hold, then the system (21) must have a periodic solution, which is globally attractive and in the attracting set Proof.The proof is similar to that of Theorem 3.1, so we omit it here. with where b k = 0, t k = k, k = 1, 2, . . . .It follows from Theorem 3.1 that Eq. ( 22) has a 4-periodic solution, which is globally attractive.

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