Existence of solutions for neutral non-autonomous evolution equations with nonlocal conditions (Q879452)

The authors study the existence of solutions of the Cauchy problem: \[ \begin{multlined} \tfrac{d} {dt}[x(t)+F(t,x(t),x(b_1(t)),\dots,x(b_n(t)))]+A(t)x(t)=G(t, x(t),x(a_1(t)),\dots,x(a_m(t))),\\ \text{for } \quad 0\leq t\leq T,\quad x(0)+g(x)=x_0,\end{multlined} \] where the family \(\{A(t):0\leq t \leq Z\}\) of linear operators generates a linear evolution system, and \(F,G\) are given functions. By using Sadovskii's fixed point theorem the existence of mild solutions and solutions are studied. The results present a generalization and continuation of recent results of this problem. Two examples are given, which show the applicability of the obtained results.