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On periodic solutions of nonlinear second order vector differential equations

Published online by Cambridge University Press:  14 November 2011

P. Habets  and
M. N. Nkashama
P. Habets
Affiliation:
U.C.L., Institut de Mathématique pure et appliquée, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium
M. N. Nkashama
Affiliation:
Department of Mathematical Sciences, Memphis State University, Memphis, Tennessee 38152, U.S.A.
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Synopsis

This paper considers existence of periodic solutions for vector Liénard differential equations

In our main result we write

where Q(t, x) is a symmetric matrix and h(t, x) is sublinear. The key assumption relates the asymptotic behaviour as x →+ ∞ of the eigenvalues of Q(t, x) to the spectrum of the linear operator −d2/dt2 Several choices for Q(t, x) are considered which lead to known theorems and extend others. In the case of the Duffing equation

the assumptions are weakened.

Our approach is based on Leray-Schauder's degree theory and a priori estimates.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 104 , Issue 1-2 , 1986 , pp. 107 - 125
Copyright
Copyright © Royal Society of Edinburgh 1986

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