Symmetric perturbations of nonlinear equations: Symmetry of small solutions

Abstract

where p(t) is a 2rc-periodic function, while CT is a small parameter. In many of the works done on this type of equation (see e.g. [l-5]), one has also p( t) = p(t), and then one restricts the problem to that of finding 2rc-periodic solutions which are even functions in t. If CJ = 0, then this restriction can easily be justified (see Section 2), and even in the case CJ # 0 this is sufficient for proving, for example, existence of solutions. The question remains, however, whether or not (1.1) has other solutions, which are not even functions in t. In [5] Hale and Rodrigues have answered this question for the Dufting equation (g(x, 2) = /zx x3), and under some conditions for p(t). They prove that if p( t) = p(t), and CJ # 0 is sufficiently small, then all sufficiently small 2n-periodic solutions of (1.1) will be even functions in t. The main idea of their proof is to use the fact that for 0 = 0 the equation (1.1) is autonomous: a phase shift t I-+ t + 4 leaves the equation invariant. They use the Liapunov-Schmidt method, and exploit the invariance of (1.1) under the transformation x(t) H x( t). Another important fact in their proof is that each element in the kernel of the operator L, defined by Lx = X + x, can be written as I cos(t + 4); i.e. each element in this kernel can be obtained by applying a phase shift on some element in the kernel, which is an even function in t. It is the aim of this paper to generalize this result of Hale and Rodrigues to a large class of nonlinear equations having some symmetry properties. We will consider equations between Banach spaces of the form