Periodic forcing for some difference equations in Hilbert spaces

Abstract

Let $H$ be a real Hilbert space and let $A:D(A)\subset H\rightarrow H$ be a (possibly multivalued) maximal monotone operator. We are concerned with the difference equation \begin{equation*} \Delta u_{n} + c_{n} A u_{n+1}\ni f_{n},\qquad n=0,1, ... , \end{equation*} where $(c_{n})\subset (0,+\infty)$, $(f_{n})\subset H$ are $p$-periodic sequences for a positive integer $p$. We investigate the existence of periodic solutions to this equation as well as the weak or strong convergence of solutions to $p$-periodic solutions. The first result of this paper (Theorem 1) is a discrete analogue of the 1977 result by Baillon and Haraux (on the periodic forcing problem for the continuous counterpart of the above equation) and was essentially stated by Djafari Rouhani and Khatibzadeh in a recent paper. Here we provide a simpler proof of this result that is based on old existing results due to Browder and Petryshyn and Opial. A strong convergence result is also given and some examples are discussed to illustrate the theoretical results.