Periodic Problems for Doubly Nonlinear Evolution Equations

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. G. Akagi and U. Stefanelli [Weighted energy-dissipation functionals for doubly nonlinear evolution, J. Funct. Analysis 260/9 (2011) 2541–2578] considered the Cauchy problem of the same equation via the so-called WED functional approach. The 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 [loc.cit]. Because of the difference in nature between the 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.