A continuation result for forced oscillations of constrained motion problems with infinite delay

TL;DR

This paper establishes a global continuation theorem for periodic solutions of parametrized retarded functional differential equations on manifolds, with applications to forced oscillations in constrained motion systems.

Contribution

It introduces a new continuation result for periodic solutions of delayed differential equations on manifolds using fixed point index theory.

Findings

Proves existence of forced oscillations under nonzero frictional coefficient.

Applies topological methods to constrained motion equations.

Suggests potential extension to frictionless cases.

Abstract

We prove a global continuation result for $T$-periodic solutions of a $T$-periodic parametrized second order retarded functional differential equation on a boundaryless compact manifold with nonzero Euler-Poincare' characteristic. The approach is based on the fixed point index theory for locally compact maps on ANRs. As an application, we prove the existence of forced oscillations of retarded functional motion equations defined on topologically nontrivial compact constraints. This existence result is obtained under the assumption that the frictional coefficient is nonzero, and we conjecture that it is still true in the frictionless case.