1 Introduction

Study on the stability of stochastic differential equations has gained lots of attention over the last years. The results and methods have been improved from time to time. Very recently, Taniguchi [1] studied the exponential stability for stochastic delay partial differential equations by use of the energy method which overcomes the difficulty of constructing the Liapunov functional on delay differential equations. Wan and Duan [2] extended the result of Taniguchi [1] to be applied to more general stochastic partial differential equations with memory. Another important method is about the fixed-point theory. It was first used to consider the exponential stability for stochastic partial differential equations with delays by Luo [3], where the conditions do not require the monotone decreasing behavior of the delays. This method also employed in Sakthivel and Luo [4, 5] to study the asymptotic stability of the nonlinear impulsive stochastic differential equations and the impulsive stochastic partial differential equations with infinite delays.

On considering Volterra equations, there is a significant literature devoted to the asymptotic stability of the zero solutions of Volterra integro-differential equations. In the known literature, the properties of linear scalar Voterra equation play an important role. The equation is

where the kernel k(t) is continuous, integrable and of a single sign. Brauer [6] showed that the solution could not be stable if , Burton and Mahfoud [7] proved the zero solution is asymptotically stable if , Kordonis and Philos [8] discussed the stability of the solution under condition . Therefore, a necessary condition for for all solutions is that . About exponential asymptotic stability, Murakami [9] showed that the uniform asymptotic stability and the exponential asymptotic stability of the zero solution of this equation are equivalent if and only if for some γ > 0. Hence if it fails to hold, a uniformly asymptotically stable solution cannot be exponentially asymptotically stable. Some deeper related work on deterministic equations by Appleby can be found in [10–12], including the so-called "non-exponential decay rate" and "subexponential solution". Mao [13] investigated the mean square stability of the generalized equation

On some special stochastic volterra equations without impulse, we highlight here the contribution of Appely [14–19]. However real-world systems can be modeled to include random effects, including stochastic perturbations and impulses. It is natural to ask how the presence of such random effects can influence the stability of the systems. Based on the generalized equation [13], in this article, we consider the effect of the impulse intensity and the impulse time on the mean square exponential and non-exponential asymptotic stability of impulsive stochastic Volterra equation

for all i ∈ N = {0, 1, 2, ·····} by using Liapunov function, which show that both the presence of impulses and the time of the presence can influence the stability of the systems. By choosing the impulse intensity and the impulse time, We find that is not necessary condition for the exponential asymptotic stability.

The article is organized as follows: some preliminary notations and useful lemmas are given in Sect. 2. Then, sufficient conditions of the mean square exponential asymptotic stability are shown in the first part of Sect. 3, and the second part mainly deals with the mean square non-exponential asymptotic stability of the solution. Finally, an example is given.

2 Preliminary notes

Let {τ i , i = 1, 2,...} be a series of numbers such that t0 = τ0 < ···< τk < τk+1< ··· and . We denote R+ = [0, +1). Consider the impulsive stochastic Volterra equations

(1)

where D i = (τI, τi+1) for all i ∈ N. f(t, x, y) : R+ × Rn × Rn → Rn, g(t, x, y) : R+ × Rn × Rn → Rn. ξ i = τ i - τi-1, with respect to probability distribution for all i = 1, 2,.... I i (t, x) : R+ × Rn → Rn. F (t) and G(t) are both continuous and integrable matrix-valued functions on R+. B(t) is standard n-dimensional Brownian motion on a complete filtered probability space Ω, F, (FB (t)) t ≥ 0, P), where the filtration is defined as F B(t) = σ (B(s) : 0 ≤ s ≤ t). Almost sure events are Palmost sure in this article denoted by "a.s.". Suppose f(t, 0, y) = 0, g(t, 0, y) = 0 and I i (t, 0) = 0 for t > t0, then x(t) ≡ 0 is the solution of (1), which is called zero solution of (1). In this article, we always assume there exists a unique stochastic process satisfying (1), and assume all solutions of (1) are continuous on the left and limitable on the right. We further recall the various standard notions of stability of the zero solution required.

Definition 2.1. The zero solution of (1) is said to be

  1. (i)

    mean square asymptotically stable, if for any ε > 0, there exist constants δ > 0 and T = T (t0, ε) > 0 such that E (||x (t)||2) < ε for all t > t0 + T when E (||x0||2) < δ.

  2. (ii)

    mean square exponentially asymptotically stable, if for any t0 ∈ R+ there exist λ > 0, T > 0 and C = C(x0, t0) > 0 such that E (||x (t)||2)< C exp (-λt) for t > T.

  3. (iii)

    mean square non-exponentially asymptotically stable, if and hold.

Suppose that a ∨ b = max{a, b}, E(x) is the expectation of x and ||x (t)|| is some norm in the sequel. Let C1[0, ∞) be the family of all continuous functions on [0, ∞) which are once continuously differentiable and C1,2(R+ × Rn, R+) denote the family of all nonnegative functions from R+ × Rn to Rn which are once continuously differentiable in t and twice in x. For each V ∈ C1,2(R+ × Rn, R+), we denote V (t) = E(V (t, x(t))), V (t-) = E(V (t, x(t-))) and

where and .

Before going to the main results, let's consider some lemmas about linear Volterra equation without impulses.

Lemma 2.2. Suppose k(t) > 0 is a function on R+. a > 0 is constant. Let z(t) satisfy

(2)

Then z(t) > 0 and for t ≥ s ≥ 0. Moreover, implies that z(t) ≤ 1.

Proof. Firstly we claim that z(t) > 0 for all t ∈ [0, +∞), if not, there exists t > 0 such that . Then we have z(t) > 0 for all . Since , we get that , then there is satisfying

(3)

From (2),

holds, which contradicts with (3). So we get that z(t) > 0 for all t ∈ [0, +∞). Again from (2), we get

By integrating on both sides, we get for t ≥ s ≥ 0.

If , by integrating on (2) we get

The proof is complete.

Lemma 2.3. [[20], Corollary 3.3] Under conditions in Lemma 2.2. Let z(t) be solution of (2). Suppose

Then z(t) is nonincreasing on [0, +∞).

Lemma 2.4. Suppose k(t) > 0 is a function on R+. a > 0 is constant. h(t) ≥ 0 is a function on R+. Let y(t) satisfy

for all t ∈ (τ I , τi+1) where i ∈ N. Then

(4)

is true for all t ∈ (τ I , τi+1).

Proof. If τ i = 0,

holds for

If τ i > 0, we have . By supposing for all t ∈ [τ I , τi+1), we get

(5)

where z(t) is solution of (2). Now we prove that y(t) ≤ p(t) for all t ∈ [τ i , τi+1). If it is not true, then there exist t1 ∈ (τI, τi+1) such that y(t1) > p(t1). Denote , then and

(6)

for all . Therefore holds, which implies

(7)

From (4), we have

(8)

for all . From (5), we get

(9)

for all . By combining (6)(8) and (9), we obtain which contradicts with (7). From above all, we arrive at the desired result.

3 Main results

In this section, we consider the nonlinear volterra equation with impulsive effect and denote the solution of (1) by x(t). Several sufficient conditions of mean square stability are presented by comparison method with Liapunov function, which include mean square exponential asymptotic stability and mean square non-exponential asymptotic stability.

3.1 Mean square exponential asymptotic stability

Theorem 3.1. If there exist positive numbers c1, c2 and V ∈ C1,2(R+ × Rn, R+) satisfying

  1. (i)

    c1 ||x||p ≤ V (t, x) ≤ c2 ||x||p;

  2. (ii)

    there exist two continuous and integrable functions k, h : R+ → R+ and constant a > 0 such that

for any i = 1, 2,...;

  1. (iii)

    there exist constants ω i such that for any i = 1, 2,..., we have

  2. (iv)

    and ;

  3. (v)

    there exists γ > 0 such that and .

Then zero solution of (1) is mean square exponentially asymptotically stable.

Proof. From (ii), we have

where D+ denotes the right Dini derivative. By Lemma 2.4,

(10)

for all t ∈ [τ i -1, τ i ). Now let's prove that

(11)

holds for all t∈ [τi -1, τ i ) by mathematical induction for i = 1, 2,.... We stipulate and as i = 1 here and in the sequel. (11) is true for i = 1 immediately from (10). Assume that (11) holds for any i ≥ 1, then for t = τi we get

From assumption (iii) Then by use of (10) for all t ∈ [τ i , τi+1) we get

Thus by mathematical induction (11) is true for i = 1, 2,....

By Lemma 2.2, it follows (11) that

Then the mean square exponential asymptotic stability of (1) inherits from that of solutions of (2) under assumptions (iv) and (v). The proof is complete.

Corollary 3.2. If there exist positive numbers c1, c2 and V ∈ C1,2(R+ × Rn, R+) satisfying (i)-(iii) and

(v) in Theorem 3.1 and

(H1) exp (as)ds < ∞;

(H2) there exists 0 < ρ < 1 such that .

Then zero solution of (1) is mean square exponentially asymptotically stable.

Proof. Since implies , the result is proved by Theorem 3.1.

Theorem 3.3. If there exist positive numbers c1, c2 and V ∈ C1,2(R+ × Rn, R+) satisfying

  1. (i)

    c1 ||x||p ≤ V (t, x) ≤ c2 ||x||p;

  2. (ii)

    there exist continuous and integrable function k : R+ → R+ and positive constant a such that for any i = 1, 2,...

holds when E||x (τi-1)||2 < θ for some constant θ > 0;

  1. (iii)

    there exist positive constants ω i such that for any i = 1, 2,..., we have

  2. (iv)

    there exists 0 < ρ < 1 such that ;

  3. (v)

    ;

  4. (vi)

    τi ≤ t0 + i for all i ∈ N.

Then zero solution of (1) is mean square exponentially asymptotically stable.

Proof. Assumption (ii) implies

From Lemma 2.4, let h(t) = 0, for all t ∈ [τi-1, τ i ) we get

(12)

By denoting , it can be proved that when ||x0||2 < δ0,

(13)

Holds for all i = 1,2,..., and

(14)

holds for all t ∈ [τ i-1 , τ i ) by mathematical induction. From (12), it is obviously true for i = 1. Assume that (14) is true for any i ≥ 1, then for all t ∈ [τ i-1 , τ i )), it is true that

and

From assumption (iii), we have that

Then by (ii),

for . Then by mathematical induction (14) is true for i = 1,2,....

Combining z(t) ≤ 1, (iv) (vi) and the above results,

since τi-1≤ t ≤ τ i ≤ t0 + i.

Therefore holds. The proof is complete.

Remark 1. Theorem 3.3 is not a simple corollary of Theorem 3.1, since the conditions (ii) and (v) in Theorem 3.3 is weaker than that in Theorem 3.1.

Remark 2. Theorem 3.3 shows that is not necessary condition for exponential asymptotical stability, which can also be found in Theorem 3.5.

3.2 Mean square non-exponential asymptotic stability

To show that the solution of (1) is mean square non-exponentially asymptotically stable, we have to prove that and . Now we prove the solution convergent to zero firstly.

Theorem 3.4. If there exist positive numbers c1, c2 and V ∈ C1,2(R+ × Rn, R+) satisfying

  1. (i)

    c1 ||x||p ≤ V (t, x) ≤ c2 ||x||p;

  2. (ii)

    there exist two continuous and integrable functions k, h : R+ → R+ such that for any i = 1, 2, ······

holds for some constant a > 0;

  1. (iii)

    there exist constants ω i such that for any i = 1, 2,······, we have

  2. (iv)

    there exists 0 < ρ < 1 such that ;

  3. (v)

    .

Then zero solution of (1) is mean square asymptotically stable.

Proof. From (11), by Lemma 2.2 and by Lemma 2.3,

(15)

for t ∈ [τi-1, τ i ). Noticing 0 < ρ < 1, for any ε > 0, there is k0 > 0 such that where . For h(t) is integrable, for any ε defined above, there is such that for . It follows that

for . By choosing , it follows (15) directly that for any ε > 0, we have for when . The proof is complete.

Theorem 3.5. If there exist positive numbers c1, c2 and V ∈ C1,2(R+ × Rn, R+) satisfying (i)-(v) in theorem 3.3 and

(H1) there exist continuous and integrable function satisfying and constant satisfying such that

for any i = 1, 2,...;

(H2) there exist constants such that for any i = 1, 2,..., we have

(H3) satisfies and ;

(H4) there is constant 1 > d > 0 such that ;

(H5) log (τ i - t0) ≥ i for all i = 1, 2,....

Then zero solution of (1) is mean square non-exponentially asymptotically stable.

Proof. By use of Theorem 3.3, we obtain

From (H 1), we have that

for all t ∈ [τi-1, τ i ). Consequently, we get

for all t ∈ [τi-1, τ i ) for i = 1, 2,... by mathematical induction from (H1)-(H2). From assumption

Since satisfies and , then

By L'Hospital rule

(16)

From (H5) and d < 1,

(17)

Thus, combine (16) and (17) to show

Since holds under assumptions (i)-(v) in Theorem 3.3, we get

Therefore

The proof is complete.

Remark 3. Assumption (H5) in Theorem 3.5 can be replaced by .

4 Example

Example 1. Consider a nonlinear impulsive stochastic Volterra equation of the form

(18)

for t ∈ (τ k , τ k + 1) with , where τ k = 2kand the impulse is defined as

(19)

for all k ∈ N. λ1(k) and λ2(k) are random variables on . Then the zero solution of (18) and (19) is mean square non-exponentially asymptotically stable.

Proof. By putting , we have that

(20)

It follows that

Since

we have

for all j > 0. In addition,

and

hold. By Theorem 3.4, it is true that

Next, we prove

From (20),

Since and log τ k = k, we finish the proof by Theorem 3.5.