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An implicit iteration scheme with errors for a finite family of uniformly continuous mappings. (English) Zbl 1159.47039

This article deals with iterations of the form

x n =α n x n-1 +β n T i(n) k(n) x n +γ n u n ,n1,

to a common fixed point of a finite family {T 1 ,,T N } of uniformly continuous mappings in a real uniformly convex Banach space E. It is assumed that the T i , i=1,,N, map a nonempty closed convex set CE into itself, have common fixed points, and satisfy the condition

max 1iN T i n x-T i n yx-y+d n ,x,yC, n=1 d n <;

u n is a bounded sequence in C; (α n ),(β n ),(γ n ) are sequences in [0,1], α n +β n +γ n =1, α n [s,1-s] (s(0,1/2)), n=1 γ n <; k(n), i(n)n(modN) and i(n){0,1,,N-1}.

The main result is the following: if at least one T i is semi-compact (i.e., the relations x n C and x n -Tx n 0 imply that (x n ) has a convergent subsequence), then (x n ) converges strongly to a common fixed point if T i , i=1,,N. As the authors state, their “results are generalizations of some well-known results in the current literature”.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces
54H25Fixed-point and coincidence theorems in topological spaces