Common Fixed Points of Asymptotically Quasi-Nonexpansive Mappings in Cat(0) Spaces

Abstract

In this manuscript, we investigate and approximate common fixed points of two asymptotically quasi-nonexpansive mappings in CAT(0) spaces. Suppose X is a CAT(0) space and C is a nonempty closed convex subset of X. Let T1,T2:C→C be two asymptotically quasi-nonexpansive mappings, and F=FT1∩FT2≔x∈C:T1x=T2x=x≠∅. Let αn,βn be sequences in [0,1]. If the sequence {xn} is generated iteratively by xn+1=1−αnxn⊕αnT1nyn,yn=1−βnxn⊕βnT2nxn,n≥1 and x1∈C is the initial element of the sequence (A). We prove that {xn} converges strongly to a common fixed point of T1 and T2 if and only if limn→∞dxnF=0. (B). Suppose αn and βn are sequences in ε1−ε forsome ε∈01. If X is uniformly convex and if either T2 or T1 is compact, then {xn} converges strongly to some common fixed point of T1 and T2. Our results extend and improve the related results in the literature. We also give an example in support of our main results.

1. Introduction

Let C be a nonempty subset of a real normed linear space X. Let $T:C\to C$ be a self-mapping of C. Then T is said to be.

1. nonexpansive if $\parallel \mathit{Tx}-\mathit{Ty}\parallel \le \parallel x-y\parallel$ for all $x,y\in C$;

2. quasi-nonexpansive if $F\left(T\right)\ne \varnothing$ and $\parallel \mathit{Tx}-p\parallel \le \parallel x-p\parallel$ for all $x\in C$ and $p\in F\left(T\right)$ where $F\left(T\right)=\left\{x\in C:\mathit{Tx}=x\right\}$_;

3. asymptotically nonexpansive with sequence $\left\{k_n\right\}\subset \left[0,\infty \right)$ if $\underset{n\to \infty }{lim}{k}_n=1$ and $\parallel T^{n}x-T^{n}y\parallel \le k_n\parallel x-y\parallel$ for all $x,y\in C$ and $n\ge 1$;

4. symptotically quasi-nonexpansive with sequence $\left\{k_n\right\}\subset \left[0,\infty \right)$ if $F\left(T\right)\ne \varnothing$, $\underset{n\to \infty }{lim}{k}_n=1$ and $\parallel T^{n}x-p\parallel \le k_n\parallel x-p\parallel$ for all $x\in C,p\in F\left(T\right)$ and $n\ge 1$.

It is clear that a nonexpansive mapping with $F\left(T\right)\ne \varnothing$ is quasi-nonexpansive and an asymptotically nonexpansive mapping with $F\left(T\right)\ne \varnothing$ is asymptotically quasi-nonexpansive. The converses are not true in general. The mapping T is said to be uniformly $\left(L,\gamma \right)$-Lipschitzian if there exists a constant $L>0$ and $\gamma >0$ such that $\parallel T^{n}x-T^{n}y\parallel \le L\parallel x-y{\parallel }^{\gamma }$ for all $x,y\in C$ and $n\ge 1$.

The following example shows that there is a quasi-nonexpansive mapping which is not a nonexpansive mapping.

Example 1.1. (see [1]) Let $C={\mathbb{R}}^1$ and define a mapping $T:C\to C$ by

$$\mathit{Tx}=\left\{\begin{array}{ll}\frac{x}{2}& ,\text{if}x\ne 0\\ 0& ,\text{if}x=0\end{array}\right.$$

Then T is quasi-nonexpansive but not nonexpansive.

It is easy to see that a nonexpansive mapping is an asymptotically nonexpansive mapping with the sequence $\left\{k_n\right\}=\left\{1\right\}.$

It is easy to see that a quasi-nonexpansive mapping is an asymptotically quasi-nonexpansive mapping with the sequence $\left\{k_n\right\}=\left\{1\right\}.$

In 1972, Goebel and Kirk [2] introduced the class of asymptotically nonexpansive maps as a significant generalization of the class of nonexpansive maps. They proved that if the map $T:C\to C$ is asymptotically nonexpansive and C is a nonempty closed convex bounded subset of a uniformly convex Banach space X, then T has a fixed point. In [3], Goebel and Kirk extended this result to the broader class of uniformly $\left(L,1\right)$-Lipschitzian mappings with $L<\lambda$ and, where $\lambda$ is sufficiently near 1 (but greater than 1).

Iterative approximation of fixed points of nonexpansive mappings and their generalizations (asymptotically nonexpansive mappings, etc.) have been investigated by a number of authors (see, [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] for examples) via the Mann iterates or the Ishikawa-type iteration.

Later, in 2001 Khan and Takahashi [22] studied the problem of approximating common fixed points of two asymptotically nonexpansive mappings. In 2002, Qihou [23] also established a strong convergence theorem for the Ishikawa-type iterative sequences with errors for a uniformly $\left(L,\gamma \right)$-Lipschitzian asymptotically nonexpansive self-mapping of a nonempty compact convex subset of a uniformly convex Banach space.

Recently, in 2005 Shahzad and Udomene [24] investigated the approximation of common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces. More precisely, they obtained the following results.

Theorem 1.2. [24] Let C be a nonempty closed convex subset of a real Banach space X. Let $T_1,T_2:C\to C$ be two asymptotically quasi-nonexpansive mappings with sequences $\left\{u_n\right\},\left\{v_n\right\}\subset \left[0,\infty \right)$ such that ${\sum }_{n=1}^{\infty }{u}_n<\infty$ and ${\sum }_{n=1}^{\infty }{v}_n<\infty$, and $\mathbb{F}=F\left(T_1\right)\cap F\left(T_2\right)≔\left\{x\in C:T_{1}x=T_{2}x=x\right\}\ne \varnothing$. Let $x_1\in C$ be arbitrary, define the sequence $\left\{x_n\right\}$ iteratively by the iteration

$$\begin{array}{l}{x}_{n+1}=\left(1-{\alpha }_n\right)x_n+{\alpha }_n{T}_1^n{y}_n\\ y_n=\left(1-{\beta }_n\right)x_n+{\beta }_n{T}_2^n{x}_n,\end{array}$$E1

for all $n\ge 1$, where $\left\{{\alpha }_n\right\}$ and $\left\{{\beta }_n\right\}$ are sequences in $\left[0,1\right]$. Then.

1. $\parallel x_{n+1}-p\parallel \le \left(1-{\gamma }_n\right)\parallel x-p\parallel$ for all $n\ge 1,p\in \mathbb{F}$, and for some sequence $\left\{{\gamma }_n\right\}$ of numbers with ${\sum }_{n=1}^{\infty }{\gamma }_n<\infty .$

2. There exists a constant $K>0$ such that $\parallel x_{n+m}-p\parallel \le K\parallel x_n-p\parallel$ for all $n,m\ge 1$ and $p\in \mathbb{F}$.

Theorem 1.3. [24] Let C be a nonempty closed convex subset of a real Banach space X. Let $T_1,T_2:C\to C$ be two asymptotically quasi-nonexpansive mappings with sequences $\left\{u_n\right\},\left\{v_n\right\}\subset \left[0,\infty \right)$ such that ${\sum }_{n=1}^{\infty }{u}_n<\infty$ and ${\sum }_{n=1}^{\infty }{v}_n<\infty$, and $\mathbb{F}=F\left(T_1\right)\cap F\left(T_2\right)\ne \varnothing$. Let $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\}\subset \left[0,1\right]$. Define the sequence $\left\{x_n\right\}$ as in (1) and $x_1\in C$ is the initial element of the sequence. Then $\left\{x_n\right\}$ converges strongly to a common fixed point of $T_1$ and $T_2\iff lim{inf}_{n\to \infty }d\left(x_n,\mathbb{F}\right)=0.$.

Theorem 1.4. [24] Let X be a real uniformly convex Banach space and C a nonempty closed convex subset of X. Let $T_1,T_2:C\to C$ be two uniformly continuous asymptotically quasi-nonexpansive mappings with sequences $\left\{u_n\right\},\left\{v_n\right\}\subset \left[0,\infty \right)$ such that ${\sum }_{n=1}^{\infty }{u}_n<\infty$, ${\sum }_{n=1}^{\infty }{v}_n<\infty$, and $\mathbb{F}=F\left(T_1\right)\cap F\left(T_2\right)\ne \varnothing$. Let $\left\{{\alpha }_n\right\}$ and $\left\{{\beta }_n\right\}$ be sequences in $\left[\epsilon ,1-\epsilon \right]$ forsome $\epsilon \in \left(0,1\right)$. Define the sequence $\left\{x_n\right\}$ as in (1) and $x_1\in C$ is the initial element of the sequence. Assume, in addition, that either $T_2$ or $T_1$ is compact. Then $\left\{x_n\right\}$ converges strongly to a common fixed point of $T_1$ and $T_2$.

2. Preliminaries

In this section, we present some basic facts about the CAT(0) spaces and hyperbolic spaces with some useful results which are required in the sequel. The connection between CAT(0) spaces and hyperbolic spaces presented here would help, at least for beginners, to appreciate the main results presented in this manuscript.

2.1 CAT(0) spaces

Let $\left(X,d\right)$ 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 $\omega :\left[0,a\right]\to X,\left[0,a\right]\subset R$ such that $\omega \left(0\right)=x,\omega \left(a\right)=y$, and $d\left(\omega \left(m\right),\omega \left(n\right)\right)=\mid m-n\mid$ for all $m,n\in \left[0,a\right]$. In particular, $\omega$ is an isometry and $d\left(x,y\right)=a$. The image $\alpha$ of $\omega$ is called a geodesic (or metric) segment joining x and y. A unique geodesic segment from x to y is denoted by $\left[x,y\right]$. The space $\left(X,d\right)$ is called 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 $x,y\in X$. If $Y\subseteq X$ then Y is said to be convex if Y includes every geodesic segment joining any two of its points. If $\left(X,d\right)$ is a geodesic metric space, a geodesic triangle $\mathrm{\Delta }\left(a_1,a_2,a_3\right)$ consists of three points $a_1,a_2,a_3$ 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 $\mathrm{\Delta }\left(a_1,a_2,a_3\right)$ in $\left(X,d\right)$ is a triangle $\overline{\mathrm{\Delta }}\left(a_1,a_2,a_3\right)≔\mathrm{\Delta }\left({\overline{a}}_1,{\overline{a}}_2,{\overline{a}}_3\right)$ in the Euclidean plane ${\mathbb{R}}^2$ satisfying $d_{{\mathbb{R}}^2}\left({\overline{a}}_i,{\overline{a}}_j\right)=d\left(a_i,a_j\right)$ for $i,j\in \mathrm{1,2,3}$. Such a triangle always exists (See [25]).

Definition 2.1. A geodesic space $\left(X,d\right)$ is said to be a CAT(0) space if for any geodesic triangle $\mathrm{\Delta }\subset X$ and $a,b\in \mathrm{\Delta }$ we have $d\left(a,b\right)\le d\left(\overline{a},\overline{b}\right)$ where $\overline{a},\overline{b}\in \overline{\mathrm{\Delta }}$.

Remark 2.2. Any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space. Other examples of CAT(0) spaces include pre-Hilbert spaces, R-trees, Euclidean buildings, and the complex Hilbert ball with a hyperbolic metric, (see [25, 26, 27] for example).

Definition 2.3. A geodesic triangle $\mathrm{\Delta }\left(p,q,r\right)$ in $\left(X,d\right)$ is said to satisfy the $\mathit{CAT}\left(0\right)$ inequality if for any $u,v\in \mathrm{\Delta }\left(p,q,r\right)$ and for their comparison points $\overline{u},\overline{v}\in \overline{\mathrm{\Delta }}\left(\overline{p},\overline{q},\overline{r}\right),$ one has

$$d\left(u,v\right)\le d_{{\mathbb{R}}^2}\left(\overline{u},\overline{v}\right).$$

For other equivalent definitions and basic properties of $\mathit{CAT}\left(0\right)$ spaces, we refer the readers to standard texts, such as ref. [25].

Note that if $x,a_1,a_2$ are points of $\mathit{CAT}\left(0\right)$ space and if $a_0$ is the midpoint of the segment $\left[a_1,a_2\right]$ (we write $a_0=\frac{1}{2}{a}_1\oplus \frac{1}{2}{a}_2$), then the $\mathit{CAT}\left(0\right)$ inequality implies

$$d{\left(x,a_0\right)}^2=d\left(x,\frac{1}{2}{a}_1\oplus \frac{1}{2}{a}_2\right)\le \frac{1}{2}d{\left(x,a_1\right)}^2+\frac{1}{2}d{\left(x,a_2\right)}^2-\frac{1}{4}d{\left(a_1,a_2\right)}^2$$E2

The inequality (2) is called the CN inequality of Bruhat and Tits [28]. We refer readers to some brilliant known CAT(0) space results in [29, 30, 31, 32, 33] and references therein.

We now collect some useful facts about CAT(0) spaces, which will be used frequently in the proof of our main results.

Lemma 2.4. (See [31]) Let $\left(X,d\right)$ be a CAT(0) space.

1. For $x_1,x_2\in X$ and $\alpha \in \left[0,1\right]$, there exists a unique point $y\in \left[x_1,x_2\right]$ such that

   $$d\left(x_1,y\right)=\mathit{\alpha d}\left(x_1,x_2\right)\mathit{\text{and}}d\left(x_2,y\right)=\left(1-\alpha \right)d\left(x_1,x_2\right).$$E3

   We write $y=\left(1-\alpha \right)x_1\oplus \alpha x_2$ for the unique point y satisfying (3).

2. For $x,y,z\in X$ and $\alpha \in \left[0,1\right],$ we have

   $$d\left(\left(1-\alpha \right)x\oplus \mathit{\alpha y},z\right)\le \left(1-\alpha \right)d\left(x,z\right)+\mathit{\alpha d}\left(y,z\right).$$

3. For $x,y,z\in X$ and $\alpha \in \left[0,1\right]$ we have

   $$d{\left(\left(1-\alpha \right)x\oplus \mathit{\alpha y},z\right)}^2\le \left(1-\alpha \right)d{\left(x,z\right)}^2+\mathit{\alpha d}{\left(y,z\right)}^2-\alpha \left(1-\alpha \right)d{\left(x,y\right)}^2.$$

Lemma 2.5. (See [34]) Let $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\}$ be two sequences such that.

1. $0\le {\alpha }_n,{\beta }_n<1$,

2. ${\beta }_n\to 0$ and $\sum {\alpha }_n{\beta }_n=\infty$.

Let $\left\{{\gamma }_n\right\}$ be a nonnegative real sequence such that $\sum {\alpha }_n{\beta }_n\left(1-{\beta }_n\right){\gamma }_n$ is bounded. Then $\left\{{\gamma }_n\right\}$ has a subsequence that converges to zero.

Lemma 2.6. (see, [17]). Let $\left\{{\lambda }_n\right\}$ and $\left\{{\sigma }_n\right\}$ be sequences of nonnegative real numbers such that ${\lambda }_{n+1}\le {\lambda }_n+{\sigma }_n$, $\forall$ $n\ge 1$ and ${\sum }_{n=1}^{\infty }{\sigma }_n<\infty .$ Then $\underset{n\to \infty }{lim}{\lambda }_n$ exists. Moreover, if there exists a subsequence $\left\{{\lambda }_{n_j}\right\}$ of $\left\{{\lambda }_n\right\}$ such that ${\lambda }_{n_j}\to 0$ as $j\to \infty ,$ then ${\lambda }_n\to 0$ as $n\to \infty .$.

2.2 Hyperbolic spaces

In this section, we recall some notions of hyperbolic spaces. This class of spaces contains the class of CAT(0) spaces (See [35, 36]).

Definition 2.7. (See [36]) Let $\left(X,d\right)$ be a metric space and $\mathcal{W}:X\times X\times \left[0,1\right]\to X$ be a mapping satisfying:-.

$\mathcal{W}1$. $d\left(z,\mathcal{W}\left(x,y,\alpha \right)\right)\le \left(1-\alpha \right)d\left(z,x\right)+\mathit{\alpha d}\left(z,y\right),$.

$\mathcal{W}2$. $d\left(\mathcal{W}\left(x,y,\alpha \right),\mathcal{W}\left(x,y,\beta \right)\right)=\mid \alpha -\beta \mid d\left(x,y\right)$,

$\mathcal{W}3$. $\mathcal{W}\left(x,y,\alpha \right)=\mathcal{W}\left(y,x,\left(1-\alpha \right)\right),$.

$\mathcal{W}4$. $d\left(\mathcal{W}\left(x,z,\alpha \right),\mathcal{W}\left(y,w,\alpha \right)\right)\le \left(1-\alpha \right)d\left(x,y\right)+\mathit{\alpha d}\left(z,w\right)$.

for all $x,y,z,w\in X,\alpha ,\beta \in \left[0,1\right].$ We call the triple $\left(X,d,\mathcal{W}\right)$ a hyperbolic space.

It follows from ($\mathcal{W}1$.) that, for each $x,y\in X$ and $\alpha \in \left[0,1\right]$,

$$d\left(x,\mathcal{W}\left(x,y,\alpha \right)\right)\le \mathit{\alpha d}\left(x,y\right),d\left(y,\mathcal{W}\left(x,y,\alpha \right)\right)\le \left(1-\alpha \right)d\left(x,y\right)$$E4

In fact, we can get that (see [33]),

$$d\left(x,\mathcal{W}\left(x,y,\alpha \right)\right)=\mathit{\alpha d}\left(x,y\right),d\left(y,\mathcal{W}\left(x,y,\alpha \right)\right)=\left(1-\alpha \right)d\left(x,y\right).$$E5

Similar to (3), we can also use the notation $\left(1-\alpha \right)x\oplus \mathit{\alpha y}$ for such a point $\mathcal{W}\left(x,y,\alpha \right)$ in hyperbolic space.

A mapping $\eta :\left(0,\infty \right)\times \left(0,2\right]\to \left(0,1\right]$ providing such a $\delta ≔\eta \left(r,\epsilon \right)$ forgiven $r>0$ and $\epsilon \in \left(0,2\right]$ is called a modulus of uniform convexity.

Definition 2.8. (See [37, 38]) Let $\left(X,d\right)$ be a hyperbolic metric space. X is said to be uniformly convex whenever $\delta \left(r,\epsilon \right)>0$, for any $r>0$ and $\epsilon >0$, where

$$\delta \left(r,\epsilon \right)=inf\left\{1-\frac{1}{r}d\left(\frac{1}{2}x\oplus \frac{1}{2}y,a\right):d(x,a)\le r,d(y,a)\le r,d(x,y)\ge \mathit{r\epsilon }\right\}$$

for any $a\in X$.

Note that if X is a uniformly convex hyperbolic space, then for every $s\ge 0$ and $\epsilon >0$, there exists $\eta \left(s,\epsilon \right)>0$ such that $\delta \left(r,\epsilon \right)>\eta \left(s,\epsilon \right)>0$ for any $r>s$. One can see that $\delta \left(r,0\right)=0$. Moreover $\delta \left(r,\epsilon \right)$ is an increasing function of $\epsilon$.

The following result is very useful which is an analog of Shu ([15], Lemma 1.3). It can be applied to a CAT(0) space as well.

Lemma 2.9. (See [33, 39]) Let $\left(X,d\right)$ be a uniformly convex hyperbolic space. Let $\left\{x_n\right\},\left\{y_n\right\}$ be sequences in X and $c\in \left[0,+\infty \right)$ be such that ${limsup}_{n\to \infty }d\left(x_n,a\right)\le c,{limsup}_{n\to \infty }d\left(y_n,a\right)\le c,$ and $\underset{n\to \infty }{lim}d\left(\left(1-{\alpha }_n\right)x_n\oplus {\alpha }_n{y}_n,a\right)=c,$ where ${\alpha }_n\in \left[a,b\right],$ with $0<a\le b<1$. Then $\underset{n\to \infty }{lim}d\left(x_n,y_n\right)=0.$.

Inspired and motivated by Shahzad and Udomene [24], the purpose of this paper is to establish common fixed point theorems for two asymptotically quasi-nonexpansive mappings in the setting of CAT(0) spaces. Our results significantly extend and improve the results obtained by Shahzad and Udomene in ref. [24], as well as the related results in the existing literature.

3. Main results

In this section, we let X denote a CAT(0) space and C be a nonempty closed convex subset of a CAT(0) space X. Let $T_1,T_2:C\to C$ be two asymptotically quasi-nonexpansive mappings with sequences $\left\{k_n^{\left(i\right)}\right\}\subset \left[1,\infty \right)$ satisfying ${\sum }_{n=1}^{\infty }\left(k_n^{\left(i\right)}-1\right)<\infty ,\left(i=1,2\right)$, respectively. Put $k_n=max\left\{k_n^{\left(1\right)},k_n^{\left(2\right)}\right\}$, then obviously ${\sum }_{n=1}^{\infty }\left(k_n-1\right)<\infty$. From now on we will take this sequence $\left\{k_n\right\}$ for both $T_1$ and $T_2$. Recall that $F\left(T\right)=\left\{x:\mathit{Tx}=x\right\}$ and $\mathbb{F}≔F\left(T_1\right)\cap F\left(T_2\right)=\left\{x\in C:T_{1}x=T_{2}x=x\right\}$. Following ref. [24], we introduce the following iterative scheme in the setting of CAT(0) space. Starting from arbitrary $x_1\in C,$

$$\begin{array}{ll}{x}_{n+1}& =\left(1-{\alpha }_n\right)x_n\oplus {\alpha }_n{T}_1^n{y}_n\\ y_n& =\left(1-{\beta }_n\right)x_n\oplus {\beta }_n{T}_2^n{x}_n,\end{array}$$E6

for all $n\ge 1$, where $\left\{{\alpha }_n\right\}$ and $\left\{{\beta }_n\right\}$ are sequences in $\left[0,1\right]$.

Lemma 3.1. Let $\left(\mathbb{X},d\right)$ be a CAT(0) space and C a nonempty closed convex subset of X. Let $T_1,T_2:C\to C$ be two asymptotically quasi-nonexpansive mappings and $\mathbb{F}=F\left(T_1\right)\cap F\left(T_2\right)\ne \varnothing$. Let $\left\{{\alpha }_n\right\}$ and $\left\{{\beta }_n\right\}$ be sequences in [0,1]. Define the sequence $\left\{x_n\right\}$ by iteration (6). Then.

1. $d\left(x_{n+1},p\right)\le \left(1+{\gamma }_n\right)d\left(x_n,p\right)$ for all $n\ge 1,p\in \mathbb{F}$, for some sequence of numbers $\left\{{\gamma }_n\right\}$ with ${\sum }_{n=1}^{\infty }{\gamma }_n<\infty$.

2. there exists a constant $K>0$ such that $d\left(x_{n+m},p\right)\le \mathit{Kd}\left(x_n,p\right)$ for all $n,m\ge 1$ and $p\in \mathbb{F}$.

Proof: (i). Taking $p\in \mathbb{F}$. Let $y_n=\left(1-{\beta }_n\right)x_n\oplus {\beta }_n{T}_2^n{x}_n$. From (6) and by using Lemma 2.4(ii) we get

$$\begin{array}{ll}d\left(x_{n+1},p\right)& =d\left(\left(1-{\alpha }_n\right)x_n\oplus {\alpha }_n{T}_1^n{y}_n,p\right)\\ & \le \left(1-{\alpha }_n\right)d\left(x_n,p\right)+{\alpha }_{n}d\left(T_1^n{y}_n,p\right)\\ & \le \left(1-{\alpha }_n\right)d\left(x_n,p\right)+{\alpha }_n{k}_{n}d\left(y_n,p\right)\\ & =\left(1-{\alpha }_n\right)d\left(x_n,p\right)+{\alpha }_n{k}_{n}d\left(\left(1-{\beta }_n\right)x_n\oplus {\beta }_n{T}_2^n{x}_n,p\right)\\ & \le \left(1-{\alpha }_n\right)d\left(x_n,p\right)+{\alpha }_n{k}_n\left[\left(1-{\beta }_n\right)d\left(x_n,p\right)+{\beta }_{n}d\left(T_2^n{x}_n,p\right)\right]\\ & \le \left(1-{\alpha }_n\right)d\left(x_n,p\right)+{\alpha }_n{k}_n\left[\left(1-{\beta }_n\right)d\left(x_n,p\right)+{\beta }_n{k}_{n}d\left(x_n,p\right)\right]\\ & =(\left(1-{\alpha }_n+{\alpha }_n{k}_n-{\alpha }_n{k}_n{\beta }_n+{\alpha }_n{\beta }_n{k}_n^2\right)d\left(x_n,p\right)\\ \le (\left(1+{\alpha }_n{k}_n+{\alpha }_n{\beta }_n{k}_n^2\right)d\left(x_n,p\right)\\ & =\left(1+{\gamma }_n\right)d\left(x_n,p\right)\end{array}$$E7

where ${\gamma }_n={\alpha }_n{k}_n+{\alpha }_n{\beta }_n{k}_n^2$ with ${\sum }_{n=1}^{\infty }{b}_n<\infty .$

1. We know that $1+x\le exp\left(x\right)$, for all $x\ge 0$. Notice that for any $n,m\ge 1,$

   $$\begin{array}{ll}d\left(x_{n+m},p\right)& \le \left(1+b_{n+m-1}\right)d\left(x_{n+m-1},p\right)\\ & \le exp\left(b_{n+m-1}\right)d\left(x_{n+m-1,p}\right)\\ & \le exp\left(b_{n+m-1}+b_{n+m-2}\right)d\left(x_{n+m-2,p}\right)\\ & ⋮\\ & \le exp\left(\sum _{k=n}^{n+m-1}{b}_k\right)d\left(x_n,p\right).\end{array}$$E8

Taking $K=exp\left({\sum }_{k=n}^{\infty }{b}_k\right).$ Then $0<K<\infty$, we obtain

$$d\left(x_{n+m},p\right)\le \mathit{Kd}\left(x_n,p\right)$$E9

where $p\in \mathbb{F}.$ This completes our proof.

Theorem 3.2. Let $\left(\mathbb{X},d\right)$ be a complete CAT(0) space and C a nonempty closed convex subset of X. Let $T_1,T_2:C\to C$ be two asymptotically quasi-nonexpansive mappings ($T_1$ and $T_2$ need not be continuous), and $\mathbb{F}=F\left(T_1\right)\cap F\left(T_2\right)\ne \varnothing$. Let $\left\{{\alpha }_n\right\},\left\{{\beta }_n\right\}$ be sequences in [0,1]. From arbitrary $x_1\in C$, define the sequence $\left\{x_n\right\}$ by iteration (6). Then $\left\{x_n\right\}$ converges strongly to a common fixed point of $T_1$ and $T_2$ if and only if $\underset{n\to \infty }{lim}d\left(x_n,\mathbb{F}\right)=0.$

Proof: The necessary conditions are obvious. We shall only prove the sufficient condition. By Lemma 3.1, we have $d\left(x_{n+1},p\right)\le \left(1+{\gamma }_n\right)d\left(x_n,p\right)$ for all $n\ge 1$ and $p\in \mathbb{F}.$ Therefore,

$$d\left(x_{n+1},\mathbb{F}\right)\le \left(1+{\gamma }_n\right)d\left(x_n,\mathbb{F}\right).$$

Since ${\sum }_{n=1}^{\infty }{\gamma }_n<\infty$ and $lim{inf}_{n\to \infty }d\left(x_n,\mathbb{F}\right)=0$, from Lemma 2.6 we deduce that $\underset{n\to \infty }{lim}d\left(x_n,\mathbb{F}\right)=0$. Next, we show that the sequence $\left\{x_n\right\}$ is Cauchy. Since $\underset{n\to \infty }{lim}d\left(x_n,\mathbb{F}\right)=0$, given any $\epsilon >0$, there exists a positive number $N_0$ such that $d\left(x_n,\mathbb{F}\right)<\frac{\epsilon }{4K}$ for all $n\ge N_0$, where $K>0$ is the constant in Lemma 3.1(2). So we can find $q\in \mathbb{F}$ such that $d\left(x_{N_0},q\right)\le \frac{\epsilon }{3K}$. Again by Lemma 3.1(2), we have that

$$\begin{array}{ll}d\left(x_{n+m},x_n\right)& \le d\left(x_{n+m},q\right)+d\left(x_n,q\right)\\ & \le \mathit{Kd}\left(x_{N_0},q\right)+\mathit{Kd}\left(x_{N_0},q\right)\\ & =2\mathit{Kd}\left(x_{N_0},q\right)<\epsilon .\end{array}$$E10

for all $n\ge N_0$ and $m\ge 1$. This implies that $\left\{x_n\right\}$ is Cauchy and so is convergent since $\mathbb{X}$ is complete. Hence, $\left\{x_n\right\}$ is a Cauchy sequence in a closed convex subset C of a CAT(0) space $\mathbb{X}$, therefore, it must converge to a point in $C$. Let $\underset{n\to \infty }{lim}{x}_n=q^{\prime }.$ Now, $\underset{n\to \infty }{lim}d\left(x_n,\mathbb{F}\right)=0$ yields that $d\left(q^{\prime },\mathbb{F}\right)=0$. Since the set of fixed points of asymptotically nonexpansive mappings is closed, we have $q^{\prime }\in \mathbb{F}$. This completes our proof.

Lemma 3.3. Let $\left(\mathbb{X},d\right)$ be a CAT(0) space and C a nonempty closed convex subset of X. Let $T_1,T_2:C\to C$ be two uniformly continuous asymptotically quasi-nonexpansive mappings, and $\mathbb{F}=F\left(T_1\right)\cap F\left(T_2\right)\ne \varnothing$. Let $\left\{{\alpha }_n\right\}$ and $\left\{{\beta }_n\right\}$ be sequences in $\left[\epsilon ,1-\epsilon \right]$ forsome $\epsilon \in \left(0,1\right)$. From arbitrary $x_1\in C$, define the sequence $\left\{x_n\right\}$ by iteration (6). Then

$$\underset{n\to \infty }{lim}d\left(x_n,T_2^n{x}_n\right)=\underset{n\to \infty }{lim}d\left(x_n,T_1^n{x}_n\right)=\underset{n\to \infty }{lim}d\left(x_n,T_1^n{y}_n\right)=0.$$E11

Proof: Let $p\in \mathbb{F}$. Then, by Lemma 3.1(1) and Lemma 2.6 $\underset{n\to \infty }{lim}d\left(x_n,p\right)$ exists. Suppose $\underset{n\to \infty }{lim}d\left(x_n,p\right)=r$. If $r=0$, then by the continuity of $T_1$ and $T_2$ the conclusion follows. Now suppose $r>0$. We claim

$$\underset{n\to \infty }{lim}d\left(x_n,T_1^n{y}_n\right)=\underset{n\to \infty }{lim}d\left(x_n,T_1^n{x}_n\right)=\underset{n\to \infty }{lim}d\left(x_n,T_2^n{x}_n\right)=0.$$E12

From $y_n=\left(1-{\beta }_n\right)x_n\oplus {\beta }_n{T}_2^n{x}_n$. Since $\left\{x_n\right\}$ is bounded, there exists $R>0$ such that $x_n-p,y_n-p\in B_R\left(0\right)$ for all $n\ge 1$. Using Lemma 2.4(iii), we have that

$$\begin{array}{ll}d{\left(y_n,p\right)}^2& =d{\left(\left(1-{\beta }_n\right)x_n\oplus {\beta }_n{T}_2^n{x}_n,p\right)}^2\\ & \le \left(1-{\beta }_n\right)d{\left(x_n,p\right)}^2+{\beta }_{n}d{\left(T_2^n{x}_n,p\right)}^2-{\beta }_n\left(1-{\beta }_n\right)d{\left(x_n,T_2^n{x}_n\right)}^2\\ & \le \left(1-{\beta }_n\right)d{\left(x_n,p\right)}^2+{\beta }_n{k}_n^{2}d{\left(x_n,p\right)}^2-{\beta }_n\left(1-{\beta }_n\right)d{\left(x_n,T_2^n{x}_n\right)}^2\\ & \le \left(1+{\beta }_n\left(k^2-1\right)\right)d{\left(x_n,p\right)}^2\le d{\left(x_n,p\right)}^2.\end{array}$$E13

Again by Lemma 2.4(iii), it follows that

$$\begin{array}{ll}d{\left(x_{n+1},p\right)}^2& =d{\left(\left(1-{\alpha }_n\right)x_n\oplus {\alpha }_n{T}_1^n{y}_n,p\right)}^2\\ & \le \left(1-{\alpha }_n\right)d{\left(x_n,p\right)}^2+{\alpha }_{n}d{\left(T_1^n{x}_n,p\right)}^2-{\alpha }_n\left(1-{\alpha }_n\right)d{\left(x_n,T_1^n{y}_n\right)}^2\\ & \le \left(1-{\alpha }_n\right)d{\left(x_n,p\right)}^2+{\alpha }_n{k}_n^{2}d{\left(x_n,p\right)}^2-{\alpha }_n\left(1-{\alpha }_n\right)d{\left(x_n,T_1^n{y}_n\right)}^2.\\ & \end{array}$$E14

Equivalently

$$\begin{array}{ll}{\alpha }_n\left(1-{\alpha }_n\right)d{\left(x_n,T_1^n{y}_n\right)}^2& \le \left(1+{\alpha }_n\left(k_n^2-1\right)\right)d{\left(x_n,p\right)}^2-d{\left(x_{n+1},p\right)}^2\\ & \le \left(1+\left(k_n^2-1\right)\right)d{\left(x_n,p\right)}^2-d{\left(x_{n+1},p\right)}^2\\ & =d{\left(x_n,p\right)}^2-d{\left(x_{n+1},p\right)}^2.\\ & \end{array}$$E15

Summing up the first m term of the above inequality, we get

$$\sum _{n=1}^m{\alpha }_n\left(1-{\alpha }_n\right)d{\left(x_n,T_1^n{y}_n\right)}^2\le d{\left(x_1,p\right)}^2-d{\left(x_{m+1},p\right)}^2<\infty$$E16

for all $m\ge 1.$ Now (16) implies that

$$\sum _{n=1}^{\infty }{\alpha }_n\left(1-{\alpha }_n\right)d{\left(x_n,T_1^n{y}_n\right)}^2<\infty .$$E17

Since $0\le {\alpha }_n\left(1-{\alpha }_n\right)<1$, $d{\left(x_n,T_1^n{x}_n\right)}^2\to 0$ as $n\to \infty$. Therefore, we obtain

$$\underset{n\to \infty }{lim}d\left(x_n,T_1^n{y}_n\right)=0.\left(\ast \right)$$E18

Since $T_1$ is asymptotically quasi-nonexpansive, we can get that $d\left(T_1^n{y}_n,p\right))\le k_{n}d\left(y_n,p\right)$ for all $n\in \mathrm{\mathbb{N}}$. From (13), we have that

$$\underset{n\to \infty }{limsup}d\left(T_1^n{y}_n,p\right)\le r.$$E19

Similarly, we get

$$\underset{n\to \infty }{limsup}d\left(T_2^n{x}_n,p\right)\le r.$$E20

One can see that

$$\underset{n\to \infty }{limsup}d\left(x_n,p\right)\le \underset{n\to \infty }{lim}d\left(x_n,p\right)=r.$$E21

Since $T_1$ is asymptotically quasi-nonexpansive, we get

$$\begin{array}{ll}d\left(x_n,p\right)& \le d\left(x_n,T_1^n{y}_n\right)+d\left(T_1^n{y}_n,p\right)\\ & \le d\left(x_n,T_1^n{y}_n\right)+k_{n}d\left(y_n,p\right).\end{array}$$E22

Taking the limit inferior to above inequality and from (18), we obtain

$$r\le \underset{n\to \infty }{liminf}d\left(y_n,p\right).$$E23

On the other hand, by Lemma 2.4(ii) we have

$$\begin{array}{ll}d\left(y_n,p\right)& =d\left(\left(1-{\beta }_n\right)x_n\oplus {\beta }_n{T}_2^n{x}_n,p\right)\\ & \le \left(1-{\beta }_n\right)d\left(x_n,p\right)+{\beta }_{n}d\left(T_2^n{x}_n,p\right)\\ & =\left[\right(1+{\beta }_n\left(k_n-1\right)]d\left(x_n,p\right)\end{array}$$E24

which implies

$$\underset{n\to \infty }{limsup}d\left(y_n,p\right)\le r.$$E25

This gives

$$\underset{n\to \infty }{lim}d\left(\left(1-{\alpha }_n\right)x_n\oplus {\alpha }_n{T}_2^n{x}_n,p\right)=r.$$E26

Using (20), (21), (26), and Lemma 2.9, we obtain

$$\underset{n\to \infty }{lim}d\left(x_n,T_2^n{x}_n\right)=0.\left(\ast \right)$$E27

From (23) and (25), we obtain

$$\underset{n\to \infty }{lim}d\left(y_n,p\right)=r.$$E28

On the other hand, consider

$$\begin{array}{ll}d\left(x_{n+1},p\right)& =d\left(\left(1-{\alpha }_n\right)x_n\oplus {\alpha }_n{T}_1^n{y}_n,p\right)\\ & \le \left(1-{\alpha }_n\right)d\left(x_n,p\right)+{\alpha }_n{k}_{n}d\left(y_n,p\right).\end{array}$$E29

This implies

$$\underset{n\to \infty }{lim}d\left(\left(1-{\alpha }_n\right)x_n\oplus {\alpha }_n{T}_1^n{y}_n,p\right)=r.$$E30

From (19), (21), (30), and by Lemma 2.9, we also obtain

$$\underset{n\to \infty }{lim}d\left(x_n,T_1^n{y}_n\right)=0.$$E31

Next, we show $\underset{n\to \infty }{lim}d\left(x_n,T_1^n{x}_n\right)=0.$.

Consider

$$\begin{array}{ll}d\left(x_n,y_n\right)& \le d\left(x_n,\left(1-{\beta }_n\right)x_n\oplus {\beta }_n{T}_2^n{x}_n\right)\\ & \le \left(1-{\beta }_n\right)d\left(x_n,x_n\right)+{\beta }_{n}d\left(x_n,T_2^n{x}_n\right)\to 0\mathrm{as}n\to \infty \end{array}$$E32

and

$$\begin{array}{ll}d\left(T_1^n{x}_n,x_n\right)& \le d\left(T_1^n{x}_n,T_1^n{y}_n\right)+d\left(T_1^n{y}_n,x_n\right).\\ & \end{array}$$E33

Since $T_1$ is uniformly continuous and $d\left(x_n,y_n\right)\to 0$ as $n\to \infty$, it follows from (32) and (33) that

$$\underset{n\to \infty }{lim}d\left(T_1^n{x}_n,x_n\right)=0.\left(\ast \right)$$

Our proof is finished.

Theorem 3.4. Let $\left(\mathbb{X},d\right)$ be a CAT(0) space and $C$ a nonempty closed convex subset of $\mathbb{X}$. Let $T_1,T_2:C\to C$ be two uniformly continuous asymptotically quasi-nonexpansive mappings, and $\mathbb{F}=F\left(T_1\right)\cap F\left(T_2\right)\ne \varnothing$. Let $\left\{{\alpha }_n\right\}$ and $\left\{{\beta }_n\right\}$ be sequences in $\left[\epsilon ,1-\epsilon \right]$ forsome $\epsilon \in \left(0,1\right)$. From arbitrary $x_1\in C$, define the sequence $\left\{x_n\right\}$ by the recursion (6). Assume, in addition, that either $T_2$ or $T_1$ is compact. Then $\left\{x_n\right\}$ converges strongly to some common fixed point of $T_1$ and $T_2$.

Proof: By Lemma 3.3, we have

$$\underset{n\to \infty }{lim}d\left(x_n,T_1^n{x}_n\right)=0=\underset{n\to \infty }{lim}d\left(x_n,T_2^n{x}_n\right)$$E34

and also

$$\underset{n\to \infty }{lim}d\left(x_n,T_1^n{y}_n\right)=0.$$E35

If $T_2$ is compact, then there exists a subsequence $\left\{T_2^{n_k}{x}_{n_k}\right\}$ of $\left\{T_2^n{x}_n\right\}$ such that $T_2^{n_k}{x}_{n_k}\to p$ as $k\to \infty$ for some $p\in C$ and so $T_2^{n_k+1}{x}_{n_k}\to T_{2}p$ as $k\to \infty$. From (34), we have $x_{n_k}\to p$ as $k\to \infty$. Also, by (35) we get that $T_1^{n_k}{y}_{n_k}\to p$ as $k\to \infty$. Consider

$$\begin{array}{l}d\left(x_{n_k+1},x_{n_k}\right)=d\left(\left(1-{\alpha }_{n_k}\right)x_{n_k}\oplus {\alpha }_{n_k}{T}_1^{n_k}{y}_{n_k},x_{n_k}\right)\\ \le d\left(x_{x_k},T_1^{n_k}{y}_{n_k}\right).\end{array}$$E36

From (35) and (36), it follows that $x_{n_k+1}\to p$ as $k\to \infty$. Again, from (35), we have $T_1^{n_k+1}{y}_{n_k}\to T_{1}p$.

Next, we show that $p\in \mathbb{F}$. Notice that

$$\begin{array}{l}d\left(p,T_{2}p\right)\le d\left(p,x_{n_k+1}\right)+d\left(x_{n_k+1},T_2^{n_k+1}{x}_{n_k+1}\right)\\ +d\left(T_2^{n_k+1}{x}_{n_k+1},T_2^{n_k+1}{x}_{n_k}\right)+d\left(T_2^{n_k+1}{x}_{n_k},T_{2}p\right)\\ & \end{array}$$E37

Since $T_2$ is uniformly continuous, taking the limit as $k\to \infty$, and using (34) we obtain that $p=T_{2}p$ and so $p\in F\left(T_2\right).$ Notice that

$$\begin{array}{l}d\left(p,T_{1}p\right)\le d\left(p,x_{n_k+1}\right)+d\left(x_{n_k+1},T_1^{n_k+1}{x}_{n_k+1}\right)\\ +d\left(T_1^{n_k+1}{x}_{n_k+1},T_1^{n_k+1}{x}_{n_k}\right)+d\left(T_1^{n_k+1}{x}_{n_k},T_{1}p\right).\end{array}$$E38

Letting $k\to \infty$, we also obtain that $p=T_{1}p$ and hence $p\in F\left(T_1\right)$. Therefore $p\in \mathbb{F}$. Hence, by Lemma 2.6, $x_n\to p\in \mathbb{F}$, since $\underset{n\to \infty }{lim}d\left(x_n,p\right)$ exists. If $T_1$ is compact, then essentially the same arguments as above our result follow. This completes the proof.

We give the following example in support of our main results.

Example 3.5. Let $\mathbb{X}={\mathbb{R}}^1.$ and $C=\left[0,1\right]$_, a closed convex subset of $\mathbb{X}$ and define $T_1,T_2:C\to C$ by

$$T_{1}x=\left\{\begin{array}{ll}\frac{x}{2}& ,\text{if}x\in \left[0,\frac{1}{2}\right]\\ 0& ,\text{if}x\in \left(\frac{1}{2},1\right]\end{array}\right.$$

and

$$T_{2}x=\left\{\begin{array}{ll}x& ,\text{if}x\in \left[0,\frac{1}{2}\right]\\ \frac{1}{2}& ,\text{if}x\in \left(\frac{1}{2},1\right].\end{array}\right.$$

Then, $T_1,T_2$ are asymptotically quasi-nonexpansive but not nonexpansive with $\mathbb{F}=\left\{0\right\}\ne \varnothing$. For

$$d\left(T_1^{n}x,T_1^{n}y\right)\le \frac{1}{2^n}d\left(x,y\right)\le d\left(x,y\right),\forall x,y\in \left[0,\frac{1}{2}\right].$$

And

$$d\left(T_1^{n}x,T_1^{n}y\right)=0\le d\left(x,y\right),\forall x,y\in \left(\frac{1}{2},1\right].$$

Hence, $T_1$ is asymptotically quasi-nonexpansive. Similarly, we can show that $T_2$ is asymptotically quasi-nonexpansive.

Define a sequence $\left\{x_n\right\}$ as in (6) by starting from arbitrary $x_1\in C,$

$$\begin{array}{ll}{x}_{n+1}& =\left(1-{\alpha }_n\right)x_n+{\alpha }_n{T}_1^n{y}_n\\ y_n& =\left(1-{\beta }_n\right)x_n+{\beta }_n{T}_2^n{x}_n,\end{array}$$E39

for all $n\ge 1$, where $\left\{{\alpha }_n\right\}$ and $\left\{{\beta }_n\right\}$ are sequences in $\left[0,1\right]$. Taking ${\alpha }_n=\frac{1}{2}={\beta }_n.$

Next, we construct a sequence $\left\{x_n\right\}$. Starting from $x_1=1$, we get

$$y_1=\frac{1}{2}{x}_1+\frac{1}{2}{T}_2{x}_1=\frac{1}{2}\left(1\right)+\frac{1}{2}{T}_2\left(1\right)=\frac{1}{2}\left(1\right)+\frac{1}{2}\left(\frac{1}{2}\right)=\frac{3}{4},$$

we get

$$x_2=\frac{1}{2}{x}_1+\frac{1}{2}{T}_1{y}_1=\frac{1}{2}\left(1\right)+\frac{1}{2}{T}_1\left(\frac{3}{4}\right)=\frac{1}{2}+\frac{1}{2}\left(0\right)=\frac{1}{2}=0.5.$$

$$y_2=\frac{1}{2}{x}_2+\frac{1}{2}{T}_2^2{x}_2=\frac{1}{2}\left(\frac{1}{2}\right)+\frac{1}{2}{T}_2^2\left(\frac{1}{2}\right)=\frac{1}{4}+\frac{1}{2}\left(\frac{1}{2}\right)=\frac{1}{2},$$

we get

$$x_3=\frac{1}{2}{x}_2+\frac{1}{2}{T}_1^2{y}_2=\frac{1}{2}\left(\frac{1}{2}\right)+\frac{1}{2}{T}_1^2\left(\frac{1}{2}\right)=\frac{1}{4}+\frac{1}{2}\left(\frac{1}{8}\right)=\frac{5}{16}=0.3125.$$

$$y_3=\frac{1}{2}{x}_3+\frac{1}{2}{T}_2^3{x}_3=\frac{1}{2}\left(\frac{5}{16}\right)+\frac{1}{2}{T}_2^3\left(\frac{5}{16}\right)=\frac{5}{32}+\frac{1}{2}\left(\frac{5}{16}\right)=\frac{5}{16},$$

we get

$$x_4=\frac{1}{2}{x}_3+\frac{1}{2}{T}_1^3{y}_3=\frac{1}{2}\left(\frac{5}{16}\right)+\frac{1}{2}{T}_1^3\left(\frac{5}{16}\right)=\frac{5}{32}+\frac{1}{2}\left(\frac{5}{128}\right)=\frac{45}{256}=0.1757.$$

$$y_4=\frac{1}{2}{x}_4+\frac{1}{2}{T}_2^4{x}_4=\frac{1}{2}\left(\frac{45}{256}\right)+\frac{1}{2}{T}_2^4\left(\frac{45}{256}\right)=\frac{45}{512}+\frac{1}{2}\left(\frac{45}{256}\right)=\frac{45}{256},$$

we get

$$x_5=\frac{1}{2}{x}_4+\frac{1}{2}{T}_1^4{y}_4=\frac{1}{2}\left(\frac{45}{256}\right)+\frac{1}{2}{T}_1^4\left(\frac{45}{256}\right)=\frac{45}{512}+\frac{1}{2}\left(\frac{45}{4096}\right)=\frac{765}{8192}=0.0933.$$

Proceeding in a similar method, we will get a sequence $\left\{x_n\right\}$ that converges to 0, the common fixed point of $T_1$ and $T_2$, that is, we obtain the sequence

$$1,\frac{1}{2},\frac{5}{16},\frac{45}{256},\frac{765}{8192},\dots ,x_n\to 0.$$

Corollary 3.6. Let $\mathbb{X}$ be a CAT(0) space and C a nonempty compact convex subset of X. Let $T_1,T_2:C\to C$ be two uniformly continuous asymptotically quasi-nonexpansive mappings, and $\mathbb{F}=F\left(T_1\right)\cap F\left(T_2\right)\ne \varnothing$. Let $\left\{{\alpha }_n\right\}$ and $\left\{{\beta }_n\right\}$ be sequences in $\left[\epsilon ,1-\epsilon \right]$ forsome $\epsilon \in \left(0,1\right)$. From arbitrary $x_1\in C$, define the sequence $\left\{x_n\right\}$ by iteration (6). Assume, in addition, that either $T_2$ or $T_1$ is compact. Then, $\left\{x_n\right\}$ converges strongly to some common fixed point of $T_1$ and $T_2$.

Corollary 3.7. Let $\mathbb{X}$ be a CAT(0) space and $C$ a nonempty compact convex subset of $\mathbb{X}$. Let $T:C\to C$ be two uniformly continuous asymptotically quasi-nonexpansive mappings with sequences $\left\{k_n\right\}\subset \left[0,\infty \right)$ such that ${\sum }_{n=1}^{\infty }{k}_n<\infty$. Let $\left\{{\alpha }_n\right\}$ and $\left\{{\beta }_n\right\}$ be sequences in $\left[\epsilon ,1-\epsilon \right]$ forsome $\epsilon \in \left(0,1\right)$. From arbitrary $x_1\in K$, define the sequence $\left\{x_n\right\}$ by the iteration

$$\begin{array}{ll}{x}_{n+1}& =\left(1-{\alpha }_n\right)x_n\oplus {\alpha }_n{T}^n{y}_n\\ y_n& =\left(1-{\beta }_n\right)x_n\oplus {\beta }_n{T}^n{x}_n,\end{array}$$E40

with $n\ge 1.$ Then, $\left\{x_n\right\}$ converges strongly to some fixed point of T.

Corollary 3.8. Let X be a Hibert space and C a nonempty closed convex subset of X. Let $T_1,T_2:C\to C$ be two uniformly continuous asymptotically quasi-nonexpansive mappings, and $\mathbb{F}=F\left(T_1\right)\cap F\left(T_2\right)≔\left\{x\in K:T_{1}x=T_{2}x=x\right\}\ne \varnothing$. Let $\left\{{\alpha }_n\right\}$ and $\left\{{\beta }_n\right\}$ be sequences in $\left[\epsilon ,1-\epsilon \right]$ forsome $\epsilon \in \left(0,1\right)$. From arbitrary $x_1\in C$, define the sequence $\left\{x_n\right\}$ by the iterative scheme (6). Assume, in addition, that either $T_2$ or $T_1$ is compact. Then, $\left\{x_n\right\}$ converges strongly to some common fixed point of $T_1$ and $T_2$.

4. Conclusions

In this chapter, we establish strong convergence results for two asymptotically quasi-nonexpansive mappings $T_1,T_2$ in the setting of CAT(0) spaces via the sequence $\left\{x_n\right\}$ generated iteratively by arbitrary $x_1\in C$, $x_{n+1}=\left(1-{\alpha }_n\right)x_n\oplus {\alpha }_n{T}_1^n{y}_n$, $y_n=\left(1-{\beta }_n\right)x_n\oplus {\beta }_n{T}_2^n{x}_n,n\ge 1.$ We obtained the following results:-.

1. Lemma 3.1, an extension of Theorem 1.2 (See [24], Theorem 3.1).

2. Theorem 3.2, it is proved that the sequence $\left\{x_n\right\}$ converges strongly to a common fixed point of $T_1$ and $T_2$ if and only if $\underset{n\to \infty }{lim}d\left(x_n,\mathbb{F}\right)=0.$ (Note that $T_1$ and $T_2$ need not be continuous). This theorem extends and improves Theorem 1.3 (See [24] Theorem 3.2).

3. Lemma 3.3, it is proved that

   $$\underset{n\to \infty }{lim}d\left(x_n,T_2^n{x}_n\right)=\underset{n\to \infty }{lim}d\left(x_n,T_1^n{x}_n\right)=\underset{n\to \infty }{lim}d\left(x_n,T_1^n{y}_n\right)=0.$$

   This lemma extends and improves Theorem 3.3 in [24].

4. Theorem 3.4, it is proved that If $T_1,T_2:C\to C$ be two uniformly continuous asymptotically quasi-nonexpansive mappings. Suppose, in addition, that either $T_2$ or $T_1$ is compact. Then, $\left\{x_n\right\}$ converges strongly to some common fixed point of $T_1$ and $T_2$. This theorem significantly extends and improves Theorem 1.3 (See [24], Theorem 3.4).

As consequence, we obtain Corollaries 3.6, 3.7, and 3.8. All of our results remain true for the subclass of asymptotically nonexpansive mappings.

Acknowledgments

The authors would like to thank the referee for his/her useful comments on the improvement of this chapter. The second author wishes to thank the faculty of Science of Maejo University for moral support in the writing of this manuscript.

Competing interests

The authors declare that they have no competing interests.