Periodicity in the $\alpha$-Norm for Some Partial Functional Differential Equations with Infinite Delay

Abstract

The aim of this work is to study the existence of periodic solutions in the $\alpha$-norm for some partial differential equations with infinite delay. A linear part of equations is assumed to generate an analytic semigroup. The delayed part is assumed to be continuous with respect to the fractional norm of the linear part and $\sigma$-periodic with respect to the first argument. Using Massera's approach we prove the existence of periodic solutions in the linear case. In the nonlinear case, a fixed point theorem for multivalued mapping is used to prove the existence of periodic solutions. We use also Horn's fixed point theorem to get the existence of periodic solutions when solutions are ultimate bounded. For illustration an example is provided for some reactiondiffusion equation involving infinite delay.