This article deals with the following iterative scheme
in a smooth, strictly convex, and reflexive Banach space with the Kadec–Klee property ( is a metric projection in ). It is assumed that is a countable family of mappings of a nonempty closed convex subset of E into itself such that and that satisfies the condition
for some , (here, ).
It is proved the strong convergence of the sequence to under some additional assumptions about and (in particular, that the relations , , , and imply that ). The case when is real Hilbert space is considered as a particular case. Furthermore, the case when is a family of maximal monotone operators and an application to the feasibility problem are considered.