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Small Almost Periodic and Almost Automorphic Oscillations in Forced Liénard Equations

Volume 1, Number 1 (2010), 75 - 85

Small Almost Periodic and Almost Automorphic Oscillations in Forced Liénard Equations

Communicated By: 
Alain Haraux
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Abstract

We consider forced Li´enard differential equations in the form x00(t)+ f (x(t), p) · x0(t)+g(x(t), p) = ep(t), where p is a parameter in a Banach space. Starting from the zero solution when p = 0, we prove the existence of almost periodic (respectively almost automorphic) solutions xp when ep is almost periodic (respectively almost automorphic), when p belongs to a neighborhood of 0. We also consider the equations x00(t)+ f (x(t)) · x0(t)+g(x(t)) = e(t) and x00(t)+ f (x(t),q) · x0(t)+g(x(t),q) = e(t) which are special cases of the previous equations.