Iterative construction of a common fixed point of finite families of nonlinear mappings

Let K be a nonempty closed convex subset of a real reflexive Banach space E with uniformly Gâteuax differentiable norm. Let T1, T2, ..., Tm : K —> K be m Lipschitz mappings (for some m ∈ N) such that (wzór). We construct a new iteration process and prove that the iteration process converges strongly to a common fixed point of these mappings provided at least one of the mappings is pseudocontractive. We also obtain as easy corollaries convergence results for finite families of Lipschitz pseudocontractive mappings and nonexpansive mappings. Furthermore, We prove that a slight modification of our iteration process converges strongly to a common zero of a finite family of Lipschitz accretive operators. Our new iteration process and our method of proof are of independent interest.