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# Approximating common fixed points in hyperbolic spaces

- Hafiz Fukhar-ud-din
^{1, 2}and - Mohamed Amine Khamsi
^{3}Email author

**2014**:113

https://doi.org/10.1186/1687-1812-2014-113

© Fukhar-ud-din and Khamsi; licensee Springer. 2014

**Received:**14 February 2014**Accepted:**16 April 2014**Published:**8 May 2014

## Abstract

We establish strong convergence and Δ-convergence theorems of an iteration scheme associated to a pair of nonexpansive mappings on a nonlinear domain. In particular we prove that such a scheme converges to a common fixed point of both mappings. Our results are a generalization of well-known similar results in the linear setting. In particular, we avoid assumptions such as smoothness of the norm, necessary in the linear case.

**MSC:**47H09, 46B20, 47H10, 47E10.

## Keywords

- Δ-convergence
- fixed point
- Ishikawa iterations
- nonexpansive mapping
- strong convergence
- uniformly convex hyperbolic space

## 1 Introduction

*C*be a nonempty subset of a metric space $(X,d)$ and $T:C\to C$ be a mapping. Denote the set of fixed points of

*T*by $F(T)$. Then

*T*is (i) nonexpansive if $d(Tx,Ty)\le d(x,y)$ for $x,y\in C$ (ii) quasi-nonexpansive if $F(T)\ne \mathrm{\varnothing}$ and $d(Tx,y)\le d(x,y)$ for $x\in C$ and $y\in F(T)$. For an initial value ${x}_{1}\in C$, Das and Debata [1] studied the strong convergence of Ishikawa iterates $\{{x}_{n}\}$ defined by

for two quasi-nonexpansive mappings *S*, *T* on a nonempty closed and convex subset of a strictly convex Banach space. Takahashi and Tamura [2] proved weak convergence of (1.1) to a common fixed point of two nonexpansive mappings in a uniformly convex Banach space which satisfies Opial’s condition or whose norm is Fréchet differentiable and strong convergence in a strictly convex Banach space (see also [3, 4]). Mann and Ishikawa iterative procedures are well-defined in a vector space through its built-in convexity. In the literature, several mathematicians have introduced the notion of convexity in metric spaces; for example [5–8]. In this work, we follow the original metric convexity introduced by Menger [9] and used by many authors like Kirk [5, 6] and Takahashi [8]. Note that Mann iterative procedures were also investigated in hyperbolic metric spaces [10, 11].

In this paper we investigate the results published in [2] and generalize them to uniformly convex hyperbolic spaces. A particular example of such metric spaces is the class of $CAT(0)$-spaces (in the sense of Gromov) and ℝ-trees (in the sense of Tits). Heavy use of the linear structure of Banach spaces in [2] presents some difficulties when extending these results to metric spaces. For example, a key assumption in many of their results is the smoothness of the norm which is hard to extend to metric spaces.

## 2 Menger convexity in metric spaces

*x*and

*y*in

*X*, there exists a unique metric segment $[x,y]$, which is an isometric copy of the real line interval $[0,d(x,y)]$. Note by ℱ the family of the metric segments in

*X*. For any $\beta \in [0,1]$, there exists a unique point $z\in [x,y]$ such that

*convex metric spaces*[9]. Moreover, if we have

*p*,

*q*,

*x*,

*y*in

*X*and $\alpha \in [0,1]$, then

*X*is said to be a

*hyperbolic metric space*(see [11–13]). For $q=y$, the hyperbolic inequality reduces to the convex structure inequality [8]. Throughout this paper, we will assume

for any $\alpha \in [0,1]$ and any $x,y\in X$.

An example of hyperbolic spaces is the family of Banach vector spaces or any normed vector spaces. Hadamard manifolds [14], the Hilbert open unit ball equipped with the hyperbolic metric [15], and the $CAT(0)$ spaces [6, 16–20] (see Example 2.1) are examples of nonlinear structures which play a major role in recent research in metric fixed point theory. A subset *C* of a hyperbolic space *X* is said to be convex if $[x,y]\subset C$, whenever $x,y\in C$ (see also [21]).

for any $a\in X$. *X* is said to be uniformly convex whenever $\delta (r,\epsilon )>0$, for any $r>0$ and $\epsilon >0$.

*X*is a uniformly convex hyperbolic space, then for every $s\ge 0$ and $\epsilon >0$, there exists $\eta (s,\epsilon )>0$ such that

**Remark 2.1**

- (i)
We have $\delta (r,0)=0$. Moreover, $\delta (r,\epsilon )$ is an increasing function of

*ε*. - (ii)For ${r}_{1}\le {r}_{2}$, we have$1-\frac{{r}_{2}}{{r}_{1}}(1-\delta ({r}_{2},\epsilon \frac{{r}_{1}}{{r}_{2}}))\le \delta ({r}_{1},\epsilon ).$

Next we give a very important example of uniformly convex hyperbolic metric space.

**Example 2.1** [16]

Let $(X,d)$ be a metric space. A *geodesic* from *x* to *y* in *X* is a mapping *c* from a closed interval $[0,l]\subset \mathbb{R}$ 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 *α* of *c* is called a geodesic (or metric) *segment* joining *x* and *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* and *y* for each $x,y\in X$, which will be denoted by $[x,y]$, and called the segment joining *x* to *y*.

A *geodesic triangle* $\mathrm{\Delta}({x}_{1},{x}_{2},{x}_{3})$ in a geodesic metric space *X* consists of three points ${x}_{1}$, ${x}_{2}$, ${x}_{3}$ in *X* (the *vertices* of Δ) and a geodesic segment between each pair of vertices (the *edges* of Δ). A *comparison triangle* for geodesic triangle $\mathrm{\Delta}({x}_{1},{x}_{2},{x}_{3})$ in $(X,d)$ is a triangle $\overline{\mathrm{\Delta}}({x}_{1},{x}_{2},{x}_{3}):=\mathrm{\Delta}({\overline{x}}_{1},{\overline{x}}_{2},{\overline{x}}_{3})$ in ${\mathbb{R}}^{2}$ such that ${d}_{{\mathbb{R}}^{2}}({\overline{x}}_{i},{\overline{x}}_{j})=d({x}_{i},{x}_{j})$ for $i,j\in \{1,2,3\}$. Such a triangle always exists (see [18]).

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

*X*and let $\overline{\mathrm{\Delta}}\subset {\mathbb{R}}^{2}$ be a comparison triangle for Δ. Then Δ is said to satisfy the $CAT(0)$

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

*Hadamard spaces*(see [16]). If

*x*, ${y}_{1}$, ${y}_{2}$ are points of a $CAT(0)$ space, then the $CAT(0)$ inequality implies

*X*, (1.1) is written as

*X*and

*C*be a nonempty subset. Define $r(\cdot ,\{{x}_{n}\}):C\to [0,\mathrm{\infty})$, by

*C*is given by

*ρ* will denote the asymptotic radius of $\{{x}_{n}\}$ with respect to *X*. A point $\xi \in C$ is said to be an asymptotic center of $\{{x}_{n}\}$ with respect to *C* if $r(\xi ,\{{x}_{n}\})=r(C,\{{x}_{n}\})=min\{r(x,\{{x}_{n}\}):x\in C\}$. We denote with $A(C,\{{x}_{n}\})$, the set of asymptotic centers of $\{{x}_{n}\}$ with respect to *C*. When $C=X$, we call *ξ* an asymptotic center of $\{{x}_{n}\}$ and we use the notation $A(\{{x}_{n}\})$ instead of $A(X,\{{x}_{n}\})$. In general, the set $A(C,\{{x}_{n}\})$ of asymptotic centers of a bounded sequence $\{{x}_{n}\}$ may be empty or may even contain infinitely many points. Note that in the study of the geometry of Banach spaces, the function $r(\cdot ,\{{x}_{n}\})$ is also known as a type. For more on types in metric spaces, we refer to [25].

The Δ-convergence, introduced independently several years ago by Kuczumow [26] and Lim [27], is shown in $CAT(0)$ spaces to behave similarly as the weak convergence in Banach spaces.

**Definition 2.2** A bounded sequence $\{{x}_{n}\}$ in *X* is said to Δ-converge to $x\in X$ if *x* is the unique asymptotic center of every subsequence $\{{u}_{n}\}$ of $\{{x}_{n}\}$. We write ${x}_{n}\stackrel{\mathrm{\Delta}}{\to}x$ ($\{{x}_{n}\}$ Δ-converges to *x*).

In this paper, we study the iteration schemes (2.1)-(2.2) for nonexpansive mappings. We study strong convergence of these iterates in strictly convex hyperbolic spaces and prove Δ-convergence results in uniformly convex hyperbolic spaces without requiring any condition similar to norm Fréchet differentiability.

In the sequel, the following results will be needed.

*Let* *X* *be a hyperbolic metric spaces*. *Assume that* *X* *is uniformly convex*. *Let* *C* *be a nonempty*, *closed and convex subset of* *X*. *Then every bounded sequence* $\{{x}_{n}\}\in X$ *has a unique asymptotic center with respect to* *C*.

*Let* *X* *be a hyperbolic metric spaces*. *Assume that* *X* *is uniformly convex*. *Let* *C* *be a nonempty*, *closed and convex subset of* *X*. *Let* *C* *be a nonempty closed and convex subset of* *X*, *and* $\{{x}_{n}\}$ *be a bounded sequence in* *C* *such that* $A(\{{x}_{n}\})=\{y\}$ *and* $r(\{{x}_{n}\})=\rho $. *If* $\{{y}_{m}\}$ *is a sequence in* *C* *such that* ${lim}_{m\to \mathrm{\infty}}r({y}_{m},\{{x}_{n}\})=\rho $, *then* ${lim}_{m\to \mathrm{\infty}}{y}_{m}=y$.

The following lemma [29] will be useful in studying the sequence generated by (2.1) in uniformly convex metric spaces. Here we give a proof based on the ideas developed in [25].

**Lemma 2.3**

*Let*

*X*

*be a uniformly convex hyperbolic space*.

*Then for arbitrary positive numbers*$\epsilon >0$

*and*$r>0$,

*and*$\alpha \in [0,1]$,

*we have*

*for all* $a,x,y\in X$, *such that* $d(z,x)\le r$, $d(z,y)\le r$, *and* $d(x,y)\ge r\epsilon $.

*Proof*Without loss of generality, we may assume $\alpha <\frac{1}{2}$. In this case, we have $min\{\alpha ,1-\alpha \}=\alpha $. Let $a\in X$ be fixed and $x,y\in X$. Set $\overline{x}=2\alpha x\oplus (1-2\alpha )y$. Since

*X*will imply $\frac{1}{2}\overline{x}\oplus \frac{1}{2}y=\alpha x\oplus (1-\alpha )y$. Using the uniform convexity of

*X*, we get

□

**Remark 2.2**If $(X,d)$ is uniformly convex, then $(X,d)$ is strictly convex,

*i.e.*, whenever

for $\alpha \in (0,1)$ and any $x,y,a\in X$, then we must have $x=y$.

The following result is very useful.

**Lemma 2.4** [25]

*Let*$(X,d)$

*be a uniformly convex hyperbolic space*.

*Let*$R\in [0,+\mathrm{\infty})$

*be such that*

*Then we have*

But since we use convex combinations other than the middle point, we will need the following generalization obtained by using Lemma 2.3.

**Lemma 2.5**

*Let*$(X,d)$

*be a uniformly convex hyperbolic space*.

*Let*$R\in [0,+\mathrm{\infty})$

*be such that*${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}d({x}_{n},a)\le R$, ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}d({y}_{n},a)\le R$,

*and*

*where*${\alpha}_{n}\in [a,b]$,

*with*$0<a\le b<1$.

*Then we have*

A subset *C* of a metric space *X* is Chebyshev if for every $x\in X$, there exists ${c}_{0}\in C$ such that $d({c}_{0},x)<d(c,x)$ for all $c\in C$, $c\ne {c}_{0}$. In other words, for each point of the space, there is a well-defined nearest point of *C*. We can then define the nearest point projection $P:X\to C$ by sending *x* to ${c}_{0}$. We have the following result.

**Lemma 2.6** [25]

*Let*$(X,d)$

*be a complete uniformly convex hyperbolic space*.

*Let*

*C*

*be nonempty*,

*convex and closed subset of*

*X*.

*Let*$x\in X$

*be such that*$d(x,C)<\mathrm{\infty}$.

*Then there exists a unique best approximant of*

*x*

*in*

*C*,

*i*.

*e*.,

*there exists a unique*${c}_{0}\in C$

*such that*

*i*.*e*., *C* *is Chebyshev*.

## 3 Convergence in strictly convex hyperbolic space

*C*be a nonempty closed convex subset of

*X*. Let $S,T:C\to C$ be two nonexpansive mappings. Throughout the paper, assume that $F=F(S)\cap F(T)$. Let ${x}_{1}\in C$ and $p\in F$ (assuming

*F*is not empty). Set $r=d({x}_{1},p)$. Then

*S*and

*T*. Therefore one may always assume that

*C*is bounded provided that

*S*and

*T*have a common fixed point. Moreover, if $\{{x}_{n}\}$ is the sequence generated by (2.1), then we have

The first result of this work discusses the convergence behavior of the sequence generated by (2.1).

**Theorem 3.1**

*Let*

*X*

*be a strictly convex hyperbolic space*.

*Let*

*C*

*be a nonempty bounded*,

*closed and convex subset of*

*X*.

*Let*$S,T:C\to C$

*be two nonexpansive mappings*.

*Assume that*$F\ne \mathrm{\varnothing}$.

*Let*${x}_{1}\in C$

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

*be given by*(2.1).

*Then the following hold*:

- (i)
*if*${\alpha}_{n}\in [a,b]$*and*${\beta}_{n}\in [0,b]$,*with*$0<a\le b<1$,*then*${x}_{{n}_{i}}\to y$*implies*$y\in F(S)$; - (ii)
*if*${\alpha}_{n}\in [a,1]$*and*${\beta}_{n}\in [a,b]$,*with*$0<a\le b<1$,*then*${x}_{{n}_{i}}\to y$*implies*$y\in F(T)$; - (iii)
*if*${\alpha}_{n},{\beta}_{n}\in [a,b]$,*with*$0<a\le b<1$,*then*${x}_{{n}_{i}}\to y$*implies*$y\in F$.*In this case*,*we have*${x}_{n}\to y$.

*Proof*Assume that ${x}_{{n}_{i}}\to y$. Let $p\in F$. Without loss of generality, we may assume ${lim}_{n\to \mathrm{\infty}}{\alpha}_{{n}_{i}}=\alpha $ and ${lim}_{n\to \mathrm{\infty}}{\beta}_{{n}_{i}}=\beta $. Since $\{d({x}_{n},p)\}$ is decreasing, we get

- (1)If $\alpha \in (0,1)$ and $\beta >0$, then$d(p,S(\beta Ty\oplus (1-\beta )y))=d(\alpha S(\beta Ty\oplus (1-\beta )y)\oplus (1-\alpha )y,p)=r.$
The strict convexity of

*X*will imply $S(\beta Ty\oplus (1-\beta )y)=y$. - (2)If $\alpha \in (0,1)$ and $\beta =0$, then$d(p,y)=d(p,S(y))=d(\alpha S(y)\oplus (1-\alpha )y,p).$

*X*will imply $S(y)=y$.

- (3)If $\beta \in (0,1)$ and $\alpha >0$, then$d(p,y)=d(p,T(y))=d(p,\beta Ty\oplus (1-\beta )y).$

*X*will imply $T(y)=y$.

- (4)
If $\alpha ,\beta \in (0,1)$, then $T(y)=y$ and $S(\beta Ty\oplus (1-\beta )y)=y$. Hence $S(y)=y$.

Let us finish the proof of Theorem 3.1. Note that (i) implies $\alpha \in [a,b]$ and $\beta \in [0,b]$. If $\beta =0$, then the conclusion (2) above implies $y\in F(S)$. Otherwise the conclusion (4) will imply $y\in F$. This proves (i).

For (ii), notice that $\alpha \in [a,1]$ and $\beta \in [a,b]$. Hence the conclusion (3) will imply $y\in F(T)$ which proves (ii).

we get ${x}_{n}\to y$, which completes the proof of (iii). □

If we assume compactness, Theorem 3.1 will imply the following result.

**Theorem 3.2** *Let* *X* *be a strictly convex hyperbolic space*. *Let* *C* *be a nonempty bounded*, *closed and convex subset of* *X*. *Let* $S,T:C\to C$ *be two nonexpansive mappings*. *Assume that* $F\ne \mathrm{\varnothing}$. *Fix* ${x}_{1}\in C$. *Assume that* $\overline{co}\{\{{x}_{1}\}\cup S(C)\cup T(C)\}$ *is a compact subset of* *C*. *Define* $\{{x}_{n}\}$ *as in* (2.1) *where* ${\alpha}_{n},{\beta}_{n}\in [a,b]$, *with* $0<a\le b<1$, *and* ${x}_{1}$ *is the initial element of the sequence*. *Then* $\{{x}_{n}\}$ *converges strongly to a common fixed point of* *S* *and* *T*.

*Proof* We have ${x}_{n}\in \overline{co}\{\{{x}_{1}\}\cup S(C)\cup T(C)\}$. Since $\overline{co}\{\{{x}_{1}\}\cup S(C)\cup T(C)\}$ is compact, $\{{x}_{n}\}$ has a convergent subsequence $\{{x}_{{n}_{i}}\}$, *i.e.*, ${x}_{{n}_{i}}\to z$. By Theorem 3.1, we have $z\in F$ and ${x}_{n}\to z$. □

*T*and

*S*is crucial. If one assumes that

*T*and

*S*commute,

*i.e.*, $S\circ T=T\circ S$, then a common fixed point exists under the assumptions of Theorem 3.2. Indeed, fix ${x}_{0}\in C$ and define

*C*. Since the closure of $T(C)$ is compact, there exists a subsequence $\{T{u}_{{n}_{i}}\}$ of $\{T{u}_{n}\}$ such that $T{u}_{{n}_{i}}\to u$. Since $T(C)$ is bounded and

we have $d({u}_{n},T{u}_{n})\to 0$. In particular, we have ${u}_{{n}_{i}}\to u$. Continuity of *T* implies $Tu=u$. Since *X* is strictly convex, then $F(T)$ is a nonempty convex subset of *X*. Since *T* and *S* commute, we have $S(F(T))\subset F(T)$. Moreover, since the closure of $T(C)$ is compact, we see that $F(T)$ is compact. The above proof shows that *S* has a fixed point in $F(T)$, *i.e.*, $F=F(S)\cap F(T)\ne \mathrm{\varnothing}$.

The case $S=T$ gives the following result.

**Theorem 3.3** *Let* *C* *be a nonempty closed and convex subset of a complete strictly convex hyperbolic space* *X*. *Let* $T:C\to C$ *be a nonexpansive mapping such that* $\overline{co}\{\{{c}_{0}\}\cup T(C)\}$ *is a compact subset of* *C*, *where* ${c}_{0}\in C$. *Define* $\{{x}_{n}\}$ *by* (2.2), *where* ${x}_{1}={c}_{0}$, ${\alpha}_{n}\in [a,b]$ *and* ${\beta}_{n}\in [0,b]$ *or* ${\alpha}_{n}\in [a,1]$ *and* ${\beta}_{n}\in [a,b]$, *with* $0<a\le b<1$. *Then* $\{{x}_{n}\}$ *converges strongly to a fixed point of* *T*.

*Proof* We saw that in this case, we have $F(T)\ne \mathrm{\varnothing}$. Since ${x}_{n}\in \overline{co}\{\{{x}_{1}\}\cup T(C)\}$. Then there exists a subsequence $\{{x}_{{n}_{i}}\}$ of $\{{x}_{n}\}$ such that ${x}_{{n}_{i}}\to z\in C$. By Theorem 3.1, we have $Tz=z$ and ${x}_{n}\to z$. □

## 4 Convergence in uniformly convex hyperbolic spaces

The following result is similar to the well-known demi-closedness principle discovered by Göhde in uniformly convex Banach spaces [30].

**Lemma 4.1** *Let* *C* *be a nonempty*, *closed and convex subset of a complete uniformly convex hyperbolic space* *X*. *Let* $T:C\to C$ *be a nonexpansive mapping*. *Let* $\{{x}_{n}\}\in C$ *be an approximate fixed point sequence of* *T*, *i*.*e*., ${lim}_{n\to \mathrm{\infty}}d({x}_{n},T{x}_{n})=0$. *If* $x\in C$ *is the asymptotic center of* $\{{x}_{n}\}$ *with respect to* *C*, *then* *x* *is a fixed point of* *T*, *i*.*e*., $x\in F(T)$. *In particular*, *if* $\{{x}_{n}\}\in C$ *is an approximate fixed point sequence of* *T*, *such that* ${x}_{n}\stackrel{\mathrm{\Delta}}{\to}x$, *then* $x\in F(T)$.

*Proof*Let $\{{x}_{n}\}$ be an approximate fixed point sequence of

*T*. Let $x\in C$ be the unique asymptotic center of $\{{x}_{n}\}$ with respect to

*C*. Since

By the uniqueness of the asymptotic center, we get $Tx=x$. □

The following theorem is necessary to discuss the behavior of the iterates in (2.1).

**Theorem 4.1**

*Let*

*C*

*be a nonempty*,

*closed and convex subset of a complete uniformly convex hyperbolic space*

*X*.

*Let*$S,T:C\to C$

*be nonexpansive mappings such that*$F\ne \mathrm{\varnothing}$.

*Fix*${x}_{1}\in C$

*and generate*$\{{x}_{n}\}$

*by*(2.1).

*Set*

*for any*$n\ge 1$.

- (i)
*If*${\alpha}_{n}\in [a,b]$,*where*$0<a\le b<1$,*then*$\underset{n\to \mathrm{\infty}}{lim}d({x}_{n},S{y}_{n})=0.$ - (ii)
*If*${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}>0$*and*${\beta}_{n}\in [a,b]$,*with*$0<a\le b<1$,*then*$\underset{n\to \mathrm{\infty}}{lim}d({x}_{n},T{x}_{n})=0.$ - (iii)
*If*${\alpha}_{n},{\beta}_{n}\in [a,b]$,*with*$0<a\le b<1$,*then*$\underset{n\to \mathrm{\infty}}{lim}d({x}_{n},S{x}_{n})=0\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\underset{n\to \mathrm{\infty}}{lim}d({x}_{n},T{x}_{n})=0.$

*Proof*Let $p\in F$. Then the sequence $\{d({x}_{n},p)\}$ is decreasing. Set $c={lim}_{n\to \mathrm{\infty}}d({x}_{n},p)$. If $c=0$, then all the conclusions are trivial. Therefore we will assume that $c>0$. Note that we have

Using Lemma 2.5, we get ${lim}_{n\to \mathrm{\infty}}d(S{y}_{n},{x}_{n})=0$.

Using Lemma 2.5, we get ${lim}_{n\to \mathrm{\infty}}d(T{x}_{n},{x}_{n})=0$.

we conclude that ${lim}_{n\to \mathrm{\infty}}d({x}_{n},S{x}_{n})=0$. □

The conclusion of Theorem 4.1(iii) is amazing because the sequence generated by (2.1) gives an approximate fixed point sequence for both *S* and *T* without assuming that these mappings commute.

**Remark 4.1**If we assume that $0\le {\beta}_{n}\le b<1$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}>0$, then

As a direct consequence to Theorem 4.1 and Remark 4.1, we get the following result which discusses the Δ-convergence of the iterative sequence defined by (2.1).

**Theorem 4.2**

*Let*

*C*

*be a nonempty*,

*closed and convex subset of a complete uniformly convex hyperbolic space*

*X*.

*Let*$S,T:C\to C$

*be two nonexpansive mappings such that*$F\ne \mathrm{\varnothing}$.

*Fix*${x}_{1}\in C$

*and generate*$\{{x}_{n}\}$

*by*(2.1).

*Then*

- (i)
*if*${\alpha}_{n}\in [a,b]$*and*${\beta}_{n}\in [0,b]$,*with*$0<a\le b<1$,*then*${x}_{n}\stackrel{\mathrm{\Delta}}{\to}y$*and*$y\in F(S)$; - (ii)
*if*${\alpha}_{n}\in [a,1]$*and*${\beta}_{n}\in [a,b]$,*with*$0<a\le b<1$,*then*${x}_{n}\stackrel{\mathrm{\Delta}}{\to}y$*and*$y\in F(T)$; - (iii)
*if*${\alpha}_{n},{\beta}_{n}\in [a,b]$,*with*$0<a\le b<1$,*then*${x}_{n}\stackrel{\mathrm{\Delta}}{\to}y$*and*$y\in F$.

*Proof*Let us prove (i). Assume ${\alpha}_{n}\in [a,b]$ and ${\beta}_{n}\in [0,b]$, with $0<a\le b<1$. Theorem 4.1 and Remark 4.1 imply that $\{{x}_{n}\}$ is an approximate fixed point sequence of

*S*,

*i.e.*,

*y*be the unique asymptotic center of $\{{x}_{n}\}$. Then Lemma 4.1 implies that $y\in F(S)$. Let us prove that in fact $\{{x}_{n}\}$ Δ-converges to

*y*. Let $\{{x}_{{n}_{i}}\}$ be a subsequence of $\{{x}_{n}\}$. Let

*z*be the unique asymptotic center of $\{{x}_{{n}_{i}}\}$. Again since $\{{x}_{{n}_{i}}\}$ is an approximate fixed point sequence of

*S*, we get $z\in F(S)$. Hence

Since *y* is the unique asymptotic center of $\{{x}_{n}\}$, we get $y=z$. This proves that $\{{x}_{n}\}$ Δ-converges to *y*.

*T*,

*i.e.*,

Following the same proof as given above for (i), we get $\{{x}_{n}\}$ Δ-converges to its unique asymptotic center which is a fixed point of *T*.

The conclusion (iii) follows easily from (i) and (ii). □

As a corollary to Theorem 4.2, we get the following result when $S=T$.

**Corollary 4.1** *Let* *C* *be a nonempty*, *closed and convex subset of a complete uniformly convex hyperbolic space* *X*. *Let* $T:C\to C$ *be a nonexpansive mapping with a fixed point*. *Suppose that* $\{{x}_{n}\}$ *is given by* (2.2), *where* ${\alpha}_{n}\in [a,b]$ *and* ${\beta}_{n}\in [0,b]$ *or* ${\alpha}_{n}\in [a,1]$ *and* ${\beta}_{n}\in [a,b]$, *with* $0<a\le b<1$. *Then* ${x}_{n}\stackrel{\mathrm{\Delta}}{\to}p$, *with* $p\in F(T)$.

Using the concept of near point projection, we establish the following amazing convergence result.

**Theorem 4.3** *Let* *C* *be a nonempty*, *closed and convex subset of a complete uniformly convex hyperbolic space* *X*. *Let* $S,T:C\to C$ *be nonexpansive mappings such that* $F\ne \mathrm{\varnothing}$. *Let* *P* *be the nearest point projection of* *C* *onto* *F*. *For an initial value* ${x}_{1}\in C$, *define* $\{{x}_{n}\}$ *as given in* (2.1), *where* ${\alpha}_{n},{\beta}_{n}\in [a,b]$, *with* $0<a\le b<1$. *Then* $\{P{x}_{n}\}$ *converges strongly to the asymptotic center of* $\{{x}_{n}\}$.

*Proof*First, we claim that

*y*, which is in

*F*. Let us prove that $\{P{x}_{n}\}$ converges strongly to

*y*. Assume not,

*i.e.*, there exist $\epsilon >0$ and a subsequence $\{P{x}_{{n}_{i}}\}$ such that $d(P{x}_{{n}_{i}},y)\ge \epsilon $, for any ${n}_{i}\ge 1$. It is clear that we must have $R=d({x}_{1},y)>0$, otherwise $\{{x}_{n}\}$ is a constant sequence. Since

*P*, we get

*n*and fixed ${n}_{i}$). Hence

*y*is the asymptotic center of $\{{x}_{n}\}$, we get

Since $\epsilon \le d({x}_{{n}_{i}},P{x}_{{n}_{i}})\le d({x}_{{n}_{i}},y)$, we conclude that $\epsilon \le {lim}_{n\to \mathrm{\infty}}d({x}_{n},y)$, which implies $1\le 1-\eta $ which is our desired contradiction. Therefore $\{P{x}_{n}\}$ converges strongly to *y*. □

## Declarations

### Acknowledgements

The authors are grateful to King Fahd University of Petroleum and Minerals for supporting research project IN121055.

## Authors’ Affiliations

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