On two-point boundary value problems

Abstract

where the (n x n) matrix A and the n-vector f are continuous functions of t on [a, b], A4 and N are (m x n) constant matrices with rank [M N-j = P < min(m, n), tx is an m-vector, and all scalars are assumed to be real. Problem (1. l), (1.2) is invertible if m = n and the homogeneous boundary value problem corresponding to (1.1 ), (1.2) withf= 0 and a = 0 has only the trivial solution solution y = 0. Otherwise (1.1 ), (1.2) is called noninvertible. It is well known that in the invertible case, problem (l.l), (1.2) has a unique solution for every choice of g and a, while in the alternative noninvertible case problem (l.l), (1.2) can have either no solution or many solutions depending on the choice of g and a. In this paper, we establish existence and uniqueness of solutions to (l.l), (1.2) in the noninvertible case. However, standard methods are not directly applicable in the noninvertible case.