Existence of mild solution for impulsive stochastic differential equations with nonlocal conditions
type: Item
instance of: scholarly article
title: Existence of mild solution for impulsive stochastic differential equations with nonlocal conditions
zbMATH Open document ID: 1254.34113
author: Lijun Pan
publication date: 2012-09-21T00:00:00Z
full work available at URL: dea-04-28.pdf
Mathematics Subject Classification ID: 34K50
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 34K45
zbMATH DE Number: 6085111
zbMATH Keywords: mild solution
zbMATH Keywords: impulsive stochastic differential equations
zbMATH Keywords: nonlocal conditions
zbMATH Keywords: semigroup
MaRDI profile type: Publication
OpenAlex ID: W2322970939
DOI: 10.7153/DEA-04-28
published in: Differential Equations and Applications
review text: The nonlocal stochastic initial value problem with impulses NEWLINE\[NEWLINE\left\{\begin{aligned} & dx(t)=[Ax(t)+F(t,x(t),x(a_1(t)),\dotsc, x(a_{\nu}(t)))]dt+\\ & \qquad\qquad +G(t,x(t),x(b_1(t)),\dotsc, x(b_m(t)))d\omega(t),\quad t\in J=[0,T], \;t\neq t_k,\\ & x(t_k^+) -x(t_k)=I_k(x(t_k)),\quad k=1,2,\dotsc, q,\\ &x(0)=x_0+g(x) \end{aligned}\right.NEWLINE\]NEWLINE is considered, where \(F: J\times H^{\nu+1}\to H\) is continuous, \(G: J\times H^{m+1}\to L(K,H),\) \(a_i, b_j: J\to J\), \(i=1,2,\dotsc,\nu\), \(j=1,2,\dotsc, m\) are continuous, \(A\) is the infinitesimal generator of an analytic semigroup of bounded operators and \(H\), \(K\) are Hilbert spaces. Existence and uniqueness results are obtained by using Krasnosel'skii's and Banach's fixed point theorems.
reviewed by: Sotiris K. Ntouyas
description: scientific article; zbMATH DE number 6085111