Global existence of solutions for stochastic impulsive differential equations

Shen, Li

doi:10.1007/s10114-011-8650-9

Cited by 10 publications

(6 citation statements)

References 23 publications

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“…Similar to Lemma 3.1 in Shen and Sun, 31 for any

t \in false[0, T false]

and

P_{0} false(a false) = ξ false(0 false)

, the impulsive stochastic population equations can be rewritten as follows:

({, \begin{matrix} array \\ array P (a, t) = ξ (0) - \int_{0}^{t} \frac{\partial P (a, s)}{\partial a} d s - \int_{0}^{t} μ (a, s, C^{0} (s)) P (a, s) d s + \int_{0}^{t} g (t, P (a, s)) d W (s) \\ array + \sum_{0 \leq t_{i} \leq t} I_{i} (P (a, t_{i})), \\ array C^{0} (t) = C^{0} (0) + \int_{0}^{t} k C^{e} (s) d s - \int_{0}^{t} (l + m) C^{0} (s) d s + \sum_{0 \leq t_{i} \leq t} J_{i} (C^{0} (t_{i})), \\ array C^{e} (t) = C^{e} (0 ... \end{matrix})

…”

Section: Preliminaries and Basic Assumptionmentioning

confidence: 56%

A periodic averaging method for impulsive stochastic age‐structured population model in a polluted environment

Zhang

et al. 2022

Math Methods in App Sciences

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In this paper, we present a periodic averaging (PA) method for impulsive stochastic age‐structured population model (ISASPM) in a polluted environment. By using Young's inequality and Gronwall's lemma and under the stochastic Lipschitz condition, we reveal that both the standard ISASPM and the averaged ISASPM have a unique solution. We also study the mean‐square convergence criteria of the numerical solutions produced by the PA method. Finally, asimulation example is given to demonstrate that the PA method is efficient for our results.

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“…Similar to Lemma 3.1 in Shen and Sun, 31 for any

t \in false[0, T false]

and

P_{0} false(a false) = ξ false(0 false)

, the impulsive stochastic population equations can be rewritten as follows:

({, \begin{matrix} array \\ array P (a, t) = ξ (0) - \int_{0}^{t} \frac{\partial P (a, s)}{\partial a} d s - \int_{0}^{t} μ (a, s, C^{0} (s)) P (a, s) d s + \int_{0}^{t} g (t, P (a, s)) d W (s) \\ array + \sum_{0 \leq t_{i} \leq t} I_{i} (P (a, t_{i})), \\ array C^{0} (t) = C^{0} (0) + \int_{0}^{t} k C^{e} (s) d s - \int_{0}^{t} (l + m) C^{0} (s) d s + \sum_{0 \leq t_{i} \leq t} J_{i} (C^{0} (t_{i})), \\ array C^{e} (t) = C^{e} (0 ... \end{matrix})

…”

Section: Preliminaries and Basic Assumptionmentioning

confidence: 56%

A periodic averaging method for impulsive stochastic age‐structured population model in a polluted environment

Zhang

et al. 2022

Math Methods in App Sciences

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“…Next, according to Lemma 3.1 in [37], the solution of impulsive stochastic dynamical system (1) can be given by the following integral equation:…”

Section: Lemma 27mentioning

confidence: 99%

Periodic averaging method for impulsive stochastic dynamical systems driven by fractional Brownian motion under non-Lipschitz condition

2019

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This paper presents the periodic averaging principle for impulsive stochastic dynamical systems driven by fractional Brownian motion (fBm). Under non-Lipschitz condition, we prove that the solutions to impulsive stochastic differential equations (ISDEs) with fBm can be approximated by the solutions to averaged SDEs without impulses both in the sense of mean square and probability. Finally, an example is provided to illustrate the theoretical results.

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“…The goal of this paper is to study a large class of systems, which is a generalization of the classical Itô stochastic differential systems or the systems studied by Shen and Sun . Because the nonlinear terms are dependent on

double-struckE double-struckx

and the state is influenced by impulse, some new issues must be considered, including how to weaken the requirement of impulsive functions and how to describe the global existence by some other function and conclude the rule influenced by impulse.…”

Section: Introductionmentioning

confidence: 99%

“…Existence and uniqueness of systems without expectation in nonlinear terms have been reported in . And these topics have been developed mainly by differential inequalities or converting the problem into a fixed point one (for example, ). In , by using the Lyapunov method, the author has investigated the existence and stability of SISs.…”

Section: Introductionmentioning

confidence: 99%

Existence results of stochastic impulsive systems with expectations‐dependent nonlinear terms

Shen

2014

Math Methods in App Sciences

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In this paper, sufficient conditions are established for the existence and uniqueness of global solutions to stochastic impulsive systems with expectations in the nonlinear terms. The maximal interval and the estimate of mild solutions are also discussed. These results are obtained by using the fixed point theorem, interval partition, and Lyapunov‐like technique. Finally, examples are given to illustrate the theory. Copyright © 2014 John Wiley & Sons, Ltd.

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Global existence of solutions for stochastic impulsive differential equations

Cited by 10 publications

References 23 publications

A periodic averaging method for impulsive stochastic age‐structured population model in a polluted environment

A periodic averaging method for impulsive stochastic age‐structured population model in a polluted environment

Periodic averaging method for impulsive stochastic dynamical systems driven by fractional Brownian motion under non-Lipschitz condition

Existence results of stochastic impulsive systems with expectations‐dependent nonlinear terms

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