# On the strong convergence of a modified S-iteration process for asymptotically quasi-nonexpansive mappings in a $\mathrm{CAT}(0)$ space

- Aynur Şahin
^{1}Email author and - Metin Başarır
^{1}

**2013**:12

https://doi.org/10.1186/1687-1812-2013-12

© Şahin and Başarır; licensee Springer 2013

**Received: **27 August 2012

**Accepted: **21 December 2012

**Published: **16 January 2013

## Abstract

In this paper, we give strong convergence theorems for the modified S-iteration process of asymptotically quasi-nonexpansive mappings on a $CAT(0)$ space which extend and improve many results in the literature.

**MSC:** Primary 47H09; secondary 47H10.

## Keywords

## 1 Introduction

A metric space *X* is a $CAT(0)$ *space* if it is geodesically connected and if every geodesic triangle in *X* is at least as ‘thin’ as its comparison triangle in the Euclidean plane. The initials of the term ‘CAT’ are in honor of E. Cartan, A. D. Alexanderov and V. A. Toponogov. A $CAT(0)$ space is a generalization of the Hadamard manifold, which is a simply connected, complete Riemannian manifold such that the sectional curvature is non-positive. In fact, it is very well known that any complete simply connected Riemannian manifold with non-positive sectional curvature is a $CAT(0)$ space. The complex Hilbert ball with a hyperbolic metric is a $CAT(0)$ space (see [1]). Other examples include Pre-Hilbert spaces, R-trees (see [2]) and Euclidean buildings (see [3]). A $CAT(0)$ space plays a fundamental role in various areas of mathematics (see Bridson and Haefliger [2], Burago, Burago and Ivanov [4], Gromov [5]). Moreover, there are applications in biology and computer science as well [6, 7].

Fixed point theory in a $CAT(0)$ space has been first studied by Kirk (see [8, 9]). He showed that every nonexpansive mapping defined on a bounded closed convex subset of a complete $CAT(0)$ space always has a fixed point. Since then the fixed point theory in a $CAT(0)$ space has been rapidly developed and many papers have appeared (see, *e.g.*, [8–14]). It is worth mentioning that the results in a $CAT(0)$ space can be applied to any $CAT(k)$ space with $k\le 0$ since any $CAT(k)$ space is a $CAT({k}^{\mathrm{\prime}})$ space for every ${k}^{\mathrm{\prime}}\ge k$ (see [[2], p.165]). Throughout the paper, ℕ and ℝ denote the set of natural numbers and the set of real numbers, respectively.

where $\{{a}_{n}\}$ is a sequence in $(0,1)$.

where $\{{a}_{n}\}$ and $\{{b}_{n}\}$ are the sequences in $(0,1)$. This iteration process reduces to the Mann iteration process when ${b}_{n}=0$ for all $n\in \mathbb{N}$.

where $\{{a}_{n}\}$ and $\{{b}_{n}\}$ are the sequences in $(0,1)$. Note that (1.3) is independent of (1.2) (and hence of (1.1)). They showed that their process is independent of those of Mann and Ishikawa and converges faster than both of these (see [[15], Proposition 3.1]).

where $\{{a}_{n}\}$ is a sequence in $(0,1)$.

where the sequences $\{{a}_{n}\}$ and $\{{b}_{n}\}$ are in $(0,1)$. This iteration process reduces to the modified Mann iteration process when ${b}_{n}=0$ for all $n\in \mathbb{N}$.

where the sequences $\{{a}_{n}\}$ and $\{{b}_{n}\}$ are in $(0,1)$. Note that (1.6) is independent of (1.5) (and hence of (1.4)). Also, (1.6) reduces to (1.3) when $n=1$.

We now modify (1.6) in a $CAT(0)$ space as follows.

*K*be a nonempty closed convex subset of a complete $CAT(0)$ space

*X*and $T:K\to K$ be an asymptotically quasi-nonexpansive mapping with $F(T)\ne \mathrm{\varnothing}$. Suppose that $\{{x}_{n}\}$ is a sequence generated iteratively by

where and throughout the paper $\{{a}_{n}\}$, $\{{b}_{n}\}$ are the sequences such that $0\le {a}_{n},{b}_{n}\le 1$ for all $n\in \mathbb{N}$.

In this paper, we study the modified S-iteration process for asymptotically quasi-nonexpansive mappings on the $CAT(0)$ space and generalize some results of Khan and Abbas [14] which studied the S-iteration process in a $CAT(0)$ space for nonexpansive mappings. This paper contains three sections. In Section 2, we first collect some known preliminaries and lemmas that will be used in the proofs of our main theorems. We give the main results related to the strong convergence theorems of the modified S-iteration process in a $CAT(0)$ space in Section 3. Under some suitable condition, we obtain the main theorems which state that $\{{x}_{n}\}$ converges strongly to a fixed point of *T*. Our results can be applied to an S-iteration process since the modified S-iteration process reduces to the S-iteration process when $n=1$.

## 2 Preliminaries and lemmas

Let us recall some definitions and known results in the existing literature on this concept.

Let $(X,d)$ be a metric space and *K* its nonempty subset. Let $T:K\to K$ be a mapping. A point $x\in K$ is called a fixed point of *T* if $Tx=x$. We will also denote by $F(T)$ the set of fixed points of *T*, that is, $F(T)=\{x\in K:Tx=x\}$.

The concept of quasi-nonexpansiveness was introduced by Diaz and Metcalf [18] in 1967, the concept of asymptotically nonexpansiveness was introduced by Goebel and Kirk [19] in 1972. The iterative approximation problems for asymptotically quasi-nonexpansive mapping were studied by Liu [20], Fukhar-ud-din *et al*. [21], Khan *et al.* [22] and Beg *et al.* [23] in a Banach space and a $CAT(0)$ space.

**Definition 1**Let $(X,d)$ be a metric space and

*K*be its nonempty subset. Then $T:K\to K$ is said to be

- (1)
nonexpansive if $d(Tx,Ty)\le d(x,y)$ for all $x,y\in K$,

- (2)
asymptotically nonexpansive if there exists a sequence $\{{u}_{n}\}\in [0,\mathrm{\infty})$ with the property ${lim}_{n\to \mathrm{\infty}}{u}_{n}=0$ and such that $d({T}^{n}x,{T}^{n}y)\le (1+{u}_{n})d(x,y)$ for all $x,y\in K$,

- (3)
quasi-nonexpansive if $d(Tx,p)\le d(x,p)$ for all $x\in K$, $p\in F(T)$,

- (4)
asymptotically quasi-nonexpansive if there exists a sequence $\{{u}_{n}\}\in [0,\mathrm{\infty})$ with the property ${lim}_{n\to \mathrm{\infty}}{u}_{n}=0$ and such that $d({T}^{n}x,p)\le (1+{u}_{n})d(x,p)$ for all $x\in K$, $p\in F(T)$,

- (5)
semi-compact if for a sequence $\{{x}_{n}\}$ in

*K*with ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$, there exists a subsequence $\{{x}_{{n}_{k}}\}$ of $\{{x}_{n}\}$ such that ${x}_{{n}_{k}}\to p\in K$.

**Remark 1** From Definition 1, it is clear that the class of quasi-nonexpansive mappings and asymptotically nonexpansive mappings includes nonexpansive mappings, whereas the class of asymptotically quasi-nonexpansive mappings is larger than that of quasi-nonexpansive mappings and asymptotically nonexpansive mappings. The reverse of these implications may not be true.

Let $(X,d)$ be a metric space. A *geodesic path* joining $x\in X$ to $y\in X$ (or more briefly, a *geodesic* from *x* to *y*) is a map *c* from a closed interval $[0,l]\subset R$ to *X* such that $c(0)=x$, $c(l)=y$ and $d(c(t),c({t}^{\mathrm{\prime}}))=|t-{t}^{\mathrm{\prime}}|$ for all $t,{t}^{\mathrm{\prime}}\in [0,l]$. In particular, *c* is an isometry and $d(x,y)=l$. The image of *c* is called a *geodesic* (or *metric*) *segment* joining *x* and *y*. When it is unique, this geodesic is denoted by $[x,y]$. The space $(X,d)$ is said to be a *geodesic space* if every two points of *X* are joined by a geodesic, and *X* is said to be *uniquely geodesic* if there is exactly one geodesic joining *x* to *y* for each $x,y\in X$. A subset $Y\subset X$ is said to be *convex* if *Y* includes every geodesic segment joining any two of its points.

*geodesic triangle*$\u25b3({x}_{1},{x}_{2},{x}_{3})$ in a geodesic metric space $(X,d)$ consists of three points in

*X*(the vertices of △) and a geodesic segment between each pair of vertices (the edges of △). A

*comparison*

*triangle*for a geodesic triangle $\u25b3({x}_{1},{x}_{2},{x}_{3})$ in $(X,d)$ is a triangle $\overline{\u25b3}({x}_{1},{x}_{2},{x}_{3})=\u25b3({\overline{x}}_{1},{\overline{x}}_{2},{\overline{x}}_{3})$ in the Euclidean plane ${\mathbb{R}}^{2}$ such that

for $i,j\in \{1,2,3\}$. Such a triangle always exists (see [2]).

A geodesic metric space is said to be a $CAT(0)$ space [2] if all geodesic triangles of an appropriate size satisfy the following comparison axiom.

*X*and let $\overline{\u25b3}$ be a comparison triangle for △. Then △ is said to satisfy the $CAT(0)$

*inequality*if for all $x,y\in \u25b3$ and all comparison points $\overline{x},\overline{y}\in \overline{\u25b3}$,

A complete $CAT(0)$ space is often called *Hadamard space* (see [24]).

*x*, ${y}_{1}$, ${y}_{2}$ are points of a $CAT(0)$ space and if ${y}_{0}$ is the midpoint of the segment $[{y}_{1},{y}_{2}]$, which we will denote by $\frac{{y}_{1}\oplus {y}_{2}}{2}$, then the $CAT(0)$ inequality implies

The equality holds for the Euclidean metric. In fact (see [[2], p.163]), a geodesic metric space is a $CAT(0)$ space if and only if it satisfies inequality (2.1) (which is known as the *CN* inequality of Bruhat and Tits [25]).

From now on, we will use the notation $(1-t)x\oplus ty$ for the unique point *z* satisfying (2.2). By using this notation, Dhompongsa and Panyanak [12] obtained the following lemma which will be used frequently in the proof of our main results.

**Lemma 1**

*Let*

*X*

*be a*$CAT(0)$

*space*.

*Then*

*for all* $t\in [0,1]$ *and* $x,y,z\in X$.

The following lemma can be found in [26].

**Lemma 2**

*Let*$\{{a}_{n}\}$

*and*$\{{u}_{n}\}$

*be two sequences of positive real numbers satisfying*

*for all* $n\in \mathbb{N}$. *If* ${\sum}_{n=1}^{\mathrm{\infty}}{u}_{n}<\mathrm{\infty}$, *then* ${lim}_{n\to \mathrm{\infty}}{a}_{n}$ *exists*.

## 3 Main results

In this section we prove the strong convergence theorems of the modified S-iteration process in a $CAT(0)$ space.

**Theorem 1**

*Let*

*K*

*be a nonempty closed convex subset of a complete*$CAT(0)$

*space*

*X*, $T:K\to K$

*be asymptotically quasi*-

*nonexpansive mapping with*$F(T)\ne \mathrm{\varnothing}$

*and*$\{{u}_{n}\}$

*be a nonnegative real sequence with*${\sum}_{n=1}^{\mathrm{\infty}}{u}_{n}<\mathrm{\infty}$.

*Suppose that*$\{{x}_{n}\}$

*is defined by the iteration process*(1.7).

*If*

*where* $d(x,F(T))={inf}_{z\in F(T)}d(x,z)$, *then the sequence* $\{{x}_{n}\}$ *converges strongly to a fixed point of* *T*.

*Proof*Let $p\in F(T)$. Since

*T*is an asymptotically quasi-nonexpansive mapping, there exists a sequence $\{{u}_{n}\}\in [0,\mathrm{\infty})$ with the property ${lim}_{n\to \mathrm{\infty}}{u}_{n}=0$ and such that

*K*. Since ${lim}_{n\to \mathrm{\infty}}d({x}_{n},F(T))=0$, for each $\u03f5>0$, there exists ${n}_{1}\in \mathbb{N}$ such that

*K*. Since the set

*K*is complete, the sequence $\{{x}_{n}\}$ must be convergence to a point in

*K*. Let ${lim}_{n\to \mathrm{\infty}}{x}_{n}=p\in K$. Here after, we show that

*p*is a fixed point. By ${lim}_{n\to \mathrm{\infty}}{x}_{n}=p$, for all ${\u03f5}_{1}>0$, there exists ${n}_{2}\in \mathbb{N}$ such that

Since ${\u03f5}_{1}$ is arbitrary, so $d(Tp,p)=0$, *i.e.*, $Tp=p$. Therefore, $p\in F(T)$. This completes the proof. □

**Remark 2** Let the hypothesis of Theorem 1 be satisfied and $T:K\to K$ be an asymptotically nonexpansive or quasi-nonexpansive mapping. From Remark 1, the class of asymptotically quasi-nonexpansive mappings includes quasi-nonexpansive mappings and asymptotically nonexpansive mappings. Then the sequence $\{{x}_{n}\}$ converges strongly to a fixed point of *T*.

Now, we give the following corollaries which have been proved by Theorem 1.

**Corollary 1**

*Under the hypothesis of Theorem*1,

*T*

*satisfies the following conditions*:

- (1)
${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$.

- (2)
*If the sequence*$\{{z}_{n}\}$*in**K**satisfies*${lim}_{n\to \mathrm{\infty}}d({z}_{n},T{z}_{n})=0$,*then*$\underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}inf}d({z}_{n},F(T))=0\phantom{\rule{1em}{0ex}}\mathit{\text{or}}\phantom{\rule{1em}{0ex}}\underset{n\to \mathrm{\infty}}{lim\hspace{0.17em}sup}d({z}_{n},F(T))=0.$

*Then the sequence* $\{{x}_{n}\}$ *converges strongly to a fixed point of* *T*.

*Proof*It follows from the hypothesis that ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$. From (2),

Therefore, the sequence $\{{x}_{n}\}$ must converge to a fixed point of *T* by Theorem 1. □

**Corollary 2**

*Under the hypothesis of Theorem*1,

*T*

*satisfies the following conditions*:

- (1)
${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$.

- (2)
*There exists a function*$f:[0,\mathrm{\infty})\to [0,\mathrm{\infty})$*which is right*-*continuous at*0, $f(0)=0$*and*$f(r)>0$*for all*$r>0$*such that*$d(x,Tx)\ge f\left(d(x,F(T))\right)\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}x\in K,$

*where* $d(x,F(T))={inf}_{z\in F(T)}d(x,z)$.

*Then the sequence* $\{{x}_{n}\}$ *converges strongly to a fixed point of* *T*.

*Proof*It follows from the hypothesis that

Thus, ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}d({x}_{n},F(T))={lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}d({x}_{n},F(T))=0$. By Theorem 1, the sequence $\{{x}_{n}\}$ converges strongly to *q*, a fixed point of *T*. This completes the proof. □

Finally, we give the following theorem which has a different hypothesis from Theorem 1.

**Theorem 2** *Let* *K* *be a nonempty closed convex subset of a complete* $CAT(0)$ *space* *X*, $T:K\to K$ *be an asymptotically quasi*-*nonexpansive mapping with* $F(T)\ne \mathrm{\varnothing}$ *and* $\{{u}_{n}\}$ *be a nonnegative real sequence with* ${\sum}_{n=1}^{\mathrm{\infty}}{u}_{n}<\mathrm{\infty}$. *Suppose that* $\{{x}_{n}\}$ *is defined by the iteration process* (1.7). *If* *T* *is semi*-*compact and* ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$, *then the sequence* $\{{x}_{n}\}$ *converges strongly to a fixed point of* *T*.

*Proof*From the hypothesis, we have ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$. Also, since

*T*is semi-compact, there exists a subsequence $\{{x}_{{n}_{k}}\}$ of $\{{x}_{n}\}$ such that ${x}_{{n}_{k}}\to p\in K$. Hence,

Since ${\sum}_{n=1}^{\mathrm{\infty}}{u}_{n}<\mathrm{\infty}$, we have ${\sum}_{n=1}^{\mathrm{\infty}}(2{u}_{n}+{u}_{n}^{2})<\mathrm{\infty}$. By Lemma 2, ${lim}_{n\to \mathrm{\infty}}d({x}_{n},p)$ exists and ${x}_{{n}_{k}}\to p\in F(T)$ gives that ${x}_{n}\to p\in F(T)$. This completes the proof. □

## 4 Conclusions

The class of quasi-nonexpansive mappings and asymptotically nonexpansive mappings includes nonexpansive mappings, where as the class of asymptotically quasi-nonexpansive mappings is larger than that of quasi-nonexpansive mappings and asymptotically nonexpansive mappings. Then these results presented in this paper extend and generalize some works for a $CAT(0)$ space in the literature.

## Declarations

### Acknowledgements

This paper has been presented in The ‘International Conference on Applied Analysis and Algebra (ICAAA2012)’ in Yıldız Technical University, 20-24 June 2012, Istanbul, Turkey.

## Authors’ Affiliations

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