Open Access Research

Global continuation of periodic solutions for retarded functional differential equations on manifolds

Pierluigi Benevieri12, Alessandro Calamai3, Massimo Furi1 and Maria Patrizia Pera1*

Author Affiliations

1 Dipartimento di Matematica e Informatica, Università degli Studi di Firenze, Via S. Marta 3, Firenze, I-50139, Italy

2 Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, São Paulo, 05508-090, Brasil

3 Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, Ancona, I-60131, Italy

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Boundary Value Problems 2013, 2013:21  doi:10.1186/1687-2770-2013-21

Published: 11 February 2013

Abstract

We consider T-periodic parametrized retarded functional differential equations, with infinite delay, on (possibly) noncompact manifolds. Using a topological approach, based on the notions of degree of a tangent vector field and of the fixed point index, we prove a global continuation result for T-periodic solutions of such equations.

Our main theorem is a generalization to the case of retarded equations of a global continuation result obtained by the last two authors for ordinary differential equations on manifolds. As corollaries we obtain a Rabinowitz-type global bifurcation result and a continuation principle of Mawhin type.

MSC: 34K13, 34C40, 37C25, 70K42.

Keywords:
retarded functional differential equations; global bifurcation; fixed point index; degree of a vector field