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Some results for asymptotically pseudo-contractive mappings and asymptotically nonexpansive mappings. (English) Zbl 0968.47017

Let E be a real Banach space, E * the topological dual space of E, ·,· the dual pair between E and E * , D(T), F(T) the domain of T and the set of all fixed points of T, respectively, and J:E2 E * the normalized duality mapping defined by J(x)={fE * :x,f=x,f,f=x}, xE. A mapping T:D(T)EE is said to be

(1) asymptotically nonexpansive if there exists a sequence {k n } in (0,) with lim n k n =1 such that T n x-T n yk n x-y for all x,yD(T) and n=1,2, ,

(2) asymptotically pseudo-contractive if there exists a sequence {k n } in (0,) with lim n k=1 and for any x,yD(T) there exists j(x-y)J(x-y) such that T n x-T n y,j(x-y)k n x-y 2 for all n=1,2, .

In this paper some convergence theorems of modified Ishikawa and Mann iterative sequence with errors for asymptotically pseudo-contractive and asymptotically nonexpansive mappings in Banach spaces are obtained.

Let T:D(T)EE be a mapping, let D(T) be a nonempty convex subset of E, let x 0 D(T) be a given point, and let α n , β n , γ n and δ n be four sequences in [0,1]. Then the sequence {x n } defined by x n+1 =(1-α n -γ n )x n +α n T n y n +γ n u n , y n =(1-β n -δ n )x n +β n T n x n +δ n v n for all n0 is called the modifies Ishikawa iterative sequence with errors of T, where u n and v n are two bounded sequences in D(T).

If β n =0 and δ n =0, n=0,1,2,, then y n =x n . The sequence x n+1 =(1-α n -γ n )x n +α n T n x n +γ n u n , n0, is called the modified Mann iterative sequences with errors of T. Main result is the following:

Theorem 1. Let E be a real uniformly smooth Banach space, let D be a non-empty bounded closed convex subset of E, let T:DD be an asymptotically pseudo-contractive mapping with a sequence {k n }(0,), lim n k n =1, and let F(T). Let α n , β n , γ n , and δ n be four sequences in [0,1] satisfying the following conditions:

(i) α n +γ n 1, β n +δ n 1;

(ii) α n 0, β n 0, δ n 0 (n);

(iii) n=0 α n =, n=0 γ n <.

Let x 0 D be any given point and let {x n }, {y n } be the modified Ishikawa iterative sequence with errors. Then:

(1) If {x n } converges strongly to a fixed point q of T in D, there exists a nondecreasing function :[0,)[0,), (0)=0 such that T n y n -q,J(y n -q)k n y n -q 2 -(y n -q), for all n0.

(2) Conversely, if there exists a strictly increasing function :[0,)[0,), (0)=0 satisfying preceding inequality, then x n qF(T).

A similar result for asymptotically nonexpansive mappings is proved. If {x n } is the modified Mann iterative sequence with errors, a similar result for asymptotically pseudo-contractive mappings is obtained.

This paper generalizes the results by [K. Goebel and W. A. Kirk, Proc. Am. Math. Soc. 35, No. 1, 171-174 (1972; Zbl 0256.47045), W. A. Kirk, Am. Math. Mont. 72, 1004-1006 (1965; Zbl 0141.32402), Q. H. Liu, Nonlinear Anal. Theory Methods Appl. 26, No. 11, 1835-1842 (1996; Zbl 0861.47047), J. Schu, J. Math. Anal. Appl. 15, No. 2, 407-413 (1991; Zbl 0734.47036)].

Reviewer: V.Popa (Bacau)

MSC:
47H05Monotone operators (with respect to duality) and generalizations
47J25Iterative procedures (nonlinear operator equations)
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties