Existence of solutions to neutral functional integrodifferential equations in Banach spaces (Q1306816)

The authors prove the existence of a mild solution to the neutral functional integrodifferential equation \[ {d \over dt} [x(t)-g(t,x_t)] = Ax(t) + \int_0^t f(s,x_s) ds,\quad t\in [0,b], \] with initial condition \(x_0=\varphi\) on \([-r,0]\), in a Banach space \(X\). Here \(x_t(\theta) = x(t+\theta)\) and \(A\) is the generator of a strongly continuous semigroup. It is assumed that the operator \(G\) defined by \((G\varphi)(t) = g(t,\varphi)\) is compact and some restrictions on the growth of the function \(f\) are assumed as well. The proof relies on a fixed point theorem due to Schaefer.