A global bifurcation theorem with applications to functional differential equations

Abstract

We are interested here in studying the closure S of the set {(x, α):F(x, α) = x and x ≠ 0} for a certain class of nonlinear operators F such that F(0, α) = 0 for all α in an interval of real numbers. Our main abstract theorem establishes under certain conditions on F the existence of a closed, connected unbounded subset S0 of S which contains a point (0, α0) and no points of the form (0, α) for α ≠ α0. The novelty of this result is that it is obtained under hypotheses of the type used in asymptotic fixed point theorems; in our applications F is not, in general, Frechet differentiable at points (0, α) and may not even be continuous at such points. The abstract theorem is then applied to study the structure of the set of periodic solutions of the equation x′(t) = −αf(x(t − 1)) and to obtain sharp results on the range of periods as α varies.