The Mann algorithm in a complete geodesic space with curvature bounded above

Abstract

The purpose of this paper is to prove two Δ-convergence theorems of the Mann algorithm to a common fixed point for a countable family of mappings in the case of a complete geodesic space with curvature bounded above by a positive number. The first one for nonexpansive mappings improves the recent result of He et al. (Nonlinear Anal. 75:445-452, 2012). The last one is proved for quasi-nonexpansive mappings and applied to the problem of finding a common fixed point of a countable family of quasi-nonexpansive mappings.

MSC:49J53.

1 Introduction

For a real number κ, a $CAT(\kappa )$ space is defined by a geodesic metric space whose geodesic triangle is sufficiently thinner than the corresponding comparison triangle in a model space with curvature κ. The concept of these spaces has been studied by a large number of researchers. We know that any $CAT(\kappa )$ space is a $CAT({\kappa }^{\prime })$ space for ${\kappa }^{\prime }>\kappa$ (see [1]), thus all results for $CAT(0)$ spaces can immediately be applied to any $CAT(\kappa )$ with $\kappa \le 0$. Moreover, $CAT(\kappa )$ spaces with positive κ can be treated as $CAT(1)$ spaces by changing the scale of the space. So we are interested in $CAT(1)$ spaces.

One of the most important analytical problems is the existence of fixed points for nonlinear mappings. In the case for nonexpansive mappings in a $CAT(\kappa )$ space was proved by Kirk [2, 3] for $\kappa \le 0$, and by Espánola and Fernández-León [4] for $\kappa >0$. In the cases when at least one fixed point exists, it is natural to wonder whether such a fixed point can be approximated by iterations. There are many methods for approximating fixed points of a nonexpansive mapping T. One of the most successful methods is the Mann algorithm [5] which is defined in a geodesic space X by $x_1\in X$ and

$$x_{n+1}=t_n{x}_n\oplus (1-t_n)T{x}_n\text{for all }n\in \mathbb{N},$$

(1.1)

where $\{t_n\}$ is a sequence in $[0,1]$. By using this algorithm, He et al. [6] proved the following result.

Theorem 1.1 Let X be a complete $CAT(1)$ space and $T:X\to X$ be a mapping. Let $\{x_n\}$ be the Mann algorithm (1.1) in X. Suppose that

• (C0′) T is nonexpansive with $\mathrm{F}(T)\ne \mathrm{\varnothing }$;

• (C1′) $d(x_1,\mathrm{F}(T))<\pi /4$;

• (C2) ${\sum }_{n=1}^{\mathrm{\infty }}{t}_n(1-t_n)=\mathrm{\infty }$.

Then the sequence $\{x_n\}$ Δ-converges to a fixed point of T.

Motivated by these results, we prove two Δ-convergence theorems of the Mann algorithm to a common fixed point for a countable family of mappings in complete $CAT(1)$ spaces. The first one for nonexpansive mappings improves Theorem 1.1. The last one is proved for quasi-nonexpansive mappings and applied to the problem of finding a common fixed point of a countable family of quasi-nonexpansive mappings.

2 Preliminaries

Let X be a metric space with a metric d and let $x,y\in X$ with $d(x,y)=l$. A geodesic path from x to y is an isometry $c:[0,l]\to X$ such that $c(0)=x$ and $c(l)=y$. The image of a geodesic path from x to y is called a geodesic segment joining x and y. Let $r\in (0,\mathrm{\infty }]$. If for every $x,y\in X$ with $d(x,y)<r$, a geodesic from x to y exists, then we say that X is r-geodesic. Moreover, if such a geodesic is unique for each pair of points, then X is said to be r-uniquely geodesic.

A geodesic segment joining x and y is not necessarily unique in general. When it is unique, this geodesic segment is denoted by $[x,y]$. We write $z\in [x,y]$ if and only if there exists $t\in [0,1]$ such that $d(z,x)=(1-t)d(x,y)$ and $d(z,y)=td(x,y)$. In this case, we will write $z=tx\oplus (1-t)y$ for simplicity. A geodesic triangle $\mathrm{△}(x,y,z)$ consists of three points $x,y,z\in X$ and geodesic segments $[y,z]$, $[z,x]$ and $[x,z]$ joining two of them. We write $w\in \mathrm{△}(x,y,z)$ if $w\in [y,z]\cup [z,x]\cup [x,y]$.

To define a $CAT(\kappa )$ space, we use the following notation called model space. For $\kappa =0$, the two-dimensional model space $M_{\kappa }^2=M_0^2$ is the Euclidean space ${\mathbb{R}}^2$ with the metric induced from the Euclidean norm. For $\kappa >0$, $M_{\kappa }^2$ is the two-dimensional sphere $(1/\sqrt{\kappa }){\mathbb{S}}^2$ whose metric is a length of a minimal great arc joining each two points. For $\kappa <0$, $M_{\kappa }^2$ is the two-dimensional hyperbolic space $(1/\sqrt{-\kappa }){\mathbb{H}}^2$ with the metric defined by a usual hyperbolic distance.

The diameter of $M_{\kappa }^2$ is denoted by $D_{\kappa }$, that is, $D_{\kappa }=\pi /\sqrt{\kappa }$ if $\kappa >0$ and $D_{\kappa }=\mathrm{\infty }$ if $\kappa \le 0$. We know that $M_{\kappa }^2$ is a $D_{\kappa }$-uniquely geodesic space for each $\kappa \in \mathbb{R}$.

Let $\kappa \in \mathbb{R}$. For $\mathrm{△}(x,y,z)$ in a geodesic space X satisfying that $d(x,y)+d(y,z)+d(x,z)<2D_{\kappa }$, there exist points $\overline{x},\overline{y},\overline{z}\in M_{\kappa }^2$ such that $d(x,y)=d_{M_{\kappa }^2}(\overline{x},\overline{y})$, $d(y,z)=d_{M_{\kappa }^2}(\overline{y},\overline{z})$ and $d(x,z)=d_{M_{\kappa }^2}(\overline{x},\overline{z})$. We call the triangle having vertices $\overline{x}$, $\overline{y}$ and $\overline{z}$ in $M_{\kappa }^2$ a comparison triangle of $\mathrm{△}(x,y,z)$. Notice that it is unique up to an isometry of $M_{\kappa }^2$. For a specific choice of comparison triangles, we denote it by $\mathrm{△}(\overline{x},\overline{y},\overline{z})$. A point $\overline{p}\in [\overline{x},\overline{y}]$ is called a comparison point for $p\in [x,y]$ if $d(x,p)=d_{M_{\kappa }^2}(\overline{x},\overline{p})$.

Let $\kappa \in \mathbb{R}$ and X be a $D_{\kappa }$-geodesic space. If for any $x,y,z\in X$ with $d(x,y)+d(y,z)+d(x,z)<2D_{\kappa }$, for any $p,q\in \mathrm{△}(x,y,z)$, and for their comparison points $\overline{p},\overline{q}\in \mathrm{△}(\overline{x},\overline{y},\overline{z})$, the inequality

$$d(p,q)\le d_{M_{\kappa }^2}(\overline{p},\overline{q})$$

holds, then we call X a $CAT(\kappa )$ space. It is easy to see that all $CAT(\kappa )$ spaces are $D_{\kappa }$-uniquely geodesic; consider the triangle such that two of its vertices are identical.

Let $\{x_n\}$ be a bounded sequence in a metric space X. For $x\in X$, let $r(x,\{x_n\}):={lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}d(x,x_n)$ and define the asymptotic radius $r(\{x_n\})$ of $\{x_n\}$ by

$$r(\{x_n\}):=\underset{x\in X}{inf}r(x,\{x_n\}).$$

An element z of X is said to be an asymptotic center of $\{x_n\}$ if $r(z,\{x_n\})=r(\{x_n\})$. We say that $\{x_n\}$ is Δ-convergent to $x\in X$ if x is the unique asymptotic center of any subsequence of $\{x_n\}$. The concept of Δ-convergence introduced by Lim in 1976 was shown by Kirk and Panyanak [7] in $CAT(0)$ spaces to be very similar to the weak convergence in Banach space setting.

Remark 2.1

• (1) If $\{x_n\}$ is a sequence in a complete $CAT(\kappa )$ space such that $r(\{x_n\})<D_{\kappa }/2$, then its asymptotic center consists of exactly one point.

• (2) Every sequence $\{x_n\}$ whose asymptotic radius is less than $D_{\kappa }/2$ has a Δ-convergent subsequence (see [4, 7, 8]), that is, ${\omega }_{\mathrm{\Delta }}(\{x_n\}):=\{x\in X:\text{there exists }\{x_{n_k}\}\subset \{x_n\}\text{such that }\{x_{n_k}\}\mathrm{\Delta }\text{-converges to }x\}\ne \mathrm{\varnothing }$.

We note that Remark 2.1(1) was proved by Dhompongsa et al. [9] for the case $\kappa \le 0$, and by Espánola and Fernández-León [4] for the case $\kappa >0$.

Let X be a metric space with a metric d. A mapping $T:X\to X$ is called nonexpansive if

$$d(Tx,Ty)\le d(x,y)\text{for all }x,y\in X.$$

A point $x\in X$ is called a fixed point of T if $x=Tx$. We denote by $\mathrm{F}(T)$ the set of fixed points of T. The mapping T is called quasi-nonexpansive if $\mathrm{F}(T)\ne \mathrm{\varnothing }$ and

$$d(Tx,p)\le d(x,p)\text{for all }x\in X\text{ and }p\in \mathrm{F}(T).$$

The following lemmas are essentially needed for our main results.

Lemma 2.2 ([[10], Lemma 1])

Let $\{a_n\}$ and $\{b_n\}$ be two sequences of nonnegative real numbers such that

$$a_{n+1}\le a_n+b_n\mathit{\text{for all }}n\in \mathbb{N}.$$

If ${\sum }_{n=1}^{\mathrm{\infty }}{b}_n<\mathrm{\infty }$, then ${lim}_{n\to \mathrm{\infty }}{a}_n$ exists.

Lemma 2.3 ([[11], Lemma 5.4])

Let $\mathrm{△}(x,y,z)$ be a geodesic triangle in a $CAT(1)$ space such that $d(x,y)+d(x,z)+d(y,z)<2\pi$. Let $u=tz\oplus (1-t)x$ and $v=tz\oplus (1-t)y$ for some $t\in [0,1]$. If $d(x,z)\le M$, $d(y,z)\le M$, and $sin((1-t)M)\le sinM$ for some $M\in (0,\pi )$, then

$$d(u,v)\le \frac{sin(1-t)M}{sinM}d(x,y).$$

Lemma 2.4 ([[12], Corollary 2.2])

Let $\mathrm{△}(x,y,z)$ be a geodesic triangle in a $CAT(1)$ space such that $d(x,y)+d(x,z)+d(y,z)<2\pi$. Let $u=tx\oplus (1-t)y$ for some $t\in [0,1]$. Then

$$cosd(u,z)sind(x,y)\ge cosd(x,z)sin(td(x,y))+cosd(y,z)sin((1-t)d(x,y)).$$

Lemma 2.5 ([[11], Lemma 3.1])

Let $\mathrm{△}(x,y,z)$ be a geodesic triangle in a $CAT(1)$ space such that $d(x,y)+d(x,z)+d(y,z)<2\pi$, and let $t\in [0,1]$. Then

$$cosd(tx\oplus (1-t)y,z)\ge tcosd(x,z)+(1-t)cosd(y,z).$$

3 Main results

We start with some propositions which are common tools for proving the main results in the next two subsections.

Proposition 3.1 Let $\{x_n\}$ be a sequence of a complete $CAT(1)$ space X such that $r(\{x_n\})<\pi /2$. Suppose that ${lim}_{n\to \mathrm{\infty }}d(x_n,z)$ exists for all $z\in {\omega }_{\mathrm{\Delta }}(\{x_n\})$. Then $\{x_n\}$ Δ-converges to an element of ${\omega }_{\mathrm{\Delta }}(\{x_n\})$. Moreover, ${\omega }_{\mathrm{\Delta }}(\{x_n\})$ consists of exactly one point.

Proof Let x be the asymptotic center of $\{x_n\}$ and let $\{x_{n_k}\}$ be any subsequence of $\{x_n\}$ with the asymptotic center y. We show that $y=x$ and hence $\{x_n\}$ Δ-converges to x as desired. Since $r(\{x_{n_k}\})\le r(\{x_n\})<\pi /2$, there exists a subsequence $\{x_{n_{k_l}}\}$ of $\{x_{n_k}\}$ such that $\{x_{n_{k_l}}\}$ Δ-converges to z for some $z\in X$. Clearly, $z\in {\omega }_{\mathrm{\Delta }}(\{x_n\})$ and it follows from the assumption that ${lim}_{n\to \mathrm{\infty }}d(x_n,z)$ exists. Let $u\in X$. Then

$$\begin{array}{rl}\underset{n\to \mathrm{\infty }}{lim}d(x_n,z)& =\underset{k\to \mathrm{\infty }}{lim}d(x_{n_k},z)\\ =\underset{l\to \mathrm{\infty }}{lim}d(x_{n_{k_l}},z)\\ \le \underset{l\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(x_{n_{k_l}},u)\\ \le \underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(x_{n_k},u)\\ \le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(x_n,u).\end{array}$$

Since

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,z)\le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(x_n,u)\text{for all }u\in X,$$

we have $z=x$. Since

$$\underset{k\to \mathrm{\infty }}{lim}d(x_{n_k},z)\le \underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(x_{n_k},u)\text{for all }u\in X,$$

we have $z=y$. This implies that $y=x$. □

Proposition 3.2 Let X be a complete $CAT(1)$ space and $\{T_n\}:X\to X$ be a countable family of quasi-nonexpansive mappings with $\mathrm{F}:={\bigcap }_{n=1}^{\mathrm{\infty }}\mathrm{F}(T_n)\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be a sequence in X such that $d(x_1,\mathrm{F})<\pi /2$ and

$$x_{n+1}=t_n{x}_n\oplus (1-t_n)T_n{x}_n\mathit{\text{for all }}n\in \mathbb{N},$$

where $\{t_n\}$ is a sequence in $[0,1]$. Then

• (i) $\{x_n\}$ is well defined and $r(\{x_n\})<\pi /2$. In particular, ${\omega }_{\mathrm{\Delta }}(\{x_n\})\ne \mathrm{\varnothing }$ and $d(x_{n+1},p)\le d(x_n,p)$ for all $n\in \mathbb{N}$ whenever $d(x_1,p)<\pi /2$ and $p\in \mathrm{F}$.

• (ii) If ${\omega }_{\mathrm{\Delta }}(\{x_n\})\subset \mathrm{F}$, then $\{x_n\}$ Δ-converges to an element of F.

Proof (i) Let $p\in \mathrm{F}$ be such that $d(x_1,p)<\pi /2$. Since $d(T_1{x}_1,p)\le d(x_1,p)<\pi /2$, we have that $d(x_1,T_1{x}_1)<\pi$. This implies that $x_2$ is well defined. It follows from Lemma 2.5 that

$$\begin{array}{rl}cosd(x_2,p)& =cosd(t_1{x}_1\oplus (1-t_1)T_1{x}_1,p)\\ \ge t_{1}cosd(x_1,p)+(1-t_1)cosd(T_1{x}_1,p)\\ \ge t_{1}cosd(x_1,p)+(1-t_1)cosd(x_1,p)\\ =cosd(x_1,p),\end{array}$$

and we have

$$\begin{array}{rl}d(x_2,p)& \le min\{d(x_1,x_2)+d(x_1,p),d(T{x}_1,x_2)+d(T{x}_1,p)\}\\ =min\{(1-t_1)d(x_1,T{x}_1)+d(x_1,p),t_{1}d(x_1,T{x}_1)+d(T{x}_1,p)\}\\ \le min\{1-t_1,t_1\}d(x_1,T{x}_1)+d(x_1,p)\\ \le \frac{1}{2}d(x_1,T{x}_1)+d(x_1,p)\\ <\frac{\pi }{2}+\frac{\pi }{2}=\pi .\end{array}$$

Therefore, $d(x_2,p)\le d(x_1,p)<\pi /2$. Using mathematical induction, we can conclude that the sequence $\{x_n\}$ is well defined and

$$d(x_{n+1},p)\le d(x_n,p)\le d(x_1,p)<\pi /2\text{for all }n\in \mathbb{N}.$$

Then ${lim}_{n\to \mathrm{\infty }}d(x_n,p)$ exists which is less than $\pi /2$, and so $r(\{x_n\})<\pi /2$. This implies that ${\omega }_{\mathrm{\Delta }}(\{x_n\})\ne \mathrm{\varnothing }$.

(ii) Let $u\in {\omega }_{\mathrm{\Delta }}(\{x_n\})\subset \mathrm{F}$. Then there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ such that $\{x_{n_k}\}$ Δ-converges to u. Notice that $r(u,\{x_{n_k}\})=r(\{x_{n_k}\})\le r(\{x_n\})<\pi /2$. Thus there is $N\in \mathbb{N}$ such that $d(x_N,u)<\pi /2$. Similar to the first step, we have that $d(x_{n+1},u)\le d(x_n,u)$ for all $n\ge N$. This implies that ${lim}_{n\to \mathrm{\infty }}d(x_n,u)$ exists. It follows immediately from Proposition 3.1 that $\{x_n\}$ Δ-converges to an element of F and the proof is finished. □

3.1 Countable nonexpansive mappings

The following concept is introduced by Aoyama et al. [13]. Let X be a complete metric space and $\{T_n\}$ be a countable family of mappings from X into itself with $\mathrm{F}:={\bigcap }_{n=1}^{\mathrm{\infty }}\mathrm{F}(T_n)\ne \mathrm{\varnothing }$. We say that $(\{T_n\},T)$ satisfies AKTT-condition if

• ${\sum }_{n=1}^{\mathrm{\infty }}sup\{d(T_{n+1}y,T_{n}y):y\in Y\}<\mathrm{\infty }$ for each bounded subset Y of X;

• $Tx:={lim}_{n\to \mathrm{\infty }}{T}_{n}x$ for all $x\in X$ and $\mathrm{F}(T)=\mathrm{F}$.

Remark 3.3 Assume that $(\{T_n\},T)$ satisfies AKTT-condition.

• (1) For each $x\in X$, we have $\{T_{n}x\}$ is a Cauchy sequence and hence the mapping T above is well defined.

• (2) If $\{x_n\}$ is bounded, then ${\sum }_{n=1}^{\mathrm{\infty }}d(T_{n+1}{x}_n,T_n{x}_n)<\mathrm{\infty }$.

Theorem 3.4 Let X be a complete $CAT(1)$ space and $\{T_n\}$ be a countable family of mappings from X into itself. Let $\{x_n\}$ be a sequence in X defined by $x_1\in X$ and

$$x_{n+1}=t_n{x}_n\oplus (1-t_n)T_n{x}_n\mathit{\text{for all }}n\in \mathbb{N},$$

where $\{t_n\}$ is a sequence in $[0,1]$. Suppose that

• ($\text{C0}{n}^{\prime }$) $T_n$ is nonexpansive for all $n\in \mathbb{N}$ and $\mathrm{F}:={\bigcap }_{n=1}^{\mathrm{\infty }}\mathrm{F}(T_n)\ne \mathrm{\varnothing }$;

• (C1) $d(x_1,\mathrm{F})<\pi /2$;

• (C2) ${\sum }_{n=1}^{\mathrm{\infty }}{t}_n(1-t_n)=\mathrm{\infty }$;

• (C3′) $(\{T_n\},T)$ satisfies AKTT-condition.

Then the sequence $\{x_n\}$ Δ-converges to a common fixed point of $\{T_n\}$.

Proof We first show that ${lim}_{n\to \mathrm{\infty }}d(x_n,T_n{x}_n)$ exists. Using the nonexpansiveness of $T_n$ and the definition of $\{x_n\}$, we obtain that

$$\begin{array}{c}d(x_{n+1},T_{n+1}{x}_{n+1})\hfill \\ \le d(x_{n+1},T_n{x}_n)+d(T_n{x}_n,T_n{x}_{n+1})+d(T_n{x}_{n+1},T_{n+1}{x}_{n+1})\hfill \\ \le d(x_{n+1},T_n{x}_n)+d(x_n,x_{n+1})+d(T_n{x}_{n+1},T_{n+1}{x}_{n+1})\hfill \\ =d(x_n,T_n{x}_n)+d(T_n{x}_{n+1},T_{n+1}{x}_{n+1})\hfill \end{array}$$

for all $n\in \mathbb{N}$. It follows from Lemma 2.2 and ${\sum }_{n=1}^{\mathrm{\infty }}d(T_n{x}_{n+1},T_{n+1}{x}_{n+1})<\mathrm{\infty }$ that

$$\underset{n\to \mathrm{\infty }}{lim}d(x_n,T_n{x}_n)$$

exists.

Next, we show that ${lim}_{n\to \mathrm{\infty }}d(x_n,T_n{x}_n)=0$. Assume that ${lim}_{n\to \mathrm{\infty }}d(x_n,T_n{x}_n)>0$. Thus, without loss of generality, there is a positive real number A such that

$$A\le d(x_n,T_n{x}_n)<\pi \text{for all }n\in \mathbb{N}.$$

To get the right inequality of the preceding expression, let $p\in \mathrm{F}$ be such that $d(x_1,p)<\pi /2$. By Proposition 3.2(i), we have

$$d(x_n,T_n{x}_n)\le d(x_n,p)+d(T_n{x}_n,p)\le 2d(x_n,p)<2d(x_1,p)<\pi .$$

Put $A_n:=d(x_n,T_n{x}_n)$ for all $n\in \mathbb{N}$. By elementary trigonometry and Lemma 2.4, we get that

$$\begin{array}{c}cosd(x_{n+1},p)sin{A}_n\hfill \\ =cosd(t_n{x}_n\oplus (1-t_n)T_n{x}_n,p)sin{A}_n\hfill \\ \ge cosd(x_n,p)sin(t_n{A}_n)+cosd(T_n{x}_n,p)sin((1-t_n)A_n)\hfill \\ \ge cosd(x_n,p)(sin(t_n{A}_n)+sin((1-t_n)A_n)),\hfill \end{array}$$

and it follows that

$$\begin{array}{c}cosd(x_{n+1},p)-cosd(x_n,p)\hfill \\ \ge cosd(x_n,p)(\frac{sin(t_n{A}_n)+sin((1-t_n)A_n)}{sin{A}_n}-1)\hfill \\ =\frac{2cosd(x_n,p)sin(t_n{A}_n/2)sin((1-t_n)A_n/2)}{cos(A_n/2)}\hfill \\ \ge \frac{2cosd(x_1,p)sin(t_{n}A/2)sin((1-t_n)A/2)}{cos(A/2)}\hfill \\ \ge \frac{2t_n(1-t_n)cosd(x_1,p){sin}^2(A/2)}{cos(A/2)}\hfill \end{array}$$

for all $n\in \mathbb{N}$. Notice that $cosd(x_1,p)$, $cos(A/2)$, and $sin(A/2)$ are positive. Consequently,

$$\sum _{n=1}^{\mathrm{\infty }}{t}_n(1-t_n)\le \frac{cos(A/2)}{2cosd(x_1,p){sin}^2(A/2)}\sum _{n=1}^{\mathrm{\infty }}(cosd(x_{n+1},p)-cosd(x_n,p))<\mathrm{\infty },$$

which is a contradiction. Then we get that ${lim}_{n\to \mathrm{\infty }}d(x_n,T_n{x}_n)=0$ and hence

$$\begin{array}{rl}d(x_n,T{x}_n)& \le d(x_n,T_n{x}_n)+d(T_n{x}_n,T{x}_n)\\ \le d(x_n,T_n{x}_n)+sup\{d(T_{n}y,Ty):y\in Y\}\to 0,\end{array}$$

where $Y:=\{x_n\}$.

Finally, we show that $\{x_n\}$ Δ-converges to an element of $\mathrm{F}(T)$. To apply Proposition 3.2(ii), we show that ${\omega }_{\mathrm{\Delta }}(\{x_n\})\subset \mathrm{F}(T)$. Let $u\in {\omega }_{\mathrm{\Delta }}(\{x_n\})$. Then there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ such that $\{x_{n_k}\}$ Δ-converges to u. Clearly, u is the unique asymptotic center of $\{x_{n_k}\}$. Using the nonexpansiveness of T and $d(x_n,T{x}_n)\to 0$, we get that

$$\begin{array}{rl}\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(x_{n_k},Tu)& \le \underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(x_{n_k},T{x}_{n_k})+\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(T{x}_{n_k},Tu)\\ \le \underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}d(x_{n_k},u).\end{array}$$

This implies that $Tu=u$, that is, ${\omega }_{\mathrm{\Delta }}(\{x_n\})\subset \mathrm{F}(T)$. This completes the proof. □

As an immediate consequence of Theorem 3.4, we obtain the following result.

Corollary 3.5 Let X be a complete $CAT(1)$ space and $T:X\to X$ be a mapping. Let $\{x_n\}$ be the Mann algorithm (1.1) in X. Suppose that

• (C0′) T is nonexpansive with $\mathrm{F}(T)\ne \mathrm{\varnothing }$;

• (C1) $d(x_1,\mathrm{F}(T))<\pi /2$;

• (C2) ${\sum }_{n=1}^{\mathrm{\infty }}{t}_n(1-t_n)=\mathrm{\infty }$.

Then the sequence $\{x_n\}$ Δ-converges to a fixed point of T.

Remark 3.6 Our Corollary 3.5 improves Theorem 3.1 of He et al. [6] (see Theorem 1.1) because (C1′) of Theorem 1.1 implies (C1) of Corollary 3.5. Moreover, (C1) is sharp in the sense that if $d(x_1,\mathrm{F}(T))=\pi /2$, then we may construct the Mann algorithm for a nonexpansive mapping which is not Δ-convergent.

Example 3.7 Let ${\mathbb{S}}^2$ be the unit sphere of the Euclidean space ${\mathbb{R}}^3$ with the geodesic metric. Let $T:{\mathbb{S}}^2\to {\mathbb{S}}^2$ be defined by

$$T(x,y,z)=(-y,x,z)\text{for all }(x,y,z)\in {\mathbb{S}}^2.$$

Then T is nonexpansive and $\mathrm{F}(T)=\{(0,0,1),(0,0,-1)\}$. Let $\{x_n\}$ be a sequence in ${\mathbb{S}}^2$ defined by $x_1=(1,0,0)$ and

$$x_{n+1}=\frac{1}{2}{x}_n\oplus \frac{1}{2}T{x}_n\text{for all }n\in \mathbb{N}.$$

Then $d(x_1,\mathrm{F}(T))=\pi /2$ and

$$x_{n+1}=(cos\frac{n\pi }{4},sin\frac{n\pi }{4},0)\text{for all }n\in \mathbb{N}.$$

It is easy to see that $\{x_{8n+1}\}$ has the unique asymptotic center which is $\{(1,0,0)\}$ and $\{x_{8n+3}\}$ has the unique asymptotic center which is $\{(0,1,0)\}$. Hence, $\{x_n\}$ is not Δ-convergent.

3.2 Countable quasi-nonexpansive mappings

In this subsection, we give a supplement result to Theorem 3.4. Obviously, every nonexpansive mapping with a fixed point is quasi-nonexpansive. Moreover, if T is nonexpansive, then T is Δ-demiclosed [11], that is, if for any Δ-convergent sequence $\{x_n\}$ in X, its Δ-limit belongs to $\mathrm{F}(T)$ whenever ${lim}_{n\to \mathrm{\infty }}d(x_n,T{x}_n)=0$.

In the following theorem, we deal with quasi-nonexpansive mappings satisfying Δ-demiclosedness. This interesting class of mappings includes the metric projections [11]. However, there are many metric projections such that they are not nonexpansive.

Theorem 3.8 Let X be a complete $CAT(1)$ space and $\{T_n\}$ be a countable family of mappings from X into itself. Let $\{x_n\}$ be a sequence in X defined by $x_1\in X$ and

$$x_{n+1}=t_n{x}_n\oplus (1-t_n)T_n{x}_n\mathit{\text{for all }}n\in \mathbb{N},$$

where $\{t_n\}$ is a sequence in $(0,1)$. Suppose that

• (C0n) $T_n$ is quasi-nonexpansive for all $n\in \mathbb{N}$ and $\mathrm{F}={\bigcap }_{n=1}^{\mathrm{\infty }}\mathrm{F}(T_n)\ne \mathrm{\varnothing }$;

• (C1) $d(x_1,\mathrm{F})<\pi /2$;

• (C2′) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{t}_n(1-t_n)>0$;

• (C3) there exists a mapping $T:X\to X$ such that

  • $\{T_n\}$converges uniformly to T on each bounded subset of X;

  • $\mathrm{F}(T)=\mathrm{F}$;

• (C4) T is Δ-demiclosed.

Then the sequence $\{x_n\}$ Δ-converges to a common fixed point of $\{T_n\}$.

Remark 3.9 Let us compare Theorems 3.4 and 3.8:

• (1) ($\text{C0}{n}^{\prime }$) ⇒ (C0n);

• (2) (C3′) ⇒ (C3);

• (3) (C0′) and (C3′) ⇒ (C4);

• (4) (C2′) ⇒ (C2).

Proof of Theorem 3.8 We first show that ${lim}_{n\to \mathrm{\infty }}d(x_n,T_n{x}_n)=0$. Let $p\in \mathrm{F}(T)$ be such that $d(x_1,p)<\pi /2$. We have that $A:={lim}_{n\to \mathrm{\infty }}d(x_n,p)$ exists which is less than $\pi /2$ and $r(\{x_n\})<\pi /2$ by Proposition 3.2(i). Put $B:={lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}d(x_n,T_n{x}_n)$. Notice that $B<\pi$. Since ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{t}_n(1-t_n)>0$, we may assume that there exists a subsequence $\{n_k\}$ of ℕ such that ${lim}_{k\to \mathrm{\infty }}d(x_{n_k},T_{n_k}{x}_{n_k})=B$ and $t_{n_k}\to t\in (0,1)$. Using the quasi-nonexpansiveness of $T_n$ and Lemma 2.4, we get that

$$\begin{array}{c}cosd(x_{n_k+1},p)sind(x_{n_k},T_{n_k}{x}_{n_k})\hfill \\ =cosd(t_{n_k}{x}_{n_k}\oplus (1-t_{n_k})T_{n_k}{x}_{n_k},p)sind(x_{n_k},T_{n_k}{x}_{n_k})\hfill \\ \ge cosd(x_{n_k},p)sin(t_{n_k}d(x_{n_k},T_{n_k}{x}_{n_k}))+cosd(T_{n_k}{x}_{n_k},p)sin((1-t_{n_k})d(x_{n_k},T_{n_k}{x}_{n_k}))\hfill \\ \ge cosd(x_{n_k},p)(sin(t_{n_k}d(x_{n_k},T_{n_k}{x}_{n_k}))+sin((1-t_{n_k})d(x_{n_k},T_{n_k}{x}_{n_k}))).\hfill \end{array}$$

Letting $k\to \mathrm{\infty }$ yields

$$cosAsinB\ge cosA(sintB+sin(1-t)B).$$

Using elementary trigonometry, we get that $B=0$. Hence it follows that

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

By condition (C3), we get that

$$\begin{array}{rl}d(x_n,T{x}_n)& \le d(x_n,T_n{x}_n)+d(T_n{x}_n,T{x}_n)\\ \le d(x_n,T_n{x}_n)+sup\{d(T_{n}y,Ty):y\in Y\}\to 0,\end{array}$$

where $Y:=\{x_n\}$.

Finally, we show that ${\omega }_{\mathrm{\Delta }}(\{x_n\})\subset \mathrm{F}(T)$. Let $u\in {\omega }_{\mathrm{\Delta }}(\{x_n\})$. Then there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ such that $\{x_{n_k}\}$ Δ-converges to u. It follows from the Δ-demiclosedness of T and $d(x_n,T{x}_n)\to 0$ that $Tu=u$, that is, ${\omega }_{\mathrm{\Delta }}(\{x_n\})\subset \mathrm{F}(T)$. Hence the result follows from Proposition 3.2(ii). The proof is now finished. □

As an immediate consequence of Theorem 3.8, we obtain the following result.

Corollary 3.10 Let X be a complete $CAT(1)$ space and $T:X\to X$ be a mapping with $\mathrm{F}(T)\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be the Mann algorithm (1.1) in X. Suppose that

• (C0) T is quasi-nonexpansive and Δ-demiclosed;

• (C1) $d(x_1,\mathrm{F}(T))<\pi /2$;

• (C2′) ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}{t}_n(1-t_n)>0$.

Then the sequence $\{x_n\}$ Δ-converges to a fixed point of T.

Question 3.11 We do not know whether the conclusion of Corollary 3.10 holds if (C2′) is replaced by the more general condition (C2).

Let $\{T_n\}:X\to X$ be a countable family of quasi-nonexpansive mappings with $\mathrm{F}:={\bigcap }_{n=1}^{\mathrm{\infty }}\mathrm{F}(T_n)\ne \mathrm{\varnothing }$. We next show how to generate a family $\{W_n\}$ and a mapping W satisfying (C3′) and (C4), and hence Theorem 3.8 is applicable.

Theorem 3.12 Let X be a complete $CAT(1)$ space such that $d(u,v)<\pi /2$ for all $u,v\in X$, and let $\{T_n\}:X\to X$ be a countable family of quasi-nonexpansive mappings with $\mathrm{F}:={\bigcap }_{n=1}^{\mathrm{\infty }}\mathrm{F}(T_n)\ne \mathrm{\varnothing }$. Then there exist a family of quasi-nonexpansive mappings $\{W_n\}:X\to X$ and a quasi-nonexpansive mapping $W:X\to X$ such that

• (i) $(\{W_n\},W)$ satisfies AKTT-condition and $\mathrm{F}(W)={\bigcap }_{n=1}^{\mathrm{\infty }}\mathrm{F}(W_n)=\mathrm{F}$;

• (ii) W is Δ-demiclosed whenever $T_n$ is Δ-demiclosed for all $n\in \mathbb{N}$.

To prove Theorem 3.12, we need the following lemmas.

Lemma 3.13 ([14])

Let X be a complete $CAT(1)$ space such that $d(u,v)<\pi /2$ for all $u,v\in X$, and let $S,T:X\to X$ be quasi-nonexpansive mappings with $\mathrm{F}(S)\cap \mathrm{F}(T)\ne \mathrm{\varnothing }$. Then, for each $0<t<1$, $\mathrm{F}(S)\cap \mathrm{F}(T)=\mathrm{F}(tS\oplus (1-t)T)$ and the mapping $tS\oplus (1-t)T$ is quasi-nonexpansive.

The following lemma is essentially proved in [11]. For the sake of completeness, we show the proof.

Lemma 3.14 Let X be a complete $CAT(1)$ space such that $d(u,v)<\pi /2$ for all $u,v\in X$, and let $S,T:X\to X$ be quasi-nonexpansive mappings with $\mathrm{F}(S)\cap \mathrm{F}(T)\ne \mathrm{\varnothing }$. Let $\{x_n\}$ be a sequence of X. If $d(x_n,\frac{1}{2}S{x}_n\oplus \frac{1}{2}T{x}_n)\to 0$, then $d(x_n,S{x}_n)\to 0$ and $d(x_n,T{x}_n)\to 0$.

Proof Put $W:=\frac{1}{2}S\oplus \frac{1}{2}T$. By Lemma 3.13, we have that W is quasi-nonexpansive and $\mathrm{F}(W)=\mathrm{F}(S)\cap \mathrm{F}(T)$. Let $p\in \mathrm{F}(W)$. By Lemma 2.4 and the quasi-nonexpansiveness of S and T, we get that

$$\begin{array}{c}2cosd(W{x}_n,p)sin\frac{d(S{x}_n,T{x}_n)}{2}cos\frac{d(S{x}_n,T{x}_n)}{2}\hfill \\ =cosd(\frac{1}{2}S{x}_n\oplus \frac{1}{2}T{x}_n,p)sind(S{x}_n,T{x}_n)\hfill \\ \ge cosd(S{x}_n,p)sin\frac{d(S{x}_n,T{x}_n)}{2}+cosd(T{x}_n,p)sin\frac{d(S{x}_n,T{x}_n)}{2}\hfill \\ \ge 2cosd(x_n,p)sin\frac{d(S{x}_n,T{x}_n)}{2}.\hfill \end{array}$$

This implies that

$$d(S{x}_n,T{x}_n)=0\text{or}cos\frac{d(S{x}_n,T{x}_n)}{2}\ge \frac{cosd(x_n,p)}{cosd(W{x}_n,p)}.$$

It follows from $d(x_n,W{x}_n)\to 0$ that $\frac{cosd(x_n,p)}{cosd(W{x}_n,p)}\to 1$, that is, $d(S{x}_n,T{x}_n)\to 0$. Thus

$$\begin{array}{rl}d(x_n,S{x}_n)& \le d(x_n,\frac{1}{2}S{x}_n\oplus \frac{1}{2}T{x}_n)+d(\frac{1}{2}S{x}_n\oplus \frac{1}{2}T{x}_n,S{x}_n)\\ =d(x_n,\frac{1}{2}S{x}_n\oplus \frac{1}{2}T{x}_n)+\frac{d(S{x}_n,T{x}_n)}{2}\to 0.\end{array}$$

Hence $d(x_n,S{x}_n)\to 0$. Similarly, $d(x_n,T{x}_n)\to 0$ and the proof is finished. □

We are now ready to prove Theorem 3.12.

Proof of Theorem 3.12 Put $S_n:=\frac{1}{2}I\oplus \frac{1}{2}{T}_n$ for all $n\in \mathbb{N}$. We define a family of mappings $\{W_n\}:X\to X$ by

$$\begin{array}{c}{W}_{1}x=S_{1}x;\hfill \\ W_{2}x=\frac{1}{2}{S}_{1}x\oplus \frac{1}{2}{S}_{2}x;\hfill \\ W_{3}x=\frac{1}{2}{S}_{1}x\oplus \frac{1}{2}(\frac{1}{2}{S}_{2}x\oplus \frac{1}{2}{S}_{3}x);\hfill \\ ⋮\hfill \\ W_{n}x=\frac{1}{2}{S}_{1}x\oplus \frac{1}{2}(\frac{1}{2}{S}_{2}x\oplus \frac{1}{2}(\cdots \oplus \frac{1}{2}(\frac{1}{2}{S}_{n-1}x\oplus \frac{1}{2}{S}_{n}x)\cdots ));\hfill \\ ⋮\hfill \end{array}$$

It follows from Lemma 3.13 that $W_n$ is quasi-nonexpansive for all $n\in \mathbb{N}$ and ${\bigcap }_{n=1}^{\mathrm{\infty }}\mathrm{F}(W_n)={\bigcap }_{n=1}^{\mathrm{\infty }}\mathrm{F}(T_n)$.

We first show that ${\sum }_{n=1}^{\mathrm{\infty }}sup\{d(W_{n+1}x,W_{n}x):x\in X\}<\mathrm{\infty }$. For $k\in \mathbb{N}$, put $V_k^{(k)}:=S_k$ and

$$V_n^{(k)}:=\frac{1}{2}{S}_k\oplus \frac{1}{2}(\frac{1}{2}{S}_{k+1}\oplus \frac{1}{2}(\cdots \oplus \frac{1}{2}(\frac{1}{2}{S}_{n-1}\oplus \frac{1}{2}{S}_n)\cdots ))$$

for all $n>k$. By Lemma 2.3, we have that

$$\begin{array}{rl}d(W_{n+1}x,W_{n}x)& =d(\frac{1}{2}{S}_{1}x\oplus \frac{1}{2}{V}_{n+1}^{(2)}x,\frac{1}{2}{S}_{1}x\oplus \frac{1}{2}{V}_n^{(2)}x)\\ \le \left(\frac{\sqrt{2}}{2}\right)d(\frac{1}{2}{S}_{2}x\oplus \frac{1}{2}{V}_{n+1}^{(3)}x,\frac{1}{2}{S}_{2}x\oplus \frac{1}{2}{V}_n^{(3)}x)\\ \le {\left(\frac{\sqrt{2}}{2}\right)}^{2}d(\frac{1}{2}{S}_{3}x\oplus \frac{1}{2}{V}_{n+1}^{(4)}x,\frac{1}{2}{S}_{3}x\oplus \frac{1}{2}{V}_n^{(4)}x)\\ \le \cdots \\ \le {\left(\frac{\sqrt{2}}{2}\right)}^{n-1}d(\frac{1}{2}{S}_{n}x\oplus \frac{1}{2}{S}_{n+1}x,S_{n}x)\\ \le {\left(\frac{\sqrt{2}}{2}\right)}^{n-1}\cdot \frac{\pi }{2}\end{array}$$

for all $x\in X$ and $n\in \mathbb{N}$. Then

$$sup\{d(W_{n+1}x,W_{n}x):x\in X\}\le {\left(\frac{\sqrt{2}}{2}\right)}^{n-1}\cdot \frac{\pi }{2}$$

and the result follows. In particular, $\{W_{n}x\}$ is a Cauchy sequence for each $x\in X$. We now define the mapping $W:X\to X$ by

$$Wx:=\underset{n\to \mathrm{\infty }}{lim}{W}_{n}x\text{for all }x\in X.$$

Next, we show that $\mathrm{F}(W)={\bigcap }_{n=1}^{\mathrm{\infty }}\mathrm{F}(T_n)$. It is easy to see that ${\bigcap }_{n=1}^{\mathrm{\infty }}\mathrm{F}(T_n)\subset \mathrm{F}(W)$. On the other hand, let $q\in {\bigcap }_{n=1}^{\mathrm{\infty }}\mathrm{F}(T_n)={\bigcap }_{n=1}^{\mathrm{\infty }}\mathrm{F}(S_n)$ and $p\in \mathrm{F}(W)$. We prove that $p\in \mathrm{F}(T_k)$ for all $k\in \mathbb{N}$. Let $k\in \mathbb{N}$ be given. For any $n>k$, it follows from Lemma 2.5 that

$$\begin{array}{c}cosd(q,W_{n}p)\hfill \\ =cosd(q,\frac{1}{2}{S}_{1}p\oplus \frac{1}{2}(\frac{1}{2}{S}_{2}p\oplus \frac{1}{2}(\cdots \oplus \frac{1}{2}(\frac{1}{2}{S}_{n-1}p\oplus \frac{1}{2}{S}_{n}p)\cdots )))\hfill \\ \ge \frac{1}{2}cosd(q,S_{1}p)+\frac{1}{2}cosd(q,\frac{1}{2}{S}_{2}p\oplus \frac{1}{2}(\cdots \oplus \frac{1}{2}(\frac{1}{2}{S}_{n-1}p\oplus \frac{1}{2}{S}_{n}p)\cdots ))\hfill \\ \ge \frac{1}{2}cosd(q,S_{1}p)+\cdots +\frac{1}{2^k}cosd(q,S_{k}p)+\cdots +\frac{1}{2^{n-1}}cosd(q,S_{n-1}p)\hfill \\ +\frac{1}{2^{n-1}}cosd(q,S_{n}p)\hfill \\ \ge (\frac{1}{2}+\cdots +\frac{1}{2^{k-1}}+\frac{1}{2^{k+1}}+\cdots +\frac{1}{2^{n-1}}+\frac{1}{2^{n-1}})cosd(q,p)+\frac{1}{2^k}cosd(q,S_{k}p)\hfill \\ =(1-\frac{1}{2^k})cosd(q,p)+\frac{1}{2^k}cosd(q,S_{k}p).\hfill \end{array}$$

Letting $n\to \mathrm{\infty }$ yields $W_{n}p\to Wp=p$ and

$$cosd(q,p)=cosd(q,Wp)\ge (1-\frac{1}{2^k})cosd(q,p)+\frac{1}{2^k}cosd(q,S_{k}p).$$

This implies that $cosd(q,p)=cosd(q,S_{k}p)$. It follows from Lemma 2.4 that

$$\begin{array}{rl}cosd(q,p)sind(p,T_{k}p)& =cosd(q,S_{k}p)sind(p,T_{k}p)\\ =cosd(q,\frac{1}{2}p\oplus \frac{1}{2}{T}_{k}p)sind(p,T_{k}p)\\ \ge cosd(q,p)sin\frac{d(p,T_{k}p)}{2}+cosd(q,T_{k}p)sin\frac{d(p,T_{k}p)}{2}\\ \ge 2cosd(q,p)sin\frac{d(p,T_{k}p)}{2}.\end{array}$$

Using elementary trigonometry, we get that $d(p,T_{k}p)=0$. Since k is arbitrary, we have $p\in {\bigcap }_{n=1}^{\mathrm{\infty }}\mathrm{F}(T_n)$. Hence (i) is proved.

Finally, we prove (ii). We assume that $T_n$ is Δ-demiclosed for all $n\in \mathbb{N}$. We show that W is Δ-demiclosed. Let $\{x_n\}\subset X$ be such that ${lim}_{n\to \mathrm{\infty }}d(x_n,W{x}_n)=0$ and $\{x_n\}$ Δ-converges to $x\in X$. It follows from the definitions of $\{W_n\}$ and $\{V_n^{(2)}\}$ that

$$W_n=\frac{1}{2}{S}_1\oplus \frac{1}{2}{V}_n^{(2)}\text{for all }n\ge 2.$$

Similar to the proof of the first and the second steps, we can define the quasi-nonexpansive mapping $V:X\to X$ by

$$Vx:=\underset{n\to \mathrm{\infty }}{lim}{V}_n^{(2)}x\text{for all }x\in X$$

and $\mathrm{F}(V)={\bigcap }_{n=2}^{\mathrm{\infty }}\mathrm{F}(T_n)$. This implies that

$$W=\frac{1}{2}{S}_1\oplus \frac{1}{2}V\text{and}\mathrm{F}(W)=\mathrm{F}(S_1)\cap \mathrm{F}(V).$$

Then, by Lemma 3.14, we obtain that $d(x_n,S_1{x}_n)\to 0$ and $d(x_n,V{x}_n)\to 0$. Thus

$$d(x_n,T_1{x}_n)=2d(x_n,\frac{1}{2}{x}_n\oplus \frac{1}{2}{T}_1{x}_n)=2d(x_n,S_1{x}_n)\to 0.$$

Since $T_1$ is Δ-demiclosed, we have $x\in \mathrm{F}(T_1)$. Continuing this procedure gives $x\in {\bigcap }_{n=1}^{\mathrm{\infty }}\mathrm{F}(T_n)=\mathrm{F}(W)$. This completes the proof. □