On second-order boundary value problems in Banach spaces: a bound sets approach

Abstract

The existence and localization of strong (Carathéodory) solutions is obtained for a second-order Floquet problem in a Banach space. The combination of applied degree arguments and bounding (Liapunov-like) functions allows some solutions to escape from a given set. The problems concern both semilinear differential equations and inclusions. The main theorem for upper-Carathéodory inclusions is separately improved for Marchaud inclusions (i.e. for globally upper semicontinuous right-hand sides) in the form of corollary. Three illustrative examples are supplied.