An infinite dimensional bifurcation problem with application to a class of functional differential equations of neutral type

Jean-François  Couchouron; Mikhail Kamenskii; Paolo Nistri

doi:10.3934/cpaa.2013.12.1845

Article Contents

2013, Volume 12, Issue 5: 1845-1859. Doi: 10.3934/cpaa.2013.12.1845

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An infinite dimensional bifurcation problem with application to a class of functional differential equations of neutral type

1.
Université de Metz, Mathématiques, LMAM, Ile du Saulcy, 57045 Metz
2.
Voronezh State University, Department of Mathematics, Universitetskaya pl. 1, 394006 Voronezh
3.
Dipartimento di Ingegneria dell' Informazione, Università di Siena, Via Roma 56, 53100, Siena

Received: May 31, 2010

Revised: July 31, 2012

Published: August 2013

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Abstract

In this paper we consider an infinite dimensional bifurcation equation depending on a parameter $ \varepsilon>0 . $ By means of the theory of condensing operators, we prove the existence of a branch of solutions, parametrized by $ \varepsilon , $ bifurcating from a curve of solutions of the bifurcation equation obtained for $\varepsilon =0 . $ We apply this result to a specific problem, namely to the existence of periodic solutions bifurcating from the limit cycle of an autonomous functional differential equation of neutral type when it is periodically perturbed by a nonlinear perturbation term of small amplitude.

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Mathematics Subject Classification: Primary: 47H08, 34K18, 34K40; Secondary: 34K13, 34K27.

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