Periodic Problem for Doubly Nonlinear Evolution Equation

TL;DR

This paper establishes the existence of time-periodic solutions for doubly nonlinear evolution equations in Banach spaces, extending previous work on Cauchy problems by employing compactness methods due to the different nature of periodic problems.

Contribution

It proves the existence of periodic solutions for a class of doubly nonlinear equations using compactness techniques, adapting methods beyond the WED functional approach.

Findings

Existence of time-periodic solutions under specified growth conditions.

Extension of previous Cauchy problem results to periodic setting.

Application of compactness methods for doubly nonlinear equations.

Abstract

We are concerned with the time-periodic problem of some doubly nonlinear equations governed by differentials of two convex functionals over uniformly convex Banach spaces. Akagi--Stefanelli (2011) considered Cauchy problem of the same equation via the so-called WED functional approach. Main purpose of this paper is to show the existence of the time-periodic solution under the same growth conditions on functionals and differentials as those imposed in Akagi--Stefanelli (2011). Because of the difference of nature between Cauchy problem and the periodic problem, we can not apply the WED functional approach directly, so we here adopt standard compactness methods with suitable approximation procedures.