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Existence of Solutions of Boundary Value Problems for Nonlinear Second Order Systems in a Banach Space

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Existence of Solutions of Boundary Value Problems for Nonlinear Second Order Systems in a Banach Space

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Title: Existence of Solutions of Boundary Value Problems for Nonlinear Second Order Systems in a Banach Space
Author: Mitchell, A. Richard; Chandra, V.
Abstract: This paper is concerned with the existence of solutions of boundary value problems (BVP, for short) for nonlinear second order ordinary differential equations of the type (1.1) [see pdf for notation] (1.2) [see pdf for notation] where [see pdf for notation] is a real Banach space. In case [see pdf for notation], existence was proved by first obtaining a priori bounds for [see pdf for notation] of a solution of (1.1) and (1.2) and then employing a theorem of Scorza-Dragoni [3,7,16]. The methods involve assuming inequalities in terms of the second derivative of Lyapunov-like functions relative to H, using comparison theorems for scalar second order equations and utilizing Leray,Schauder's alternative or equivalently the modified function approach [2,3,6,7,8,11]. In this paper, we wish to extend this fruitful method to the case when X is an arbitrary Banach space. First of all, this necessitates extending the basic result of Scorza-Dragoni. If we assume that H is compact operator as in [14], this extension is relatively easy. Since interest in abstract BVP's is partly due to the possibility of applications to partial differential equations, assuming compactness of H excludes many interesting examples. For example, using the method of lines (see the survey paper [12]), nonlinear elliptic BVP's may be approximated by an infinite system of BVP of the type (1.1) and (1.2). Consequently, we impose compactness-like conditions on H in terms of the Kuratowski's measure of noncompactness in extending Scorza-Dragoni's theorem. Thus, in Section 1, we develop further properties of the measure of noncompactness that are needed in our work. Utilizing these properties and the fixed point theorem of Darbo [5], we prove in Section 2 the generalization of Scorza-Dragoni's theorem. (Section 2 also contains a result concerning existence in the small.) Section 3 deals with extending the modified function approach to our problem (1.1) and (1.2). Here we use a new comparison result [3,1] and Lyapunov-like functions, and follow an argument similar to the one in [2,8]. To avoid monotony, we consider only one result, omitting variations as given in [2,8].
URI: http://hdl.handle.net/10106/2162
Date: 1977-03

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