Let be a closed convex subset of a Hilbert space and be a nonexpansive mapping (i.e., , ). It is well known, that for each , the contraction defined by , ( is a fix point) has a unique fixed point . F. E. Browder [Arch. Ration. Mech. Anal. 24, 82-90 (1967; Zbl 0148.13601)] proved: converges in norm, as , to a fixed point of .
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 be a closed convex subset of a Hilbert space and be nonexpansive self-mappings of such that . Let and be a sequence in such that . Then the sequence defined in the following way:
where (here the function takes values in ) converges weakly to a common fixed point of the mappings .