Sobolev type integrodifferential equation with nonlocal condition in Banach spaces (Q1811452)

The authors prove the existence of mild and strong solutions for the equation \[ (Bu(t))' + au(t) = f\left (t,u(t),\int_0^t a(t,s)k(s,u(s)) ds\right),\quad t\in (0,b], \] with a nonlocal condition of the form \[ u(0)+g(t_1,t_2,\ldots,t_p,u(t_1),u(t_2),\ldots,u(t_p))=u_0. \] Here \(A\) and \(B\) are closed linear operators with domains \(D(B)\subset D(A)\) in a Banach space \(X\) and ranges in Banach space \(Y\) and \(B^{-1}:Y\to D(B)\) is compact. The proof of existence of a mild solution relies on the fact that \(-AB^{-1}\) generates a uniformly continuous semigroup and uses the Schauder fixed point theorem. Using stronger assumptions such as reflexivity of \(Y\) and Lipschitz-conditions on \(f\) and \(a\), the authors prove that the mild solution is a strong one.