Let C be a nonempty closed convex subset of a Banach space E and let $T_1, cdots, T_N$ be nonexpansive mappings from C into itself (recall that a mapping $T : C longrightarrow C$ is nonexpansive if $left$mid$ Tx - Ty ight$mid$ leq left$mid$ x - y ight$mid$$</TEX> for all $x, y in C$). We consider the fixed point problem for nonexpansive mappings : find a common fixed point, i.e., find a point in $cap_{i=1}^N Fix(T_i)$, where $Fix(T_i) := {x in C : x = T_i x}$ denotes the set of fixed points of $T_i$.