On the existence of weak solutions of differential equations in nonreflexive banach spaces

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 [l&4]. 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 ofthe Cauchy problem in a Banach space relative to the weak topology has lagged behind. Recently Szep [4] proved Peano’s Theorem in the weak topology for differential equations in a reflexive Banach space. His main tools are the Eberlein-Smulian Theorem and the fact that a subset of a reflexive Banach space is weakly compact if and only if it is weakly closed and norm bounded. In this paper we wish to prove this theorem in arbitrary Banach spaces. Our first approach is to impose weak compactness type conditions in terms of the measure of weak noncompactness developed by De Blasi. We also impose weak dissipative type conditions and prove an existence and uniqueness theorem. Using these existence results and the partial orderings induced by cones, existence of extremal solutions and comparison results relative to the weak topology are also proved.