Existence of solutions and Kamke's theorem for functional differential equations in Banach spaces

Abstract

In this paper we shall consider the Cauchy problem for functional differential equations (FDEs) (for a simplicity, CP( 1) or CP(f, 6, cp)) where f: 52 + E, 52 c ( - co, co) x 99, is uniformly continuous. Here E is a Banach space with norm 1. IE, and g is an abstract phase space satisfying the fundamental axioms (see Section 1) introduced by Hale and Kato [6] (also refer to [8, 173). The purpose of this paper is to establish an existence theorem of solu- tions for CP(l) and Kamke’s theorem in FDEs. It is well known that the compactness condition which is described by means of the measure of non- compactness intrdoduced by Kuratowski is useful in showing the existence of solutions for ordinary and functional differential equations in Banach spaces (cf. [l, 3, 4, 10, 12, 13, 241). We note that these results on the existence of solutions are closely related to the property of the phase spaces Eand,%=C([--r,O],E),O<r<a. In this paper, we shall prove an existence theorem of solutions for CP(l) under the condition: