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Several kinds of oscillations in forced Liénard equations

Joël Blot1*, Souhila Boudjema2 and Philippe Cieutat3

Author Affiliations

1 Laboratoire SAMM EA 4543, Université Paris 1 Panthéon-Sorbonne, centre P.M.F., 90 rue de Tolbiac, Paris cedex 13, 75634, France

2 Département de Mathématiques, Faculté des Sciences, Université de Skikda, Skikda, Algérie

3 Laboratoire de Mathématiques de Versailles, UMR-CNRS 8100, Université Versailles-Saint-Quentin-en-Yvelines, 45 avenue des États-Unis, Versailles cedex, 78035, France

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Boundary Value Problems 2013, 2013:66  doi:10.1186/1687-2770-2013-66

Published: 29 March 2013

Abstract

Near an equilibrium we study the existence of asymptotically a.p. (almost periodic), asymptotically a.a. (almost automorphic), pseudo a.p., pseudo a.a., weighed pseudo a.p. and weighed pseudo a.a. solutions of Liénard differential equations in the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M1">View MathML</a>, where the forcing term possesses a similar nature, and where p is a parameter in a Banach space. We use a perturbation method around an equilibrium. We also study two special cases of the previous family of equations that are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/66/mathml/M3">View MathML</a>.

MSC: 34C27, 34C99, 47J07.

Keywords:
asymptotically almost periodic functions; asymptotically almost automorphic functions; pseudo almost periodic functions; pseudo almost automorphic functions; weighted pseudo almost periodic functions; weighted pseudo almost automorphic functions; Liénard equations