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On the Existence of Solutions of Differential Equations in a Banach Space

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On the Existence of Solutions of Differential Equations in a Banach Space

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Title: On the Existence of Solutions of Differential Equations in a Banach Space
Author: Eisenfeld, Jerome; Lakshmikantham, V.
Abstract: The study of Cauchy problem for differential equations in a Banach space has taken two different directions. One approach is to find compactness type conditions [1,2,4,9,14] to guarantee existence of solutions only and the corresponding results are extensions of the classical Peano's Theorem. The other approach is to employ accretive type conditions [9,10,11,12,15] which assure existence as well as uniqueness of solutions. In fact, this latter technic shows that uniqueness conditions imply existence of solutions [16]. In this paper we follow the first direction. Employing Lyapunov-like functions and the notion of the measure of noncompactness, we prove a local existence result which generalizes in a natural way the compactness type conditions. We also consider a global existence result under a general set of conditions so as to include existing results in this direction.
URI: http://hdl.handle.net/10106/2175
Date: 1974-03

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