Nonconvex perturbation of differential inclusions with memory (Q1373105)

Let \(E\) be a separable Hilbert space, \(\tau> 0\), \(C_0:= C([-\tau,0]; E)\) and \(\phi_0\in C_0\). For \(a> 0\), \(x\in C([t_0- \tau,t_0+ a]; E)\), \(t\in [t_0, t_0+a]\), let \(T(t)x\) be the element of \(C_0\) defined by \([T(t)x](s)= x(t+ s)\). Let \(F\) and \(G\) be multifunctions defined on an open subset of \(\mathbb{R}\times C_0\) containing \((t_0,\phi_0)\), \(F\) being upper semicontinuous with closed convex nonempty values and \(G\) being uniformly continuous with nonempty compact values. Then, with some other technical hypotheses, a solution of \(x'(t)\in F(t,T(t)x)+ G(t, T(t)x)\) and \(T(t_0)x= \phi_0\) is shown to exist.