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 x ″ (t)+f(x(t),p)⋅ x ′ (t)+g(x(t),p)= e p (t), 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 x ″ (t)+f(x(t))⋅ x ′ (t)+g(x(t))=e(t) and x ″ (t)+f(x(t),q)⋅ x ′ (t)+g(x(t),q)=e(t).MSC:34C27, 34C99, 47J07.
Highlights
), we have proven the existence of an almost periodic (respectively almost automorphic) solution xp near an equilibrium, by using the perturbation method in the setting of Nonlinear Functional Analysis
Near an equilibrium we study the existence of asymptotically a.p., asymptotically a.a., pseudo a.p., pseudo a.a., weighed pseudo a.p. and weighed pseudo a.a. solutions of Liénard differential equations in the form x (t) + f (x(t), p) · x (t) + g(x(t), p) = ep(t), where the forcing term possesses a similar nature, and where p is a parameter in a Banach space
When ep is almost periodic in the Bohr sense, in [ ], Theorem ., we have proven the existence of an almost periodic solution xp near an equilibrium, by using the perturbation method in the setting of Nonlinear Functional Analysis. We extend this result to the frameworks of asymptotically almost periodic, asymptotically almost automorphic, pseudo almost periodic, pseudo almost automorphic, weighted pseudo almost periodic and weighted pseudo almost automorphic functions
Summary
), we have proven the existence of an almost periodic (respectively almost automorphic) solution xp near an equilibrium, by using the perturbation method in the setting of Nonlinear Functional Analysis. We establish, for a linear differential equation in a Banach space, a result on the existence and uniqueness of the solutions described above.
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