Existence of mild solutions of second-order neutral functional differential inclusions with nonlocal conditions in Banach spaces (Q1774731)

The existence of mild solutions of second-order neutral functional-differential inclusions of the form \[ {d \over dt}[x'(t)-g(t,x_t)] \in Ax(t)+ F(t,x_t,x'(t)) \] are examined, where \(A\) is the generator of a strongly continuous cosine family in a Banach space \(X\), \(g:[0,T]\times C([-r,0],X)\to X\) is continuous and \(F:[0,T]\times C([-r,0],X)\times X\to 2^X\) is bounded, closed, convex. Various sufficient conditions are presented. The results are obtained by using the theory of strongly continuous cosine functions and a fixed-point theorem for condensing maps due to Martelli.