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Strong convergence theorem for asymptotically nonexpansive mappings. (English) Zbl 0861.47030

In the present paper the authors prove the following Theorem: Let C be a nonempty closed convex subset of a real Hilbert space and T:CC be an asymptotically nonexpansive mapping (i.e. for each n1 there exists a real number k n 1 such that |T n x-T n y|k n |x-y|, x,yC and lim n k n =1). Suppose that the set of fixed points of T, F(T) is nonempty. Let

b n =1 n j=1 n (1+|1-k j |+e -j ),0<a<1,x 0 C·

Then the mapping T n on C given by

T n x=a n x 0 +(1-a n )A n xforallxC


a n =b n -1 b n -1+aandA n =1 n j=1 n T j ,

has a unique fixed point u n in C and the sequence {u n } converges strongly to the element of F(T) which is nearest to x 0 .

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