Existence of mild solution for impulsive stochastic differential equations with nonlocal conditions
impulsive stochastic differential equations
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.
scientific article; zbMATH DE number 6085111