 Research
 Open Access
Stability of sets of stochastic functional differential equations with impulse effect
 Yan Xu^{1} and
 Zhimin He^{1}Email author
https://doi.org/10.1186/s1366201505512
© Xu and He 2015
Received: 5 December 2014
Accepted: 22 June 2015
Published: 4 July 2015
Abstract
In this paper, we study the stability of sets for a class of impulsive stochastic functional differential equations. By employing piecewise continuous Lyapunov functions with Razumikhin methods, some sufficient conditions are established to guarantee the stability of sets of impulsive stochastic functional differential equations and we also show that the impulses play an important role in the stability of stochastic functional differential equations. Three examples are presented to illustrate the effectiveness of the results obtained.
Keywords
 stability of sets
 Brownian motion
 stochastic functional differential equations
 impulse
 Lyapunov function
 Razumikhin methods
MSC
 34K20
 34K45
 34K50
1 Introduction
During the past few decades, the stability theory of stochastic differential equations and impulsive differential equations has been developed very quickly; see for instance [1–15]. A lot of stability criteria on impulsive stochastic differential equations have also been reported (see [16–23] and the references therein). Almost all of them mainly focus on the stability of the zero solution, but there is very little of research addressing the stability of sets.
The concept of stability of sets of nonlinear systems, which includes as a special case stability in the sense of Lyapunov (see Krasovskii [24]; Rouche et al. [25]), such as stability of the trivial solution, stability of the solution, stability with respect to part of the variables and so on, has become one of the most important issues in the stability theory of nonlinear systems [26–28]. The theoretical works of the stability of sets with respect to nonlinear ordinary differential equations may be traced back to Yoshizawa [29–31] in the previous century. The research to the stability of sets of impulsive differential equations can be found in [15, 32–35]. For stochastic differential equations and impulsive stochastic differential equations, we refer the reader to [11, 36–39] and the references therein.
In this paper, we shall extend the Razumikhin method developed in [7, 14, 40] to investigate the stability of sets for a class of impulsive stochastic functional differential equations. Meanwhile, our results show that the impulsive effects play an important part in the stability for stochastic functional differential equations, that is, an unstable stochastic delay system can be successfully stabilized by impulses.
The rest of this paper is organized as follows. Some preliminary notes are given in Section 2. Several theorems on stability of sets of impulsive stochastic functional differential equation are established in Section 3. In Section 4, three examples are presented to illustrate the applications of the results obtained.
2 Preliminaries
Throughout this paper, we use the following notations.
Let \((\Omega, \mathcal{F}, \{\mathcal{F}_{t}\}_{t\geq0}, P)\) be a complete probability space with a natural filtration \(\{\mathcal {F}_{t}\}_{t\geq0}\) satisfying the usual conditions (i.e. it is right continuous and \(\mathcal{F}_{0}\) contains all Pnull sets), and \(E[\cdot]\) stand for the correspondent expectation operator with respect to the given probability measure P. Let \(W(t)=(W_{1}(t), \ldots, W_{m}(t))^{T}\) be an mdimensional Wiener process defined on a complete probability space with a natural filtration. Let \(\cdot\) denote the Euclidean norm in \(\mathbb{R}^{n}\).
Let \(\tau>0\) and \(PC([\tau,0]; \mathbb{R}^{n})\) = {\(\phi:[\tau ,0]\rightarrow\mathbb{R}^{n}\mid\phi(t)\) is continuous everywhere except at the points \(t=t_{k}\in[t_{0},\infty)\), \(\phi(t_{k}^{+})\) and \(\phi(t_{k}^{})\) exist with \(\phi(t_{k}^{+})=\phi(t_{k})\)} with the norm \(\\phi\=\sup_{\tau\leq\theta\leq0}\phi(\theta)\), where \(\phi(t^{+})\) and \(\phi(t^{})\) denote the righthand and lefthand limits of function \(\phi(t)\) at t.
Denote \(PC_{\mathcal{F}_{0}}^{b}([\tau,0]; \mathbb{R}^{n})\) by the family of all bounded, \(\mathcal{F}_{0}\)measurable, \(PC([\tau,0]; \mathbb{R}^{n})\)valued random variables. For \(p>0\), denote by \(PC_{\mathcal{F}_{t}}^{p}([\tau,0]; \mathbb{R}^{n})\) the family of all \(\mathcal{F}_{t}\)measurable \(PC([\tau,0]; \mathbb{R}^{n})\)valued random variables ϕ such that \(E\\phi\^{p}<\infty\).
Definition 2.1
 (i)
x: \([\sigma\tau, \sigma+\beta)\) for some β (\(0<\beta\leq \infty\)) is continuous for \(t\in[\sigma\tau, \sigma+\beta)\backslash\{ t_{k}: k=1,2,\ldots\}\), \(x(t^{+}_{k})\) and \(x(t^{}_{k})\) exist with \(x(t^{+}_{k})=x(t_{k})\) for \(t_{k}\in[\sigma\tau, \sigma+\beta)\), and \(\{x_{t}\}_{t\geq t_{0}}\) is \(\mathcal{F}_{t}\)adapted;
 (ii)
\(\{f(t,x_{t})\}\in L^{1}([t_{0},\infty];\mathbb{R}^{n})\) and \(\{ g(t,x_{t})\}\in L^{2}([t_{0},\infty];\mathbb{R}^{n\times m})\);
 (iii)
\(x(t)\) satisfies (2.1).
We denote the solution of the initial problem (2.1) by \(x(t;\sigma,\xi)\), and we denote by \([\sigma\tau, \sigma+\beta)\) the maximal right interval in which the solution \(x(t;\sigma,\xi)\) is defined.
 (H_{1}):

For all \(\psi\in PC([\tau,0]; \mathbb{R}^{n})\) and \(k\in\mathbb{Z}^{+}\), the limitsexist.$$\lim_{(t,\varphi)\rightarrow(t^{}_{k},\psi)}f(t,\varphi )=f\bigl(t^{}_{k}, \psi\bigr), \qquad \lim_{(t,\varphi)\rightarrow(t^{}_{k},\psi )}g(t,\varphi)=g\bigl(t^{}_{k}, \psi\bigr) $$
 (H_{2}):

f and g satisfy the locally Lipschitz condition in ϕ on each compact set in \(PC([\tau,0]; \mathbb{R}^{n})\). More precisely, for every \(a\in[t_{0}, \sigma+\beta)\) and every compact set \(G\in PC([\tau,0]; \mathbb{R}^{n})\), there exists a constant \(L=L(a,G)\) such thatwhenever \(t\in[t_{0}, a)\) and \(\varphi, \psi\in G\).$$\bigl\vert f(t,\varphi)f(t,\psi)\bigr\vert \vee\bigl\vert g(t, \varphi)g(t,\psi)\bigr\vert \leq L\\varphi \psi\, $$
 (H_{3}):

For any \(\rho>0\) there exists \(0<\rho_{1}\leq\rho\), such thatfor all \(k\in\mathbb{Z}^{+}\).$$x\in M(t,\rho_{1}) \quad \mbox{implies that} \quad x+I_{k}(t_{k},x)\in M(t,\rho) $$
 (H_{4}):

\(f(t,x_{t}), g(t,x_{t})\in PC([t_{0}, \infty), \mathbb {R}^{n})\) for \(x_{t}\in PC([\sigma\tau, \infty), \mathbb{R}^{n})\).
For any \(t\geq t_{0}\) and \(\kappa\geq0\), let \(PC_{\kappa}=\{\phi\in PC([\tau,0]; \mathbb{R}^{n}): \\phi\\leq\kappa\}\).
 (A_{1}):

for each \(t\in[t_{0}, \infty)\) the set \(M(t)\) is not empty;
 (A_{2}):

for any compact subset F of \([t_{0}, \infty)\times \mathbb{R}^{n}\) there exists a constant \(K>0\) depending on F such that if \((t,x), (t',x)\in F\), then the following inequality holds:$$\bigl\vert d\bigl(x,M(t)\bigr)d\bigl(x,M\bigl(t'\bigr)\bigr) \bigr\vert \leq K\bigl\vert tt'\bigr\vert ; $$
 (A_{3}):

if for solution \(x(t;\sigma,\xi)\) there exists \(h>0\) satisfyingwhere ρ is a constant, then \(x(t;\sigma,\xi)\) is defined in the interval \([\sigma,\infty)\).$$d\bigl(x(t;\sigma,\xi),M(t,\rho)\bigr)\leq h< \infty\quad \mbox{for } t\in [ \sigma,\sigma+\beta), $$
Definition 2.2
 (B_{1}):

V is continuous on each of the set \(([t_{0}\tau ,t_{0}]\cup[t_{k1},t_{k}))\times M(t,\rho)\) for all \(x\in M(t,\rho)\) and for \(k\in\mathbb{Z}^{+}\), the limit \(\lim_{(t,y)\rightarrow (t^{}_{k},x)}V(t,y)=V(t^{}_{k},x)\) exists;
 (B_{2}):

V is locally Lipschitz in \(x\in M(t,\rho)\), \(V(t,0)=0\) for \((t,x)\in M\) and \(V(t,x)>0\) for \((t,x)\notin M\).
Definition 2.3
We shall give the definitions of stability of the set M with respect to system (2.1).
Definition 2.4
 (S_{1}):

stable, if for any \(\sigma\geq t_{0}\), \(\alpha>0\), and \(\epsilon>0\), there is a \(\delta(\sigma,\epsilon,\alpha)>0\) such that \(\xi\in PC_{\alpha}\cap M_{0}(\sigma,\delta)\) implies that \(x(t,\sigma ,\xi)\in M(t,\epsilon)\) for \(t\geq\sigma\);
 (S_{2}):

uniformly stable, if the δ in (S_{1}) is independent of σ;
 (S_{3}):

asymptotically stable, if it is stable and for any \(\sigma \geq t_{0}\) and \(\alpha>0\), there exists a \(\delta=\delta(\sigma,\alpha )\) such that \(\xi\in PC_{\alpha}\cap M_{0}(\sigma,\delta)\) implies that \(x(t,\sigma,\xi)\rightarrow M(t)\) as \(t\rightarrow\infty\);
 (S_{4}):

uniformly asymptotically stable, if it is uniformly stable, and for any \(\alpha>0\) there exists a \(\delta(\alpha)>0\), such that for any \(\epsilon>0\) there is a \(T(\epsilon,\alpha,\delta)>0\) such that \(\sigma\geq t_{0}\) and \(\xi\in PC_{\alpha}\cap M_{0}(\sigma,\delta )\) implies that \(x(t,\sigma,\xi)\in M(t,\epsilon)\) for \(t\geq\sigma+T\).
3 Main results
In this section, we present and prove our main results on uniform stability and asymptotic stability of the sets of system (2.1) by utilizing piecewise continuous Lyapunov functions with Razumickhin methods.
Theorem 3.1
 (i)
\(a(d(x,M(t)))\leq EV(t,x)\leq b(d(x,M(t)))\) for all \((t,x)\in[t_{0}\tau, \infty)\times M(t,\rho)\);
 (ii)
\(ELV(t,x(t))\leq\eta(t)c(EV(t,x(t)))\), \(t\neq t_{k}\), whenever \(EV(t+s,x(t+s))\leq P(EV(t,x(t)))\) for \(\tau\leq s\leq0\), where \(x(t)\) is any solution of system (2.1), and \(\eta:[t_{0},\infty)\rightarrow\mathbb{R}^{+}\) is locally integrable;
 (iii)
\(EV(t_{k},x+I_{k}(t_{k},x))\leq P^{1}(EV(t^{}_{k},x))\) for each \(k\in\mathbb{Z}^{+}\), and all \(x\in M(t, \rho_{1})\), where \(P^{1}\) is the inverse of the function P;
 (iv)
\(\sup_{k\in\mathbb{Z}^{+}}\{t_{k}t_{k1}\}<\infty \), and \(\int_{P^{1}(\mu)}^{\mu}\frac{ds}{c(s)}\int_{t_{k1}}^{t_{k}}\eta (s)\, ds>0\) for all \(\mu\in(0,\infty)\), \(k\in\mathbb{Z}^{+}\).
Then the set M is uniformly stable with respect to the solution of system (2.1).
Proof
For any given \(\epsilon>0\), \(\alpha>0\), without loss of generality, we assume that \(\epsilon\leq\rho_{1}\). We can choose \(\delta=\delta(\epsilon,\alpha)>0\) such that \(P(b(\delta ))<\alpha(\epsilon)\) and \(\delta<\alpha\). From \(b(\delta)< P(b(\delta))<\alpha(\epsilon)<b(\epsilon)\) we know that \(\delta<\epsilon\).
Next, we will prove \(d(x(t),M(t))<\epsilon\) for \(t\in[\sigma,\sigma +\beta)\). Suppose, on the contrary, that \(d(x(t),M(t))>\epsilon\) for some \(t\in [\sigma,\sigma+\beta)\). Then let \(\hat{t}=\inf\{\sigma\leq t\leq\sigma+\beta \mid d(x(t),M(t))>\epsilon\}\). Note that \(d(x(\sigma),M(\sigma))<\epsilon\), we see that \(\hat{t}>\sigma\), \(d(x(t),M(t))\leq\epsilon\leq\rho_{1}\), for \(t\in[\sigma\tau,\hat{t})\) and either \(d(x(\hat{t}),M(\hat {t}))=\epsilon\) or \(d(x(\hat{t}),M(\hat{t}))>\epsilon\) and \(\hat {t}=t_{k}\) for some k.
Now let us first consider the case \(t_{m1}\leq\tilde{t}< t_{m}\). Let \(\bar{t}=\sup\{t\in[\sigma,\tilde{t}] \mid EV(t)\leq P^{1}(a(\epsilon))\}\). Since \(EV(\sigma)< P^{1}(a(\epsilon))\), \(EV(\tilde {t})=a(\epsilon)>P^{1}(a(\epsilon))\), and \(EV(t)\) is continuous on \([\sigma,\tilde{t}]\), we have \(\bar{t}\in(\sigma,\tilde{t})\), \(EV(\bar {t})=P^{1}(a(\epsilon))\), and \(EV(t)\geq P^{1}(a(\epsilon))\) for \(t\in [\bar{t},\tilde{t}]\).
Remark 3.1
From Theorem 3.1, we know that impulsive perturbations may cause uniform stability even if the unperturbed system is unstable.
The following result on the asymptotical stability of sets will reveal that impulsive perturbation make stable systems asymptotically stable.
Theorem 3.2
 (i)
\(a(d(x,M(t)))\leq EV(t,x)\leq b(d(x,M(t)))\) for all \((t,x)\in[t_{0}\tau, \infty)\times M(t,\rho)\);
 (ii)
\(EV(t_{k},x+I_{k}(t_{k},x))EV(t^{}_{k},x)\leq h_{k}(EV(t^{}_{k},x))\) for all \(k\in\mathbb{Z}^{+}\) and \(x\in M(t,\rho_{1})\);
 (iii)
for any solution \(x(t)\) of system (2.1), \(ELV(t,x)\leq0\);, and for any \(\sigma\geq t_{0}\), and \(r>0\), there exists \(\{r_{k}\}\) such that \(EV(t,x)\geq r\) for \(t\geq \sigma\) implies that \(h_{k}(EV(t^{}_{k},x))\geq r_{k}\); where \(r_{k}\geq0\) with \(\sum_{k=1}^{\infty}r_{k}=\infty\).
Then the set M with respect to the solution of system (2.1) is uniformly stable and asymptotically stable.
Proof
At first, we show that the set M is uniform stability.
For given \(\epsilon>0 \) (\(\epsilon\leq\rho_{1}\)), \(\alpha>0\), we choose a \(\delta(\epsilon,\alpha)>0\) such that \(b(\delta)\leq a(\epsilon )\) and \(\delta<\alpha\). For any \(\sigma\geq t_{0}\) and \(\xi\in PC_{\alpha}\cap M_{0}(\sigma ,\delta)\), let \(x(t)=x(t;\sigma,\xi)\) be the solution of system (2.1). We will show that \(x(t)\in M(t,\epsilon)\) for \(t\in[\sigma ,\sigma+\beta)\).
Set \(EV(t)=EV(t,x(t))\), where \(\sigma\in[t_{m1},t_{m})\) for some \(m\in\mathbb{Z}^{+}\). Then condition (iii) implies that \(ELV(t)\leq0\) for \(t\in[\sigma,\sigma +\beta)\cap([\sigma,t_{m})\cup(\bigcup_{k=m}^{\infty}[t_{k1},t_{k})))\), \(k\in\mathbb{Z}^{+}\).
Next we shall prove that the set M is asymptotically stable.
From conditions (ii), (iii), and \(EV(t)\geq0\), we note that \(EV(t)\) is nonincreasing on the interval \([\sigma,\infty)\). So the limit \(\lim_{t\rightarrow\infty}EV(t)\) exists.
Assume \(\sigma\in[t_{m1},t_{m}]\) for some \(m\in\mathbb{Z}^{+}\). Set \(\lim_{t\rightarrow\infty}EV(t)=r\geq0\), one can easily see that \(EV(t)\geq r\) for \(t\geq\sigma\). Then by condition (iii), it follows that there is a sequence \(\{r_{k}\}\) with \(r_{k}\geq0\) for \(k\in\mathbb {Z}^{+}\), which implies that \(h_{k}(EV(t^{}_{k},x))\geq r_{k}\) with \(\sum_{k=1}^{\infty}r_{k}=\infty\).
Theorem 3.3
 (i)
\(a(d(x,M(t)))\leq EV(t,x)\leq b(d(x,M(t)))\) for all \((t,x)\in[t_{0}\tau, \infty)\times M(t,\rho)\);
 (ii)
\(EV(t_{k},x+I_{k}(t_{k},x))\leq\psi _{k}(EV(t^{}_{k},x))\), for all \(K\in\mathbb{Z}^{+}\), and \(x\in M(t,\rho)\);
 (iii)for any solution \(x(t)\) of system (2.1), \(ELV(t,x)\leq\theta(t)C(EV(t,x))\) for \(t\neq t_{k}\), where \(\theta:[t_{0},\infty)\rightarrow\mathbb{R}^{+}\) is locally intergrade, and there exists \(\mu_{0}\), such that for any \(\mu\in(0,\mu _{0})\),where \(\gamma_{k}\geq0\) with \(\sum_{k=1}^{\infty}\gamma_{k}=\infty\).$$\int_{\mu}^{\psi_{k}(\mu)}\frac{ds}{C(s)}\int _{t_{k1}}^{t_{k}}\theta (s)\, ds\leq\gamma_{k}, $$
Then the set M with respect to the solution of system (2.1) is uniformly stable and asymptotically stable.
Proof
Without loss of generality, for any given \(\epsilon >0\), \(\alpha>0\), we can assume that \(\epsilon\leq\rho_{1}\). We choose a \(\beta:0<\beta<\min\{a(\epsilon),\mu_{0}\}\) such that \(\psi _{k}(s)< a(\epsilon)\) for \(0\leq s\leq\beta\) and for all \(k\in\mathbb{Z}^{+}\).
Since \(EV(t)\leq EV(t_{k})\) for \(t_{k}\leq t< t_{k+1}\), it follows that \(\lim_{t\rightarrow\infty}EV(t)=0\), which yields \(\lim_{t\rightarrow \infty}d(x(t,M(t)))=0\). The proof of Theorem 3.3 is complete. □
4 Illustrative examples
As an application, we consider the following examples.
Example 4.1

\(t_{k}t_{k1}<\frac{\ln0.5}{1.6}\), for \(k\in \mathbb{Z}^{+}\), where \(t_{0}\geq0\).
Example 4.2

\(0< u<1\) and \(m+nu^{1}+\frac {1}{2}p^{2}+pqu^{1}+\frac{1}{2}q^{2}u^{2}<0\).
We check that for any \(\sigma\geq t_{0}\), and \(r>0\), there exists \(\{r_{k}\}\) such that \(EV(t,x)\geq r\) for \(t\geq\sigma\) implies that \(h_{k}(EV(t^{}_{k},x))\geq r_{k}\); where \(r_{k}\geq0\) with \(\sum_{k=1}^{\infty}r_{k}=\infty\). Since \(h_{k}(EV(t^{}_{k},x))=(1u^{2})EV(t^{}_{k},x)\), when \(EV(t,x)\geq r\) for \(t\geq\sigma\), then we have \(h_{k}(EV(t^{}_{k},x))=(1u^{2})EV(t^{}_{k},x)\geq(1u^{2})r\). We take \(r_{k}=(1u^{2})r\) and we have \(\sum_{k=1}^{\infty}r_{k}=\infty\). Thus all of the conditions in Theorem 3.2 are satisfied. Therefore, it follows from Theorem 3.2 that the set M is uniformly stable and asymptotically stable with respect to the solution of the system (4.2).
Example 4.3
 (i)
\(0< h<1\) and \(a+bh^{1}+\frac {1}{2}c^{2}+crh^{1}+\frac{1}{2}r^{2}h^{2}<0\);
 (ii)
\(t_{k}t_{k1}>\frac{\ln h}{a+bh^{1}+\frac {1}{2}c^{2}+crh^{1}+\frac{1}{2}r^{2}h^{2}}\), for \(k\in\mathbb {Z}^{+}\), where \(t_{0}\geq0\).
Declarations
Acknowledgements
The authors would like to thank the referee for his/her careful reading and valuable suggestions, which lead to improvement of the manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Bao, JH, Hou, ZT, Yuan, CG: Stability in distribution of mild solutions to stochastic partial differential equations. Proc. Am. Math. Soc. 138, 21692180 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Bao, JH, Truman, A, Yuan, CG: Stability in distribution of mild solutions to stochastic partial differential delay equations with jumps. Proc. R. Soc. Lond. Ser. A 465, 21112134 (2009) MathSciNetView ArticleGoogle Scholar
 Bao, JH, Hou, ZT, Yuan, CG: Stability in distribution of neutral stochastic differential delay equations with Markovian switching. Stat. Probab. Lett. 79, 16631673 (2009) MathSciNetView ArticleGoogle Scholar
 Khasminskii, R: Stochastic Stability of Differential Equations, 2nd edn. Springer, Berlin (2012) View ArticleGoogle Scholar
 Ladde, GS, Lakshmikantham, V: Random Differential Inequalities. Academic Press, New York (1980) MATHGoogle Scholar
 Ladde, GS, Sambandham, M: Stochastic versus Deterministic Systems of Differential Equations. Dekker, New York (2004) MATHGoogle Scholar
 Lakshmikantham, V, Bainov, DD, Simeonov, DS: Theory of Impulsive Differential Equation. World Scientific, Singapore (1989) View ArticleGoogle Scholar
 Lakshmikantham, V, Liu, X: Stability of impulsive differential systems in terms of two measures. Appl. Math. Comput. 29, 8998 (1989) MathSciNetView ArticleMATHGoogle Scholar
 Liu, X, Wang, Q: On stability in terms of two measures for impulsive systems of functional differential equations. J. Math. Anal. Appl. 326, 252265 (2007) MathSciNetView ArticleMATHGoogle Scholar
 Mao, X: Stochastic Differential Equations and Their Applications, 2nd edn. Horwood, Chichester (2007) Google Scholar
 Mao, X: Attraction, stability and boundedness for stochastic differential delay equations. Nonlinear Anal. 47, 47954806 (2001) MathSciNetView ArticleGoogle Scholar
 Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations. World Scientific, Singapore (1995) MATHGoogle Scholar
 Shaikhet, L: Lyapunov Functionals and Stability of Stochastic Functional Differential Equations. Springer, New York (2013) View ArticleMATHGoogle Scholar
 Shen, J, Yan, J: Razumikhin type stability theorems for impulsive functional differential equations. Nonlinear Anal. 33, 519531 (1998) MathSciNetView ArticleMATHGoogle Scholar
 Stamova, I: Stability Analysis of Impulsive Functional Differential Equations. de Gruyter, New York (2009) View ArticleMATHGoogle Scholar
 Caro, EA, Rao, ANV: Stability analysis of impulsive stochastic differential systems in terms of two measures. In: Conference Proceedings COM, pp. 132135. IEEE Press, New York (1996) Google Scholar
 Pan, L, Cao, J: Exponential stability of impulsive stochastic functional differential equations. J. Math. Anal. Appl. 382, 672685 (2011) MathSciNetView ArticleGoogle Scholar
 Liu, J, Liu, X, Xie, W: Impulsive stabilization of stochastic functional differential equations. Appl. Math. Lett. 24, 264269 (2011) MathSciNetView ArticleGoogle Scholar
 Liu, ZM, Peng, J: pMoment stability of stochastic nonlinear delay systems with impulsive jump and Markovian switching. Stoch. Anal. Appl. 27, 911923 (2009) MathSciNetView ArticleMATHGoogle Scholar
 Peng, S, Jia, B: Some criteria on pth moment stability of impulsive stochastic functional differential equations. Stat. Probab. Lett. 80, 10851092 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Rao, ANV, Tsokos, CP: Stability behavior of impulse stochastic differential systems. Dyn. Syst. Appl. 4, 317327 (1995) MathSciNetMATHGoogle Scholar
 Zhang, S, Sun, J, Zhang, Y: Stability of impulsive stochastic differential equations in terms of two measures via perturbing Lyapunov functions. Appl. Math. Comput. 218, 51815186 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Xu, Y, He, ZM: Stability of impulsive stochastic differential equations with Markovian switching. Appl. Math. Lett. 35, 3540 (2014) MathSciNetView ArticleMATHGoogle Scholar
 Krasovskii, NN: Stability of Motion. Stanford University Press, Stanford (1963) MATHGoogle Scholar
 Rouche, H, Habets, P, Laloy, M: Stability Theory by Lyapunov’s Direct Method. Springer, New York (1977) View ArticleGoogle Scholar
 Lakshmikantham, V, Leela, S, Martynyuk, AA: Stability Analysis of Nonlinear Systems. Dekker, New York (1989) MATHGoogle Scholar
 Lakshmikantham, V, Leela, S, Martynyuk, AA: Practical Stability Analysis of Nonlinear Systems. World Scientific, Singapore (1990) View ArticleGoogle Scholar
 Lakshmikantham, V, Liu, X: Stability Analysis in Terms of Two Measures. World Scientific, Singapore (1993) View ArticleGoogle Scholar
 Yoshizawa, T: Stability of sets and perturbed system. Funkc. Ekvacioj 5, 3169 (1962) MathSciNetGoogle Scholar
 Yoshizawa, T: Some notes on stability of sets and perturbed system. Funkc. Ekvacioj 6, 111 (1964) MathSciNetMATHGoogle Scholar
 Yoshizawa, T: Stability Theory by Lyapunov’s Second Method. The Mathematical Society of Japan, Tokyo (1966) Google Scholar
 Kulev, GK, Bainov, DD: Stability of sets for impulsive systems. Int. J. Theor. Phys. 28(2), 195207 (1989) MathSciNetView ArticleMATHGoogle Scholar
 Kulev, GK, Bainov, DD: Global stability of sets for impulsive differential systems by Lyapunov’s direct method. Comput. Math. Appl. 19, 1728 (1990) MathSciNetView ArticleGoogle Scholar
 Xie, S: Stability of sets of functional differential equations with impulse effect. Appl. Math. Comput. 218, 592597 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Xie, S, Shen, J: Stability of sets for impulsive functional differential equations via Razumikhin method. J. Math. Sci. 177, 474486 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Long, S: Attracting and invariant sets of nonlinear stochastic neutral differential equations with delays. Results Math. 63, 745762 (2013) MathSciNetView ArticleGoogle Scholar
 Samoilenko, AM, Stanzhytskyi, O: Qualitative and Asymptotic Analysis of Differential Equations with Random Perturbations. World Scientific, Singapore (2011) MATHGoogle Scholar
 Luo, JW: Stability of invariant sets of Itô stochastic differential equations with Markovian switching. J. Appl. Math. Stoch. Anal. 2006, Article ID 59032 (2006) View ArticleGoogle Scholar
 Xu, LG, Xu, DY: PAttracting and pinvariant sets for a class of impulsive stochastic functional differential equations. Comput. Math. Appl. 57, 5461 (2009) MathSciNetView ArticleGoogle Scholar
 Mao, X: Razumikhin type theorems on exponential stability of stochastic functional differential equations. Stoch. Process. Appl. 65, 233250 (1996) View ArticleGoogle Scholar