  In this article, we propose a Krasnosel'ski\v{\i}-Mann-type algorithm for
finding a common fixed point of a countably infinite family of nonexpansive
operators $(T_n)_{n \geq 0}$ in Hilbert spaces. We formulate an asymptotic
property which the family $(T_n)_{n \geq 0}$ has to fulfill such that the
sequence generated by the algorithm converges strongly to the element in
$\bigcap_{n \geq 0} \operatorname{Fix} T_n$ with minimum norm. Based on this,
we derive a forward-backward algorithm that allows variable step sizes and
generates a sequence of iterates that converge strongly to the zero with
minimum norm of the sum of a maximally monotone operator and a cocoercive one.
We demonstrate the superiority of the forward-backward algorithm with variable
step sizes over the one with constant step size by means of numerical
experiments on variational image reconstruction and split feasibility problems
in infinite dimensional Hilbert spaces.
