Application of Pettis integration to differential inclusions with three-point boundary conditions in Banach spaces

Abstract

This paper provide some applications of Pettis integration to differential inclusions in Banach spaces with three point boundary conditions of the form $$ddot{u}(t) in F(t,u(t),dot u(t))+H(t,u(t),dot u(t)),quad hbox{a.e. } t in [0,1],$$ where $F$ is a convex valued multifunction upper semicontinuous on $Eimes E$ and $H$ is a lower semicontinuous multifunction. The existence of solutions is obtained under the non convexity condition for the multifunction $H$, and the assumption that $F(t,x,y)subset Gamma_{1}(t)$, $H(t,x,y)subset Gamma_{2}(t)$, where the multifunctions $Gamma_{1},Gamma_{2}:[0,1] ightrightarrows E$ are uniformly Pettis integrable