Let

$K$ be a closed convexed subset of a Banach space

$X$, and let

$F$ be a nonempty closed subset of

$K$. The authors consider complete metric spaces of self-mappings of

$K$ which fix all the points of

$F$ and are relatively nonexpansive with respect to a given convex function

$f$ on

$X$. The aim of this paper is to prove that under quite mild conditions on

$F$ strong convergence of the sequences

${\left\{{T}^{k}x\right\}}_{k=1}^{\infty}$ generated by relatively nonexpansive mappings is the rule and that weak, but not strong convergence is the exception.