A compatibility theorem for two point boundary problems

Abstract

where A and B are n by n matrices the elements of which are continuous complex valued functions on R. The compatibility of (1), (2) at XCR is defined to be the maximum number of linearly independent solutions of (1), (2) corresponding to X. Let yi, * * *, yn be a fundamental set of solutions of (1) continuous in (t, X) for tG [a, b] and XCR, hence uniformly continuous there. A necessary and sufficient condition for the compatibility of (1), (2) at X to be k is that the rank of the n by n matrix V(X) with elements Uiyj(X) be of rank n-k. It is known [I], [2] that if the compatibility of (1), (2) is constant in some neighborhood of Xo and x(t) is a solution of (1), (2) for X=Xo, then there exists a solution x(t, X) of (1), (2) which is uniformly continuous in (t, X) for tC [a, b] and X in some neighborhood of X0 and which is such that x(t, Xo) =x(t) so that x(t, X)x(t) uniformly on [a, b]. The question, a natural one, as to what can be said in case the compatibility is not constant in any neighborhood of Xo was brought to the attention of the author by W. M. Whyburn [3]. An answer to the question is in the following theorem.