# On the strong and Δ-convergence of new multi-step and *S*-iteration processes in a $CAT(0)$ space

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

**2013**:482

https://doi.org/10.1186/1029-242X-2013-482

© Başarır and Şahin; licensee Springer. 2013

**Received: **1 August 2013

**Accepted: **19 September 2013

**Published: **7 November 2013

## Abstract

In this paper, we introduce a new class of mappings and prove the demiclosedness principle for mappings of this type in a $CAT(0)$ space. Also, we obtain the strong and Δ-convergence theorems of new multi-step and *S*-iteration processes in a $CAT(0)$ space. Our results extend and improve the corresponding recent results announced by many authors in the literature.

**MSC:**47H09, 47H10, 54E40, 58C30.

### Keywords

$CAT(0)$ space contractive-like mapping strong convergence Δ-convergence new multi-step iteration*S*-iteration fixed point

## 1 Introduction

Contractive mappings and iteration processes are some of the main tools in the study of fixed point theory. There are many contractive mappings and iteration processes that have been introduced and developed by several authors to serve various purposes in the literature (see [1–6]).

Imoru and Olantiwo [7] gave the following contractive definition.

**Definition 1**Let

*T*be a self-mapping on a metric space

*X*. The mapping

*T*is called a contractive-like mapping if there exist a constant $\delta \in [0,1)$ and a strictly increasing and continuous function $\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ with $\phi (0)=0$ such that, for all $x,y\in X$,

This mapping is more general than those considered by Berinde [8, 9], Harder and Hicks [10], Zamfirescu [11], Osilike and Udomene [12].

By taking $\delta =1$ in (1.1), we define a new class of mappings as follows.

**Definition 2**The mapping

*T*is called a generalized nonexpansive mapping if there exists a non-decreasing and continuous function $\phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ with $\phi (0)=0$ such that, for all $x,y\in X$,

**Remark 1**For $x\in F(T)$ in (1.2), we have

If *X* is an interval of ℝ, then $F(T)$ is convex. The same is also true in each space with unique geodesic for each pair of points (*e.g.*, metric trees or $CAT(0)$ spaces).

In the case $\phi (t)=0$ for all $t\in [0,\mathrm{\infty})$, it is easy to show that every nonexpansive mapping satisfies (1.2), but the inverse is not necessarily true.

**Example 1**Let $X=[0,2]$, $d(x,y)=|x-y|$, $\phi (t)=t$ and define

*T*by

Therefore *T* is a generalized nonexpansive mapping, but *T* is not nonexpansive mapping.

*X*is complete. For example, let $X=[0,\mathrm{\infty})$, $d(x,y)=|x-y|$ and define

*T*by

It is proved in Gürsoy *et al.* [13] that *T* is a contractive-like mapping. Similarly, one can prove that *T* is a generalized nonexpansive mapping. But the mapping *T* has no fixed point.

*T*is a nonexpansive mapping. In fact, if $x,y\in [0,1]$ or $x,y\in [-1,0)$, then we have

This implies that *T* is a nonexpansive mapping and so *T* is a generalized nonexpansive mapping with $\phi (t)=0$ for all $t\in [0,\mathrm{\infty})$. But $F(T)=\{x\in K;0\le x\le 1\}$.

*et al.*[1] introduced the

*S*-iteration process which is independent of those of Mann [3] and Ishikawa [2] and converges faster than both of these. We apply this iteration process in a $CAT(0)$ space as

Gürsoy *et al.* [13] introduced a new multi-step iteration process in a Banach space. We modify this iteration process in a $CAT(0)$ space as follows.

By taking $k=3$ and $k=2$ in (1.4), we obtain the SP-iteration process of Phuengrattana and Suantai [4] and the two-step iteration process of Thianwan [6], respectively.

In this paper, motivated by the above results, we prove demiclosedness principle for a new class of mappings and the Δ-convergence theorems of the new multi-step iteration and the *S*-iteration processes for mappings of this type in a $CAT(0)$ space. Also, we present the strong convergence theorems of these iteration processes for contractive-like mappings in a $CAT(0)$ space.

## 2 Preliminaries on a $CAT(0)$ space

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. Fixed point theory in a $CAT(0)$ space was first studied by Kirk (see [14, 15]). 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 [14–20]). 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 $CAT({k}^{\prime})$ space for every ${k}^{\prime}\ge k$ (see [[21], p.165]).

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 $d(c(t),c({t}^{\prime}))=|t-{t}^{\prime}|$ for all $t,{t}^{\prime}\in [0,l]$. 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 a *uniquely geodesic* if there is exactly one geodesic joining *x* to *y* for each $x,y\in X$.

*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 the 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 [21]).

A geodesic metric space is said to be a $CAT(0)$ space [21] if all geodesic triangles of 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 $\overline{x},\overline{y}\in \overline{\u25b3}$,

*x*, ${y}_{1}$, ${y}_{2}$ are points of a $CAT(0)$ space and if ${y}_{0}$ is the midpoint of the segment $[{y}_{1},{y}_{2}]$, then the $CAT(0)$ inequality implies that

The equality holds for the Euclidean metric. In fact (see [[21], p.163]), a geodesic metric space is a $CAT(0)$ space if and only if it satisfies the inequality (2.1) (which is known as the *CN* inequality of Bruhat and Tits [22]).

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 [18] obtained the following lemmas 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* $x,y,z\in X$.

**Lemma 2**

*Let*

*X*

*be a*$CAT(0)$

*space*.

*Then*

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

## 3 Demiclosedness principle for a new class of mappings

In 1976 Lim [23] introduced the concept of convergence in a general metric space setting which is called △-convergence. Later, Kirk and Panyanak [24] used the concept of △-convergence introduced by Lim [23] to prove on a $CAT(0)$ space analogs of some Banach space results which involve weak convergence. Also, Dhompongsa and Panyanak [18] obtained the △-convergence theorems for the Picard, Mann and Ishikawa iterations in a $CAT(0)$ space for nonexpansive mappings under some appropriate conditions.

Now, we recall some definitions.

*X*. For $x\in X$, we set

*asymptotic radius*$r(\{{x}_{n}\})$ of $\{{x}_{n}\}$ is given by

*asymptotic radius*${r}_{K}(\{{x}_{n}\})$ of $\{{x}_{n}\}$ with respect to $K\subset X$ is given by

*asymptotic center*$A(\{{x}_{n}\})$ of $\{{x}_{n}\}$ is the set

*asymptotic center*${A}_{K}(\{{x}_{n}\})$ of $\{{x}_{n}\}$ with respect to $K\subset X$ is the set

**Proposition 1** ([[25], Proposition 3.2])

*Let* $\{{x}_{n}\}$ *be a bounded sequence in a complete* $CAT(0)$ *space* *X*, *and let* *K* *be a closed convex subset of* *X*, *then* $A(\{{x}_{n}\})$ *and* ${A}_{K}(\{{x}_{n}\})$ *are singletons*.

**Definition 3** ([[24], Definition 3.1])

A sequence $\{{x}_{n}\}$ in a $CAT(0)$ space *X* is said to be △-convergent to $x\in X$ if *x* is the unique asymptotic center of $\{{u}_{n}\}$ for every subsequence $\{{u}_{n}\}$ of $\{{x}_{n}\}$. In this case, we write $\u25b3\text{-}{lim}_{n}{x}_{n}=x$ and *x* is called the △-limit of $\{{x}_{n}\}$.

**Lemma 3**

- (i)
*Every bounded sequence in a complete*$CAT(0)$*space always has a*△-*convergent subsequence*(*see*[[24],*p*.3690]). - (ii)
*Let**K**be a nonempty closed convex subset of a complete*$CAT(0)$*space*,*and let*$\{{x}_{n}\}$*be a bounded sequence in**K*.*Then the asymptotic center of*$\{{x}_{n}\}$*is in**K*(*see*[[17],*Proposition*2.1]).

**Lemma 4** ([[18], Lemma 2.8])

*If* $\{{x}_{n}\}$ *is a bounded sequence in a complete* $CAT(0)$ *space with* $A(\{{x}_{n}\})=\{x\}$, $\{{u}_{n}\}$ *is a subsequence of* $\{{x}_{n}\}$ *with* $A(\{{u}_{n}\})=\{u\}$ *and the sequence* $\{d({x}_{n},u)\}$ *converges*, *then* $x=u$.

*X*, and let

*K*be a closed convex subset of

*X*which contains $\{{x}_{n}\}$. We denote the notation

where $\mathrm{\Phi}(x)={lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}d({x}_{n},x)$.

We note that $\{{x}_{n}\}\rightharpoonup w$ if and only if ${A}_{K}(\{{x}_{n}\})=\{w\}$ (see [25]).

Nanjaras and Panyanak [25] gave a connection between the ‘⇀’ convergence and Δ-convergence.

**Proposition 2** ([[25], Proposition 3.12])

*Let* $\{{x}_{n}\}$ *be a bounded sequence in a* $CAT(0)$ *space X*, *and let* *K* *be a closed convex subset of* *X* *which contains* $\{{x}_{n}\}$. *Then* $\mathrm{\Delta}\text{-}{lim}_{n\to \mathrm{\infty}}{x}_{n}=p$ *implies that* $\{{x}_{n}\}\rightharpoonup p$.

By using the convergence defined in (3.1), we obtain *the* *demiclosedness principle for the new class of mappings in a* $CAT(0)$ *space*.

**Theorem 1** *Let* *K* *be a nonempty closed convex subset of a complete* $CAT(0)$ *space* *X*, *and let* $T:K\to K$ *be a generalized nonexpansive mapping with* $F(T)\ne \mathrm{\varnothing}$. *Let* $\{{x}_{n}\}$ *be a bounded sequence in* *K* *such that* $\{{x}_{n}\}\rightharpoonup w$ *and* ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$. *Then* $Tw=w$.

*Proof*By the hypothesis, $\{{x}_{n}\}\rightharpoonup w$. Then we have ${A}_{K}(\{{x}_{n}\})=\{w\}$. By Lemma 3(ii), we obtain $A(\{{x}_{n}\})=\{w\}$. Since ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$, then we have

The rest of the proof closely follows the pattern of Proposition 3.14 in Nanjaras and Panyanak [25]. Hence $Tw=w$ as desired. □

Now, we prove the Δ-convergence of the new multi-step iteration process for the new class of mappings in a $CAT(0)$ space.

**Theorem 2** *Let* *K* *be a nonempty closed convex subset of a complete* $CAT(0)$ *space* *X*, *let* $T:K\to K$ *be a generalized nonexpansive mapping with* $F(T)\ne \mathrm{\varnothing}$, *and let* $\{{x}_{n}\}$ *be a sequence defined by* (1.4) *such that* $\{{\alpha}_{n}\},\{{\beta}_{n}^{i}\}\subset [0,1]$, $i=1,2,\dots ,k-2$ *and* $\{{\beta}_{n}^{k-1}\}\subset [a,b]$ *for some* $a,b\in (0,1)$. *Then the sequence* $\{{x}_{n}\}$ Δ-*converges to the fixed point of* *T*.

*Proof*Let $p\in F(T)$. From (1.2), (1.4) and Lemma 1, we have

*T*, we prove that

and ${W}_{\u25b3}({x}_{n})$ consists of exactly one point. Let $u\in {W}_{\u25b3}({x}_{n})$. Then there exists a subsequence $\{{u}_{n}\}$ of $\{{x}_{n}\}$ such that $A(\{{u}_{n}\})=\{u\}$. By Lemma 3, there exists a subsequence $\{{v}_{n}\}$ of $\{{u}_{n}\}$ such that $\u25b3\text{-}{lim}_{n\to \mathrm{\infty}}{v}_{n}=v\in K$. By Proposition 2 and Theorem 1, $v\in F(T)$. By Lemma 4, we have $u=v\in F(T)$. This shows that ${W}_{\u25b3}({x}_{n})\subseteq F(T)$. Now, we prove that ${W}_{\u25b3}({x}_{n})$ consists of exactly one point. Let $\{{u}_{n}\}$ be a subsequence of $\{{x}_{n}\}$ with $A(\{{u}_{n}\})=\{u\}$, and let $A(\{{x}_{n}\})=\{x\}$. We have already seen that $u=v$ and $v\in F(T)$. Finally, since $\{d({x}_{n},v)\}$ converges, by Lemma 4, $x=v\in F(T)$. This shows that ${W}_{\u25b3}({x}_{n})=\{x\}$. This completes the proof. □

We give the following theorem related to the Δ-convergence of the *S*-iteration process for the new class of mappings in a $CAT(0)$ space.

**Theorem 3** *Let* *K* *be a nonempty closed convex subset of a complete* $CAT(0)$ *space* *X*, *let* $T:K\to K$ *be a generalized nonexpansive mapping with* $F(T)\ne \mathrm{\varnothing}$, *and let* $\{{x}_{n}\}$ *be a sequence defined by* (1.3) *such that* $\{{\alpha}_{n}\},\{{\beta}_{n}\}\subset [a,b]$ *for some* $a,b\in (0,1)$. *Then the sequence* $\{{x}_{n}\}$ Δ-*converges to the fixed point of* *T*.

*Proof*Let $p\in F(T)$. Using (1.2), (1.3) and Lemma 1, we have

Thus ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$. The rest of the proof follows the pattern of the above theorem and is therefore omitted. □

## 4 Strong convergence theorems for a contractive-like mapping

Now, we prove the strong convergence of the new multi-step iteration process for a contractive-like mapping in a $CAT(0)$ space.

**Theorem 4** *Let* *K* *be a nonempty closed convex subset of a complete* $CAT(0)$ *space* *X*, *let* $T:K\to K$ *be a contractive*-*like mapping with* $F(T)\ne \mathrm{\varnothing}$, *and let* $\{{x}_{n}\}$ *be a sequence defined by* (1.4) *such that* $\{{\alpha}_{n}\}\subset [0,1)$, ${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$ *and* $\{{\beta}_{n}^{i}\}\subset [0,1)$, $i=1,2,\dots ,k-1$. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to the unique fixed point of* *T*.

*Proof*Let

*p*be the unique fixed point of

*T*. From (1.1), (1.4) and Lemma 1, we have

Consequently, ${x}_{n}\to p\in F(T)$ and this completes the proof. □

**Remark 2** In Theorem 4 the condition ${\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty}$ may be replaced with ${\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}^{i}=\mathrm{\infty}$ for a fixed $i=1,2,\dots ,k-1$.

Finally, we give the strong convergence theorem of the *S*-iteration process for a contractive-like mapping in a $CAT(0)$ space as follows.

**Theorem 5** *Let* *K* *be a nonempty closed convex subset of a complete* $CAT(0)$ *space* *X*, *let* $T:K\to K$ *be a contractive*-*like mapping with* $F(T)\ne \mathrm{\varnothing}$, *and let* $\{{x}_{n}\}$ *be a sequence defined by* (1.3) *such that* $\{{\alpha}_{n}\},\{{\beta}_{n}\}\subset [0,1]$. *Then the sequence* $\{{x}_{n}\}$ *converges strongly to the unique fixed point of* *T*.

*Proof*Let

*p*be the unique fixed point of

*T*. From (1.1), (1.3) and Lemma 1, we have

Thus we have ${x}_{n}\to p\in F(T)$. If $\delta =0$, the result is clear. This completes the proof. □

## 5 Conclusions

The new multi-step iteration reduces to the two-step iteration and the SP-iteration processes. Also, the class of generalized nonexpansive mappings includes nonexpansive mappings. Then these results presented in this paper extend and generalize some works for a $CAT(0)$ space in the literature.

## Declarations

### Acknowledgements

The authors are grateful to the referee for his/her careful reading and valuable comments and suggestions which led to the present form of the paper. This paper has been presented in International Conference ‘Anatolian Communications in Nonlinear Analysis (ANCNA 2013)’ in Abant Izzet Baysal University, Bolu, Turkey, July 03-06, 2013. This paper was supported by Sakarya University Scientific Research Foundation (Project number: 2013-02-00-003).

## Authors’ Affiliations

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