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An implicit iteration process for nonexpansive mappings. (English) Zbl 0999.47043

Let C be a closed convex subset of a Hilbert space H and T:CC be a nonexpansive mapping (i.e., Tx-Tyx-y, x,yC). It is well known, that for each t(0,1), the contraction T t :CC defined by T t (x)=tu+(1-t)Tx, xC (uC is a fix point) has a unique fixed point x t . F. E. Browder [Arch. Ration. Mech. Anal. 24, 82-90 (1967; Zbl 0148.13601)] proved: {x t } converges in norm, as t0, to a fixed point of T.

In the present paper the authors study the convergence of an implicit iteration process to a fixed point of a finite family of nonexpansive mappings. The main result of the paper is the following:

Theorem 2. Let C be a closed convex subset of a Hilbert space H and T 1 ,T 2 ,,T N be N nonexpansive self-mappings of C such that i=1 N Fix(T i ). Let x 0 C and {t n } be a sequence in (0,1) such that lim n t n =0. Then the sequence {x n } defined in the following way:

x n =t n x n-1 +(1-t n )T n x n ,n1,

where T k =T kmodN (here the modN function takes values in {1,2,,N}) converges weakly to a common fixed point of the mappings T 1 ,T 2 ,,T N .


MSC:
47H09Mappings defined by “shrinking” properties
65J15Equations with nonlinear operators (numerical methods)
47H10Fixed point theorems for nonlinear operators on topological linear spaces