Existence of solutions of two-point boundary value problems for nonlinear systems

Abstract

Let I‘.” denote the inner product and 1 x / will denote the Euclidean norm. For a matrix A we will say A is positive dejnite and write A > 0 if x . (Ax) > 0 for all x (X f 0) in Rd. We say that A > 0 if either A > 0 or A is identically 0. We assume in our theorems either that A,, A, >, 0 or that A,, A, > 0. Our two-point boundary conditions are analogs of the boundary conditions studied in dimension d = 1 by Keller [9] and Bebernes and Gaines [2], (and the other conditions of our Theorem 3 generalize their conditions). On the other hand, problems (I .I), (1.3), and (1.4) include (by letting A, = A, = 0) the classical two-point boundary value problems (1.1) (1.2)