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 ScorzaDragoni [3,7,16]. The methods involve assuming inequalities in terms of the second derivative of Lyapunovlike 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 ScorzaDragoni. 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 compactnesslike conditions on H in terms of the Kuratowski's measure of noncompactness in extending ScorzaDragoni'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 ScorzaDragoni'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 Lyapunovlike 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]. 