Nonzero solutions of nonlinear systems of differential equations via fixed point theorems for multivalued maps

Abstract

IN THE present paper we consider a nonlinear System of differential equations (.) x’ = A(t, x)x + F(t, x) and a condition (. .) x E B, x(t) $ 0. where A(t, x) is a n x n real matrix defined and continuous on A x R” (A denotes a compact interval of R), F : A x R” + R” is continuous and B is a Banach space of continuous functions. To solve this problem, we use a comparison technique and we reduce the existence of solutions for this problem to that of a fixed point of a suitably defined multivalued map S (see also Anichini [l], Kartsatos [3] and Schmitt [4]). In these papers the Eilenberg-Montgomery futed point theorem is applied. First we consider the problem (. . .) x’ = A(& x)x and the condition (..). Using the Eilenberg-Montgomery Theorem 3.1 of [3]. fixed point theorem, we improve