Demiclosed principle and convergence theorems for total asymptoticallynonexpansive nonself mappings in hyperbolic spaces
© Wan; licensee Springer. 2015
Received: 18 June 2014
Accepted: 26 November 2014
Published: 16 January 2015
In this paper, we prove the demiclosed principle for total asymptoticallynonexpansive nonself mappings in hyperbolic spaces. Then we obtain convergencetheorems of the mixed Agarwal-O’Regan-Sahu type iteration for totalasymptotically nonexpansive nonself mappings. Our results extend some results inthe literature.
MSC: 47H09, 49M05.
One of the fundamental and celebrated results in the theory of nonexpansive mappingsis Browder’s demiclosed principle which states that if X is a uniformly convex Banach space,C is a nonempty closed convex subset of X, and if is a nonexpansive nonself mapping, then is demiclosed at 0, that is, for any sequence in C if weakly and , then (where I is the identity mapping inX). Later, Chidume et al. proved the demiclosed principle for asymptotically nonexpansive nonselfmappings in uniformly convex Banach spaces. Recently, Chang et al. proved the demiclosed principle for total asymptotically nonexpansivenonself mappings in CAT(0) spaces. It is well known that the demiclosed principleplays an important role in studying the asymptotic behavior for nonexpansivemappings. The purpose of this paper is to extend Chang’s result from CAT(0)spaces to the general setup of uniformly convex hyperbolic spaces. We also apply ourresult to approximate common fixed points of total asymptotically nonexpansivenonself mappings in hyperbolic spaces, using the mixed Agarwal-O’Regan-Sahutype iterative scheme . Our results extend and improve the corresponding results of Chang etal., Nanjaras and Panyanak , Chang et al., Zhao et al., Khan et al. and many other recent results.
If a space satisfies only (I), it coincides with the convex metric space introducedby Takahashi . The concept of hyperbolic spaces in  is more restrictive than the hyperbolic type introduced by Goebel andKirk  since (I)-(III) together are equivalent to being a space of hyperbolic type in . But it is slightly more general than the hyperbolic space defined inReich and Shafrir  (see ). This class of metric spaces in  covers all normed linear spaces, ℝ-trees in the sense of Tits, theHilbert ball with the hyperbolic metric (see ), Cartesian products of Hilbert balls, Hadamard manifolds (see [12, 14]), and CAT(0) spaces in the sense of Gromov (see ). A thorough discussion of hyperbolic spaces and a detailed treatment ofexamples can be found in  (see also [11–13]).
where P is a nonexpansive retraction of X onto C. It iswell known that each nonexpansive mapping is an asymptotically nonexpansive mappingand each asymptotically nonexpansive mapping is a -total asymptotically nonexpansive mapping.
Let be a complete uniformly convex hyperbolic space with monotone modulus of uniformconvexity andCa nonempty closed convex subset ofX. Then every bounded sequence inXhas a unique asymptotic center with respect toC.
3 Main results
Theorem 1 (Demiclosed principle for total asymptotically nonexpansive nonselfmappings in hyperbolic spaces)
Let be a complete uniformly convex hyperbolic space with monotone modulus of uniformconvexityη. LetCbe a nonempty closed and convex subset ofX. Let be a uniformlyL-Lipschitzian and -total asymptotically nonexpansive nonselfmapping. Pis a nonexpansive retraction ofXontoC. Let be a bounded approximate fixed point sequence, i.e., and . Then we have .
Proof We divide our proof into three steps.
Consequently, a combination of (10), (11), and Lemma 3 shows that (7) isproved.
Therefore, (12) holds.
Step 3. Now we are in a position to prove the △-convergence of . Since is bounded, by Lemma 1, it has a uniqueasymptotic center . Let be any subsequence of with . Since , it follow from Theorem 1 that . By the uniqueness of asymptotic centers, we get . It implies that is the unique asymptotic center of for each subsequence of , that is, △-converges to . The proof is completed. □
Therefore, the conditions of Theorem 2 are fulfilled.
It is proved in [, Example 1] that both and are total asymptotically nonexpansive mappings with , , . Moreover, they are uniformly L-Lipschitzianmappings with . and . Let , be the same as in (22). Therefore, the conditions ofTheorem 2 are fulfilled.
then the sequence defined by (6) converges strongly(i.e., in the metric topology) to a common fixedpoint in ℱ.
Thus is a Cauchy sequence in C. C iscomplete for it is a closed subset in a complete hyperbolic space. Without loss ofgenerality, we can assume that converges strongly to some point . It is easy to prove that ℱ is closed. Itfollows from (24) that . The proof is completed. □
Supported by General Project of Educational Department in Sichuan (No. 13ZB0182)and National Natural Science Foundation of China (No. 11426190).
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