On the existence of nontrivial solutions of differential equations subject to linear constraints

Abstract

The purpose of this paper is to consider boundary value problems for second order ordinary diff erential equations where the solutions sought are subject to a host of linear constraints (such as multipoint constraints) and to present a unifying framework for studying such. We show how Leray-Schauder continuation techniques may be used to obtain existence results for nontrivial solutions of a variety of nonlinear second order diff erential equations. A typical example may be found in studies of the four-point boundary value problem for the diff erential equation y’’(t)+a(t)f(y(t)) = 0 on [0, 1], where the values of y at 0 and 1 are each some multiple of y(t) at two interior points of (0, 1). The techniques most often used in such studies have their origins in fixed point theory. By embedding such problems into parameter dependent ones, we show that detailed information may be obtained via global bifurcation theory. Of course, such techniques, as they are consequences of properties of the topological degree, are similar in nature.