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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.


MSC:
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