Existence results for nonlinear functional evolution equations with delay conditions (Q1763989)

Let \(X\) be a real Banach space, \(A(t): D(A(t))\subset X\to 2^X\), \(G: [0,T]\times L^p((- r,0), X)\to X\). The following abstract nonlinear evolution equation with delay condition is considered \[ du(t)/dt+ A(t)u(t)\in G(t,u_t, L_t u),\;t\in [0,T],\;u(0)= \Phi_0(t)\text{ for }t\in [-r, 0].\tag{1} \] By using Schauder's fixed-point theorem in the space \(L^p((0, T),X)\), \(1\leq p\leq\infty\), a local existence theorem for the problem (1) is established. Here, it is assumed, among others, that the evolution operator \(U(t,s)\) generated by \(\{A(t)\}\) is equicontinuous.