Strong convergence of an Ishikawa-type algorithm in $CAT(0)$ spaces

Abstract

We study strong convergence of an Ishikawa-type algorithm of two asymptotically nonexpansive type maps to their common fixed point on a $CAT(0)$ space. Our work provides an affirmative answer to the question of Tan and Xu (Proc. Am. Math. Soc. 122:733-739, 1994); in particular, strong convergence of an Ishikawa-type algorithm of two asymptotically nonexpansive maps without the rate of convergence condition is obtained on a nonlinear domain.

MSC:47H09, 47H10, 49M05.

1 Introduction

A $CAT(0)$ space is simply a geodesic metric space whose each geodesic triangle is at least as thin as its comparison triangle in the Euclidean plane. In 2004, Kirk [1] proved a fixed point theorem for a nonexpansive map defined on a subset of a $CAT(0)$ space. Since then, approximation of fixed points of nonlinear maps on a $CAT(0)$ space has rapidly developed (see, e.g., [2–5]).

We describe briefly the needed details for a $CAT(0)$ space. A metric space $(X,d)$ is said to be a length space if any two points of X are joined by a rectifiable path (that is, a path of finite length) and the distance between any two points of X is taken to be the infimum of the lengths of all rectifiable paths joining them. In this case, d is said to be a length metric (otherwise known as an inner metric or intrinsic metric). In case no rectifiable path joins two points of the space, the distance between them is taken to be ∞.

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 $[0,l]\subset \mathbb{R}$ 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 α of c is called a geodesic (or metric) segment joining x and y. We say that X is: (i) a geodesic space if any two points of X are joined by a geodesic, and (ii) uniquely geodesic if there is exactly one geodesic joining x and y for each $x,y\in X$, which we will denote by $[x,y]$, called the segment joining x to y.

A geodesic triangle $\mathrm{\Delta }(x_1,x_2,x_3)$ in a geodesic metric space $(X,d)$ consists of three points 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 [6]).

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.

Let Δ be a geodesic triangle in 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 $\overline{x},\overline{y}\in \overline{\mathrm{\Delta }}$,

$$d(x,y)\le d(\overline{x},\overline{y}).$$

If x, $y_1$, $y_2$ are points of a $CAT(0)$ space and $y_0$ 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

$$\begin{array}{rcl}d{(x,\frac{y_1\oplus y_2}{2})}^2& \le & \frac{1}{2}d{(x,y_1)}^2+\frac{1}{2}d{(x,y_2)}^2\\ -\frac{1}{4}d{(y_1,y_2)}^2.\end{array}$$

The above inequality is the (CN) inequality of Bruhat and Titz [7] and it was extended in [8] as follows:

$$\begin{array}{rcl}d{(z,\alpha x\oplus (1-\alpha )y)}^2& \le & \alpha d{(z,x)}^2+(1-\alpha )d{(z,y)}^2\\ -\alpha (1-\alpha )d{(x,y)}^2\end{array}$$

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

Let us recall that a geodesic metric space is a $CAT(0)$ space if and only if it satisfies the (CN) inequality (see [6], p. 163). Moreover, if X is a $CAT(0)$ metric space and $x,y\in X$, then for any $\alpha \in [0,1]$, there exists a unique point $\alpha x\oplus (1-\alpha )y\in [x,y]$ such that

$$d(z,\alpha x\oplus (1-\alpha )y)\le \alpha d(z,x)+(1-\alpha )d(z,y)$$

for any $z\in X$ and $[x,y]=\{\alpha x\oplus (1-\alpha )y:\alpha \in [0,1]\}$.

A subset C of a $CAT(0)$ space X is convex if for any $x,y\in C$, we have $[x,y]\subset C$.

Complete $CAT(0)$ spaces are known as Hadamard spaces (see [9]). The reader interested in a more general nonlinear domain, namely 2-uniformly convex hyperbolic space containing a $CAT(0)$ space as a special case, is referred to Dehaish [10] and Dehaish et al. [11].

Let C be a nonempty subset of a metric space $(X,d)$. Then a selfmap T on C is:

• (i) uniformly L-Lipschitzian if for some $L>0$, $d(T^{n}x,T^{n}y)\le Ld(x,y)$ for $x,y\in C$, $n\ge 1$;

• (ii) uniformly Hölder continuous if for some positive constants L and α, $d(T^{n}x,T^{n}y)\le Ld{(x,y)}^{\alpha }$ for $x,y\in C$, $n\ge 1$;

• (iii) uniformly equicontinuous if for any $\epsilon >0$, there exists $\delta >0$ such that $d(T^{n}x,T^{n}y)\le \epsilon$ whenever $d(x,y)\le \delta$ for $x,y\in C$, $n\ge 1$ or, equivalently, T is uniformly equicontinuous if and only if $d(T^n{x}_n,T^n{y}_n)\to 0$ whenever $d(x_n,y_n)\to 0$ as $n\to \mathrm{\infty }$;

• (iv) asymptotically nonexpansive if there is a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with ${lim}_{n\to \mathrm{\infty }}{k}_n=1$ such that $d(T^{n}x,T^{n}y)\le k_{n}d(x,y)$ for $x,y\in C$, $n\ge 1$;

• (v) asymptotically nonexpansive in the intermediate sense provided T is uniformly continuous and $lim{sup}_{n\to \mathrm{\infty }}{sup}_{x,y\in C}\{d(T^{n}x,T^{n}y)-d(x,y)\}\le 0$ for $n\ge 1$, and

• (vi) of asymptotically nonexpansive type in the sense of Xu [12] if $lim{sup}_{n\to \mathrm{\infty }}{sup}_{x\in C}\{d(T^{n}x,T^{n}y)-d(x,y)\}\le 0$ for each $y\in C$, $n\ge 1$;

• (vii) of asymptotically nonexpansive type in the sense of Chang et al. [13] if $lim{sup}_{n\to \mathrm{\infty }}{sup}_{x\in C}\{d{(T^{n}x,T^{n}y)}^2-d{(x,y)}^2\}\le 0$ for each $y\in C$, $n\ge 1$.

The map T is semi-compact if for any bounded sequence $\{x_n\}$ in C with $d(x_n,T{x}_n)\to 0$ as $n\to \mathrm{\infty }$, there is a subsequence $\{x_{n_i}\}$ of $\{x_n\}$ such that $x_{n_i}\to x^{\ast }\in C$ as $n_i\to \mathrm{\infty }$.

It is not difficult to see that nonexpansive map, asymptotically nonexpansive map, asymptotically nonexpansive map in the intermediate sense and asymptotically nonexpansive type map in the sense of Xu [12] all are special cases of asymptotically nonexpansive type map in the sense of Chang et al. [13]. Moreover, a uniformly L-Lipschitzian map is uniformly Hölder continuous, and a uniformly Hölder continuous map is uniformly equicontinuous. However, the converse statements are not true as indicated below.

Example 1.1 Take $X=\mathbb{R}$ and $C=[0,1]$. Define $T:C\to C$ by $Tx={(1-x^{\frac{3}{2}})}^{\frac{2}{3}}$ for all $x\in C$. Then T is uniformly equicontinuous, but it is neither uniformly L-Lipschitzian nor uniformly Hölder continuous.

In uniformly convex Banach spaces, the convergence of an Ishikawa-type algorithm and a Mann-type algorithm of nonexpansive maps, asymptotically nonexpansive maps and asymptotically nonexpansive maps in the intermediate sense to their fixed points have been studied by a number of researchers [12, 14–24]. For the iterative construction of fixed points of some other classes of nonlinear maps, see [25–27].

The sequence $\{k_n\}$ in definition (iv) satisfies the rate of convergence condition if ${\sum }_{n=1}^{\mathrm{\infty }}(k_n-1)<\mathrm{\infty }$. This condition has been extensively used in iterative construction of fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces and $CAT(0)$ spaces (see, e.g., [4, 5, 21, 28, 29]).

Chang et al. [13] established strong convergence of an Ishikawa-type algorithm as well as a Mann-type algorithm to a fixed point of an asymptotically nonexpansive type map.

We shall follow the idea of a geodesic path, namely, there exists a unique point $\alpha x\oplus (1-\alpha )y$ for any $x,y\in C$ and $\alpha \in [0,1]$, to construct an Ishikawa-type algorithm of two asymptotically nonexpansive type maps on a nonempty subset C of a $CAT(0)$ space.

$$\begin{array}{r}{x}_1\in C,\\ x_{n+1}=(1-{\alpha }_n)x_n\oplus {\alpha }_n{S}^n{y}_n,\\ y_n=(1-{\beta }_n)x_n\oplus {\beta }_n{T}^n{x}_n,n\ge 1,\end{array}$$

(1.1)

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

When $T=I$ (the identity map) in (1.1), it reduces to the following Mann-type algorithm:

$$\begin{array}{r}{x}_1\in C,\\ x_{n+1}=(1-{\alpha }_n)x_n\oplus {\alpha }_n{T}^n{y}_n,n\ge 1,\end{array}$$

(1.2)

where $0\le {\alpha }_n\le 1$.

The purpose of this paper is to approximate a common fixed point of asymptotically nonexpansive type maps in a special kind of a metric space, namely a $CAT(0)$ space. Our work is a significant generalization of the corresponding results in [5], and it provides analogues of the related results of Chang et al. [13] in uniformly convex Banach spaces. One of our results (Theorem 2.4) gives an affirmative answer to a famous question of Tan and Xu [30] on a nonlinear domain for common fixed points.

2 Fixed point approximation

We begin with the following asymptotic regularity result.

Lemma 2.1 Let C be a nonempty bounded closed convex subset of a $CAT(0)$ space X. Let $S,T:C\to C$ be uniformly equicontinuous. Then for the sequence $\{x_n\}$ in (1.1) satisfying

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,S^n{x}_n)=0=\underset{n\to \mathrm{\infty }}{lim}d(x_n,T^n{x}_n),$$

we have that

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,S{x}_n)=0=\underset{n\to \mathrm{\infty }}{lim}d(x_n,T{x}_n).$$

Proof Since S is uniformly equicontinuous and

$$\begin{array}{rcl}d(x_n,y_n)& =& d(x_n,(1-{\beta }_n)x_n\oplus {\beta }_n{T}^n{x}_n)\\ \le & (1-{\beta }_n)d(x_n,x_n)+{\beta }_{n}d(x_n,T^n{x}_n)\\ =& {\beta }_{n}d(x_n,T^n{x}_n)\to 0,\end{array}$$

therefore,

$$d(S^n{x}_n,S^n{y}_n)\to 0.$$

Now

$$\begin{array}{rcl}d(x_n,x_{n+1})& =& d(x_n,(1-{\alpha }_n)x_n\oplus {\alpha }_n{S}^n{y}_n)\\ \le & {\alpha }_{n}d(x_n,S^n{y}_n)\\ \le & d(x_n,S^n{x}_n)+d(S^n{x}_n,S^n{y}_n)\end{array}$$

gives that

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,x_{n+1})=0.$$

(2.1)

Clearly,

$$\begin{array}{rl}d(x_n,S{x}_n)\le & d(x_n,x_{n+1})+d(x_{n+1},S^{n+1}{x}_{n+1})\\ +d(S^{n+1}{x}_{n+1},S^{n+1}{x}_n)+d(S^{n+1}{x}_n,S{x}_n),\end{array}$$

(2.2)

applying limsup to both sides of (2.2), using the uniformly equicontinuous property of S and (2.1), we get that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(x_n,S{x}_n)\le 0$$

and hence

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,S{x}_n)=0.$$

Similarly,

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,T{x}_n)=0.$$

That is,

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,S{x}_n)=0=\underset{n\to \mathrm{\infty }}{lim}d(x_n,T{x}_n).$$

 □

Our main result is as follows.

Theorem 2.2 Let C be a nonempty, bounded, closed and convex subset of a $CAT(0)$ space X. Let $S,T:C\to C$ be uniformly equicontinuous and asymptotically nonexpansive type maps such that $F(S)\cap F(T)\ne \mathrm{\varnothing }$. Suppose that $0<\delta \le {\alpha }_n,{\beta }_n\le 1-\delta$ for some $\delta \in (0,1)$, where $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are the control parameters of the iteration scheme $\{x_n\}$ in (1.1). If S or T is semi-compact, then $\{x_n\}$ converges strongly to a common fixed point of S and T.

Proof For any $p\in F(S)\cap F(T)$, by the (CN)-inequality, we have

$$\begin{array}{rcl}d{(x_{n+1},p)}^2& =& d{({\alpha }_n{x}_n\oplus {\alpha }_n{S}^n{y}_n,p)}^2\\ \le & (1-{\alpha }_n)d{(x_n,p)}^2+{\alpha }_{n}d{(S^n{y}_n,p)}^2\\ -{\alpha }_n(1-{\alpha }_n)d{(x_n,S^n{y}_n)}^2\\ =& d{(x_n,p)}^2+{\alpha }_n\{d{(S^n{y}_n,p)}^2-d{(y_n,p)}^2\}\\ +{\alpha }_n\{d{(y_n,p)}^2-d{(x_n,p)}^2\}\\ -{\alpha }_n(1-{\alpha }_n)d{(x_n,S^n{y}_n)}^2.\end{array}$$

That is,

$$\begin{array}{rl}d{(x_{n+1},p)}^2\le & d{(x_n,p)}^2+{\alpha }_n\{d{(S^n{y}_n,p)}^2-d{(y_n,p)}^2\}\\ +{\alpha }_n\{d{(y_n,p)}^2-d{(x_n,p)}^2\}\\ -{\alpha }_n(1-{\alpha }_n)d{(x_n,S^n{y}_n)}^2.\end{array}$$

(2.3)

Next we consider the third term on the right side of (2.3):

$$\begin{array}{rcl}d{(y_n,p)}^2-d{(x_n,p)}^2& =& d{((1-{\beta }_n)x_n\oplus {\beta }_n{T}^n{x}_n,p)}^2-d{(x_n,p)}^2\\ \le & (1-{\beta }_n)d{(x_n,p)}^2+{\beta }_{n}d{(T^n{x}_n,p)}^2-d{(x_n,p)}^2\\ -{\beta }_n(1-{\beta }_n)d{(x_n,T^n{x}_n)}^2\\ =& {\beta }_n\{d{(T^n{x}_n,p)}^2-d{(x_n,p)}^2\}\\ -{\beta }_n(1-{\beta }_n)d{(x_n,T^n{x}_n)}^2.\end{array}$$

That is,

$$\begin{array}{rl}{\alpha }_n\{d{(y_n,p)}^2-d{(x_n,p)}^2\}\le & {\alpha }_n{\beta }_n\{d{(T^n{x}_n,p)}^2-d{(x_n,p)}^2\}\\ -{\alpha }_n{\beta }_n(1-{\beta }_n)d{(x_n,T^n{x}_n)}^2.\end{array}$$

(2.4)

Substituting (2.4) into (2.3) and using $0<\delta \le {\alpha }_n,{\beta }_n\le 1-\delta$, we have

$$\begin{array}{r}d{(x_{n+1},p)}^2\\ \le d{(x_n,p)}^2-\frac{{\alpha }_n(1-{\alpha }_n)}{2}d{(S^n{y}_n,p)}^2\\ -\frac{{\alpha }_n{\beta }_n(1-{\beta }_n)}{2}d{(x_n,T^n{x}_n)}^2\\ +{\alpha }_n\{d{(S^n{y}_n,p)}^2-d{(y_n,p)}^2-\frac{(1-{\alpha }_n)}{2}d{(S^n{y}_n,p)}^2\}\\ +{\alpha }_n{\beta }_n\{d{(T^n{x}_n,p)}^2-d{(x_n,p)}^2-\frac{(1-{\beta }_n)}{2}d{(x_n,T^n{x}_n)}^2\}\\ \le d{(x_n,p)}^2-\frac{{\delta }^2}{2}d{(S^n{y}_n,p)}^2-\frac{{\delta }^3}{2}d{(x_n,T^n{x}_n)}^2\\ +(1-\delta )\{d{(S^n{y}_n,p)}^2-d{(y_n,p)}^2-\frac{\delta }{2}d{(S^n{y}_n,p)}^2\}\\ +{(1-\delta )}^2\{d{(T^n{x}_n,p)}^2-d{(x_n,p)}^2-\frac{\delta }{2}d{(x_n,T^n{x}_n)}^2\}.\end{array}$$

(2.5)

Next we prove that

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,S^n{y}_n)=0=\underset{n\to \mathrm{\infty }}{lim}d(x_n,T^n{x}_n).$$

Assume that ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}d(x_n,S^n{y}_n)>0$ and ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}d(x_n,T^n{x}_n)>0$.

Then there exist subsequences (we use the same notation for a subsequence as well) of $\{x_n\}$, $\{y_n\}$ and ${\mu }_1>0$, ${\mu }_2>0$ such that $d(x_n,S^n{y}_n)\ge {\mu }_1>0$ and $d(x_n,T^n{x}_n)\ge {\mu }_2>0$.

Now from (2.5) it follows that

$$\begin{array}{rl}d{(x_{n+1},p)}^2\le & d{(x_n,p)}^2-\frac{{\delta }^2{\mu }_1^2}{2}-\frac{{\delta }^3{\mu }_2^2}{2}\\ +(1-\delta )\{d{(S^n{y}_n,p)}^2-d{(y_n,p)}^2-\frac{\delta {\mu }_1^2}{2}\}\\ +{(1-\delta )}^2\{d{(T^n{x}_n,p)}^2-d{(x_n,p)}^2-\frac{\delta {\mu }_2^2}{2}\}.\end{array}$$

(2.6)

For an asymptotically nonexpansive type map T, we have that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\underset{x\in C}{sup}\{d{(T^{n}x,p)}^2-d{(x,p)}^2\}\le 0.$$

That is,

$$\underset{n\to \mathrm{\infty }}{lim}\underset{m\ge n}{sup}\left\{\underset{x\in C}{sup}(d{(T^{m}x,p)}^2-d{(x,p)}^2)\right\}\le 0.$$

Hence, for given $\frac{\delta {\mu }_i^2}{2}>0$ ($i=1,2$), there exists a positive integer $n_0$ such that

$$\underset{n\ge n_0}{sup}\left\{\underset{x\in C}{sup}(d{(T^{n}x,p)}^2-d{(x,p)}^2)\right\}<\frac{\delta {\mu }_i^2}{2}.$$

Since $\{x_n\}$ and $\{y_n\}$ are sequences in C, therefore, for $n\ge n_0$, it follows that

$$d{(S^n{y}_n,p)}^2-d{(y_n,p)}^2<\frac{\delta {\mu }_1^2}{2}$$

and

$$d{(T^n{x}_n,p)}^2-d{(x_n,p)}^2<\frac{\delta {\mu }_2^2}{2}.$$

In the light of the two inequalities above, (2.6) reduces to

$$\frac{{\delta }^2{\mu }_1^2}{2}+\frac{{\delta }^3{\mu }_2^2}{2}\le d{(x_n,p)}^2-d{(x_{n+1},p)}^2\text{for all }n\ge n_0.$$

(2.7)

Let $m\ge n_0$ be any positive integer. Obtain $m-n_0$ inequalities from (2.7) and then, summing up these inequalities, we get

$$\begin{array}{rcl}(\frac{{\delta }^2{\mu }_1^2}{2}+\frac{{\delta }^3{\mu }_2^2}{2})(m-n_0)& \le & d{(x_{n_0},p)}^2-d{(x_{m+1},p)}^2\\ \le & d{(x_{n_0},p)}^2<\mathrm{\infty }.\end{array}$$

If $m\to \mathrm{\infty }$, then

$$\mathrm{\infty }=d{(x_{n_0},p)}^2<\mathrm{\infty },$$

a contradiction.

This proves that ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}d(x_n,S^n{y}_n)=0={lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}d(x_n,T^n{x}_n)$.

That is,

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,S^n{y}_n)=0=\underset{n\to \mathrm{\infty }}{lim}d(x_n,T^n{x}_n).$$

As

$$d(x_n,S^n{x}_n)\le d(x_n,S^n{y}_n)+d(S^n{x}_n,S^n{y}_n),$$

$d(x_n,y_n)\to 0$ and S is uniformly equicontinuous. So, by taking lim sup on both sides, we get

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,S^n{x}_n)=0.$$

Now, Lemma 2.1 implies that

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,S{x}_n)=0=\underset{n\to \mathrm{\infty }}{lim}d(x_n,T{x}_n).$$

(2.8)

Since T is semi-compact, therefore there exists a subsequence $\{x_{n_i}\}$ of $\{x_n\}$ and $q\in C$ such that

$$x_{n_i}\to q.$$

(2.9)

Now, by the uniform equicontinuity of S and T and hence continuity, it follows from (2.8) that

$$d(q,Sq)=0=d(q,Tq).$$

This gives that q is a common fixed point of S and T.

We now proceed to establish strong convergence of $\{x_n\}$ to q.

Since

$$d(T^{n_i}{x}_{n_i},q)\le d(T^{n_i}{x}_{n_i},x_{n_i})+d(x_{n_i},q),$$

therefore

$$T^{n_i}{x}_{n_i}\to q\text{as }{n}_i\to \mathrm{\infty }.$$

(2.10)

Clearly,

$$\begin{array}{rcl}d(y_{n_i},q)& =& d((1-{\beta }_{n_i})x_{n_i}\oplus {\beta }_{n_i}{T}^{n_i}{x}_{n_i},q)\\ \le & (1-{\beta }_{n_i})d(x_{n_i},q)+{\beta }_{n_i}d(T^{n_i}{x}_{n_i},q).\end{array}$$

Therefore, from (2.9) and (2.10), it follows that

$$y_{n_i}\to q\text{as }{n}_i\to \mathrm{\infty }.$$

Next we prove that $S^{n_i}{y}_{n_i}\to q$ as $n_i\to \mathrm{\infty }$.

Since $S:C\to C$ is of asymptotically nonexpansive type and $\{y_{n_i}\}$ is a sequence in C, therefore we have

$$\begin{array}{r}\underset{n_i\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\{d{(S^{n_i}{y}_{n_i},q)}^2-d{(y_{n_i},q)}^2\}\\ \le \underset{n_i\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\underset{x\in C}{sup}\{d{(S^{n_i}x,q)}^2-d{(x,q)}^2\}\\ \le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\underset{x\in C}{sup}\{d{(S^{n}x,q)}^2-d{(x,q)}^2\}\\ \le 0.\end{array}$$

(2.11)

As $y_{n_i}\to q$ as $n_i\to \mathrm{\infty }$, it follows from (2.11) that

$$\underset{n_i\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d{(S^{n_i}{y}_{n_i},q)}^2\le 0.$$

That is,

$$S^{n_i}{y}_{n_i}\to q\text{as }{n}_i\to \mathrm{\infty }.$$

Replace p by q in (2.5) to get

$$\begin{array}{rcl}d{(x_{n_i+1},q)}^2& \le & d{(x_{n_i},q)}^2-\frac{{\delta }^2}{2}d{(S^{n_i}{y}_{n_i},q)}^2-\frac{{\delta }^3}{2}d{(x_{n_i},T^{n_i}{x}_{n_i})}^2\\ +(1-\delta )\{d{(S^{n_i}{y}_{n_i},q)}^2-d{(y_{n_i},q)}^2-\frac{\delta }{2}d{(S^{n_i}{y}_{n_i},q)}^2\}\\ +{(1-\delta )}^2\{d{(T^{n_i}{x}_{n_i},q)}^2-d{(x_{n_i},q)}^2-\frac{\delta }{2}d{(x_{n_i},T^{n_i}{x}_{n_i})}^2\},\end{array}$$

which gives that $x_{n_i+1}\to q$ as $n_i\to \mathrm{\infty }$.

Continuing in this way, by induction, we can prove that for any $m⩾0$,

$$x_{n_i+m}\to q\text{as }{n}_i\to \mathrm{\infty }.$$

By induction, one can prove that ${\bigcup }_{m=0}^{\mathrm{\infty }}\{x_{n_i+m}\}$ converges to q as $i\to \mathrm{\infty }$; in fact ${\{x_n\}}_{n=n_1}^{\mathrm{\infty }}={\bigcup }_{m=0}^{\mathrm{\infty }}{\{x_{n_i+m}\}}_{i=1}^{\mathrm{\infty }}$ gives that $x_n\to q$ as $n\to \mathrm{\infty }$. □

We need the following lemma to approximate a common fixed point of two asymptotically nonexpansive maps.

Lemma 2.3 Every asymptotically nonexpansive selfmap T on a nonempty bounded subset C of a metric space X is uniformly equicontinuous and of asymptotically nonexpansive type.

Proof Let $T:C\to C$ be an asymptotically nonexpansive map with a sequence $\{k_n\}\subseteq [1,\mathrm{\infty })$ such that ${lim}_{n\to \mathrm{\infty }}{k}_n=1$. Let $\epsilon >0$. Then, for each $\gamma >0$, there exists a positive integer $n_0$ such that $k_n-1<\gamma$ for all $n\ge n_0$. Put $s=max\{1+\gamma ,k_1,k_2,\dots ,k_{n_0}\}$. Then $d(T^{n}x,T^{n}y)\le k_{n}d(x,y)\le sd(x,y)$ for $x,y\in C$, $n\ge 1$. Choose $\delta =\frac{\epsilon }{s}$. Then $d(T^{n}x,T^{n}y)\le \epsilon$ whenever $d(x,y)\le \delta$ for $x,y\in C$, $n\ge 1$, proving that T is uniformly equicontinuous.

The second part of the lemma follows from

$$\begin{array}{c}\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\underset{x\in C}{sup}\{d^2(T^{n}x,T^{n}y)-d^2(x,y)\}\hfill \\ \le \underset{n\to \mathrm{\infty }}{lim}(k_n-1)\underset{x\in C}{sup}{d}^2(x,y)\hfill \\ =0.\underset{x\in C}{sup}{d}^2(x,y)\hfill \\ =0.\hfill \end{array}$$

 □

By Theorem 2.2 and Lemma 2.3, we have the following result which is new in the literature and sets an analogue of Theorem 2 in [21] without the rate of convergence condition.

Theorem 2.4 Let C be a nonempty, bounded, closed and convex subset of a $CAT(0)$ space X. Let $S,T:C\to C$ be asymptotically nonexpansive maps with sequences $\{s_n\},\{t_n\}\subseteq [1,\mathrm{\infty })$, respectively and $F(S)\cap F(T)\ne \mathrm{\varnothing }$. Suppose that $0<\delta \le {\alpha }_n,{\beta }_n\le 1-\delta$ for some $\delta \in (0,1)$, where $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are the control parameters of the sequence $\{x_n\}$ in (1.1). If S or T is semi-compact, then $\{x_n\}$ converges strongly to a common fixed point of S and T.

As every uniformly equicontinuous map is uniformly L-Lipschitzian, so the following result is immediate and it unifies Theorem 2.1 and Theorem 2.2 of Chang et al. [13] in Hadamard spaces.

Theorem 2.5 Let C be a nonempty, bounded, closed and convex subset of a $CAT(0)$ space X. Let $S,T:C\to C$ be uniformly L-Lipschitzian and asymptotically nonexpansive type maps such that $F(S)\cap F(T)\ne \mathrm{\varnothing }$. Suppose that $0<\delta \le {\alpha }_n,{\beta }_n\le 1-\delta$ for some $\delta \in (0,1)$, where $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are the control parameters of the sequence $\{x_n\}$ in (1.1). If S or T is semi-compact, then $\{x_n\}$ converges strongly to a common fixed point of S and T.

For $S=T$, Theorem 2.5 sets an analogue of Theorem 2.1 in [13].

Theorem 2.6 Let C be a nonempty, bounded, closed and convex subset of a $CAT(0)$ space X. Let $T:C\to C$ be a uniformly L-Lipschitzian and asymptotically nonexpansive type map such that $F(T)\ne \mathrm{\varnothing }$. Suppose that $0<\delta \le {\alpha }_n,{\beta }_n\le 1-\delta$ for some $\delta \in (0,1)$, where $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are the control parameters of the sequence $\{x_n\}$ in (1.1) with $S=T$. If T is semi-compact, then $\{x_n\}$ converges strongly to a fixed point of T.

On taking $S=I$ (the identity map) in Theorem 2.5, we obtain an analogue of Theorem 2.2 in [13].

Theorem 2.7 Let C be a nonempty, bounded, closed and convex subset of a $CAT(0)$ space X. Let $T:C\to C$ be a uniformly L-Lipschitzian and asymptotically nonexpansive type map such that $F(T)\ne \mathrm{\varnothing }$. Suppose that $0<\delta \le {\alpha }_n,{\beta }_n\le 1-\delta$ for some $\delta \in (0,1)$, where $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are the control parameters of the sequence $\{x_n\}$ in (1.2). If T is semi-compact, then $\{x_n\}$ converges strongly to a fixed point of T.

Remark 2.8 (1) Tan and Xu [30] obtained only weak convergence theorems for asymptotically nonexpansive maps satisfying the rate of convergence condition and remarked, ‘We do not know whether our weak convergence Theorem 3.1 remains valid if $k_n$ is allowed to approach 1 slowly enough so that ${\sum }_{n=1}^{\mathrm{\infty }}(k_n-1)$ diverges’. Our Theorem 2.4 gives an affirmative answer to their question in $CAT(0)$ spaces.

(2) Our results are generalizations in $CAT(0)$ spaces of the corresponding basic results in [16, 21, 28, 29].

(3) Theorem 2.2 improves and generalizes Theorems 4.2-4.3 in [5].