Periodic solutions for semilinear partial differential equation with lack of compactness

Abstract

Through this work, we examine the periodicity of solutions for the following semilinear retarded equation defined as ddtw(t)=B˜w(t)+D˜(wt)+G˜(t,wt). Summarizing the theory of perturbation of semi-Fredholm operators and the contraction mapping theorem, we propose some adequate conditions on the bounded linear operator D˜ and the function G to establish the periodicity of solutions on the interval [0,+∞) without taking into consideration the compactness condition on the semigroup generated by the part of B˜. Furthermore, we present an application that includes numerical simulations to illustrate the applicability of the theoretical findings.