Sufficient conditions for periodic solutions to a class of second-order differential equations

Abstract

where 0 # b E R and g(0, 0,O) = 0, possess nonzero, generalized, even, T-periodic solutions x(t); i.e. x E W2V2( T, T) (defined below) such that x(t + T) = x(t), x( t) = x(t), x(t) is not identically zero and satisfies equation (1) for a.a. t E F?. Equation (1) cannot be “solved” in a useful manner for the highest order derivative. We emphasize that T is not given, but is an unknown quantity to be determined. Our method of solution is to convert the problem to an operator equation in Hilbert space and obtain results using global bifurcation theory. This bifurcation approach produces more that just existence of solutions: we obtain global information on the continuum of T-periodic solutions to equation (1). The idea of solving existence problems of periodic solutions of an autonomous ordinary differential equation around an equilibrium by transformation to a bifurcation problem is due to Krasnosel’skii [2] who worked with compact mappings. Recently, similar problems to equation (1) have been studied by Hetzer [ 11, Mawhin [5], Petryshyn and Yu [7], Webb and Welsh [9] and others. Hetzer’s results we shall discuss in our concluding remarks below; Mawhin invoked his coincidence degree theory and was, therefore, unable to consider the nonlinear term g as a function of the highest order derivative; Petryshyn and Yu studied problems analogous to equation (l), but restricted their results to existence of solutions without considering global properties; Webb and Welsh were more concerned with proving existence and uniqueness of not necessarily periodic solutions of equation (1). Equation (1) was alluded to in [lo] as an application of bifurcation theory, but the details were not given, nor were the hypotheses stated precisely. The main result in this paper (corollary 5) follows as a consequence of a bifurcation theorem for so called A-proper mappings (defined below) proved by the author in [lo]. The usefulness of this A-proper framework is that the resulting theorems apply in a more general setting than earlier bifurcation results which generally demanded that the nonlinear term g generate a