Non-autonomous evolution inclusions with nonlocal history conditions: Global integral solutions

This paper is concerned with nonlinear nonlocal differential inclusion of evolution type in Fréchet spaces, defined on right half-line. The underlying feature of the inclusion under consideration is that it is non-autonomous. We obtain some compactness characterizations of integral solution sets for the inclusion without nonlinear perturbations. Then, making use of these characterizations, we derive a new existence result of global integral solutions for the original inclusion. No invariance condition on the nonlinearity is involved. The results we obtained here extend the semilinear case of the previous related ones such as [I.I. Vrabie, Existence in the large for nonlinear delay evolution inclusions with nonlocal initial conditions, J. Funct. Anal. 262 (2012) 1363–1391] and are new even for the case of the nonlinearity being a single-valued function.