  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.
