Existence of the mild solution for impulsive neutral stochastic fractional integro-differential inclusions with nonlocal conditions (Q327337)

Consider the impulsive neutral stochastic integro-differential inclusions with nonlocal conditions in a separable Hilbert space of the form \[ \begin{aligned} & ^{c}D_{t}^{\alpha}\left[u(t)-F\left(t,u(h_1(t)),\int_0^{t}a_1(t,s,u(h_2(s)))ds\right)\right]\\ & \in Au(t)+\int_0^{t}f(t-s)u(s)ds+G(t,u(h_3(s)))((dw(t))/(dt))\end{aligned} \] \[ t\in J=[0,T],\,t\neq t_{i},\, 0<T<\infty,\tag{1} \] \[ \Delta u(t_{i})=I_{i}(u(t_{i}^-)),\quad i=1,2,\dots,m,\tag{2} \] \[ u(0)=u_0+h(u)\in H,\quad u'(0)=0,\tag{3} \] where \(^{c}D_{t}^{\alpha}\) means the Caputo fractional derivative of order \(1<\alpha<2\). Under a number of hypotheses it is proved that there exists at least one mild solution for the system (1)--(3). An example illustrating the results is given.