A Common Fixed Point Theorem for Vicinal Mappings on Geodesic Spaces

Abstract

In 2024, Kimura proposed the modified shrinking method without assuming the existence of a common fixed point for a family of nonexpansive mappings defined on a complete geodesic space with a nonpositive upper curvature bound. In this paper, we discuss this method for vicinal mappings in an admissible complete geodesic space whose upper curvature bound is an arbitrary real number. Moreover, we investigate the convex minimization problem by using the main result and a resolvent for convex functions.

1. Introduction

The methods to approximate a fixed point of mappings are considered by many researchers in various spaces. One of the most classical results in this direction is Banach’s contraction theorem, which guarantees that a contraction mapping on a complete metric space has a unique fixed point. This result has been extended in many directions; see, for example [1,2,3]. On the other hand, common fixed point problems of non-linear mappings have been investigated by numerous researchers in many kinds of spaces, for examples, Hilbert spaces, Banach spaces and CAT$\left(\kappa \right)$ spaces; see, for instance [4,5,6].

Furthermore, the shrinking projection method is an approximation method for finding a fixed point of non-linear mappings and it has been studied by many researchers in various spaces. The original one was proposed by Takahashi, Takeuchi, and Kubota [6] in 2008 in the setting of Hilbert spaces. In 2024, Kimura proved the modified shrinking projection method without assuming that the nonexpansive mappings have fixed points in a complete CAT(0) space as follows.

Theorem 1

([5]). Let X be a complete CAT$\left(0\right)$ space such that a subset $\{z\in X\mid d(u,z)\leqq d(v,z\left)\right\}$ of X is convex for any $u,v\in X$. Let $\{T_i:X\to X\mid i=1,2,\dots ,m\}$ be a finite family of nonexpansive mappings on X. Generate a sequence $\left\{x_n\right\}$ of X with a sequence $\left\{C_n\right\}$ of subsets X by the following iterative steps:

(i)

$x_1\in X$, $C_1=X$, and $n=1$;

(ii)

$C_{n+1}={\bigcap }_{i=1}^m\{z\in X\mid d(T_i{x}_n,z)\leqq d(x_n,z)\}\cap C_n$;

(iii)

• If $C_{n+1}\ne \varnothing$, then define $x_{n+1}=P_{C_{n+1}}{x}_n$, increase n by 1, and go to (i);
• If $C_{n+1}=\varnothing$, then for all $k\geqq n+1$, set $C_k=\varnothing$ and leave $x_k$ undefined and terminate the process.

Then, ${\bigcap }_{i=1}^m\mathcal{F}\left(T_i\right)\ne \varnothing$ if and only if ${\bigcap }_{k=1}^{\infty }{C}_k\ne \varnothing$.

Further, if ${\bigcap }_{i=1}^m\mathcal{F}\left(T_i\right)\ne \varnothing$, then $\left\{x_n\right\}$ is well defined and it is Δ-convergent to some $x_0\in {\bigcap }_{i=1}^m\mathcal{F}\left(T_i\right)$.

For more results of the shrinking projection method, see [7,8,9].

On the other hand, in 2018, the vicinal mapping was proposed by Kohsaka [10] in an admissible CAT$\left(1\right)$ space. A mapping T from an admissible CAT$\left(1\right)$ space to itself is said to be

• vicinal if

  $$\begin{array}{cc}\hfill & (C_x^2(1+C_y^2)+C_y^2(1+C_x^2))cosd(Tx,Ty)\hfill \\ \hfill & \geqq C_x^2(1+C_y^2)cosd(Tx,y)+C_y^2(1+C_x^2)cosd(x,Ty)\hfill \end{array}$$

  for all $x,y\in X$;
• firmly vicinal if

  $$\begin{array}{cc}\hfill & (C_x^2(1+C_y^2)C_y+C_y^2(1+C_x^2)C_x)cosd(Tx,Ty)\hfill \\ \hfill & \geqq C_x^2(1+C_y^2)cosd(Tx,y)+C_y^2(1+C_x^2)cosd(x,Ty)\hfill \end{array}$$

  for all $x,y\in X$,

where $C_z=cosd(Tz,z)$ for $z\in X$. In 2019, the authors [11] proposed a new vicinal mapping in an admissible CAT$\left(\kappa \right)$ space as follows: a self-mapping T on an admissible CAT$\left(\kappa \right)$ space is said to be

• Vicinal with $\psi$ if

  $$\begin{array}{cc}\hfill & \left(\psi \left(d\right(Tx,x\left)\right)+\psi \left(d\right(Ty,y\left)\right)\right)c_{\kappa }\left(d(Tx,Ty)\right)\hfill \\ & \leqq \psi \left(d(Ty,y)\right)c_{\kappa }\left(d(Tx,y)\right)+\psi \left(d(Tx,x)\right)c_{\kappa }\left(d(x,Ty)\right)\hfill \end{array}$$

  for all $x,y\in X$;
• Firmly vicinal with $\psi$ if

  $$\begin{array}{cc}\hfill & (\psi \left(d(Tx,x)\right)c_{\kappa }\left(d(Tx,x)\right)+\psi \left(d(Ty,y)\right)c_{\kappa }\left(d(Ty,y)\right))c_{\kappa }^{″}\left(d(Tx,Ty)\right)\hfill \\ & +\left(\psi \left(d\right(Tx,x\left)\right)+\psi \left(d\right(Ty,y\left)\right)\right)c_{\kappa }\left(d(Tx,Ty)\right)\hfill \\ & \leqq \psi \left(d(Ty,y)\right)c_{\kappa }\left(d(Tx,y)\right)+\psi \left(d(Tx,x)\right)c_{\kappa }\left(d(x,Ty)\right)\hfill \end{array}$$

  for all $x,y\in X$,

where $\psi :\left[0,D_{\kappa }/2\right[\to \left]0,\infty \right[$ is a function that is continuous at 0.

The notion of vicinal mappings summarizes common properties of various perturbation functions for the resolvent operators studied in the literature. Therefore, it provides a unified framework that contains many examples and is useful in the study of approximation methods.

In CAT$\left(0\right)$ spaces, we have valuable examples of nonexpansive mappings, including the metric projection. Therefore, Theorem 1 is convenient in CAT$\left(0\right)$ spaces. However, the metric projection is not nonexpansive in general in the setting of CAT$\left(\kappa \right)$ spaces if $\kappa >0$. On the other hand, we have many valuable examples of vicinal mappings with $\psi$ in CAT$\left(\kappa \right)$ spaces, including the metric projection and the resolvent for a convex function.

In this paper, we discuss an approximation technique to a common fixed point of vicinal mappings in an admissible complete CAT$\left(\kappa \right)$ space by using the same method as in Theorem 1. We change the underlying space and the considered mappings in Theorem 1 to a CAT$\left(\kappa \right)$ space and vicinal mappings, respectively. Moreover, we apply the main result to the convex minimization problem by using the resolvent for convex functions.

2. Preliminaries

Let X be a metric space and take $x,y\in X$. A mapping ${\gamma }_{xy}:\left[0,d(x,y)\right]\to X$ is said to be a geodesic from x to y if it satisfies the following conditions:

• ${\gamma }_{xy}\left(0\right)=x$;
• ${\gamma }_{xy}\left(d(x,y)\right)=y$;
• $d({\gamma }_{xy}\left(s\right),{\gamma }_{xy}\left(t\right))=\left|s-t\right|$ for all $s,t\in \left[0,d(x,y)\right]$.

Given $D\in \left]0,\infty \right]$, we say that X is a D-geodesic space if for any $x,y\in X$ with $d(x,y)<D$, there exists a geodesic connecting them. Such a space is said to be uniquely D-geodesic when this geodesic is uniquely determined for every pair of points. In this case, for $x,y\in X$, the image of ${\gamma }_{xy}$ is denoted by $[x,y]$ and is referred to as the geodesic segment joining x and y; that is, $[x,y]=\{z\in X\mid \exists s\in [0,d(x,y)]({\gamma }_{xy}\left(s\right)=z)\}$. Let $\kappa$ be a real number and

$$D_{\kappa }=\left\{\begin{array}{cc}\pi /\sqrt{\kappa }\hfill & (\kappa >0);\hfill \\ \infty \hfill & (\kappa \leqq 0).\hfill \end{array}\right.$$

In order to introduce CAT$\left(\kappa \right)$ spaces, we first define a function $c_{\kappa }$ from $\left[0,D_{\kappa }/2\right]$ into $\left[0,\infty \right]$ as follows:

$$\begin{array}{cc}\hfill c_{\kappa }\left(t\right)& ={\displaystyle \frac{t^2}{2}}-{\displaystyle \frac{\kappa t^4}{24}}+{\displaystyle \frac{{\kappa }^2{t}^6}{720}}+\dots \hfill \\ \hfill & =\sum _{n=1}^{\infty }{\displaystyle \frac{{(-1)}^{n-1}{\kappa }^{n-1}{t}^{2n}}{\left(2n\right)!}}\hfill \\ \hfill & =\left\{\begin{array}{cc}{\displaystyle \frac{1}{\kappa }}\left(1-cos\left(\sqrt{\kappa }t\right)\right)\hfill & (\kappa >0);\hfill \\ {\displaystyle \frac{1}{2}}{t}^2\hfill & (\kappa =0);\hfill \\ {\displaystyle \frac{1}{-\kappa }}\left(cosh\left(\sqrt{-\kappa }t\right)-1\right)\hfill & (\kappa <0),\hfill \end{array}\right.\hfill \end{array}$$

for all $t\in \left[0,D_{\kappa }/2\right[$ and $c_{\kappa }(D_{\kappa }/2)={lim}_{t\to D_{\kappa }/2}{c}_{\kappa }\left(t\right)$. Then it follows that

$$\begin{array}{c}\hfill c_{\kappa }^{\prime }\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\sqrt{\kappa }}}sin\left(\sqrt{\kappa }t\right)\hfill & (\kappa >0);\hfill \\ t\hfill & (\kappa =0);\hfill \\ {\displaystyle \frac{1}{\sqrt{-\kappa }}}sinh\left(\sqrt{-\kappa }t\right)\hfill & (\kappa <0),\hfill \end{array}\right.\end{array}$$

and that

$$\begin{array}{c}\hfill c_{\kappa }^{″}\left(t\right)=\left\{\begin{array}{cc}cos\left(\sqrt{\kappa }t\right)\hfill & (\kappa >0);\hfill \\ 1\hfill & (\kappa =0);\hfill \\ cosh\left(\sqrt{-\kappa }t\right)\hfill & (\kappa <0)\hfill \end{array}\right.\end{array}$$

for all $t\in \left[0,D_{\kappa }/2\right[$. It is known that:

• $c_{\kappa }$ is convex, increasing, and satisfies $c_{\kappa }\left(0\right)=0$;
• $c_{\kappa }^{\prime }$ is increasing and satisfies $c_{\kappa }^{\prime }\left(0\right)=0$;
• $c_{\kappa }^{″}\left(0\right)=1$.

For further details, we refer to [11]. A uniquely $D_{\kappa }$-geodesic space X is called a CAT$\left(\kappa \right)$ space if

$$\begin{array}{cc}\hfill & c_{\kappa }\left(d(tx\oplus (1-t)y,z)\right)c_{\kappa }^{\prime }\left(d(x,y)\right)\hfill \\ \hfill & \leqq c_{\kappa }\left(d(x,z)\right)c_{\kappa }^{\prime }\left(td(x,y)\right)+c_{\kappa }\left(d(y,z)\right)c_{\kappa }^{\prime }\left(d(x,y)\right)\hfill \\ \hfill & -4c_{\kappa }^{\prime }(d(x,y)/2)c_{\kappa }^{\prime }(td(x,y)/2)c_{\kappa }^{\prime }((1-t)d(x,y)/2)\hfill \end{array}$$

holds for all $x,y,z\in X$ with $d(x,y)+d(y,z)+d(z,x)<2D_{\kappa }$ and $t\in \left[0,1\right]$. We say a CAT$\left(\kappa \right)$ space is admissible if $d(x,y)<D_{\kappa }/2$ for all $x,y\in X$. For more details of CAT$\left(\kappa \right)$ spaces, see [12].

For two points $x,y$ in an admissible CAT$\left(\kappa \right)$ space X, a convex combination between x and y is defined by

$$tx\oplus (1-t)y={\gamma }_{xy}\left((1-t)d(x,y)\right)$$

for all $x,y\in X$ and $t\in \left[0,1\right]$. A function f from X into $\left]-\infty ,\infty \right]$ is said to be

• Proper if there exists $x\in X$ such that $f\left(x\right)<\infty$;
• Lower semicontinuous if

  $$f\left(x_{\infty }\right)\leqq \underset{n\to \infty }{lim\; inf}f\left(x_n\right)$$

  whenever a sequence $\left\{x_n\right\}$ of X is convergent to $x_{\infty }\in X$;
• Convex if

  $$f(tx\oplus (1-t\left)y\right)\leqq tf\left(x\right)+(1-t)f\left(y\right)$$

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

A sequence $\left\{x_n\right\}$ of X is said to be geodesically bounded if

$$\underset{y\in X}{inf}\underset{n\to \infty }{lim\; sup}d(x_n,y)<{\displaystyle \frac{D_{\kappa }}{2}}.$$

An asymptotic center of $\left\{x_n\right\}$ is defined by

$$\mathcal{A}\left(\left\{x_n\right\}\right)=\underset{y\in X}{argmin}\left\{\underset{n\to \infty }{lim\; sup}d(x_n,y)\right\}.$$

A sequence $\left\{x_n\right\}$ is $\Delta$-convergent to $x_0\in X$ if $x_0$ is a unique asymptotic center of any subsequences $\left\{x_{n_i}\right\}$ of $\left\{x_n\right\}$. If X is complete and $\left\{x_n\right\}$ is geodesically bounded, then $\left\{x_n\right\}$ has a unique asymptotic center; see [13,14]. We say a subset C of X is convex if $[x,y]\subset C$ for all $x,y\in C$. If X is complete, then the metric projection $P_C$ from X onto a nonempty closed convex subset C of X is defined by

$$P_{C}x=\underset{y\in C}{argmin}\left\{c_{\kappa }\left(d(y,x)\right)\right\}=\underset{y\in C}{argmin}\left\{d(y,x)\right\}$$

for each $x\in X$. The metric projection $P_C$ satisfies

$$c_{\kappa }d(x,P_{C}x)c_{\kappa }^{″}d(P_{C}x,u)+c_{\kappa }d(P_{C}x,u)\leqq c_{\kappa }d(x,u)$$

for all $x\in X$ and $u\in C$. For more details of the metric projection, see [15].

3. A Common Fixed Point Theorem for Vicinal Mappings

Let $\psi$ be a function from $\left[0,D_{\kappa }/2\right[$ into $\left[0,\infty \right[$ satisfying the following properties:

• $\psi$ is continuous at 0;
• If $\psi \left(t\right)=0$, then $t=0$.

A mapping T from an admissible CAT$\left(\kappa \right)$ space X into itself is said to be vicinal with $\psi$ if

$$\begin{array}{cc}\hfill & (\psi \left(d(x,Tx)\right)+\psi \left(d(y,Ty)\right))c_{\kappa }\left(d(Tx,Ty)\right)\hfill \\ \hfill & \leqq \psi \left(d(x,Tx)\right)c_{\kappa }\left(d(x,Ty)\right)+\psi \left(d(y,Ty)\right)c_{\kappa }\left(d(Tx,y)\right)\hfill \end{array}$$

holds for all $x,y\in X$. Note that if T is vicinal with $\psi$ and it has a fixed point, then T is quasi-nonexpansive; that is, $d(Tx,p)\leqq d(x,p)$ for all $x\in X$ and $p\in \mathcal{F}\left(T\right)$. For more details of the vicinal mapping, see [11]. We next discuss a common fixed point theorem for vicinal mappings in an admissible complete CAT$\left(\kappa \right)$ space.

Lemma 1

([5]). Let X be an admissible complete CAT$\left(\kappa \right)$ space and $\left\{C_n\right\}$ a sequence of nonempty closed convex subsets of X satisfying $C_{n+1}\subset C_n$ for $n\in \mathbb{N}$. Let $\left\{y_n\right\}$ be a geodesically bounded sequence of X. If $y_n\in C_n$ for all $n\in \mathbb{N}$, then $y_0=\mathcal{A}\left(\left\{y_n\right\}\right)$ belongs to ${\bigcap }_{n=1}^{\infty }{C}_n$.

Theorem 2.

For $\kappa \in \mathbb{R}$, let X be an admissible complete CAT$\left(\kappa \right)$ space satisfying the following condition: a set $\{z\in X\mid d(u,z)\leqq d(v,z\left)\right\}$ is convex for all $u,v\in X$. Let ${\psi }_i$ be functions from $\left[0,D_{\kappa }/2\right[$ to $\left[0,\infty \right[$ satisfying the following conditions for $i=1,2,\dots ,m$:

• ${\psi }_i$ is continuous at 0;
• ${\psi }_i\left(t\right)=0⟹t=0$.

Let $T_i: X\to X$ be vicinal mappings with ${\psi }_i$ for $i=1,2,\dots ,m$. Generate a sequence $\left\{x_n\right\}$ with a sequence of subsets $\left\{C_n\right\}$ of X by the following iterative method:

(i)

$x_1\in X$, $C_1=X$, and $n=1$;

(ii)

$C_{n+1}={\bigcap }_{i=1}^m\{z\in X\mid d(T_i{x}_n,z)\leqq d(x_n,z)\}\cap C_n$;

(iii)

• If $C_{n+1}\ne \varnothing$, then define $x_{n+1}=P_{C_{n+1}}{x}_n$, increment n by 1, and go to (i);
• If $C_{n+1}=\varnothing$, then for all $k\geqq n+1$, set $C_k=\varnothing$ and leave $x_k$ undefined and terminate the process.

Then,

$$\bigcap _{i=1}^m\mathcal{F}\left(T_i\right)\ne \varnothing \iff \bigcap _{k=1}^{\infty }{C}_k\ne \varnothing .$$

Further, if ${\bigcap }_{i=1}^m\mathcal{F}\left(T_i\right)\ne \varnothing$, then $\left\{x_n\right\}$ is Δ-convergent to $x_0\in {\bigcap }_{i=1}^m\mathcal{F}\left(T_i\right)$.

Proof. 

We first suppose that ${\bigcap }_{k=1}^{\infty }{C}_k\ne \varnothing$ and show ${\bigcap }_{i=1}^m\mathcal{F}\left(T_i\right)\ne \varnothing$. In this case, the iterative scheme does not stop and thus the sequence $\left\{x_n\right\}$ is well defined. Let $u\in {\bigcap }_{k=1}^{\infty }{C}_k$. Since $u\in C_{n+1}$, it follows from the property of the metric projection that

$$\begin{array}{cc}\hfill 0& \leqq c_{\kappa }d(x_n,u)-c_{\kappa }d(x_{n+1},x_n)c_{\kappa }^{″}d(x_{n+1},u)-c_{\kappa }d(x_{n+1},u)\hfill \\ \hfill & \leqq c_{\kappa }d(x_n,u)-c_{\kappa }d(x_{n+1},u).\hfill \end{array}$$

(1)

Since $c_{\kappa }$ is strictly increasing, the sequence $\left\{d(x_n,u)\right\}$ is nonincreasing; that is,

$$\begin{array}{c}\hfill d(x_{n+1},u)\leqq d(x_n,u)\leqq \dots \leqq d(x_1,u)<{\displaystyle \frac{D_{\kappa }}{2}}.\end{array}$$

(2)

Thus, $\left\{d(x_n,u)\right\}$ is convergent to some $a\in \left[0,D_{\kappa }/2\right[$, and $\left\{x_n\right\}$ is geodesically bounded. By (1), we have

$$0\leqq c_{\kappa }d(x_{n+1},x_n)\leqq {\displaystyle \frac{c_{\kappa }d(x_n,u)-c_{\kappa }d(x_{n+1},u)}{c_{\kappa }^{″}d(x_n,u)}}\to 0$$

and hence $d(x_{n+1},x_n)\to 0$ as $n\to \infty$. Since $x_{n+1}=P_{C_{n+1}}{x}_n\in C_{n+1}$, from the definition of $C_{n+1}$, we obtain

$$d(T_i{x}_n,x_{n+1})\leqq d(x_{n+1},x_n).$$

Therefore, we have

$$0\leqq d(T_i{x}_n,x_n)\leqq d(T_i{x}_n,x_{n+1})+d(x_{n+1},x_n)\leqq 2d(x_{n+1},x_n)\to 0$$

as $n\to \infty$, and thus,

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

for each $i\in \{1,2,\dots ,m\}$. Using this fact with uniform continuity of $c_{\kappa }$ on bounded sets, we have

$$\underset{n\to \infty }{lim}(c_{\kappa }d(x_n,T_i{x}_0)-c_{\kappa }d(T_i{x}_n,T_i{x}_0))=0.$$

(3)

We also obtain

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}d(T_i{x}_n,y)=\underset{n\to \infty }{lim\; sup}d(x_n,y)\end{array}$$

(4)

for all $y\in X$.

On the other hand, we know $\left\{x_n\right\}$ has a unique asymptotic center $x_0\in X$ since $\left\{x_n\right\}$ is geodesically bounded. The vicinality of $T_i$ implies that

$$\begin{array}{cc}\hfill & ({\psi }_i\left(d(x_n,T_i{x}_n)\right)+{\psi }_i\left(d(x_0,T_i{x}_0)\right))c_{\kappa }d(T_i{x}_n,T_i{x}_0)\hfill \\ \hfill & \leqq {\psi }_i\left(d(x_n,T_i{x}_n)\right)c_{\kappa }d(x_n,T_i{x}_0)+{\psi }_i\left(d(x_0,T_i{x}_0)\right)c_{\kappa }d(T_i{x}_n,x_0),\hfill \end{array}$$

or equivalently,

$$\begin{array}{cc}\hfill & {\psi }_i\left(d(x_0,T_i{x}_0)\right)c_{\kappa }d(T_i{x}_n,T_i{x}_0)\hfill \\ \hfill & \leqq {\psi }_i\left(d(x_n,T_i{x}_n)\right)(c_{\kappa }d(x_n,T_i{x}_0)-c_{\kappa }d(T_i{x}_n,T_i{x}_0))\hfill \\ \hfill & +{\psi }_i\left(d(x_0,T_i{x}_0)\right)c_{\kappa }d(T_i{x}_n,x_0).\hfill \end{array}$$

Taking the upper limit of its both sides, by (3) and the continuity of ${\psi }_i$ at 0, we get

$$\begin{array}{cc}\hfill & {\psi }_i\left(d(x_0,T_i{x}_0)\right)\underset{n\to \infty }{lim\; sup}{c}_{\kappa }d(T_i{x}_n,T_i{x}_0)\hfill \\ \hfill & \leqq {\psi }_i\left(0\right)\underset{n\to \infty }{lim\; sup}(c_{\kappa }d(x_n,T_i{x}_0)-c_{\kappa }d(T_i{x}_n,T_i{x}_0))\hfill \\ \hfill & +{\psi }_i\left(d(x_0,T_i{x}_0)\right)\underset{n\to \infty }{lim\; sup}{c}_{\kappa }d(T_i{x}_n,x_0)\hfill \\ \hfill & ={\psi }_i\left(d(x_0,T_i{x}_0)\right)\underset{n\to \infty }{lim\; sup}{c}_{\kappa }d(T_i{x}_n,x_0).\hfill \end{array}$$

(5)

If ${\psi }_i\left(d(x_0,T_i{x}_0)\right)>0$, then we get

$$\underset{n\to \infty }{lim\; sup}{c}_{\kappa }d(T_i{x}_n,T_i{x}_0)\leqq \underset{n\to \infty }{lim\; sup}{c}_{\kappa }d(T_i{x}_n,x_0)$$

by dividing both sides of inequality (5) by $\psi \left(d(x_0,T_i{x}_0)\right)$. Therefore, by using the increasingness of $c_{\kappa }$ and equality (4), we have

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; sup}d(x_n,T_i{x}_0)& =\underset{n\to \infty }{lim\; sup}d(T_i{x}_n,T_i{x}_0)\hfill \\ & \leqq \underset{n\to \infty }{lim\; sup}d(T_i{x}_n,x_0)=\underset{n\to \infty }{lim\; sup}d(x_n,x_0)\hfill \\ & =\underset{n\to \infty }{lim\; sup}d(x_n,y)\hfill \end{array}$$

(6)

for every $y\in X$, since $x_0$ is an asymptotic center of $\left\{x_n\right\}$. This inequality shows that $T_i{x}_0$ is also an asymptotic center of $\left\{x_n\right\}$. Since it is unique, we have $T_i{x}_0=x_0\in \mathcal{F}\left(T_i\right)$. If ${\psi }_i\left(d(x_0,T_i{x}_0)\right)=0$, then the assumption of ${\psi }_i$ implies that $d(T_i{x}_0,x_0)=0$ and hence $T_i{x}_0=x_0\in \mathcal{F}\left(T_i\right)$. Consequently, we have $x_0\in {\bigcap }_{i=1}^m\mathcal{F}\left(T_i\right)\ne \varnothing$.

We next show the inverse. Suppose that ${\bigcap }_{i=1}^m\mathcal{F}\left(T_i\right)\ne \varnothing$. Let $z\in {\bigcap }_{i=1}^m\mathcal{F}\left(T_i\right)$. We show $z\in C_k$ for all $k\in \mathbb{N}$ by induction. If $k=1$, then $z\in C_1=X$. Suppose that $z\in C_n$. In this case, we can define $x_n\in C_n$. Since $T_i$ is quasi-nonexpansive, we have $d(T_i{x}_n,z)\leqq d(x_n,z)$ and hence $z\in C_{n+1}$. By induction, we obtain $z\in C_k$ for all $k\in \mathbb{N}$. Therefore, we conclude that

$$\bigcap _{k=1}^{\infty }{C}_k\supset \bigcap _{i=1}^m\mathcal{F}\left(T_i\right)\ne \varnothing .$$

We finally show that $\left\{x_n\right\}$ is $\Delta$-convergent to $x_0\in {\bigcap }_{i=1}^m\mathcal{F}\left(T_i\right)$ if it is a nonempty set. Let $\left\{x_{n_j}\right\}$ be a subsequence of $\left\{x_n\right\}$ whose asymptotic center is $\left\{y_0\right\}\subset X$. Since $C_n$ is a nonempty closed convex subset of X that satisfies that

$$C_{n+1}\subset C_n$$

for all $n\in \mathbb{N}$, it follows from Lemma 1 that

$$y_0\in \bigcap _{j=1}^{\infty }{C}_{n_j}=\bigcap _{n=1}^{\infty }{C}_n$$

and that $\left\{d(x_n,y_0)\right\}$ is convergent to some real number in $\left[0,D_{\kappa }/2\right[$ from inequality (2). Then, we obtain

$$\begin{array}{cc}\hfill \underset{n\to \infty }{lim\; sup}d(x_n,y_0)& =\underset{n\to \infty }{lim}d(x_n,y_0)\hfill \\ \hfill & =\underset{j\to \infty }{lim}d(x_{n_j},y_0)\hfill \\ \hfill & \leqq \underset{j\to \infty }{lim\; sup}d(x_{n_j},x_0)\hfill \\ \hfill & \leqq \underset{n\to \infty }{lim\; sup}d(x_n,x_0).\hfill \end{array}$$

Since $x_0$ is a unique asymptotic center of $\left\{x_n\right\}$, we get $y_0=x_0$. Thus, $\left\{x_n\right\}$ is $\Delta$-convergent to $x_0\in {\bigcap }_{i=1}^m\mathcal{F}\left(T_i\right)$. □

We remark that the convexity of the set $\{z\in X\mid d(u,z)\le d(v,z\left)\right\}$ holds under suitable conditions in CAT($\kappa$) spaces. Although this property does not hold in general, it is satisfied in several important classes of spaces. Typical examples include:

• Convex subsets of a Hilbert space;
• The unit sphere of a Hilbert space with the spherical metric;
• The real Hilbert ball;
• The model spaces $M_{\kappa }^n$ for any $n\in \mathbb{N}$ and $\kappa \in \mathbb{R}$;
• The $\mathbb{R}$-tree.

Detailed discussions and sufficient conditions for this property can be found in [12,13,16,17,18].

4. A Convex Minimization Problem

The resolvent for a convex function plays an important role in discussing the convex minimization problem. For $\kappa \in \mathbb{R}$, let X be an admissible CAT$\left(\kappa \right)$ space, f a proper lower semicontinuous convex function from X into $\left]-\infty ,\infty \right]$ and $\phi$ a strictly increasing function from $\left[0,c_{\kappa }(D_{\kappa }/2)\right[$ into $\left[0,\infty \right[$ of class $C^1$. If

$$R_{f}x=\underset{y\in X}{argmin}\{f\left(y\right)+\phi \left(c_{\kappa }\left(d(y,x)\right)\right)\}$$

is a singleton, then the mapping $R_f:X\to X$ is called a resolvent for f. We know the following properties:

• ${argmin}_{X}f=\mathcal{F}\left(R_f\right)$;
• For any $x,y\in X$,

  $$\begin{array}{cc}\hfill & [{\phi }^{\prime }\left(c_{\kappa }\left(d(x,R_{f}x)\right)\right)+{\phi }^{\prime }\left(c_{\kappa }\left(d(y,R_{f}y)\right)\right)]c_{\kappa }\left(d(R_{f}x,R_{f}y)\right)\hfill \\ \hfill & \leqq {\phi }^{\prime }\left(c_{\kappa }\left(d(x,R_{f}y)\right)\right)c_{\kappa }\left(d(x,R_{f}y)\right)+{\phi }^{\prime }\left(c_{\kappa }\left(d(y,R_{f}y)\right)\right)c_{\kappa }\left(d(R_{f}x,y)\right),\hfill \end{array}$$

See [11]. By using the resolvent and Theorem 2, we obtain the following theorem.

Theorem 3.

For $\kappa \in \mathbb{R}$, let X be an admissible complete CAT$\left(\kappa \right)$ space satisfying the following condition: a set $\{z\in X\mid d(u,z)\leqq d(v,z\left)\right\}$ is convex for all $u,v\in X$. Let ${\phi }_i$ be strictly increasing functions from $\left[0,c_{\kappa }(D_{\kappa }/2)\right[$ to $\left[0,\infty \right[$ of class $C^1$ and $f_i$ proper lower semicontinuous convex functions from X into $\left]-\infty ,\infty \right]$ for $i=1,2,\dots ,m$. Suppose that ${\phi }_i^{\prime }$ is either increasing or decreasing for $i=1,2,\dots ,m$, ${\bigcap }_{i=1}^m{argmin}_X{f}_i\ne \varnothing$ and

$$R_{f_i}x=\underset{y\in X}{argmin}\{f_i\left(y\right)+{\phi }_i\left(c_{\kappa }\left(d(y,x)\right)\right)\}$$

is a singleton for $x\in X$ and $i=1,2,\dots ,m$. Generate a sequence $\left\{x_n\right\}$ and $\left\{C_n\right\}$ of X for $n\in \mathbb{N}$ as follows:

• $x_1\in X$ and $C_1=X$;
• $C_{n+1}={\bigcap }_{i=1}^m\{z\in X\mid d(R_{f_i}{x}_n,z)\leqq d(x_n,z)\}\cap C_n$;
• $x_{n+1}=P_{C_{n+1}}{x}_n$.

Then $\left\{x_n\right\}$ is Δ-convergent to an element of ${\bigcap }_{i=1}^m{argmin}_X{f}_i$.

Proof. 

By the properties of the resolvent, we know that

$$\bigcap _{i=1}^m\mathcal{F}\left(R_{f_i}\right)=\bigcap _{i=1}^m\underset{X}{argmin}{f}_i\ne \varnothing .$$

Since ${\phi }_i:\left[0,c_{\kappa }(D_{\kappa }/2)\right[\to \left[0,\infty \right[$ is a strictly increasing function of class $C^1$ and ${\phi }_i^{\prime }$ is either increasing or decreasing, we get ${\phi }_i^{\prime }\left(t\right)>0$ for all $t\in \left]0,D_{\kappa }/2\right[$. Therefore, from the inequality of the property of the resolvent, we notice that $R_{f_i}$ is vicinal with ${\psi }_i={\phi }_i^{\prime }\circ c_{\kappa }$ for each $i=1,2,\dots ,m$. Thus it follows from Theorem 2 that $\left\{x_n\right\}$ is $\Delta$-convergent to an element of ${\bigcap }_{i=1}^m\mathcal{F}\left(R_{f_i}\right)={\bigcap }_{i=1}^m{argmin}_X{f}_i$. □

We finally give an application of the common convex minimization problem in the setting of a Hilbert space.

Corollary 1.

Let H be a Hilbert space, and let $f_i$ be proper lower semicontinuous convex functions from H into $\left]-\infty ,\infty \right]$ for $i=1,2,\dots ,m$ that have a common fixed point and

$$R_{f_i}x=\underset{y\in H}{argmin}\left\{f_i\left(y\right)+{\displaystyle \frac{1}{2^i}}{∥y-x∥}^{2i}\right\}$$

for $x\in H$. Generate a sequence $\left\{x_n\right\}$ and $\left\{C_n\right\}$ of H for $n\in \mathbb{N}$ as follows:

• $x_1\in H$ and $C_1=H$;
• $C_{n+1}={\bigcap }_{i=1}^m\{z\in H\mid \parallel R_{f_i}{x}_n-z\parallel \leqq ∥x_n-z∥\}\cap C_n$;
• $x_{n+1}=P_{C_{n+1}}{x}_n$.

Then $\left\{x_n\right\}$ is weakly convergent to an element of ${\bigcap }_{i=1}^m{argmin}_X{f}_i$.

Proof. 

It is easy to check the convexity of the set

$$\{z\in H\mid ∥u-z∥\leqq ∥v-z∥\}$$

for all $u,v\in X$. Let ${\phi }_i\left(t\right)=t^i$ for $t\in \left[0,\infty \right[$. Then, we notice that ${\phi }_i$ satisfies the conditions in Theorem 3 for each $i=1,2,\dots ,m$. Moreover, it is well known that $R_{f_i}$ is a single-valued mapping. Therefore, by using Theorem 3, we get the desired result. □