Weak and strong convergence theorems for a finite family of I-asymptotically nonexpansive mappings
Introduction
We assume that X is a real normed space and K is a nonempty subset of X. A mapping is said to be nonexpansive if holds for all . A mapping is said to be asymptotically nonexpansive [1] if there exists a sequence such thatfor all and . A mapping is said to be uniformly Lipschitz with a Lipschitzian constant if holds for all and . If denotes the set of fixed points of a mapping T, and there exists a sequence such that for all , the following inequality holds:then T is said to be asymptotically quasi-nonexpansive. The class of asymptotically nonexpansive mappings is a natural generalization of the important class of nonexpansive mappings. Every asymptotically nonexpansive mapping with a fixed point is asymptotically quasi-nonexpansive, but the converse may be not true. Goebel and Kirk [1] proved that if K is a nonempty closed and bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive self-mapping has a fixed point.
Let . A mapping is said to be I-Lipschitz [2] if there exists such thatfor all . If (respectively ), then T is said to be I-contraction (respectively I-nonexpansive) [3] on K.
T is said to be I-asymptotically nonexpansive [2], [3] if there exists a sequence with such thatfor all and .
T is said to be I-asymptotically quasi-nonexpansive [2] if there exists a sequence with such thatfor all and and .
A subset K of X is said to be a retract of X if there exists a continuous map such that , for all . Every closed convex subset of a uniformly convex Banach space is a retraction. A mapping is said to be a retraction if . It follows that if a map P is a retraction, then for all y in the range of P.
For nonself nonexpansive mappings, some authors (see, e.g., [5], [6], [7], [8], [9] and the references therein) have studied the strong and weak convergence theorems in Hilbert space or uniformly convex Banach spaces. However, iterative algorithms for approximating fixed points of nonself asymptotically nonexpansive mappings have not been paid too much attention. The concept of nonself asymptotically nonexpansive mappings was introduced by Chidume et al. [10] in 2003 as the generalization of asymptotically nonexpansive self-mappings. The nonself asymptotically nonexpansive mapping is defined as follows: Definition 1.1 [10] Let K a nonempty subset of real normed linear space E. Let be the nonexpansive retraction of E onto K. A nonself-mapping is said to be asymptotically nonexpansive if there exists sequence such thatfor all and . T is said to be uniformly L-Lipschitz if there exists a constant such thatfor all and .
If T is self-mapping, then P becomes the identity mapping, so that (1.4) reduces to (1.1).
By studying the following iterative sequence:
Chidume et al. [10] established demi-closed principle, strong and weak convergence theorems for nonself asymptotically nonexpansive mapping in a uniformly convex Banach space. Recently concerning the convergence problem of an explicit iterative process to a common fixed point for some nonself asymptotically nonexpansive mappings in uniformly convex Banach spaces have been considered by several authors (see, for example, Wang [11], Yang [12], Thainwan [13] and the references therein).
The purpose in this paper first introduce the class of I-asymptotically nonexpansive nonself-maps. Then, an iteration scheme for approximating common fixed points of a finite family of -asymptotically nonexpansive nonself-mappings belonging to this class (when such common fixed points exist) is constructed; and strong and weak convergence theorems are proved. Our theorems improve and generalize important related results of the previously known results in this area.
Section snippets
Preliminaries
Let X be a Banach space with dimension . The modulus of X is the function defined by
A Banach space X is uniformly convex if and only if for all .
Let . The space X is said to be smooth ifexists for all . For any , we denote this limit by . The norm of X is said to be Fréchet differentiable if for all , the limit exists uniformly for all .
Now, we give
Main results
In this section, we shall state the weak and strong convergence of the iteration scheme (2.3) to a common fixed point for a finite family of -asymptotically nonexpansive nonself-mappings in a real uniformly convex Banach spaces. We first prove the following lemmas. Lemma 3.1 Let X be a real uniformly convex space, K be a nonempty closed convex subset of X which is also a nonexpansive retract with retraction P. Let be -asymptotically nonexpansive nonself-mappings with sequences
Acknowledgments
The second author is supported by the National Natural Science Foundation of China (Grant No. 70871028).
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