Topological Structure of the Solution Sets for a Nonlinear Delay Evolution

Abstract

Abstract We consider a nonlinear delay evolution equation with multivalued perturbation on a noncompact interval. The nonlinearity, having convex and closed values, is upper hemicontinuous with respect to the solution variable. A basic question on whether there exists a solution set carrying $R_{\delta }$-structure remains unsolved when the operator families generated by the principal part lack compactness. One of our main goals is to settle this question in the affirmative. Moreover, we prove that the solution map, having compact values, is an $R_{\delta }$-map, which maps any connected set into a connected set. It is then exploited to deal with the existence in the large for a nonlocal problem. Finally, several examples are worked out in detail, illustrating the applicability of our general results.