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September  2013, 12(5): 1845-1859. doi: 10.3934/cpaa.2013.12.1845

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

Jean-François Couchouron 1, , Mikhail Kamenskii 2, and  Paolo Nistri 3,

1. 

Université de Metz, Mathématiques, LMAM, Ile du Saulcy, 57045 Metz, France
2. 

Voronezh State University, Department of Mathematics, Universitetskaya pl. 1, 394006 Voronezh, Russian Federation
3. 

Dipartimento di Ingegneria dell' Informazione, Università di Siena, Via Roma 56, 53100, Siena

Received  June 2010 Revised  August 2012 Published  January 2013

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.
Keywords: functional differential equations, Condensing operators, periodic solutions., bifurcation.
Mathematics Subject Classification: Primary: 47H08, 34K18, 34K40; Secondary: 34K13, 34K2.
Citation: Jean-François Couchouron, Mikhail Kamenskii, Paolo Nistri. An infinite dimensional bifurcation problem with application to a class of functional differential equations of neutral type. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1845-1859. doi: 10.3934/cpaa.2013.12.1845
References:
[1]

J. Cronin, "Differential Equations: Introduction and Qualitative Theory," Pure and Applied Mathematics, A Series of Monographs and Textbooks, 54, Marcel Dekker, Inc. New York and Basel, 1980.  Google Scholar

[2]

I. C. Gohberg, S. Golberg and M. A. Kaashoek, "Classes of Linear Operators I," Operator Theory: Advances and Applications, 49, Birkhäuser Verlag, Basel, 1990.  Google Scholar

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I. C. Gohberg and M. G. Krein, The basic proposition on defect numbers, root numbers and indexes of linear operators, Amer. Math. Soc. Transl., 13 (1960), 185-264.  Google Scholar

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M. I. Kamenskii, O. Makarenkov and P. Nistri, An alternative approach to study bifurcation from a limit cycle in periodically perturbed autonomous systems, J. Dyn. Diff. Equat., 23 (2011), 425-435. doi: 10.1007/s10884-011-9207-4.  Google Scholar

[5]

S. G. Krantz and H. R. Parks, "The Implicit Function Theorem: History, Theory and Applications," Birkhäuser, Boston, 2003.  Google Scholar

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W. S. Loud, Periodic solutions of a perturbed autonomous system, Ann. Math., 70 (1959), 490-529.  Google Scholar

[7]

I. G. Malkin, "Some Problems of the Theory of Nonlinear Oscillations," (Russian) Gosudarstv. Isdat. Techn. Teor. Lit., Moscow, 1956.  Google Scholar

show all references

References:
[1]

J. Cronin, "Differential Equations: Introduction and Qualitative Theory," Pure and Applied Mathematics, A Series of Monographs and Textbooks, 54, Marcel Dekker, Inc. New York and Basel, 1980.  Google Scholar

[2]

I. C. Gohberg, S. Golberg and M. A. Kaashoek, "Classes of Linear Operators I," Operator Theory: Advances and Applications, 49, Birkhäuser Verlag, Basel, 1990.  Google Scholar

[3]

I. C. Gohberg and M. G. Krein, The basic proposition on defect numbers, root numbers and indexes of linear operators, Amer. Math. Soc. Transl., 13 (1960), 185-264.  Google Scholar

[4]

M. I. Kamenskii, O. Makarenkov and P. Nistri, An alternative approach to study bifurcation from a limit cycle in periodically perturbed autonomous systems, J. Dyn. Diff. Equat., 23 (2011), 425-435. doi: 10.1007/s10884-011-9207-4.  Google Scholar

[5]

S. G. Krantz and H. R. Parks, "The Implicit Function Theorem: History, Theory and Applications," Birkhäuser, Boston, 2003.  Google Scholar

[6]

W. S. Loud, Periodic solutions of a perturbed autonomous system, Ann. Math., 70 (1959), 490-529.  Google Scholar

[7]

I. G. Malkin, "Some Problems of the Theory of Nonlinear Oscillations," (Russian) Gosudarstv. Isdat. Techn. Teor. Lit., Moscow, 1956.  Google Scholar

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