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Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings. (English) Zbl 0971.47038

Let X be a uniformly convex Banach space and K a nonempty subset of X. A mapping T:KK is said to be asymptotically nonexpansive mapping if there exists a sequence {k n } with k n 1 and lim n k n =1 such that T n x-T n yk n x-y for all x,yK and for all n. In this paper, if K is a nonempty closed convex subset of X and T:KK is a nonexpansive asymptotically mapping with a nonempty fixed point set, weak and strong convergence theorems for the iterative approximation of fixed points of T are proved.

Furthermore, the results by this paper show that the boundedness requirement imposed on the subset K in recent results by Z. Huang [Comput. Math. Appl. 37, No. 3, 1-7 (1999; Zbl 0942.47046)]; B. E. Rhoades [J. Mat. Anal. Appl. 183, No. 1, 118-120 (1994; Zbl 0807.47045)]; J. Schu [J. Math. Anal. Appl. 158, No. 2, 407-413 (1991; Zbl 0734.47036); Bull. Aust. Math. Soc. 43, No. 1, 153-159 (1991; Zbl 0709.47051)], can be dropped. The main results are the following:

Theorem 1: Let E be a uniformly convex Banach space sastisfying Opial’s condition and let K be a nonempty closed convex subset of E. Let T:KK be an asymptotically nonexpansive mapping with F(T) and sequence {k n }[1,) such that limk n =1 and n=1 (k n -1)<. Let {u n } and {v n } be bounded sequences in K and let {a n }, {b n }, {c n }, {a n ' }, {b n ' } and {c n ' } be real sequence in [0,1] satisfying the conditions:

(i) a n +b n +c n =a n ' +b n ' +c n ' =1, n1;

(ii) a<a n <b n ' <b<1, n1;

(iii) limb n =0;

(iv) n=1 e n <, n=1 c n ' <.

Then the sequence generated from an arbitrary x 1 K by y n =a n x n +b n T n x n +c n u n , n1, x n+1 =a n ' x n +b n ' T n y n +c n ' v n , n1 converges weakly to some fixed point of T.

Theorem 2. Let E be a uniformly convex Banach space and K a nonempty closed subset of E. Let T:KK be an asymptotically nonexpansive mapping with F(T) and sequence {k n }[1,) such that limk n =1 and n=1 (k n -1)<. Suppose T n is compact for some m. Let {u n } and {v n } be bounded sequence in K and let {a n }, {b n }, {c n }, {a n ' }, {b n ' } and {c n ' } be as in Theorem 1. Then the sequence {x n } generate from an arbitrary x 1 K as in Theorem 1 converges strongly to some fixed point of T.

Reviewer: V.Popa (Bacau)

MSC:
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47J25Iterative procedures (nonlinear operator equations)
46B20Geometry and structure of normed linear spaces