A continuation principle for forced oscillations on differentiable manifolds

M. Furi and Maria Patrizia Pera

Vol. 121 (1986), No. 2, 321–338

DOI: 10.2140/pjm.1986.121.321

Abstract

In the present paper we are concerned with the existence of T-periodic solutions for the differential equation ẋ(t) = f(t,x(t)), t ∈ R, where f is a continuous time dependent T-periodic tangent vector field defined on an n-dimensional differentiable manifold M possibly with boundary. We prove that if the Euler characteristic of the average vector field w(p) = (1∕T)∫ ₀^Tf(t,p)dt is defined and nonzero and if all the possible orbits of the parametrized equation ẋ(t) = λf(t,x(t)), t ∈ R and λ ∈ (0,1], lie in a compact set and do not hit the boundary of M, then the given equation admits a T-periodic solution.

Mathematical Subject Classification

Primary: 58F22

Milestones

Received: 4 May 1984

Revised: 8 March 1985

Published: 1 February 1986