A continuation principle for forced oscillations on differentiable manifolds

Abstract

In the present paper we are concerned with the existence of Γ-periodic solutions for the differential equation x(t) = f(t, .x( r)), t e R, where / is a continuous time dependent Γ-periodic tangent vector field defined on an ^-dimensional differentiable manifold M possibly with boundary. We prove that if the Euler characteristic of the average vector field w(p) = (l/T)fίff(t,p)dt is defined and nonzero and if all the possible orbits of the parametrized equation x(t) = λf(ί,x(t)), /eR and λ £ (0,1], lie in a compact set and do not hit the boundary of M, then the given equation admits a Γ-periodic solution. 0. Introduction. Let M be an w-dimensional differentiable manifold, possibly with boundary, and let /: RxM->Γ(M) be a time dependent Γ-periodic tangent vector field on M. In this paper we give a topological result concerning the existence of Γ-periodic solutions for the differential equation