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Strong convergence to common fixed points of infinite nonexpansive mappings and applications. (English) Zbl 0993.47037

This article deals with iterations x n+1 =β n x+(1-β n )W n x n (n=0,1,), where W n (n=1,2,) are mappings generated by the scheme

W n =U n,1 ,U n,k =α k T k U n,k+1 +(1-α k )I(k=1,,n),U n,n+1 =I,

T 1 ,T 2 , are nonexpansive mappings of a convex subset of a Banach space E into itself, n=1 FixT n , α n satisfy the condition 0<α n b<1, β n satisfy the conditions 0β n 1, lim n β n =0, n=1 |β n+1 -β n |<, n=1 β n =. The basic result is the convergence of x n to Px, where P is the unique sunny nonexpansive retraction from C onto n=1 FixT n ; it is assumed that the norm in E is uniformly convex and uniformly Gâteaux differentiable.

47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47J25Iterative procedures (nonlinear operator equations)
46B04Isometric theory of Banach spaces