A continuation theorem for periodic boundary value problems with oscillatory nonlinearities

Abstract

We prove a continuation theorem for the solvability of the coincidence equationLx=Nx in normed spaces. Applications are given to the periodic boundary value problem for second order ordinary differential equations. Dealing, in particular, with the periodically forced Duffing equation $$x'' + g(x) = p(t) = p(t + T),$$ (D) we show that our main theorem can be applied to the case in whichg(x)/x crosses an arbitrary number of eigenvalues for |x| large. A typical result in this direction is the following (see Corollary 4.2):Assume g odd with limx→+∞g(x)=+∞, and, for G′(x)=g(x), suppose that\(|\sqrt {G(x)} - \sqrt {G(y)} |\) bounded for x and y positive and large, implies that |x−y| is bounded. Then (D) has at least one T-periodic solution provided that lim infx→+∞2G(x)/x2< lim supx→+∞2G(x)/x2. The technical condition on\(\sqrt G\) generalizes various growth restrictions ong previously considered in the literature and, in general, it is always satisfied for a functiong having order of growth at infinity like |x|α, with α≥1. (See Proposition 3.1 and Remark 3.1 for more precise informations about the technical condition on\(\sqrt G\)).