On the Existence of Weak Solutions of Differential Equations in Nonreflexive Banach Spaces

Abstract

**Please note that the full text is embargoed** ABSTRACT: The study of the Cauchy problem for differential equations in a Banach space relative to the strong topology has attracted much attention in recent years [2,4,5,7]. This study has taken two different directions. One direction is to impose compactness type conditions that guarantee only existence and the corresponding results are extensions of the classical Peano's Theorem. The other approach is to utilize dissipative type conditions that assure existence and uniqueness of solutions, and the corresponding results are extensions of the classical Picard's Theorem. However, a similar study of the Cauchy problem in a Banach space relative to the weak topology has lagged behind. Recently Szep [7] proved Peano's Theorem in the weak topology for differential equations in a reflexive Banach space and his main tool is the Eberlein-Smulian Theorem which assures the weak compactness of a closed set in the weak topology. In this paper we wish to prove this theorem in arbitrary Banach spaces, imposing weak compactness type conditions. For this purpose we introduce the notion of measure of weak noncompactness which is parallel to the Kuratowski measure of noncompactness and develop its properties. We also impose weak dissipative type conditions and prove an existence and uniqueness theorem.