Existence results for impulsive neutral stochastic functional integro-differential equations with infinite delays. (English) Zbl 1202.60098

The existence of mild solutions of a stochastic evolution equation \[ [x(t)-G_1(t,x)]^\prime=Ax(t)+G_2(t,x)+C(t,x),\qquad t\in [0,b]\setminus\{t_0,\dots,t_k\} \]
\[ x(t^+)-x(t^-)=I_k(x(t^-)),\quad t\in\{t_1,\dots,t_m\},\qquad x_0=\varphi \] in a Hilbert space is established via the Krasnoselskii-Schaefer fixed point theorem. Here \[ G_i(t,x)=g_i(t,x_t,\int_0^ta_i(t,s,x_s)\,ds),\qquad C(t,x)=\int_{-\infty}^t\sigma(t,s,x_s)\,dW_s, \] \(x_t(r)=x(t+r)\) for \(r\leq 0\), the deterministic entries \(g_i\), \(a_i\), \(\sigma\), \(I_k\) satisfy various expectable conditions (such as continuity, Lipschitzianity, boundedness, linear growth, compact range etc.) and several other hypotheses, \(A\) has a compact resolvent and generates a holomorphic semigroup and \(W\) is a Wiener process with a trace class covariance operator.


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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