This article deals with new, more cumbersome and freakish, approximations to a fixed point for nonexpansive mappings of a nonempty closed convex subset of a Banach space ; the authors consider the case when is reflexive and has a weakly continuous duality mapping and the norm of is uniformly Gâteaux differentiable. Moreover, they consider a finite family of nonexpansive mappings of into itself with nonempty set of common fixed points and a class of -mappings generated by . These -mappings are defined by chains , , , where are reals from . The authors’ approximations are
where is a sequence of -mappings generated by and is a sequence from such that
Theorems on the strong convergence of these approximations are proved. Based on these results, the problem of finding a common fixed point of finitely many mappings is also considered.