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Strong convergence theorems for finitely many nonexpansive mappings and applications. (English) Zbl 1123.47044

This article deals with new, more cumbersome and freakish, approximations x n to a fixed point for nonexpansive mappings of a nonempty closed convex subset of a Banach space E; the authors consider the case when E is reflexive and has a weakly continuous duality mapping and the norm of E is uniformly Gâteaux differentiable. Moreover, they consider a finite family of nonexpansive mappings 𝒯={T 1 ,,T r } of C into itself with nonempty set of common fixed points and a class of W-mappings generated by 𝒯. These W-mappings are defined by chains U 1 =α 1 T 1 +(1-α 1 )I, U 2 =α 2 T 2 U 1 +(1-α 2 )I,,U r-1 =α r-1 T r-1 U r-2 +(1-α r )I, W=U r =α r T r U r-1 +(1-α r )I, where α,,α r are reals from [0,1]. The authors’ approximations are

x n+1 =λ n y+(1-λ n )W n x n ,n=1,2,,y,x 1 C,

where W n is a sequence of W-mappings generated by 𝒯 and λ n is a sequence from (0,1) such that

lim n λ n =0, n=1 λ n =,lim n λ n-1 λ n =1·

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.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces