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Mean square exponential and nonexponential asymptotic stability of impulsive stochastic Volterra equations
Journal of Inequalities and Applications volume 2011, Article number: 9 (2011)
Abstract
In this article, some inequalities on convolution equations are presented firstly. The mean square stability of the zero solution of the impulsive stochastic Volterra equation is studied by using obtained inequalities on Liapunov function, including mean square exponential and nonexponential asymptotic stability. Several sufficient conditions for the mean square stability are presented. Results in this article indicate that not only the impulse intensity but also the time of impulse can influence the stability of the systems. At last, an example is given to show application of some obtained results.
Mathematics classification Primary(2000): 60H10, 60F15, 60J70, 34F05.
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 fixedpoint 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 integrodifferential 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 socalled "nonexponential 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 realworld 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 nonexponential 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 nonexponential 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 t_{0} = τ_{0} < ···< τ_{k} < τ_{k+1}< ··· and . We denote R^{+} = [0, +1). Consider the impulsive stochastic Volterra equations
where D_{ i } = (τ_{I}, τ_{i+1}) for all i ∈ N. f(t, x, y) : R^{+} × R^{n} × R^{n} → R^{n}, g(t, x, y) : R^{+} × R^{n} × R^{n} → R^{n}. ξ_{ i } = τ_{ i }  τ_{i1}, with respect to probability distribution for all i = 1, 2,.... I_{ i }(t, x) : R^{+} × R^{n} → R^{n}. F (t) and G(t) are both continuous and integrable matrixvalued functions on R^{+}. B(t) is standard ndimensional Brownian motion on a complete filtered probability space Ω, F, (F^{B} (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 > t_{0}, 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

(i)
mean square asymptotically stable, if for any ε > 0, there exist constants δ > 0 and T = T (t_{0}, ε) > 0 such that E (x (t)^{2}) < ε for all t > t_{0} + T when E (x_{0}^{2}) < δ.

(ii)
mean square exponentially asymptotically stable, if for any t_{0} ∈ R^{+} there exist λ > 0, T > 0 and C = C(x_{0}, t_{0}) > 0 such that E (x (t)^{2})< C exp (λt) for t > T.

(iii)
mean square nonexponentially 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 C^{1}[0, ∞) be the family of all continuous functions on [0, ∞) which are once continuously differentiable and C^{1,2}(R^{+} × R^{n}, R^{+}) denote the family of all nonnegative functions from R^{+} × R^{n} to R^{n} which are once continuously differentiable in t and twice in x. For each V ∈ C^{1,2}(R^{+} × R^{n}, 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
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
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 }, τ_{i+1}) where i ∈ N. Then
is true for all t ∈ (τ_{ I }, τ_{i+1}).
Proof. If τ_{ i }= 0,
holds for
If τ _{ i }> 0, we have . By supposing for all t ∈ [τ_{ I }, τ_{i+1}), we get
where z(t) is solution of (2). Now we prove that y(t) ≤ p(t) for all t ∈ [τ_{ i }, τ_{i+1}). If it is not true, then there exist t_{1} ∈ (τ_{I}, τ_{i+1}) such that y(t_{1}) > p(t_{1}). Denote , then and
for all . Therefore holds, which implies
From (4), we have
for all . From (5), we get
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 nonexponential asymptotic stability.
3.1 Mean square exponential asymptotic stability
Theorem 3.1. If there exist positive numbers c_{1}, c_{2} and V ∈ C^{1,2}(R^{+} × R^{n}, R^{+}) satisfying

(i)
c_{1} x^{p} ≤ V (t, x) ≤ c_{2} x^{p};

(ii)
there exist two continuous and integrable functions k, h : R^{+} → R^{+} and constant a > 0 such that
for any i = 1, 2,...;

(iii)
there exist constants ω_{ i } such that for any i = 1, 2,..., we have

(iv)
and ;

(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,
for all t ∈ [τ_{ i } _{1}, τ_{ i }). Now let's prove that
holds for all t∈ [τ_{i} _{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 = τ_{i} we get
From assumption (iii) Then by use of (10) for all t ∈ [τ_{ i }, τ_{i+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 c_{1}, c_{2} and V ∈ C^{1,2}(R^{+} × R^{n}, 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 c_{1}, c_{2} and V ∈ C^{1,2}(R^{+} × R^{n}, R^{+}) satisfying

(i)
c_{1} x^{p} ≤ V (t, x) ≤ c_{2} x^{p};

(ii)
there exist continuous and integrable function k : R^{+} → R^{+} and positive constant a such that for any i = 1, 2,...
holds when Ex (τ_{i1})^{2} < θ for some constant θ > 0;

(iii)
there exist positive constants ω_{ i } such that for any i = 1, 2,..., we have

(iv)
there exists 0 < ρ < 1 such that ;

(v)
;

(vi)
τ_{i} ≤ t_{0} + 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 ∈ [τ_{i1}, τ_{ i }) we get
By denoting , it can be proved that when x_{0}^{2} < δ_{0},
Holds for all i = 1,2,..., and
holds for all t ∈ [τ_{ i1 }, 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_{ i1 }, 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 τ_{i1}≤ t ≤ τ_{ i }≤ t_{0} + 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 nonexponential asymptotic stability
To show that the solution of (1) is mean square nonexponentially 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 c_{1}, c_{2} and V ∈ C^{1,2}(R^{+} × R^{n}, R^{+}) satisfying

(i)
c_{1} x^{p} ≤ V (t, x) ≤ c_{2} x^{p};

(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;

(iii)
there exist constants ω_{ i } such that for any i = 1, 2,······, we have

(iv)
there exists 0 < ρ < 1 such that ;

(v)
.
Then zero solution of (1) is mean square asymptotically stable.
Proof. From (11), by Lemma 2.2 and by Lemma 2.3,
for t ∈ [τ_{i1}, τ_{ i }). Noticing 0 < ρ < 1, for any ε > 0, there is k_{0} > 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 c_{1}, c_{2} and V ∈ C^{1,2}(R^{+} × R^{n}, 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 }  t_{0}) ≥ i for all i = 1, 2,....
Then zero solution of (1) is mean square nonexponentially asymptotically stable.
Proof. By use of Theorem 3.3, we obtain
From (H 1), we have that
for all t ∈ [τ_{i1}, τ_{ i }). Consequently, we get
for all t ∈ [τ_{i1}, τ_{ i }) for i = 1, 2,... by mathematical induction from (H1)(H2). From assumption
Since satisfies and , then
By L'Hospital rule
From (H5) and d < 1,
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
for t ∈ (τ_{ k }, τ_{ k } + 1) with , where τ_{ k } = 2^{k}and the impulse is defined as
for all k ∈ N. λ_{1}(k) and λ_{2}(k) are random variables on . Then the zero solution of (18) and (19) is mean square nonexponentially asymptotically stable.
Proof. By putting , we have that
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.
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Acknowledgements
The authors sincerely thank the anonymous reviewer for his careful reading, constructive comments and fruitful suggestions to improve the quality of the manuscript. This article is partially supported by NSFC (No. 11001173).
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DZ and DH conceived of the study. DZ carried out most of the analysis and drafted the manuscript. DH revised and commented on the draft. All authors read and approved the final manuscript.
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Zhao, D., Han, D. Mean square exponential and nonexponential asymptotic stability of impulsive stochastic Volterra equations. J Inequal Appl 2011, 9 (2011). https://doi.org/10.1186/1029242X20119
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DOI: https://doi.org/10.1186/1029242X20119
Keywords
 Stochastic Volterra equations
 Impulse
 Liapunov function
 Asymptotic stability