AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 251705 10.1155/2013/251705 251705 Research Article -Convergence Problems for Asymptotically Nonexpansive Mappings in CAT(0) Spaces Shi Luo Yi 1 Chen Ru Dong 1 Wu Yu Jing 2 Song Yisheng 1 Department of Mathematics Tianjin Polytechnic University Tianjin 300387 China tjpu.edu.cn 2 Tianjin Vocational Institute Tianjin 300410 China tjtc.edu.cn 2013 20 3 2013 2013 23 01 2013 17 02 2013 2013 Copyright © 2013 Luo Yi Shi 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.

New △-convergence theorems of iterative sequences for asymptotically nonexpansive mappings in CAT(0) spaces are obtained. Consider an asymptotically nonexpansive self-mapping T of a closed convex subset C of a CAT(0) space X. Consider the iteration process {xn}, where x0C is arbitrary and xn+1=αnxn(1-αn)Tnyn or xn+1=αnTnxn(1-αn)yn,yn=βnxn(1-βn)Tnxn for n1, where {αn},{βn}(0,1). It is shown that under certain appropriate conditions on αn,βn,{xn}  △-converges to a fixed point of T.

1. Introduction and Preliminaries

Let C be a nonempty subset of a metric space (X,d). A mapping T:CC is a contraction if there exists k[0,1) such that for all x,yC, we have d(Tx,Ty)<kd(x,y). It is said to be nonexpansive if for all x,yC, we have d(Tx,Ty)d(x,y). T is said to be asymptotically nonexpansive if there exists a sequence {kn}[1,) with kn1 such that d(Tnx,Tny)knd(x,y) for all integers n1 and all x,yC. Clearly, every contraction mapping is nonexpansive and every nonexpansive mapping is asymptotically nonexpansive with sequence kn=1, for all n1. There are, however, asymptotically nonexpansive mappings which are not nonexpansive (see, e.g., ). As a generalization of the class of nonexpansive mappings, the class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk  in 1972 and has been studied by several authors (see, e.g., ). Goebel and Kirk proved that if C is a nonempty closed convex and bounded subset of a uniformly convex Banach space (more general than a Hilbert space, i.e., CAT(0) space), then every asymptotically nonexpansive self-mapping of C has a fixed point. The weak and strong convergence problems to fixed points of nonexpansive and asymptotically nonexpansive mappings have been studied by many authors.

We will denote by F(T) the set of fixed points of T. In 1967, Halpern  introduced an explicit iterative scheme for a nonexpansive mapping T on a subset C of a Hilbert space by taking any point u,x1C and defined the iterative sequence {xn} by (1)xn+1=αnu+(1-αn)Txn,for  n1, where αn[0,1]. He pointed out that under certain appropriate conditions on αn,{xn} converges strongly to a fixed point of T. In 1994, Tan and Xu  introduced the following iterative scheme for asymptotically nonexpansive mapping on uniformly convex Banach space: (2)x0C,xn+1=αnf(xn)+(1-αn)Tnyn,n0,yn=γnxn+(1-γn)Tnxn,n0, where {αn},{γn}(0,1). They proved that under certain appropriate conditions on αn,γn,{xn} converges weakly to a fixed point of T.

In 2012, we  studied the viscosity approximation methods for nonexpansive mappings on CAT(0) space. For a contraction f on C, consider the iteration process {xn}, where x0C is arbitrary and (3)xn+1=αnf(xn)(1-αn)Txn, for n1, where {αn}(0,1). We proved that under certain appropriate conditions on αn,{xn} converges strongly to a fixed point of T which solves some variational inequality.

The purpose of this paper is to study the iterative scheme defined as follows: consider an asymptotically nonexpansive self-mapping T of a closed convex subset C of a CAT(0) space X with coefficient kn. consider the iteration process {xn}, where x0C is arbitrary and (4)xn+1=αnxn(1-αn)Tnyn,yn=βnxn(1-βn)Tnxn, or (5)xn+1=αnTnxn(1-αn)yn,yn=βnxn(1-βn)Tnxn, for n1, where {αn},{βn}(0,1). We show that {xn}  -converges to a fixed point of T under certain appropriate conditions on αn,βn, and kn.

We now collect some elementary facts about CAT(0) spaces which will be used in the proofs of our main results.

Lemma 1.

Let X be a CAT(0) space. Then, one has the following:

(see [9, Lemma 2.4])  for each x,y,zX and t[0,1], one has (6)d((1-t)xty,z)(1-t)d(x,z)+td(y,z),

(see ) for each x,y,zX and t,s[0,1] one has (7)d((1-t)xty,(1-s)xsy)|t-s|d(x,y),

(see [5, Lemma 3]) for each x,y,zX and t[0,1], one has (8)d((1-t)ztx,(1-t)zty)td(x,y),

(see ) for each x,y,zX and t[0,1], one has (9)d2((1-t)xty,z)td2(x,z)+(1-t)d2(y,z)-t(1-t)d2(x,y).

Let X be a complete CAT(0) space and let {xn} be a bounded sequence in a complete X and for xX set (10)r(x,{xn})=limsup  nd(x,xn).

The asymptotic radius r({xn}) of {xn} is given by (11)r({xn})=inf{r(x,{xn}):xX}, and the asymptotic center A({xn}) of {xn} is the set (12)A({xn})={xX:r(x,{xn})=r({xn})}.

It is known (see, e.g., [11, Proposition 7]) that in a CAT(0) space, A({xn}) consists of exactly one point.

A sequence {xn} in X is said to -converge to xX if x is the unique asymptotic center of {un} for every subsequence {un} of {xn}. In this case, we write -limnxn=x and call x the -limit of {xn}.

Lemma 2.

Assume that X is a CAT(0) space. Then, one has the following:

(see ) every bounded sequence in X has a -convergent subsequence;

(see ) if K is a closed convex subset of X and T:KX is an asymptotically nonexpansive mapping, then the conditions {xn}  -converge to x and d(xn,T(xn))0, imply xK and xF(T).

Lemma 3 (see [<xref ref-type="bibr" rid="B3">14</xref>, <xref ref-type="bibr" rid="B16">15</xref>]).

Let {an},{bn},and  {cn} be three nonnegative real sequences satisfying the following condition: (13)an+1(1+bn)an+cn,nn0, where n0 is some nonnegative integer, n=1bn<, n=1cn<. Then the limit limnan exists.

2. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M125"><mml:mrow><mml:mo>△</mml:mo></mml:mrow></mml:math></inline-formula>-Convergence of the Iteration Sequences

In this section, we will study the -convergence of the iteration sequence for asymptotically nonexpansive mappings in CAT(0) spaces.

Suppose that X be a CAT(0) space, C a closed convex subset of X, and T:CC an asymptotically nonexpansive mapping with coefficient kn. Firstly, we consider the iteration process: (14)x0C,xn+1=αnxn(1-αn)Tnyn,n0,yn=βnxn(1-βn)Tnxn,n0, where {αn},{βn}(0,1) and kn satisfy the following.

There exist positive integers n0, n1, and δ>0, 0<b<min{1,1/L}, where L=supnkn, such that (15)0<δ<αn<1-δ,nn0,0<1-βn<b,nn1,

Consider n=1(kn2-1)<.

We will prove that {xn}  -converges to a fixed point of T.

Lemma 4.

Let X be a CAT(0) space, C a closed convex subset of X, T:CC an asymptotically nonexpansive mapping with coefficient kn, and n=1(kn2-1)<. If F(T), {αn},{βn}(0,1). Let x0C, {xn} be generated by xn+1=αnxn(1-αn)Tnyn, yn=βnxn(1-βn)Tnxn, n0. Then the limit limnd(xn,p) exists for all pF(T).

Proof.

Taking pF(T), we have (16)d(xn+1,p)=d(αnxn(1-αn)Tnyn,p)αnd(xn,p)+(1-αn)d(Tnyn,p)αnd(xn,p)+(1-αn)knd(yn,p)αnd(xn,p)+(1-αn)kn{βnd(xn,p)+(1-βn)d(Tnxn,p)}αnd(xn,p)+(1-αn)kn{βnd(xn,p)+(1-βn)knd(xn,p)}={1+(1-αn)(kn-1)×[kn(1-βn)+1]}d(xn,p){1+(kn2-1)}d(xn,p). By Lemma 3, we can get that limnd(xn,p) exists.

Remark 5.

The above lemma implies that {xn} is bounded and so is the sequence {Txn}. Moreover, let L=supnkn, then we have (17)d(Tnxn,p)knd(xn,p)Ld(xn,p),d(yn,p)βnd(xn,p)+(1-βn)d(Tnxn,p)Ld(xn,p)d(Tnyn,p)knd(yn,p)L2d(xn,p). It follows that the sequences {Tnxn}, {yn}, {Tnyn} are bounded.

Proposition 6.

Let X be a CAT(0) space, C a closed convex subset of X, and T:CC an asymptotically nonexpansive mapping with coefficient kn. If F(T), {αn},{βn}(0,1). Let x0C, {xn} be generated by xn+1=αnxn(1-αn)Tnyn, yn=βnxn(1-βn)Tnxn, n0. Then under the hypotheses (i) and (ii), one can get that limnd(xn,Tnyn)=0.

Proof.

By the assumption, F(T) is nonempty. Take pF(T), by Lemma 1(iv), we have (18)d2(xn+1,p)=d2(αnxn(1-αn)Tnyn,p)αnd2(xn,p)+(1-αn)d2(Tnyn,p)-αn(1-αn)d2(xn,Tnyn)d2(xn,p)+(1-αn){d2(Tnyn,p)-d2(yn,p)}+(1-αn){d2(yn,p)-d2(xn,p)}-αn(1-αn)d2(xn,Tnyn),d2(yn,p)-d2(xn,p)=d2(βnxn(1-βn)Tnxn,p)-d2(xn,p)βnd2(xn,p)+(1-βn)d2(Tnxn,p)-βn(1-βn)d2(xn,Tnxn)-d2(xn,p)βnd2(xn,p)+(1-βn)d2(Tnxn,p)-d2(xn,p), which implies that (19)d2(yn,p)-d2(xn,p)(1-βn)[d2(Tnxn,p)-d2(xn,p)](1-βn)(kn2-1)d2(xn,p).

Therefore, we have (20)d2(xn+1,p)d2(xn,p)+(1-αn)(kn2-1)d2(yn,p)+(1-αn)(1-βn)(kn2-1)d2(xn,p)-αn(1-αn)d2(xn,Tnyn).

Since {xn} and {yn} are bounded and 0<δ<αn<1-δ for all nn0. we have (21)δ2d2(xn,Tnyn)d2(xn,p)-d2(xn+1,p)+(1-αn)(kn2-1)d2(yn,p)+(1-αn)(1-βn)(kn2-1)d2(xn,p).

By the conditions (i) and (ii), we have (22)n=1δ2d2(xn,Tnyn)<, which implies that (23)limnd2(xn,Tnyn)=0.

Theorem 7.

Let X be a CAT(0) space, C a closed convex subset of X, and T:CC an asymptotically nonexpansive mapping with coefficient kn. If F(T), {αn},{βn}(0,1). Let x0C, {xn} be generated by xn+1=αnxn(1-αn)Tnyn,yn=βnxn(1-βn)Tnxn,n0. Then under the hypotheses (i) and (ii), one can get that {xn}  -converges to a fix point of T.

Proof.

We first show that limnd(xn,Tnxn)=0. Indeed (24)d(xn,yn)=d(xn,βnxn(1-βn)Tnxn)(1-βn)d(xn,Tnxn)(1-βn){d(xn,Tnyn)+d(Tnyn,Tnxn)}(1-βn){d(xn,Tnyn)+Ld(yn,xn)}; it follows that (25)[1-L(1-βn)]d(xn,yn)(1-βn)d(xn,Tnyn). By the conditions (i) and (ii) and Proposition 6, we get limnd(xn,yn)=0.

And then, (26)d(xn,Tnxn)d(xn,Tnyn)+d(Tnyn,Tnxn)d(xn,Tnyn)+Ld(yn,xn). By Proposition 6, we get that limnd(xn,Tnxn)=0.

We claim that limnd(xn,Txn)=0. Indeed we have (27)d(yn,Tnxn)=d(βnxn(1-βn)Tnxn,Tnxn)βnd(xn,Tnxn)0.d(xn+1,xn)=d(αnxn(1-αn)Tnyn,xn)(1-αn)d(xn,Tnyn)0.d(xn-1,Tn-1xn)d(xn-1,Tn-1xn-1)+d(Tn-1xn-1,Tn-1xn)d(xn-1,Tn-1xn-1)+Ld(xn-1,xn)0.d(xn,Tn-1xn)d((1-αn-1)Tn-1yn-1,Tn-1xnαn-1xn-1(1-αn-1)Tn-1yn-1,Tn-1xn)αn-1d(xn-1,Tn-1xn)+(1-αn-1)d(Tn-1yn-1,Tn-1xn)αn-1d(xn-1,Tn-1xn)+(1-αn-1)Ld(yn-1,xn)αn-1d(xn-1,Tn-1xn)+(1-αn-1)L[d(yn-1,xn-1)  +d(xn-1,xn)]0. Thus, (28)d(xn,Txn)d(xn,Tnxn)+d(Tnxn,Txn)d(xn,Tnxn)+Ld(Tn-1xn,xn)0.

Since {xn} is bounded, we may assume that {xn}  -converges to a point x^. By Lemma 2, we have x^F(T).

Next we will consider another iteration process: (29)x0C,xn+1=αnTnxn(1-αn)yn,n0,yn=βnxn(1-βn)Tnxn,n0, where {αn},{βn}(0,1), and kn satisfy the following

There exist positive integers n0 and δ>0, such that (30)0<δ<αn<1-δ,nn0;1-βn0;

n=1(kn-1)<.

We will prove that {xn} also -converges to a fixed point of T.

Lemma 8.

Let X be a CAT(0) space, C a closed convex subset of X, T:CC an asymptotically nonexpansive mapping with coefficient kn, and n=1(kn-1)<. If F(T), {αn},{βn}(0,1). Let x0C, {xn} be generated by xn+1=αnTnxn(1-αn)yn,yn=βnxn(1-βn)Tnxn,n0. Then the limit limnd(xn,p) exists for all pF(T).

Proof.

Taking pF(T), we have (31)d(xn+1,p)=d(αnTnxn(1-αn)yn,p)αnknd(xn,p)+(1-αn)d(yn,p)αnknd(xn,p)+(1-αn){βnd(xn,p)+(1-βn)d(Tnxn,p)}αnknd(xn,p)+(1-αn){βnd(xn,p)+(1-βn)knd(xn,p)}={1+(kn-1)[1-(1-αn)βn]}d(xn,p). By Lemma 3, we can get that limnd(xn,p) exists.

Next, we will prove limnd(Tnxn,yn)=0.

Proposition 9.

Let X be a CAT(0) space, C a closed convex subset of X, and T:CC an asymptotically nonexpansive mapping with coefficient kn. If F(T), {αn},{βn}(0,1). Let x0C, {xn} be generated by xn+1=αnTnxn(1-αn)yn,yn=βnxn(1-βn)Tnxn,n0. Then under the hypotheses (H1) and (H2), one can get that limnd(Tnxn,yn)=0.

Proof.

By the assumption, F(T) is nonempty. Take pF(T), let L=supnkn, then we have (32)d(Tnxn,p)knd(xn,p)Ld(xn,p),d(yn,p)βnd(xn,p)+(1-βn)d(Tnxn,p)Ld(xn,p)d(Tnyn,p)knd(yn,p)L2d(xn,p). It follows that the sequences {xn},{Tnxn},{yn},{Tnyn} are bounded.

By Lemma 1, we have (33)d2(xn+1,p)=d2(αnTnxn(1-αn)yn,p)αnkn2d2(xn,p)+(1-αn)d2(yn,p)-αn(1-αn)d2(Tnxn,yn)d2(xn,p)+(1-αn){d2(yn,p)-d2(xn,p)}+αn(kn2-1)d2(xn,p)-αn(1-αn)d2(Tnxn,yn).

Similar to the proof of Proposition 6, we can get (34)d2(yn,p)-d2(xn,p)(1-βn)(kn2-1)d2(xn,p).

Therefore, we have (35)d2(xn+1,p)d2(xn,p)+(1-αn)(1-βn)×(kn2-1)d2(xn,p)+αn(kn2-1)d2(xn,p)-αn(1-αn)d2(Tnxn,yn). Since {xn},{yn} are bounded and 0<δ<αn<1-δ for all nn0. we have (36)δ2d2(Tnxn,yn)d2(xn,p)-d2(xn+1,p)+(1-αn)(1-βn)(kn2-1)d2(xn,p)+αn(kn2-1)d2(xn,p).

By the conditions (H1) and (H2), we have n=1(kn2-1)< and (37)n=1δ2d2(Tnxn,yn)<, which implies that (38)limnd2(Tnxn,yn)=0.

Theorem 10.

Let X be a CAT(0) space, C a closed convex subset of X, and T:CC an asymptotically nonexpansive mapping with coefficient kn. If F(T), {αn},{βn}(0,1). Let x0C, {xn} be generated by xn+1=αnTnxn(1-αn)yn,yn=βnxn(1-βn)Tnxn,n0. Then under the hypotheses (H1) and (H2), one can get that {xn}  -converges to a fix point of  T.

Proof.

We first show that limnd(xn,Tnxn)=0. Indeed, by Lemma 1, and βn1, we can get (39)d(xn,yn)=d(xn,βnxn(1-βn)Tnxn)(1-βn)d(xn,Tnxn)0. And then, (40)d(xn,Tnxn)d(xn,yn)+d(yn,Tnxn). By Proposition 9, we obtain that limnd(xn,Tnxn)=0.

We claim that limnd(xn,Txn)=0. Indeed we have (41)d(xn+1,xn)=d(αnTnxn(1-αn)yn,xn)αnd(Tnxn,xn)+(1-αn)d(xn,yn)0.d(xn,Tn-1xn)d(αn-1Tn-1xn-1(1-αn-1)yn-1,Tn-1xn)αn-1d(Tn-1xn-1,Tn-1xn)+(1-αn-1)d(yn-1,Tn-1xn)αn-1kn-1d(xn-1,xn)+(1-αn-1)[d(yn-1,Tn-1xn-1)+d(Tn-1xn-1,Tn-1xn)]αn-1kn-1d(xn-1,xn)+(1-αn-1)[d(yn-1,Tn-1xn-1)+kn-1d(xn-1,xn)(yn-1,Tn-1xn-1)]0. Thus, (42)d(xn,Txn)d(xn,Tnxn)+d(Tnxn,Txn)d(xn,Tnxn)+Ld(Tn-1xn,xn)0.

Since {xn} is bounded, we may assume that {xn}  -converges to a point x^. By Lemma 2, we have x^F(T).

Acknowledgment

This research was supported by NSFC Grants nos. 11071279 and 11226125.

Goebel K. Kirk W. A. Topics in metric fixed point theory Cambridge Studies In Advanced Mathematics 1990 28 Cambridge, UK Cambridge University Press Goebel K. Kirk W. A. A fixed point theorem for asymptotically nonexpansive mappings Proceedings of the American Mathematical Society 1972 35 171 174 MR0298500 10.1090/S0002-9939-1972-0298500-3 ZBL0256.47045 Schu J. Approximation of fixed points of asymptotically nonexpansive mappings Proceedings of the American Mathematical Society 1991 112 1 143 151 10.2307/2048491 MR1039264 ZBL0734.47037 Górnicki J. Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces Commentationes Mathematicae Universitatis Carolinae 1989 30 2 249 252 MR1014125 ZBL0686.47045 Kirk W. A. Geodesic geometry and fixed point theory. II International Conference on Fixed Point Theory and Applications 2004 Yokohama, Japan Yokohama Publishers 113 142 MR2144169 ZBL1083.53061 Halpern B. Fixed points of nonexpanding maps Bulletin of the American Mathematical Society 1967 73 957 961 MR0218938 10.1090/S0002-9904-1967-11864-0 ZBL0177.19101 Tan K.-K. Xu H. K. Fixed point iteration processes for asymptotically nonexpansive mappings Proceedings of the American Mathematical Society 1994 122 3 733 739 10.2307/2160748 MR1203993 ZBL0820.47071 Shi L. Y. Chen R. D. Strong convergence of viscosity approximation methods for nonexpansive mappings in CAT(0) spaces Journal of Applied Mathematics 2012 2012 11 421050 10.1155/2012/421050 MR2935524 Dhompongsa S. Panyanak B. On δ-convergence theorems in CAT(0) spaces Computers & Mathematics with Applications 2008 56 10 2572 2579 10.1016/j.camwa.2008.05.036 MR2460066 Chaoha P. Phon-on A. A note on fixed point sets in CAT(0) spaces Journal of Mathematical Analysis and Applications 2006 320 2 983 987 10.1016/j.jmaa.2005.08.006 MR2226009 ZBL1101.54040 Dhompongsa S. Kirk W. A. Sims B. Fixed points of uniformly Lipschitzian mappings Nonlinear Analysis: Theory, Methods & Applications 2006 65 4 762 772 10.1016/j.na.2005.09.044 MR2232680 ZBL1105.47050 Kirk W. A. Panyanak B. A concept of convergence in geodesic spaces Nonlinear Analysis: Theory, Methods & Applications 2008 68 12 3689 3696 10.1016/j.na.2007.04.011 MR2416076 ZBL1145.54041 Hussain N. Khamsi M. A. On asymptotic pointwise contractions in metric spaces Nonlinear Analysis: Theory, Methods & Applications 2009 71 10 4423 4429 10.1016/j.na.2009.02.126 MR2548672 ZBL1176.54031 Chang S.-s. Cho Y. J. Zhou H. Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings Journal of the Korean Mathematical Society 2001 38 6 1245 1260 MR1858763 ZBL1020.47059 Tan K.-K. Xu H. K. The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces Proceedings of the American Mathematical Society 1992 114 2 399 404 10.2307/2159661 MR1068133 ZBL0781.47045