Existence of solutions for nonlocal impulsive partial functional integrodifferential equations via fractional operators (Q629481)

The paper deals with the following partial functional integro-differential equation: \[ \begin{multlined} x'(t) = Ax(t) + F\left(t,\,x(\sigma_1(t)),\,\dots,\,x(\sigma_n(t)),\int_0^t h(t,\,s,\,x(\sigma_{n+1}(s)))ds\right),\\ t\in [0,\,b],\,t\neq t_k,\; k=\overline{1,\,m}, \end{multlined} \] where \(A\) is the infinitesimal generator of a compact, analytic semigroup, \(t_k\in [0,\,b]\), and \(F,\,h,\,\sigma_k\) are some given functions. The equation is considered here together with the conditions: \[ x(0) + g(x) = x_0\;\;\text{ and }\;\;\; x(t_k^+) - x(t_k^-) = I_k(x(t_k)),\;\; k=\overline{1,\,m}. \] It is shown that, under suitable conditions on the functions \(F,\,h,\,g,\,\sigma_k\), and for any \(x_0\in X^{\alpha}\), the above Cauchy problem has at least one mild solution on \([0,\,b]\). The proof of this result employs the Leray-Schauder alternative. The author also presents an illustrative example at the end of the paper.