Journal of Applied Mathematics

Volume 2013 (2013), Article ID 689765, 5 pages

http://dx.doi.org/10.1155/2013/689765

Research Article

Convergence Theorems for Total Asymptotically Nonexpansive Mappings in Hyperbolic Spaces

Liang-cai Zhao,¹ Shih-sen Chang,² and Xiong Rui Wang¹

¹Institute of Mathematics, Yibin University, Yibin, Sichuan 644000, China

²College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China

Received 15 July 2013; Accepted 30 September 2013

Academic Editor: Zhenyu Huang

Copyright © 2013 Liang-cai Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The purpose of this paper is to introduce the concept of total asymptotically nonexpansive mappings and to prove some -convergence theorems of the iteration process for this kind of mappings in the setting of hyperbolic spaces. The results presented in the paper extend and improve some recent results announced in the current literature.

1. Introduction and Preliminaries

Most of the problems in various disciplines of science are nonlinear in nature whereas fixed point theory proposed in the setting of normed linear spaces or Banach spaces majorly depends on the linear structure of the underlying spaces. A nonlinear framework for fixed point theory is a metric space embedded with a “convex structure.” The class of hyperbolic spaces, nonlinear in nature, is a general abstract theoretic setting with rich geometrical structure for metric fixed point theory. The study of hyperbolic spaces has been largely motivated and dominated by questions about hyperbolic groups, one of the main objects of study in geometric group theory.

Throughout this paper, we work in the setting of hyperbolic spaces introduced by Kohlenbach [1], defined below, which is more restrictive than the hyperbolic type introduced in [2] and more general than the concept of hyperbolic space in  [3].

A hyperbolic space is a metric space together with a mapping satisfying

(i); (ii); (iii); (iv),

for all and . A nonempty subset of a hyperbolic space is convex if for all and . The class of hyperbolic spaces contains normed spaces and convex subsets thereof, the Hilbert ball equipped with the hyperbolic metric [4], Hadamard manifolds, and CAT(0) spaces in the sense of Gromov (see [5]).

A hyperbolic space is uniformly convex [6] if for any and there exists a , such that, for all , we have

provided , and .

A map , which provides such a for given and , is known as a modulus of uniform convexity of . We call monotone if it decreases with (for a fixed ), that is, for all , for all .

In the sequel, let be a metric space and let be a nonempty subset of . We will denote the fixed point set of a mapping by .

A mapping is said to be nonexpansive, if

A mapping is said to be asymptotically nonexpansive if there exists a sequence with such that

A mapping is said to be uniformly L-Lipschitzian if there exists a constant such that

Definition 1. A mapping is said to be -total asymptotically nonexpansive, if there exist nonnegative sequences with ,   and a strictly increasing continuous function with such that

Remark 2. From the definitions, it is clear that each nonexpansive mapping is an asymptotically nonexpansive mapping with a sequence and each asymptotically nonexpansive mapping is a -total asymptotically nonexpansive mapping with , and .

The existence of fixed points of various nonlinear mappings has relevant applications in many branches of nonlinear analysis and topology. On the other hand, there are certain situations where it is difficult to derive conditions for the existence of fixed points for certain types of nonlinear mappings. It is worth mentioning that fixed point theory for nonexpansive mappings, a limit case of a contraction mapping when the Lipschitz constant is allowed to be 1, requires tools far beyond from metric fixed point theory. Iteration schemas are the only main tool for analysis of generalized nonexpansive mappings. Fixed point theory has a computational flavor as one can define effective iteration schemas for the computation of fixed points of various nonlinear mappings. The problem of finding a common fixed point of some nonlinear mappings acting on a nonempty convex domain often arises in applied mathematics.

The purpose of this paper is to introduce the concept of total asymptotically nonexpansive mappings and to prove some -convergence theorems of the iteration process for the approximation of total asymptotically nonexpansive mappings in hyperbolic spaces. The results presented in the paper extend and improve some recent results given in [6–18].

In order to define the concept -convergence in the general setup of hyperbolic spaces, we first collect some basic concepts.

Let be a bounded sequence in a hyperbolic space . For , we define a continuous functional by

The asymptotic radius   of is given by

The asymptotic center of a bounded sequence with respect to is the set

This is the set of minimizers of the functional . If the asymptotic center is taken with respect to , then it is simply denoted by . It is known that uniformly convex Banach spaces and CAT(0) spaces enjoy the property that “bounded sequences have unique asymptotic centers with respect to closed convex subsets.” The following lemma is due to Leuştean [19] and ensures that this property also holds in a complete uniformly convex hyperbolic space.

Lemma 3 (see [19]). Let be a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity. Then, every bounded sequence in has a unique asymptotic center with respect to any nonempty closed convex subset of .

Recall that a sequence in is said to -converge to if is the unique asymptotic center of for every subsequence of . In this case, we write - and call the - of .

A mapping is semicompact if every bounded sequence , satisfying , has a convergent subsequence.

Lemma 4 (see [8]). Let , , and be sequences of nonnegative real numbers satisfying

If and , then the limit exists. If there exists a subsequence such that , then .

Lemma 5 (see [11]). Let be a uniformly convex hyperbolic space with monotone modulus of uniform convexity . Let and be a sequence in for some . If and are sequences in such that

for some , then .

Lemma 6 (see [11]). Let be a nonempty closed convex subset of uniformly convex hyperbolic space and a bounded sequence in such that and . If is another sequence in such that , then .

2. Main Results

Theorem 7. Let be a nonempty closed convex subset of a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity . Let , be uniformly -Lipschitzian and -total asymptotically nonexpansive mappings with sequences and satisfying , , and strictly increasing function with ,  . Assume that , and for arbitrarily chosen , is defined as follows:

where , , , , , and satisfy the following conditions:

(1); (2)there exist constants with such that and ; (3)there exist a constant such that .

Then, the sequence defined by (11) -converges to a common fixed point of .

Proof. The proof of Theorem 7 is divided into four steps.

Step 1. First, we prove that exists for each .

Set , , and . Since ,  , ,  . For any , by (11) we have

where

Substituting (13) into (12), we have

Applying Lemma 4 to the inequality, we get that exist for .

Step  2. We show that .

For each , from the proof of Step 1, we know that exists. We may assume that . The case is trivial. Next, we deal with the case . From (13), we have

Taking limsup on both sides in (15), we have

In addition, since

we have

Since , it is easy to prove that

It follows from Lemma 5 that

On the other hand, since

we have . Combined with (16), it yields that

This implies that

Since

we have

So, it follows from (25) and Lemma 5 that

Observe that

where

It follows from (26) that

Thus, from (20), (27), and (29), we have

In addition, since

from (20), we have

Finally, since

it follows from (30) and (32) that

Similarly, we also can show that

Step 3. Now we prove that the sequence -converges to a common fixed point of .

In fact, since, for each , exists, this implies that the sequence is bounded, so is the sequence . Hence, by virtue of Lemma 3, has a unique asymptotic center .

Let be any subsequence of with . It follows from (34) that

Now, we show that . For this, we define a sequence in by . So, we calculate

Since is uniformly -Lipschitzian, from (37) we have

Taking limsup on both sides of the previous estimate and using (36), we have

Since , by the definition of asymptotic center of a bounded sequence with respect to and (8), this implies that , for all . Therefore, as . It follows from Lemma 6 that . As is uniformly continuous, . That is, . Similarly, we also can show that . Hence, is the common fixed point of and . Reasoning as previously mentioned by utilizing the uniqueness of asymptotic centers, we get that . Since is an arbitrary subsequence of , for all subsequence of . This proves that -converges to a common fixed point of and . This completes the proof.

The following theorem can be obtained from Theorem 7 immediately.

Theorem 8. Let be a nonempty closed convex subset of a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity . Let ,  , be asymptotically nonexpansive mappings with sequence satisfying . Assume that ; for arbitrarily chosen , is defined as follows:

where , , and satisfy the following conditions:

 ( 1) ;  ( 2)there exist constants with such that and .

Then, the sequence defined in (40) -converges to a common fixed point of .

Proof. Take in Theorem 7. Since all conditions in Theorem 7 are satisfied, it follows from Theorem 7 that the sequence -converges to a common fixed point of .

This completes the proof of Theorem 8.

Acknowledgments

The authors would like to express their thanks to the editors and the referees for their helpful comments and suggestions. This work is supported by Scientific Research Fund of Sichuan Provincial Education Department (no. 11ZA222) and the Natural Science Foundation of Yibin University (no. 2012S07).

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