Existence of solutions of nonlinear abstract neutral integrodifferential equations (Q1779573)

Existence of mild solutions of the neutral integro-differential functional equation \[ \begin{multlined}{d\over dt}\left[x(t)-g{\left(t,x_t,\int_0^tk_0(t,s,x_s)\,ds\right)}\right]\\ = Ax(t)+h(t,x_t)+\int_0^tk_1(t,s,x_s)\,ds+ f{\left(t,x_t,\int_0^tk_2(t,s,x_s)\,ds\right)}\end{multlined} \] in a Banach space with initial value \(x_0=\phi\) is obtained. Here, \(x_t(\theta)=x(t+\theta)\) for \(\theta\in[-r,0]\), and \(A\) is the generator of a compact semigroup. The essential hypothesis are compactness of \(g\) and a certain linear growth of \(f\), \(h\) and the \(k_j\)'s which allows to obtain \textit{a priori} bounds for solutions. The result is applied to a control problem of a similar form.