S-asymptotically ω-Periodic Solutions of generalized Liénard

Abstract

The S-asymptotically ω-periodic functions are a continuous and bounded functions from the real axis to a Banach space that converges to a periodic functions as tends to infinity. Starting from the zero solution, we prove in this work the existence and uniqueness of S-asymptotically ω-periodic solution of generalized Liénard's differential Equation. We stady after that the regular dependence of this solution with a certain parameter in Banach space, present in our equation, and with the forcing term hwo possesses a similar nature as the later. For this, our approach will be to use a perturbation method around an equilibrium. More precisely when the forcing is a Sasymptotically ω-periodic function we study the differentiable dependence of the S-asymptotically ω-periodic solution of Liénard equation. In this study, we changed our initial objective, which was to search for na S asymptotically ω-periodic solution for our Lienard equation, a problem relating to dynamical systems, towards an approach based on functional analysis. Concretely, we adopted a strategy consisting of using the implicit function theorem on a specific operator that we defined in our workspaces. This approach allowed us to achieve the objective stated in our main theorem. To realize our aim, we use the Nemytskii operators (also called superposition operators) and state some properties on these operators. We have also extended the well-established result on the almost periodic function of the derivative of an almost periodic function to the context of S-asymptotically ω-periodic cases. Finally, and to close our work, we give a corollary which presents a particular case of our main theorem.