Existence for nonlinear evolution inclusions with nonlocal retarded initial conditions

In this paper, we prove a sufficient condition for the global existence of bounded C0C0-solutions for a class of nonlinear functional differential evolution equation of the form {u′(t)∈Au(t)+f(t)t∈[0,+∞)f(t)∈F(t,u(t),u(t−τ1),…,u(t−τn))t∈[0,+∞)u(t)=g(u)(t)t∈[−τ,0], where XX is a real Banach space, AA is the infinitesimal generator of a nonlinear compact semigroup, F:R+×[D(A)¯]n+1⇝X is a nonempty, convex, weakly compact valued, and almost strongly–weakly u.s.c. multi-function, and g:Cb([−τ,+∞);D(A)¯)→C([−τ,0];D(A)¯) is nonexpansive.