A projection method for approximating fixed points of quasinonexpansive mappings in Hadamard spaces

Abstract

This work is devoted to analyzing the feasibility study of a Moudafi viscosity projection method with a weak contraction for a finite family of quasinonexpansive mappings in a Hadamard space. To this end, we need to construct a countable family of nonexpansive mappings satisfying AKTT condition with a weak contraction by choosing an appropriate control sequence under certain conditions.

1 Introduction

Let C be a nonempty subset of a metric space $(X,d)$. Suppose that, for each $x \in X$, there exists a unique point $P_{C}x \in C$ such that $d(x,P_{C}x)=d(x,C)=\inf_{y \in C}d(x,y)$. Then, the mapping $P_{C}$ of X onto C is called the metric projection.

The well-known Banach contraction principle is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-mappings of metric spaces. One generalization of the contraction principle for weak contractions is obtained by Alber and Guerre-Delabriere [1] in Hilbert spaces. A mapping $f:X \to X$ is called a φ-weak contraction if

$$ d\bigl(f(x),f(y)\bigr) \le d(x,y)-\varphi\bigl(d(x,y)\bigr),\quad x,y \in X, $$

where $\varphi:[0,\infty) \to[0,\infty)$ is a continuous and nondecreasing function with $\varphi(t)=0$ if and only if $t=0$.

Let $T:C \to X$ be a mapping. If $d(Tx,Ty) \le d(x,y)$ for all $x,y \in C$, then T is nonexpansive. We denote by $\mathfrak{F}(T)$ the set o fixed points of T. The mapping T is quasinonexpansive if $\mathfrak{F}(T)$ is nonempty and

$$d(Tx,y) \le d(x,y), \quad x \in C, y \in\mathfrak{F}(T). $$

A point $p \in C$ is said to be a strongly asymptotic fixed point [2] of T if there exists a sequence $\{x_{n}\}$ in C that converges strongly to p and $\lim_{n \to\infty}d(x_{n},Tx_{n})=0$. We denote by $\mathfrak{\widetilde{F}}(T)$ the set of strongly asymptotic fixed points of T. It is known that the fixed point set of a quasinonexpansive mapping defined on a $\operatorname{CAT}(0)$ space (see Section 2 for the definition) is closed and convex.

Approximation methods for finding specific fixed points of a family of nonexpansive mappings in Hilbert, Banach, and geodesic metric spaces have been studied by many researchers; see, e.g., [3–9] and the references therein. One well-known method, called the shrinking projection method, was first proposed by Takahashi et al. [10] and has been applied to a variety of approximation problems; see, e.g., [11, 12]. In particular, Kimura and Takahashi [11] applied this method to the zero-point problem for a maximal monotone operator defined in a Banach space and obtained strong convergence theorems. To generate the iterative sequence by the shrinking projection method, they use the metric projection onto a closed convex set $C_{n}$ for each $n \in\mathbb{N}$. It is noticeable that the larger the integer n, the more complicated the shape of $C_{n}$. Hence, the calculation of the projection is tedious as n gets larger. In 2011, Kimura et al. [2] overcome this difficulty and introduce the so-called averaged projection method of Halpern type for a family of quasinonexpansive mappings by combining the Halpern iteration. They still use the metric projection approach; nevertheless, the subsets corresponding to these projections have simpler shapes than the classical ones. Let us denote by $\mathfrak{F}(\mathfrak{T})$ the common fixed point set of all mappings in a family $\mathfrak{T}$. Their theorem is stated as follows.

Theorem 1.1

(Kimura et al. [2], Theorem 3.1)

Let C be a closed convex subset of a Hilbert space H, $\mathfrak{T}=\{T_{j}:j=1,\ldots,N\}$ a finite family of quasinonexpansive mappings of C into H with $\mathfrak{F}(\mathfrak{T}) \ne\emptyset$ and $\widetilde{\mathfrak{F}}(T_{j})=\mathfrak{F}(T_{j})$ for $j=1,\ldots,N$. Let $u,x_{1} \in C$ and define the sequence $\{x_{n}\}$ by

$$\begin{aligned}& y^{j}_{n} =\alpha_{n}x_{n}+(1- \alpha_{n})T_{j}x_{n}, \\& C^{j}_{n} =\bigl\{ z \in C:\bigl\Vert y^{j}_{n}-z \bigr\Vert \le \Vert x_{n}-z\Vert \bigr\} , \quad j=1,\ldots,N, \\& v^{j}_{n,k} =P_{C^{j}_{k}}x_{n}, \quad k=1, \ldots,n, j=1,\ldots,N, \\& w_{n,k} =\sum_{j=1}^{N} \beta^{j}_{k}v^{j}_{n,k}, \quad k=1, \ldots,n, \\& x_{n+1} = \delta_{n}u+(1-\delta_{n})\sum _{k=1}^{n}\gamma_{n,k}w_{n,k}, \end{aligned}$$

where $\{\alpha_{n}\}$, $\{\beta^{j}_{n}:j=1,\ldots,N\}$, $\{\gamma_{n,k}:k \le n\}$, and $\{\delta_{n}\}$ are sequences in $[0,1]$ satisfying the following conditions:

• (i) $\liminf_{n \to\infty}\alpha_{n}<1$,

• (ii) $\beta^{j}_{n}>0$ for $j=1,\ldots,N$, and $\sum_{j=1}^{N}\beta^{j}_{n}=1$ for $n \in\mathbb{N}$,

• (iii) $\sum_{k=1}^{n}\gamma_{n,k}=1$ for $n \in\mathbb{N}$, $\lim_{n \to\infty}\gamma_{n,k}>0$ for $k \in\mathbb{N}$, and $\sum_{n=1}^{\infty}\sum_{k=1}^{n}|\gamma_{n+1,k}-\gamma _{n,k}|<\infty$,

• (iv) $\lim_{n \to\infty}\delta_{n}=0$, $\sum_{n=1}^{\infty}\delta_{n}=\infty$, and $\sum_{n=1}^{\infty}|\delta_{n+1}-\delta_{n}|<\infty$.

Then $\{x_{n}\}$ converges strongly to the point $P_{\mathfrak{F}(\mathfrak{T})}u$.

The problem of whether or not we can construct a shrinking projection method analogous to that given in Theorem 1.1 for solving a common fixed point problem for a finite family of quasinonexpansive mappings in a geodesic metric space is still open. The purpose of this paper is to analyze the feasibility study of Moudafi viscosity type of projection method with a weak contraction for a finite family of quasinonexpansive mappings in a complete $\operatorname{CAT}(0)$ space, also known as a Hadamard space.

This paper is organized as follows. In Section 2 we recall the definition of geodesic metric spaces and summarize some useful lemmas and the main properties of $\operatorname{CAT}(0)$ spaces. Besides, without vector addition as in a Banach space, we present an inequality to estimate the distance between two elements defined by finite convex combination ‘⊕’ in a $\operatorname{CAT}(0)$ space; see Lemma 2.2. In Section 3 we construct a sequence of nonexpansive mappings satisfying AKTT condition by choosing an appropriate control sequence under certain conditions; see Theorem 3.2. Therefore, a convergence theorem of a new Moudafi viscosity approximation follows from Theorem 3.2; see Theorem 3.3. Using Theorem 3.3, we also derive a strong convergence theorem by a Moudafi type viscosity approximation with a weak contraction for a family of quasinonexpansive mappings; see Theorem 3.4. As a particular case where a weak contraction is constant in Theorem 3.4, a strong convergence theorem by the averaged projection method of Halpern type is then obtained; see Theorem 3.5.

2 Preliminaries

Let $(X,d)$ be a metric space. For $x,y \in X$, a geodesic path joining x to y (or a geodesic from x to y) is an isometric mapping $c:[0,\ell] \subset\mathbb{R} \to X$ such that $c(0)=x$, $c(\ell)=y$, that is, $d(c(t),c(t'))=|t-t'|$ for all $t,t' \in[0,\ell]$. Therefore, $d(x,y)=\ell$. The image of c is called a geodesic (segment) from x to y, and we shall denote a definite choice of this geodesic segment by $[x,y]$. A point $z=c(t)$ in the geodesic $[x,y]$ will be written as $z=(1-\lambda)x \oplus\lambda y$, where $\lambda=t/\ell$, and so $d(z,x)=\lambda d(x,y)$ and $d(z,y)=(1-\lambda)d(x,y)$. A subset C of X is convex if every pair of points $x,y \in C$ can be joined by a geodesic in X and the image of every such geodesic is contained in C.

A geodesic triangle $\triangle(x_{1},x_{2},x_{3})$ in $(X,d)$ consists of three points $x_{i} \in X$ ($i=1,2,3$), its vertices, and a geodesic segment between each pair of vertices, its sides. If a point $x \in X$ lies in the union of $[x_{i},x_{j}]$, $i,j \in\{ 1,2,3\}$, then we write $x \in\triangle(x_{1},x_{2},x_{3})$. A comparison triangle for the geodesic triangle $\triangle (x_{1},x_{2},x_{3})$ in X is a triangle $\triangle(\bar{x}_{1},\bar{x}_{2},\bar{x}_{3})$ in the Euclidean plane $\mathbb{E}^{2}$ such that $d_{\mathbb{E}^{2}}(\bar{x}_{i},\bar{x}_{j})=d(x_{i},x_{j})$ for $i,j \in\{1,2,3\}$.

A geodesic triangle △ in X is said to satisfy the $\operatorname{CAT}(0)$ inequality if, given a comparison triangle △̅ in $\mathbb{E}^{2}$ for △,

$$d(x,y) \le d_{\mathbb{E}^{2}}(\bar{x},\bar{y})\quad \mbox{for }x,y \in \triangle, $$

where $\bar{x},\bar{y} \in\overline{\triangle}$ are the corresponding comparison points of x, y. The geodesic metric space X is called a $\operatorname{CAT}(0)$ space if all geodesic triangles in X satisfy the $\operatorname{CAT}(0)$ inequality. Note that Hilbert spaces are $\operatorname{CAT}(0)$.

Lemma 2.1

Let $(X,d)$ be a $\operatorname{CAT}(0)$ space, and let $\alpha,\beta\in [0,1]$. Then:

• (i) For $x,y \in X$, we have

  $$d\bigl(\alpha x \oplus(1-\alpha)y,\beta x \oplus(1-\beta)y\bigr)=|\alpha- \beta|d(x,y). $$

• (ii) ([13], Chapter II.2. Proposition 2.2) For $x,y,p,q \in X$, we have

  $$d\bigl(\alpha x \oplus(1-\alpha)y,\alpha p \oplus(1-\alpha)q\bigr) \le\alpha d(x,p)+(1-\alpha) d(y,q). $$

  In particular, if $p=q$, this reduces to

  $$d\bigl(\alpha x \oplus(1-\alpha)y,p\bigr) \le\alpha d(x,p)+(1-\alpha)d(y,p). $$

• (iii) ([14], Lemma 2.5) For $x,y,z \in X$, we have

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

We will extend the equality in Lemma 2.1(i) to any finitely many elements in X. First, we recall the notion of a finite sum ‘⊕’ defined by Butsan et al. [4]. Fix $n \in\mathbb{N}$ with $n \ge2$ and let $\{\alpha_{1},\ldots,\alpha _{n}\} \subset(0,1)$ with $\sum_{k=1}^{n}\alpha_{k}=1$ and $\{x_{1},\ldots,x_{n}\} \subset X$. By induction we define

$$ \bigoplus_{k=1}^{n}\alpha_{k}x_{k} =(1-\alpha_{n}) \biggl(\frac{\alpha_{1}}{1-\alpha_{n}}x_{1}\oplus\cdots \oplus\frac {\alpha_{n-1}}{1-\alpha_{n}}x_{n-1} \biggr) \oplus\alpha_{n}x_{n}. $$

(2.1)

The definition of ⨁ in (2.1) is an ordered one in the sense that it depends on the order of points $x_{1},\ldots,x_{n}$. However, we occasionally use the notation $\alpha_{1}x_{1} \oplus\alpha_{2}x_{2} \oplus\cdots\oplus\alpha_{n}x_{n}$ for such a point. Lemma 2.1(ii) assures that, for $y \in X$,

$$ d \Biggl(\bigoplus_{k=1}^{n} \alpha_{k}x_{k},y \Biggr) \le\sum _{k=1}^{n}\alpha _{k}d(x_{k},y). $$

(2.2)

Lemma 2.2

Let $(X,d)$ be a $\operatorname{CAT}(0)$ space, and for $n \in\mathbb{N}$ with $n \ge2$, let $\{\alpha_{k}\}_{k=1}^{n}$ and $\{\beta_{k}\}_{k=1}^{n} \subset(0,1)$ be two sequences such that $\sum_{k=1}^{n}\alpha_{k}=\sum_{k=1}^{n}\beta_{k}=1$. Then, for $x_{1},\ldots,x_{n} \in X$, we have

$$\begin{aligned}& d \Biggl(\bigoplus_{k=1}^{n} \alpha_{k}x_{k},\bigoplus_{k=1}^{n} \beta_{k}x_{k} \Biggr) \\& \quad \le\biggl\vert \frac{\alpha_{2}}{\alpha_{1}+\alpha_{2}}-\frac{\beta_{2}}{\beta _{1}+\beta_{2}}\biggr\vert ( \alpha_{1}+\alpha_{2})d(x_{1},x_{2}) \\& \qquad {}+\biggl\vert \frac{\alpha_{3}}{\sum_{k=1}^{3}\alpha_{k}}-\frac{\beta_{3}}{\sum_{k=1}^{3}\beta_{k}}\biggr\vert \sum _{k=1}^{3}\alpha_{k}\cdot\sum _{k=1}^{2}\frac{\beta_{k}}{\beta_{1}+\beta _{2}}d(x_{k},x_{3}) \\& \qquad {}+\cdots+\biggl\vert \frac{\alpha_{j}}{\sum_{k=1}^{j}\alpha_{k}}-\frac{\beta _{j}}{\sum_{k=1}^{j}\beta_{j}}\biggr\vert \sum_{k=1}^{j}\alpha_{j}\cdot\sum _{k=1}^{j-1}\frac{\beta_{k}}{\beta_{1}+\cdots +\beta_{j-1}}d(x_{k},x_{j}) \\& \qquad {}+\cdots+\vert \alpha_{n}-\beta_{n}\vert \sum _{k=1}^{n-1}\frac{\beta_{k}}{1-\beta _{n}}d(x_{k},x_{n}). \end{aligned}$$

Proof

We will prove the result by induction.

Step 1. According to Lemma 2.1(ii), (2.1), and (2.2), we derive

$$\begin{aligned}& d \Biggl(\bigoplus_{k=1}^{n} \alpha_{k}x_{k},\bigoplus_{k=1}^{n} \beta_{k}x_{k} \Biggr) \\& \quad \le d \Biggl((1-\alpha_{n}) \Biggl(\bigoplus _{k=1}^{n-1}\frac{\alpha _{k}}{1-\alpha_{n}}x_{k} \Biggr) \oplus\alpha_{n}x_{n}, (1-\alpha_{n}) \Biggl( \bigoplus_{k=1}^{n-1}\frac{\beta_{k}}{1-\beta _{n}}x_{k} \Biggr)\oplus\alpha_{n}x_{n} \Biggr) \\& \qquad {}+d \Biggl((1-\alpha_{n}) \Biggl(\bigoplus _{k=1}^{n-1}\frac{\beta_{k}}{1-\beta _{n}}x_{k} \Biggr) \oplus\alpha_{n}x_{n}, (1-\beta_{n}) \Biggl( \bigoplus_{k=1}^{n-1}\frac{\beta_{k}}{1-\beta _{n}}x_{k} \Biggr)\oplus\beta_{n}x_{n} \Biggr) \\& \quad \le (1-\alpha_{n})d \Biggl(\bigoplus _{k=1}^{n-1}\frac{\alpha_{k}}{1-\alpha _{n}}x_{k},\bigoplus _{k=1}^{n-1}\frac{\beta_{k}}{1-\beta_{n}}x_{k} \Biggr) + |\alpha_{n}-\beta_{n}|d \Biggl(\bigoplus _{k=1}^{n-1}\frac{\beta_{k}}{1-\beta _{n}}x_{k},x_{n} \Biggr) \\& \quad \le (1-\alpha_{n})d \Biggl(\bigoplus _{k=1}^{n-1}\frac{\alpha_{k}}{1-\alpha _{n}}x_{k},\bigoplus _{k=1}^{n-1}\frac{\beta_{k}}{1-\beta_{n}}x_{k} \Biggr) \\& \qquad {}+ |\alpha_{n}-\beta_{n}|\sum _{k=1}^{n-1}\frac{\beta_{k}}{1-\beta_{n}}d(x_{k},x_{n}). \end{aligned}$$

Step 2. Apply the inequality in Step 1 for the case $n-1$ to obtain

$$\begin{aligned}& d \Biggl(\bigoplus_{k=1}^{n-1} \frac{\alpha_{k}}{1-\alpha_{n}}x_{k},\bigoplus_{k=1}^{n-1} \frac{\beta_{k}}{1-\beta_{n}}x_{k} \Biggr) \\& \quad \le\frac{1-\alpha_{n-1}-\alpha_{n}}{1-\alpha_{n}} d \Biggl(\bigoplus_{k=1}^{n-2} \frac{\alpha_{k}}{1-\alpha_{n-1}-\alpha _{n}}x_{k},\bigoplus_{k=1}^{n-2} \frac{\beta_{k}}{1-\beta_{n-1}-\beta _{n}}x_{k} \Biggr) \\& \qquad {}+\biggl\vert \frac{\alpha_{n-1}}{1-\alpha_{n}}-\frac{\beta _{n-1}}{1-\beta_{n}}\biggr\vert \sum _{k=1}^{n-2}\frac{\beta_{k}}{1-\beta _{n-1}-\beta_{n}}d(x_{k},x_{n-1}). \end{aligned}$$

Step 3. Recall that $\sum_{k=1}^{n}\alpha_{k}=\sum_{k=1}^{n}\beta_{k}=1$. Hence, the two inequalities in Step 1 and Step 2 imply that

$$\begin{aligned}& d \Biggl(\bigoplus_{k=1}^{n} \alpha_{k}x_{k},\bigoplus_{k=1}^{n} \beta_{k}x_{k} \Biggr) \\& \quad \le(1-\alpha_{n-1}-\alpha_{n})d \Biggl(\bigoplus _{k=1}^{n-2}\frac{\alpha _{k}}{1-\alpha_{n-1}-\alpha_{n}}x_{k}, \bigoplus_{k=1}^{n-2}\frac{\beta _{k}}{1-\beta_{n-1}-\beta_{n}}x_{k} \Biggr) \\& \qquad {}+\biggl\vert \frac{\alpha_{n-1}}{\sum_{k=1}^{n-1}\alpha_{k}}-\frac{\beta _{n-1}}{\sum_{k=1}^{n-1}\beta_{k}}\biggr\vert \sum _{k=1}^{n-1}\alpha_{k}\cdot\sum _{k=1}^{n-2}\frac{\beta_{k}}{1-\beta _{n-1}-\beta_{n}}d(x_{k},x_{n-1}) \\& \qquad {}+|\alpha_{n}-\beta_{n}|\sum _{k=1}^{n-1}\frac{\beta_{k}}{1-\beta_{n}}d(x_{k},x_{n}). \end{aligned}$$

Continuing the process in Step 1 to estimate the first term of this inequality on the right-hand side, after $n-2$ steps, we have

$$\begin{aligned}& d \Biggl(\bigoplus_{k=1}^{n} \alpha_{k}x_{k},\bigoplus_{k=1}^{n} \beta_{k}x_{k} \Biggr) \\& \quad \le\biggl\vert \frac{\alpha_{2}}{\alpha_{1}+\alpha_{2}}-\frac{\beta_{2}}{\beta _{1}+\beta_{2}}\biggr\vert ( \alpha_{1}+\alpha_{2})d(x_{1},x_{2}) \\& \qquad {}+\biggl\vert \frac{\alpha_{3}}{\sum_{k=1}^{3}\alpha_{k}}-\frac{\beta_{3}}{\sum_{k=1}^{3}\beta_{k}}\biggr\vert \sum _{k=1}^{3}\alpha_{k}\cdot\sum _{k=1}^{2}\frac{\beta_{k}}{\beta_{1}+\beta _{2}}d(x_{k},x_{3}) \\& \qquad {}+\cdots+\biggl\vert \frac{\alpha_{j}}{\sum_{k=1}^{j}\alpha_{k}}-\frac{\beta _{j}}{\sum_{k=1}^{j}\beta_{j}}\biggr\vert \sum_{k=1}^{j}\alpha_{j}\cdot\sum _{k=1}^{j-1}\frac{\beta_{k}}{\beta_{1}+\cdots +\beta_{j-1}}d(x_{k},x_{j}) \\& \qquad {}+\cdots+|\alpha_{n}-\beta_{n}|\sum _{k=1}^{n-1}\frac{\beta_{k}}{1-\beta_{n}}d(x_{k},x_{n}). \end{aligned}$$

 □

Let $\{\alpha_{n}\}_{n=1}^{\infty}$ be a sequence in $(0,1)$ such that $\sum_{n=1}^{\infty}\alpha_{n}=1$. For notational convenience, let

$$ \bar{\alpha}_{k}=\frac{\alpha_{k}}{\sum_{j=1}^{k}\alpha_{j}},\qquad \alpha'_{k}= \sum_{j=k+1}^{\infty}\alpha_{j} \quad \text{for }k \in \mathbb{N}, $$

The following result is an immediate consequence of Lemma 2.2.

Lemma 2.3

Let $(X,d)$ be a $\operatorname{CAT}(0)$ space, and for $n \in\mathbb{N}$ ($n \ge2$), let $\{\alpha_{k}\}_{k=1}^{n},\{\beta _{k}\}_{k=1}^{n} \subset(0,1)$ be such that $\sum_{k=1}^{n}\alpha_{k}=\sum_{k=1}^{n}\beta_{k}=1$. Then for $x_{1},\ldots,x_{n} \in X$, we have

$$ d \Biggl(\bigoplus_{k=1}^{n} \alpha_{k}x_{k},\bigoplus_{k=1}^{n} \beta_{k}x_{k} \Biggr) \le M\sum_{k=1}^{n}| \bar{\alpha}_{k}-\bar{\beta}_{k}|, $$

where $M=\max\{d(x_{i},x_{j}):i,j=1,\ldots,n\}$.

It is remarkable that Dhompongsa et al. [5] define an infinite sum ‘⊕’ as follows. Let $\{\alpha_{n}\} \subset(0,1)$ with $\sum_{n=1}^{\infty}\alpha_{n}=1$, and let $\{x_{n}\}$ be a bounded sequence in a complete metric space X. Choose arbitrary $u \in X$. Suppose that $\lim_{n \to\infty}\sum_{k=n}^{\infty}\alpha'_{k}=0$. Define the sequence $\{y_{n}\}$ in X by

$$y_{n}=\alpha_{1}x_{1} \oplus \alpha_{2}x_{2} \oplus\cdots\oplus\alpha_{n}x_{n} \oplus\alpha'_{n}u. $$

Then, according to (2.1),

$$ y_{n}= \Biggl(\sum_{k=1}^{n} \alpha_{k} \Biggr)z_{n} \oplus\alpha'_{n}u, $$

(2.3)

where

$$z_{n}=\frac{\alpha_{1}}{\sum_{k=1}^{n}\alpha_{k}}x_{1} \oplus\cdots\oplus \frac {\alpha_{n}}{\sum_{k=1}^{n}\alpha_{k}}x_{n}. $$

Recall that $\{y_{n}\}$ is a Cauchy sequence [5] and therefore converges to some point $x \in X$. We can write

$$x=\bigoplus_{n=1}^{\infty}\alpha_{n}x_{n}. $$

By (2.3), $d(y_{n},z_{n})=\alpha'_{n}d(z_{n},u)$. Hence, $\{z_{n}\}$ also converges to x, and the limit x is independent of the choice of u.

To verify our main results in Section 3, the following property is required and crucial.

Lemma 2.4

(Dhompongsa et al. [5], Lemma 3.8)

Let C be a closed convex subset of a complete $\operatorname{CAT}(0)$ space X, $\{T_{n}\}$ a sequence of nonexpansive mappings on C with $\bigcap_{n=1}^{\infty}\mathfrak{F}(T_{n}) \ne\emptyset$, and $\{\alpha_{n}\}$ a sequence in $(0,1)$ such that $\sum_{n=1}^{\infty}\alpha_{n}=1$ and $\lim_{n \to\infty}\sum_{k=n}^{\infty}\alpha'_{k}=0$. Define the mapping $S:C \to C$ by $Sx=\bigoplus_{n=1}^{\infty}\alpha _{n}T_{n}x$, $x \in C$. Then S is nonexpansive, and $\mathfrak{F}(S)=\bigcap_{n=1}^{\infty}\mathfrak{F}(T_{n})$.

3 Projection method

Let C be a closed convex subset of a complete metric space X. A family $\{T_{n}\}$ of nonexpansive self-mappings of C is said to satisfy AKTT condition [3] if for every bounded subset B of C,

$$\sum_{n=1}^{\infty}\sup\bigl\{ d(T_{n+1}x,T_{n}x):x \in B\bigr\} < \infty. $$

In this case, the sequence $\{T_{n}x\}$ is Cauchy for each $x \in C$ and so converges in X. We recall the following convergence theorem with a weak contraction for a sequence of nonexpansive mappings with AKTT condition.

Theorem 3.1

(Huang [15], Theorem 4.11)

Let X be a complete $\operatorname{CAT}(0)$ space, C a closed convex subset of X, $\{T_{n}\}$ a family of nonexpansive mappings on C satisfying AKTT condition such that $\bigcap_{n=1}^{\infty}\mathfrak{F}(T_{n})\ne\emptyset$, f a φ-weak contraction on C, where φ is strictly increasing, and $\{\alpha_{n}\}$ is a sequence in $(0,1]$ satisfying

• (C1) $\lim_{n \to\infty}\alpha_{n}=0$;

• (C2) $\sum_{n=1}^{\infty}\alpha_{n}=\infty$;

• (C3) either $\sum_{n=1}^{\infty}|\alpha_{n+1}-\alpha_{n}|<\infty$, or $\lim_{n \to\infty}(\alpha_{n+1}/\alpha_{n})=1$.

Define the mapping $S:C \to C$ by $Sx=\lim_{n \to\infty}T_{n}x$ for $x \in C$. Suppose that $\mathfrak{F}(S)=\bigcap_{n=1}^{\infty}\mathfrak{F}(T_{n})$. Then the sequence $\{x_{n}\}$ defined by $x_{1} \in C$ and

$$ x_{n+1}=\alpha_{n}f(x_{n}) \oplus(1- \alpha_{n})T_{n}x_{n} $$

converges strongly to a point $\hat{x} \in C$ such that $\hat{x}=P_{\mathfrak{F}(S)}f(\hat{x})$.

We now construct a sequence of nonexpansive mappings satisfying AKTT condition by choosing an appropriate control sequence under certain conditions.

Theorem 3.2

Let C be a closed convex subset of a complete $\operatorname{CAT}(0)$ space X, $\mathfrak{T}=\{T_{n}\}$ a family of nonexpansive mappings on C with $\mathfrak{F}(\mathfrak{T}) \ne\emptyset$, and $\{\gamma_{n,k}:k \le n\} \subset(0,1)$ a sequence satisfying

• (D1) $\sum_{k=1}^{n}\gamma_{n,k}=1$, $\forall n \in\mathbb{N}$;

• (D2) $\lambda_{k}=\lim_{n \to\infty}\gamma_{n,k}>0$, $\forall k \in\mathbb {N}$, and $\lim_{n \to\infty}\sum_{k=n}^{\infty}\lambda'_{k}=0$;

• (D3) $\sum_{n=1}^{\infty}\sum_{k=1}^{n+1}|\bar{\gamma}_{n+1,k}-\bar{\gamma }_{n,k}|<\infty$, where $\gamma_{n,n+1}=0$ and

  $$\bar{\gamma}_{n,k}=\frac{\gamma_{n,k}}{\gamma_{n,1}+\cdots+\gamma _{n,k}}, \quad k=1,\ldots,n+1. $$

For each $n \in\mathbb{N}$, define the mapping $S_{n}:C \to C$ by

$$ S_{n}x=\bigoplus_{k=1}^{n} \gamma_{n,k}T_{k}x. $$

Then $\{S_{n}\}$ is a family of nonexpansive mappings satisfying AKTT condition and

$$\bigcap_{n=1}^{\infty}\mathfrak{F}(S_{n})=\bigcap_{n=1}^{\infty}\mathfrak {F}(T_{n}). $$

Moreover, the mapping $S:C \to C$ defined by $Sx=\lim_{n \to\infty}S_{n}x$ is also nonexpansive, and $\mathfrak{F}(S)=\bigcap_{n=1}^{\infty}\mathfrak {F}(S_{n})$.

Proof

Fix any $n \in\mathbb{N}$. We may assume that $\gamma_{n,k}=0$ for all $k>n$. Then Lemma 2.4 states that $S_{n}$ is nonexpansive and $\mathfrak {F}(S_{n})=\bigcap_{k=1}^{n}\mathfrak{F}(T_{k})$. Thus,

$$\bigcap_{n=1}^{\infty}\mathfrak{F}(S_{n})= \bigcap_{k=1}^{\infty}\mathfrak {F}(T_{k}) \ne\emptyset. $$

For every bounded subset B of C, the set $\{T_{k}x:x \in B, k \in \mathbb{N}\}$ is bounded since $\bigcap_{k=1}^{\infty}\mathfrak{F}(T_{k}) \ne\emptyset$. Let

$$M=\operatorname{diam}\{T_{k}x:x \in B, k \in\mathbb{N}\}, $$

so that by Lemma 2.3, for $x \in B$ and $n \in\mathbb{N}$, we have

$$\begin{aligned} d(S_{n+1}x,S_{n}x) \le& d \Biggl(\bigoplus _{k=1}^{n}\gamma_{n+1,k}T_{k}x, \bigoplus_{k=1}^{n}\gamma _{n,k}T_{k}x \Biggr) +\gamma_{n+1,n+1}d \Biggl(T_{n+1}x,\bigoplus _{k=1}^{n}\gamma _{n,k}T_{k}x \Biggr) \\ \le& M\sum_{k=1}^{n}|\bar{ \gamma}_{n+1,k}-\bar{\gamma}_{n,k}|+M\gamma _{n+1,n+1}\sum _{k=1}^{n}\gamma_{n,k} \\ =& M\sum_{k=1}^{n}|\bar{ \gamma}_{n+1,k}-\bar{\gamma}_{n,k}|+M\bar{\gamma }_{n+1,n+1} \\ =& M\sum_{k=1}^{n+1}|\bar{ \gamma}_{n+1,k}-\bar{\gamma}_{n,k}|. \end{aligned}$$

It follows that

$$\sum_{n=1}^{\infty}\sup\bigl\{ d(S_{n+1}x,S_{n}x):x \in B\bigr\} \le M\sum _{n=1}^{\infty}\sum_{k=1}^{n+1}| \bar{\gamma}_{n+1,k}-\bar{\gamma }_{n,k}|< \infty. $$

Therefore, $\{S_{n}\}$ is a family of nonexpansive mappings on C satisfying AKTT condition such that $\bigcap_{n=1}^{\infty}\mathfrak {F}(S_{n}) \ne\emptyset$. It follows that $\{S_{n}x\}$ converges for all $x \in C$, and thus S is well defined.

If $m,n \in\mathbb{N}$ and $m > n$, then we get

$$\begin{aligned} \begin{aligned} \sum_{k=1}^{n}|\bar{\gamma}_{m,k}- \bar{\gamma}_{n,k}| & \le\sum_{k=1}^{n} \bigl(\vert \bar{\gamma}_{n+1,k}-\bar{\gamma}_{n,k}\vert +| \bar {\gamma}_{n+2,k}-\bar{\gamma}_{n+1,k}|+\cdots +|\bar{\gamma}_{m,k}-\bar{\gamma}_{m-1,k}|\bigr) \\ & = \sum_{j=n}^{m-1}\sum _{k=1}^{n}|\bar{\gamma}_{j+1,k}-\bar{\gamma }_{j,k}| \\ & \le\sum_{j=n}^{m-1}\sum _{k=1}^{j+1}|\bar{\gamma}_{j+1,k}-\bar{ \gamma}_{j,k}|. \end{aligned} \end{aligned}$$

Recall that $\bar{\lambda}_{k}=\lim_{n \to\infty}\bar{\gamma}_{n,k}$ for $k \in\mathbb{N}$. We take the limit as $m \to\infty$ to obtain

$$\sum_{k=1}^{n}|\bar{\lambda}_{k}- \bar{\gamma}_{n,k}| \le\sum_{j=n}^{\infty}\sum_{k=1}^{j+1}|\bar{\gamma}_{j+1,k}- \bar{\gamma}_{j,k}| $$

and then take the limit as $n \to\infty$ to obtain

$$ \lim_{n \to\infty}\sum_{k=1}^{n}| \bar{\lambda}_{k}-\bar{\gamma}_{n,k}|=0. $$

(3.1)

On the other hand, the absolute convergence of the series

$$\sum_{n=1}^{\infty}\sum _{k=1}^{n+1}(\bar{\gamma}_{n+1,k}-\bar{ \gamma}_{n,k}) $$

implies the convergence of its partial sums

$$\sum_{n=1}^{m}\sum _{k=1}^{n+1}(\bar{\gamma}_{n+1,k}-\bar{ \gamma}_{n,k}) = \Biggl(\sum_{k=1}^{m+1} \bar{\gamma}_{m+1,k} \Biggr)-\bar{\gamma}_{1,1} = \Biggl(\sum _{k=1}^{m+1}\bar{\gamma}_{m+1,k} \Biggr)-1. $$

Hence, by (3.1), $\sum_{k=1}^{\infty}\bar{\lambda}_{k}$ converges (in fact, to $\sum_{k=1}^{\infty}\bar{\gamma}_{n,k}$), and so does $\sum_{k=1}^{\infty}\lambda_{k}$ because $\lambda_{k} \le\bar {\lambda}_{k}$. Let $\lambda=\sum_{k=1}^{\infty}\lambda_{k}$. Define the mapping $W:C \to C$ by

$$Wx=\bigoplus_{n=1}^{\infty}\frac{\lambda_{n}}{\lambda}T_{n}x. $$

Then by (D2) Lemma 2.4 guarantees that W is nonexpansive and $\mathfrak{F}(W)=\bigcap_{n=1}^{\infty}\mathfrak{F}(T_{n})$. If

$$W_{n}x=\bigoplus_{k=1}^{n} \frac{\lambda_{k}}{\sum_{j=1}^{n}\lambda_{j}}T_{k}x,\quad x \in C, $$

then $\{W_{n}x\}$ converges to Wx. Recall that

$$\overline{ \biggl(\frac{\lambda_{k}}{\sum_{j=1}^{n}\lambda_{j}} \biggr)}=\bar {\lambda}_{k}\quad \text{for }k=1,\ldots,n. $$

Fix any $x \in C$. Then by Lemma 2.3 and (3.1) we get

$$ d(S_{n}x,W_{n}x) \le K\sum_{k=1}^{n}| \bar{\gamma}_{n,k}-\bar{\lambda}_{k}| \to0\quad \text{as }n \to \infty, $$

where $K=\max\{d(T_{i}x,T_{j}x):i,j=1,\ldots,n\}$. This shows that $Wx=Sx$ for all $x \in C$, as required. □

The following result follows immediately from Theorems 3.1 and 3.2.

Theorem 3.3

Let C be a closed convex subset of a complete $\operatorname{CAT}(0)$ space X, $\mathfrak{T}=\{T_{n}\}$ a family of nonexpansive mappings on C such that $\mathfrak{F}(\mathfrak{T}) \ne\emptyset$, and f a φ-weak contraction on C, where φ is strictly increasing. Let $\{\alpha_{n}\} \subset(0,1]$ and $\{\gamma_{n,k}:k \le n\} \subset (0,1)$ be two sequences such that $\{\alpha_{n}\}$ satisfies (C1)-(C3) and $\{\gamma_{n,k}:k \le n\}$ satisfies (D1)-(D3). Let $x_{1} \in C$ and define the sequence $\{x_{n}\}$ by

$$x_{n+1}=\alpha_{n}f(x_{n}) \oplus(1- \alpha_{n})\bigoplus_{k=1}^{n} \gamma_{n,k}T_{k}x_{n}. $$

Then $\{x_{n}\}$ converges strongly to a point $\hat{x} \in C$ such that $\hat{x}=P_{\mathfrak{F}(\mathfrak{T})}f(\hat{x})$.

Proof

For each $n \in\mathbb{N}$, let $S_{n}:C \to C$ be the mapping defined by

$$ S_{n}x=\bigoplus_{k=1}^{n} \gamma_{n,k}T_{k}x. $$

Then by Theorem 3.2, $\{S_{n}\}$ is a family of nonexpansive mappings satisfying the AKTT condition and $\bigcap_{n=1}^{\infty}\mathfrak{F}(S_{n})=\bigcap_{n=1}^{\infty}\mathfrak {F}(T_{n})$. We can write

$$x_{n+1}=\alpha_{n}f(x_{n}) \oplus(1- \alpha_{n})S_{n}x_{n}. $$

Define the mapping $S:C \to C$ by $Sx=\lim_{n \to\infty}S_{n}x$ for $x \in C$, so that S is nonexpansive and $\mathfrak{F}(S)=\bigcap_{n=1}^{\infty}\mathfrak{F}(S_{n})$. Consequently, Theorem 3.1 assures the strong convergence of $\{ x_{n}\}$ with limit x̂, say, such that $\hat{x}=P_{\mathfrak{F}(S)}f(\hat{x})$. □

Using Theorem 3.3, we establish a strong convergence theorem by a Moudafi type of shrinking projection method for a family of quasinonexpansive mappings as follows.

Theorem 3.4

Let C be a closed convex subset of a complete $\operatorname{CAT}(0)$ space X such that $\{z \in C: d(u, z) \leq d(v,z)\}$ is a convex subset of C for every $u, v \in C$. Let $\mathfrak{T}=\{T_{j}:j=1,\ldots,N\}$ be a finite family of quasinonexpansive mappings of C into X with $\mathfrak{F}(\mathfrak{T}) \ne\emptyset$ and $\widetilde{\mathfrak{F}}(T_{j})=\mathfrak{F}(T_{j})$ for $j=1,\ldots,N$, and f a φ-weak contraction on C, where φ is strictly increasing. Let $\{\alpha_{n}\}$, $\{\delta_{n}\}$ be sequences in $(0,1]$, and $\{\beta^{j}_{n}:j=1,\ldots,N\}$ and $\{\gamma_{n,k}:k \le n\}$ be sequences in $(0,1)$. Let $x_{1} \in C$ and define the sequence $\{x_{n}\}$ by

$$\begin{aligned}& y^{j}_{n} =\delta_{n}x_{n} \oplus(1- \delta_{n})T_{j}x_{n}, \\& C^{j}_{n} =\bigl\{ z \in C:d\bigl(y^{j}_{n},z \bigr) \le d(x_{n},z)\bigr\} ,\quad j=1,\ldots,N, \\& v^{j}_{n,k} =P_{C^{j}_{k}}x_{n}, \quad k=1, \ldots,n, j=1,\ldots,N, \\& w_{n,k} =\bigoplus_{j=1}^{N} \beta^{j}_{k}v^{j}_{n,k}, \quad k=1, \ldots,n, \\& x_{n+1} = \alpha_{n}f(x_{n}) \oplus(1- \alpha_{n})\bigoplus_{k=1}^{n} \gamma _{n,k}w_{n,k}, \end{aligned}$$

where $\{\alpha_{n}\}$ satisfies (C1)-(C3), $\{\gamma_{n,k}:k \le n\}$ satisfies (D1)-(D3), and $\{\delta_{n}\}$, $\{\beta^{j}_{n}\}$ satisfy the following conditions:

• (i) $\liminf_{n \to\infty}\delta_{n}<1$;

• (ii) $\sum_{j=1}^{N}\beta^{j}_{n}=1$ for $n \in\mathbb{N}$.

Then $\{x_{n}\}$ converges strongly to a point $\hat{x} \in C$ such that $\hat{x}=P_{\mathfrak{F}(\mathfrak{T})}f(\hat{x})$.

Proof

First, we can see that every $C^{j}_{n}$ is closed and convex by the assumption on the space. To prove that the metric projection $P_{C^{j}_{k}}$ is well defined, let $z \in\mathfrak{F}(\mathfrak{T})$. Since $T_{j}$ is quasinonexpansive, we have

$$d\bigl(y^{j}_{n},z\bigr) \le\delta_{n}d(x_{n},z)+(1- \delta_{n})d(T_{j}x_{n},z) \le d(x_{n},z), $$

and so $z \in C^{j}_{n}$. This implies that

$$\emptyset\ne\mathfrak{F}(\mathfrak{T}) \subset C^{j}_{n}, \quad j=1,\ldots,N, n \in \mathbb{N}. $$

Thus, the metric projection onto $C^{j}_{n}$ is well defined. For $n \in\mathbb{N}$, define $Q_{n}:C \to C$ by

$$Q_{n}x=\bigoplus_{j=1}^{N} \beta^{j}_{n}P_{C^{j}_{n}}x, \quad x \in C. $$

It follows from Lemma 2.4 and condition (ii) that $Q_{n}$ is nonexpansive and $\mathfrak{F}(Q_{n})=\bigcap_{j=1}^{N}C^{j}_{n}$. According to our construction, we can write

$$\begin{aligned}& w_{n,k}=Q_{k}x_{n}, \quad k=1,\ldots,n, \\& x_{n+1}=\alpha_{n}f(x_{n}) \oplus(1- \alpha_{n})\bigoplus_{j=1}^{n} \gamma _{n,k}Q_{k}x_{n}, \quad n \in\mathbb{N}. \end{aligned}$$

Hence, Theorem 3.3 and conditions (C1)-(C3) and (D1)-(D3) assure the strong convergence of $\{x_{n}\}$ to a point $\hat{x} \in C$ such that $\hat{x}=P_{F}f(\hat{x})$, where

$$F=\bigcap_{n=1}^{\infty}\mathfrak{F}(Q_{n})= \bigcap_{n=1}^{\infty}\bigcap _{j=1}^{N}C^{j}_{n}=\bigcap _{j=1}^{N}\bigcap _{n=1}^{\infty}C^{j}_{n}. $$

Notice that $\mathfrak{F}(\mathfrak{T}) \subset F$. Condition (i) asserts that there exists a convergent subsequence $\{ \delta_{n_{i}}\}$ of $\{\delta_{n}\}$ such that $\lim_{i \to\infty}\delta_{n_{i}}<1$. Since $\hat{x} \in C^{j}_{n}$ for all $j=1,\ldots,N$ and $n \in\mathbb{N}$, we obtain

$$\begin{aligned} d(x_{n_{i}},\hat{x}) & \ge d(y_{n_{i}},\hat{x}) \\ & =d\bigl(\delta_{n_{i}}x_{n_{i}} \oplus(1-\delta_{n_{i}})T_{j}x_{n_{i}}, \hat{x}\bigr) \\ & \ge d\bigl(x_{n_{i}},\delta_{n_{i}}x_{n_{i}} \oplus(1- \delta _{n_{i}})T_{j}x_{n_{i}}\bigr)-d(x_{n_{i}}, \hat{x}) \\ & =(1-\delta_{n_{i}})d(x_{n_{i}},T_{j}x_{n_{i}})-d(x_{n_{i}}, \hat{x}), \end{aligned}$$

which yields

$$\frac{2}{1-\delta_{n_{i}}}d(x_{n_{i}},\hat{x}) \ge d(x_{n_{i}},T_{j}x_{n_{i}}). $$

We then take the limit as $i \to\infty$ and get

$$\lim_{i \to\infty}d(x_{n_{i}},T_{j}x_{n_{i}})=0, \quad j=1,\ldots,N. $$

This shows that $\hat{x} \in\widetilde{\mathfrak{F}}(T_{j})=\mathfrak {F}(T_{j})$ for $j=1,\ldots,N$, that is, $\hat{x} \in\mathfrak{F}(\mathfrak{T})$. Since $\mathfrak{F}(\mathfrak{T}) \subset F$, we then have $\hat{x}=P_{F}f(\hat{x})=P_{\mathfrak{F}(\mathfrak{T})}f(\hat{x})$, which completes the proof. □

Consequently, when f is constant in Theorem 3.4, we obtain the following strong convergence theorem by a new Halpern type of shrinking projection method.

Theorem 3.5

Let X, C, $\mathfrak{T}=\{T_{j}:j=1, \ldots,N\}$, and the sequences $\{\alpha_{n}\}$, $\{\delta_{n}\}$, $\{\beta^{j}_{n}:j=1,\ldots ,N\}$, $\{\gamma_{n,k}:k \le n\}$ be as in Theorem  3.4. Let $u,x_{1} \in C$ and define the sequence $\{x_{n}\}$ by

$$\begin{aligned}& y^{j}_{n} =\delta_{n}x_{n} \oplus(1- \delta_{n})T_{j}x_{n}, \\& C^{j}_{n} =\bigl\{ z \in C:d\bigl(y^{j}_{n},z \bigr) \le d(x_{n},z)\bigr\} , \quad j=1,\ldots,N, \\& v^{j}_{n,k} =P_{C^{j}_{k}}x_{n}, \quad k=1, \ldots,n, j=1,\ldots,N, \\& w_{n,k} =\bigoplus_{j=1}^{N} \beta^{j}_{k}v^{j}_{n,k},\quad k=1, \ldots,n, \\& x_{n+1} = \alpha_{n}u \oplus(1-\alpha_{n})\bigoplus _{k=1}^{n}\gamma_{n,k}w_{n,k}. \end{aligned}$$

Then $\{x_{n}\}$ converges strongly to the point $P_{\mathfrak{F}(\mathfrak{T})}u$.