Periodic Solutions of Semilinear Parabolic Equations

Abstract

This chapter describes the methods of nonlinear functional analysis, namely, fixed-point theorems in ordered Banach spaces, to prove existence and multiplicity result for periodic solutions of semilinear parabolic differential equations of the second order. The oldest method for the study of periodic solutions of differential equations is to find fixed points of the Poincaré operator. Subsequently in the case of parabolic equations, it turns out that the Poincaré operator is compact in suitable function spaces. Even in the case of the general semilinear parabolic equations, this operator is strongly increasing. Having seen that the Poincaré operator is strongly increasing, it is clear that the problem can be included in the general framework of nonlinear equations in ordered Banach spaces. Hence, by applying other general fixed-point theorems for equations of this type, it is possible to obtain further existence and multiplicity results.