Ultimate boundedness and periodicity for some partial functional differential equations with infinite delay

Abstract

In this work we study the existence of periodic solutions for some partial functional differential equation with infinite delay. We assume that the linear part is not necessarily densely defined and satisfies the known Hille-Yosida condition. Firstly, we give some estimates of the solutions. Secondly, we prove that the Poincaré map is condensing which allows us to prove the existence of periodic solutions when the solutions are ultimately bounded.

Massera type theorems for abstract non-autonomous evolution equations

“…The existence of periodic solutions to (1.1), (1.2) and their variants has been of great interest for many authors (cf. [1,3,4,5,7,9,8,11,12,13,14,18,19]). To establish the existence of periodic solutions to (1.2), one of the key steps is to consider first the existence of periodic solutions to the linear equation (1.1).…”

On the periodic solutions for some retarded partial differential equations by the use of semi-Fredholm operators

“…The authors in [4,12] proved the periodicity of solutions when the solutions of periodic system are just bounded and ultimately bounded by the use of the Horn's fixed point theorem. Especially, in infinite dimensional spaces, the authors in [8], used the Poincaré map approach to get the periodicity of solutions for a class of retarded differential equation, they supposed that the infinitesimal generator satisfies the Hille-Yosida condition and generates a compact semigroup (T (t)) t≥0 by applying an appropriate fixed point theorem. In [22], the authors proved the periodicity for a nonhomogeneous linear differential equation when the linear part generates a C 0 -semigroup on E and they obtained the existence and uniqueness of periodic solutions for this class of equations.…”

“…The present work would be a continuation and extension of the work [8] for inhomogeneous linear retarded PDE, we establish the periodicity of solution for Equation (1.1) by using the perturbation theory of semi-Fredholm operators and without considering the compactness condition of (T (t)) t≥0 .…”