Journal of Applied Mathematics

Volume 2015, Article ID 510798, 7 pages

http://dx.doi.org/10.1155/2015/510798

## Fixed Point Approximation for Asymptotically Nonexpansive Type Mappings in Uniformly Convex Hyperbolic Spaces

^{1}Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea^{2}Department of Applied Mathematics, Shri Shankaracharya Group of Institutions, Junwani, Bhilai 490020, India^{3}Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea

Received 8 July 2014; Revised 15 September 2014; Accepted 17 September 2014

Academic Editor: Wei-Shih Du

Copyright © 2015 Shin Min Kang 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

We use a modified *S*-iterative process to prove some strong and -convergence results for asymptotically nonexpansive type mappings in uniformly convex hyperbolic spaces, which includes Banach spaces and CAT(0) spaces. Thus, our results can be viewed as extension and generalization of several known results in Banach spaces and CAT(0) spaces (see, e.g., Abbas et al. (2012), Abbas et al. (2013), Bruck et al. (1993), and Xin and Cui (2011)) and improve many results in the literature.

#### 1. Introduction

Let be a nonempty subset of a metric space and let be a mapping. Then, is called(i)*nonexpansive* if , for all ,(ii)*asymptotically nonexpansive* [1] if, for each , there exists a constant with such that
for all ,(iii)*nearly Lipschitzian *with respect to a fixed sequence , introduced by Sahu [2], if, for each , there exists a constant such that
for all , where for each and . The infimum of constants satisfying (2) is called the nearly Lipschitz constant of and is denoted by ,(iv)*asymptotically nonexpansive in the intermediate sense* [3] provided that is uniformly continuous and
(v)* a mapping of asymptotically nonexpansive types* [4] if
for all .

In [3], Bruck et al. introduced the class of asymptotically nonexpansive mappings in the intermediate sense which is essentially wider than that of asymptotically nonexpansive ones [1]. It is known that [4] that if is a nonempty closed convex bounded subset of and is asymptotically nonexpansive in the intermediate sense, then has a fixed point. Since then, many authors have studied the existence and convergence theorems of fixed points for these two classes of mappings in Banach spaces, for example, Kaczor et al. [5], Xu [6], and references in their.

On the other hand, if , then (3) reduces to relation for all and .

*Remark 1. *The class of nearly asymptotically nonexpansive mappings are intermediate classes between the class of asymptotically nonexpansive mappings that of asymptotically nonexpansive in the intermediate sense mappings.

Throughout in this paper, we have worked in the setting of hyperbolic spaces introduced by Kohlenbach [7]. It is noted that they are different from Gromov hyperbolic spaces [8] or from other notions of hyperbolic spaces that can be found in literature (see, e.g., [9–11]).

A* hyperbolic space * is a metric space together with a convexity mapping satisfying,,,, for all and .

A metric space is said to be a* convex metric space* in the sense of Takahashi [12], where a triple satisfies only . We get the notion of the space of hyperbolic type in the sense of Goebel and Kirk [13], where a triple satisfies . The was already considered by Itoh [14] under the name of “condition III” and it is used by Reich and Shafrir [11] and Kirk [10] to define their notions of hyperbolic spaces.

The class of hyperbolic spaces includes normed space and convex subsets thereof, the Hilbert space ball equipped with the hyperbolic metric [9] and Hadrmard manifold as well as the CAT(0) spaces in the sense of Gromov (see [8]).

If and , then we use the notation for . The following holds even for the more general setting of convex metric space [12]: for all and ,

A hyperbolic space is* uniformly convex* [15] if, for any and , there exists such that, for all ,
Provided that , , and .

A mapping , providing such a for given and , is called a* modulus of uniform convexity.* We call * monotone* if it decreases with (for fixed ).

In 1976, Lim [16] introduced a concept of convergence in a general metric space setting which he called “-convergence.” In 2008, Kirk and Panyanak [17] specialized Lim’s concept to CAT(0) spaces and showed that many Banach space results involving weak convergence have precise analogs in this setting.

In [15], Leustean proved that CAT(0) spaces are uniformly convex hyperbolic spaces with modulus of uniform convexity quadratic in . Thus, the classes of uniformly convex hyperbolic spaces are a natural generalization of both uniformly convex Banach spaces and CAT(0) spaces.

In the view of the above facts, many researchers have paid attention to the direction of existence and approximation of fixed points via different iterative schemes for nonexpansive, asymptotically nonexpansive, asymptotically nonexpansive type mappings, and total asymptotically nonexpansive mappings in the surrounding work of uniformly convex hyperbolic spaces (see, e.g., [10, 18–26]).

The -iteration process was introduced by Agarwal et al. [27] and it has proved that the rate of convergence of -iteration process is faster than that of Picard iteration process and Picard iteration process is faster than Mann iteration process for contraction mapping (see, e.g., [28], page 307).

The purpose of the paper is to establish a -convergence and strong convergence theorem for a modified -iteration process for asymptotically nonexpansive type mappings in uniformly convex hyperbolic spaces. Our results extend and improve the corresponding ones announced by [3, 18, 19, 26] in the sense of a modified -iteration process.

#### 2. Preliminaries

Let be a nonempty subset of metric space and let be any bounded sequence in . Consider a continuous functional defined by
Then, consider the following:(a)the infimum of over is said to be the* asymptotic radius* of with respect to and is denoted by ;(b)a point is said to be an* asymptotic center* of the sequence with respect to if
the set of all asymptotic centers of with respect to is denoted by ;(c)this set may be empty, a singleton, or certain infinitely many points;(d)if the asymptotic radius and the asymptotic center are taken with respect to , then these are simply denoted by and , respectively;(e)for .

It is known that uniformly convex Banach spaces and even 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 Leustean [29] and ensures that this property also holds in a complete uniformly convex hyperbolic space.

Lemma 2 (see [29, Proposition 3.3]). *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 -limit of .

Lemma 3 (see [22]). *Let be a nonempty closed convex subset of uniformly convex hyperbolic space and let be a bounded sequence in such that . If is any other sequence in such that , then .*

Lemma 4 (see [22]). *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 5 (see [30]). *Let and be two nonnegative real sequences such that
**
If , then exists.*

Let be a nonempty convex subset of uniformly convex subset of hyperbolic space and let be a mapping with the set of fixed points and let be sequence in ; we say that has()limited existence property for , existing for all ;()approximate fixed point property for , if .

Proposition 6. *Let be a complete uniformly convex hyperbolic space. Let be a nonempty, closed, convex subset of and let be asymptotically nonexpansive in the intermediate sense (provided that is uniformly continuous). Put
**
If and if is a bounded sequence in such that holds (i.e., approximate fixed property), then has a fixed point.*

*Proof. *Let be a nonempty, closed, and convex subset of a uniformly convex hyperbolic space and let be a bounded sequence in ; therefore, by Lemma 2, consists exactly of one point (say). We now show that is a fixed point of . Since has an approximate sequence for , therefore, by the uniform continuity of , it implies that
We define a sequence in by , . For integers , we have
Taking limit as superior as on both sides, using (13) and (14), we have
Hence,
This implies that
It follows from Lemma 3 that as . Since is closed, therefore, . By continuity of , we have
that is, has fixed point.

#### 3. -Convergence and Strong Convergence Theorems in Hyperbolic Space

Now we establish -convergence and strong convergence theorems for a modified -iteration process in uniformly convex hyperbolic spaces.

In [27], Agarwal et al. introduced a modified -iteration process in the setting of a Banach space. Now, we define a modified -iteration process in the notion of a uniformly convex hyperbolic space as follows.

Let be a nonempty closed convex subset of a uniformly convex hyperbolic space and let be total asymptotically nonexpansive mappings. Then, for arbitrarily chosen , we construct a sequence in such that
where and are sequences in which is called a* modified **-iteration process.*

Lemma 7. *Let be a nonempty closed convex subset of a uniformly convex hyperbolic space . Let be uniformly continuous asymptotically nonexpansive in the intermediate sense. Put
**
If , let be the modified -iteration process in defined by (19) having limited existence property for , where and are real sequences in such that .*

*Proof. *Let . From (19), we have
from (21) and (22), we have
It follows that
for some . Hence, by Lemma 5, we observe that exists for each . Hence, sequence has limited existence property for map . This completes the proof.

Lemma 8. *Let be a nonempty closed and convex subset of a uniformly convex hyperbolic space with monotone modulus of uniform convexity and let be uniformly continuous asymptotically nonexpansive in the intermediate sense. Put
**
If , let be the modified -iteration process in defined by (19) having approximate fixed point property, where and are real sequences in such that .*

*Proof. *It follows from Lemma 7 that exists and is bounded, so, without loss of generality, we can assume that for some . If , then we immediately obtain
and, hence, by uniform continuity of , we have . If , then, from definition of , we get
for all . Taking limit as superior as on both sides, we get
From (22), we have
for all . Taking limit as superior as on both sides, we get
Hence, from (30), we have
Since
using Lemma 4, (28), and (31), it follows that
From (19) and (33), we get
Hence,
Now, we observe that
which gives from (36) that
The estimates of (30) and (37) imply that
It follows from Lemma 4 that we have
From (33) and (39), we get
Thus, we have
By (34), (40), and the uniform continuity of , as implies that as ; we conclude that
It shows that sequence has an approximate fixed point property for map (i.e., holds).

Theorem 9. *Let be a nonempty closed, convex subset of a complete uniformly convex hyperbolic space with monotone modulus of uniform convexity and let be uniformly continuous asymptotically nonexpansive in the intermediate sense. Put
**
If . Let be the modified -iteration process in defined by (19), where and are real sequences in such that . Then, is -convergent to an element of .*

*Proof. *It follows from Lemma 7 that is bounded. Therefore, by Lemma 2, one has unique asymptotic center that . Let be any subsequence of such that and, hence, by Lemma 8, we have . Hence, by Proposition 6, we have, .

Next, we claim that is the unique asymptotic center for each subsequence of . Assume contrarily that . Since exists by Lemma 7, therefore, by the uniqueness of asymptotic centers, we have
a contradiction and, hence, . Since is an arbitrary subsequence of , therefore, for all subsequences of of . This proves that -converges to a fixed point of .

Theorem 10. *Let , , , and be defined as in Theorem 9. Then, converges strongly to a fixed point of if and only if .*

*Proof. *Necessity is obvious. We only prove the sufficiency.

Suppose that , from (24), we have
for some . It follows, from Lemma 5, that exists. It follows that . Next, we show that is a Cauchy sequence. Now, we can choose a subsequence of such that
for all integer and some on . Again, from (24), applying Lemma 5, we have
and, hence,
which shows that is a Cauchy sequence in closed subset of a complete uniformly convex hyperbolic space and so it must converge strongly to a point in . It is readily seen that and converge strongly to .

Recall that a mapping from a subset of a metric space into itself with is said to satisfy* condition * (see [31]) if there exists a nondecreasing function with and for such that
for all .

Theorem 11. *Let , , , and be defined as in Theorem 9. Suppose that satisfies condition . Then, converges strongly to a fixed point of .*

*Proof. *Note that sequence has approximate fixed point property for ; that is, . Further, by condition ,
It follows that . Therefore, the result follows from Theorem 10.

*Remark 12. *In the view of Remark 1, Theorems 9, 10, and 11 generalize and extend the results of [3, 18, 19, 26] in the sense a modified -iteration process which is faster than other iteration processes (see, e.g., Mann and Ishikawa) in the setting of unifromly convex hyperbolic spaces.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors would like to thank the referees for useful comments and suggestions. This study was supported by research funds from Dong-A University.

#### References

- K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,”
*Proceedings of the American Mathematical Society*, vol. 35, pp. 171–174, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. R. Sahu, “Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces,”
*Commentationes Mathematicae Universitatis Carolinae*, vol. 46, no. 4, pp. 653–666, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Bruck, T. Kuczumow, and S. Reich, “Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform opial property,”
*Colloquium Mathematicum*, vol. 65, no. 2, pp. 169–179, 1993. View at Google Scholar · View at MathSciNet - W. A. Kirk, “Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type,”
*Israel Journal of Mathematics*, vol. 17, pp. 339–346, 1974. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - W. Kaczor, T. Kuczumow, and S. Reich, “A mean ergodic theorem for mappings which are asymptotically nonexpansive in the intermediate sense,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 47, no. 4, pp. 2731–2742, 2001. View at Publisher · View at Google Scholar · View at Scopus - H.-K. Xu, “Existence and convergence for fixed points of mappings of asymptotically nonexpansive type,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 16, no. 12, pp. 1139–1146, 1991. View at Publisher · View at Google Scholar · View at Scopus - U. Kohlenbach, “Some logical metatheorems with applications in functional analysis,”
*Transactions of the American Mathematical Society*, vol. 357, no. 1, pp. 89–128, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. R. Bridson and A. Haefliger,
*Metric Spaces of Non-Positive Curvature*, Springer, 1999. View at Publisher · View at Google Scholar · View at MathSciNet - K. Goebel and S. Reich,
*Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings*, vol. 83 of*Monographs and Textbooks in Pure and Applied Mathematics*, Marcel Dekker, New York, NY, USA, 1984. View at MathSciNet - W. A. Kirk, “Krasnosel'sk's iteration process in hyperbolic space,”
*Numerical Functional Analysis and Optimization*, vol. 4, no. 4, pp. 371–381, 1981/82. View at Publisher · View at Google Scholar · View at MathSciNet - S. Reich and I. Shafrir, “Nonexpansive iterations in hyperbolic spaces,”
*Nonlinear Analysis*, vol. 15, no. 6, pp. 537–558, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - W. A. Takahashi, “A convexity in metric space and nonexpansive mappings. I,”
*Kodai Mathematical Seminar Reports*, vol. 22, no. 2, pp. 142–149, 1970. View at Publisher · View at Google Scholar · View at MathSciNet - K. Goebel and W. A. Kirk, “Iteration processes for nonexpansive mappings,” in
*Topological Methods in Nonlinear Functional Analysis (Toronto, 1982)*, S. P. Singh, S. Thomeier, and B. Watson, Eds., vol. 21 of*Contemporary Mathematics*, pp. 115–123, American Mathematical Society, Providence, RI, USA, 1983. View at Google Scholar - S. Itoh, “Some fixed-point theorems in metric spaces,”
*Fundamenta Mathematicae*, vol. 102, no. 2, pp. 109–117, 1979. View at Google Scholar · View at MathSciNet - L. Leustean, “A quadratic rate of asymptotic regularity for CAT(0)-spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 325, no. 1, pp. 386–399, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - T. C. Lim, “Remarks on some fixed point theorems,”
*Proceedings of the American Mathematical Society*, vol. 60, pp. 179–182, 1976. View at Publisher · View at Google Scholar · View at MathSciNet - W. A. Kirk and B. Panyanak, “A concept of convergence in geodesic spaces,”
*Nonlinear Analysis: Theory, Methods and Applications*, vol. 68, no. 12, pp. 3689–3696, 2008. View at Publisher · View at Google Scholar · View at Scopus - M. Abbas, Z. Kadelburg, and D. R. Sahu, “Fixed point theorems for Lipschitzian type mappings in CAT(0) spaces,”
*Mathematical and Computer Modelling*, vol. 55, no. 3-4, pp. 1418–1427, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Abbas, B. S. Thakur, and D. Thakur, “Fixed points of asymptotically nonexpansive mappings in the intermediate sense in CAT(0) space,”
*Communications of the Korean Mathematical Society*, vol. 28, no. 1, pp. 107–121, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - H. Fukhar-ud-din and M. A. A. Khan, “Convergence analysis of a general iteration schema of nonlinear mappings in hyperbolic spaces,”
*Fixed Point Theory and Applications*, vol. 2013, article 238, 2013. View at Publisher · View at Google Scholar - H. Fukhar-Ud-Din and A. Kalsoom, “Fixed point approximation of asymptotically nonexpansive mappings in hyperbolic spaces,”
*Fixed Point Theory and Applications*, vol. 2014, article 64, 2014. View at Publisher · View at Google Scholar · View at Scopus - A. R. Khan, H. Fukhar-ud-din, and M. A. A. Khan, “An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces,”
*Fixed Point Theory and Applications*, vol. 2012, article 54, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. R. Khan, H. Fukhar-ud-din, A. Kalsoom, and B. S. Lee, “Convergence of a general algorithm of asymptotically nonexpansive maps in uniformly convex hyperbolic spaces,”
*Applied Mathematics and Computation*, vol. 238, pp. 547–556, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - L. L. Wan, “Δ-convergence for mixed-type total asymptotically nonexpansive mappings in hyperbolic spaces,”
*Journal of Inequalities and Applications*, vol. 2013, no. 553, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - L.-C. Zhao, S.-S. Chang, and J. K. Kim, “Mixed type iteration for total asymptotically nonexpansive mappings in hyperbolic spaces,”
*Fixed Point Theory and Applications*, vol. 2013, article 353, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - J. Xin and Y. Cui, “Existence and convergence of fixed points for mappings of asymptotically nonexpansive type in uniformly convex W-hyperbolic spaces,”
*Fixed Point Theory and Applications*, vol. 2011, article 39, 12 pages, 2011. View at Publisher · View at Google Scholar - R. P. Agarwal, D. O'Regan, and D. R. Sahu, “Iterative construction of fixed points of nearly asymptotically nonexpansive mappings,”
*Journal of Nonlinear and Convex Analysis*, vol. 8, no. 1, pp. 61–79, 2007. View at Google Scholar · View at MathSciNet - R. P. Agarwal, D. O'Regan, and D. R. Sahu,
*Fixed Point Theory for Lipschitizian-Type Mappings with Applications*, vol. 6 of*Topological Fixed Point Theory and Its Applications*, Springer, New York, NY, USA, 2009. View at Publisher · View at Google Scholar - L. Leustean, “Nonexpansive iteration in uniformly convex $W$-hyperbolic spaces,” in
*Nonlinear Analysis and Optimization I: Nonlinear Analysis*, A. Leizarowitz, B. S. Mordukhovich, I. Shafrir, and A. Zaslavski, Eds., vol. 513, pp. 193–210, American Mathematical Society, Providence, RI, USA, 2010. View at Google Scholar - K. K. Tan and H. K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process,”
*Journal of Mathematical Analysis and Applications*, vol. 178, no. 2, pp. 301–308, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - H. F. Senter and J. Dotson, “Approximating fixed points of nonexpansive mappings,”
*Proceedings of the American Mathematical Society*, vol. 44, pp. 375–380, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet