Existence of periodic solutions in abstract semilinear equations and applications to biological models

Abstract

In this paper, we study the existence of mild periodic solutions of abstract semilinear equations in a setting that includes several other types of equations such as delay differential equations, first-order hyperbolic partial differential equations, and reaction-diffusion equations. Under different assumptions on the linear operator and the nonhomogeneous function, sufficient conditions are derived to ensure the existence of mild periodic solutions in the abstract semilinear equations. When the semigroup generated by the linear operator is not compact, Banach fixed point theorem is used whereas when the semigroup generated by the linear operator is compact, Schauder fixed point theorem is employed. In applications, we apply the main results to establish the existence of periodic solutions in delayed red-blood cell models, age-structured models with periodic harvesting, and the diffusive logistic equation with periodic coefficients.