Let E be a uniformly convex Banach space with a uniformly G$hat{a}teaux differentiable norm, C a nonempty closed convex subset of $E, T : C o E$ a nonexpansive mapping, and Q a sunny nonexpansive retraction of E onto C. For $u in C$ and $t in (0,1)$, let $x_t$ be a unique fixed point of a contraction $R_t : C o C$, defined by $R_tx = Q(tTx + (1-t)u), x in C$. It is proved that if ${x_t}$ is bounded, then the strong $lim_{t o1}x_t$ exists and belongs to the fixed point set of T. Furthermore, the strong convergence of ${x_t}$ in a reflexive and strictly convex Banach space with a uniformly G$hat{a}$teaux differentiable norm is also given in case that the fixed point set of T is nonempty.