Existence of solutions for nonlinear functional integrodifferential evolution equations with nonlocal conditions (Q623393)

The paper is concerned with the first-order nonlinear functional integro-differential equation with nonlocal conditions \[ \begin{multlined} x'(t)=A(t)\left[x(t)+\int_0^tH(t,s)x(s)\,ds\right]\\ +F\left(t,x(\sigma_1(t)),\dots,x(\sigma_n(t)),\int_0^th(t,s,x(\sigma_{n+1}(s)))\,ds\right), \quad t\in J=[0,b],\end{multlined} \] \[ x(0)+g(x)=x_0\in X, \] where \(X\) is a Banach space, \(A(t)\) is a closed linear operator on \(X\) with a dense domain \(D(A)\) independent of \(t\), \(H(t,s)\), \(t,\,s\in J\), is a bounded operator in \(X\), and the nonlinear operators \(F:J\times X^{n+1}\to X\), \(h:J\times J\times X\to X\), \(g:C(J,X)\to X\) \(\sigma_i:J\to J\), \(i=1,\dots,n+1\) are given functions. By using the theory of resolvent operators and the Leray-Schauder nonlinear alternative, the authors prove the existence of mild solutions for the above problem.