Some convergence results for multivalued quasi-nonexpansive mappings in $\operatorname{CAT}(\kappa)$ spaces

Abstract

In this paper, we prove some strong and △-convergence theorems for a finite family of multivalued quasi-nonexpansive mappings satisfying condition $(E)$ in $\operatorname{CAT}(\kappa)$ spaces. Our results extend the corresponding results of Abkar and Eslamian (Nonlinear Anal. 75:1895-1903, 2012), Panyanak (Fixed Point Theory Appl. 2014:1, 2014), Shahzad and Zegeye (Nonlinear Anal. 71:838-844, 2009) and many others.

1 Introduction

Fixed point theory for multivalued contractions and nonexpansive mappings using the Hausdorff metric was first studied by Markin [1] and Nadler [2]. Since then different iterative processes have been used to approximate fixed points of multivalued nonexpansive mappings. Sastry and Babu [3] defined Mann and Ishikawa iterates for a multivalued map T in a Hilbert space. Panyanak [4] and Song and Wang [5] generalized the results of Sastry and Babu [3] to uniformly convex Banach spaces. Later, Shahzad and Zegeye [6] defined two types of Ishikawa iteration processes and extended the results of [3–5]. The reader may consult [7] for more detail. Recently, Abkar and Eslamian [8] established strong and △-convergence theorems for the following iterative process for a finite family of multivalued quasi-nonexpansive mappings satisfying condition $(E)$ in $\operatorname{CAT}(0)$ spaces:

$$ \begin{cases} y_{n,1}=(1-\alpha_{n,1})x_{n}\oplus\alpha_{n,1}z_{n,1},\\ y_{n,2}=(1-\alpha_{n,2})x_{n}\oplus\alpha_{n,2}z_{n,2},\\ \cdots\\ y_{n,m-1}=(1-\alpha_{n,m-1})x_{n}\oplus\alpha_{n,m-1}z_{n,m-1},\\ x_{n+1}=(1-\alpha_{n,m})x_{n}\oplus\alpha_{n,m}z_{n,m},\quad n\geq1, \end{cases} $$

(1)

where $z_{n,1}\in T_{1}(x_{n})$ and $z_{n,k}\in T_{k}(y_{n,k-1})$ for $k=2,\ldots,m$. It is easy to see that if $m=2$ and $T_{1}=T_{2}=T$, then the sequence $\{ x_{n}\}$ defined by (1) is the Ishikawa iteration:

$$\begin{cases} y_{n}=(1-\alpha_{n,1})x_{n}\oplus\alpha_{n,1}z_{n},\\ x_{n+1}=(1-\alpha_{n,2})x_{n}\oplus\alpha_{n,2}z'_{n},\quad n\geq1, \end{cases} $$

where $z_{n}\in Tx_{n}$ and $z'_{n}\in Ty_{n}$.

The purpose of the paper is to extend and improve the corresponding results of Abkar and Eslamian [8] to the general setting of $\operatorname{CAT}(\kappa)$ spaces, which are geodesic spaces of bounded curvature, where $\kappa\in\mathbb{R}$ is the curvature bound. For example, the n-dimensional hyperbolic space $\mathbb{H}^{n}$ is a $\operatorname{CAT}(-1)$ space and the n-dimensional unit sphere $\mathbb{S}^{n}$ is a $\operatorname{CAT}(1)$ space (see Section 2 for details). It is worth mentioning that any $\operatorname{CAT}(\kappa )$ space is a $\operatorname{CAT}(\kappa')$ space for $\kappa'\geq\kappa$. Thus all results for $\operatorname{CAT}(\kappa)$ spaces with $\kappa>0$ immediately apply to any $\operatorname{CAT}(0)$ space.

Let D be a subset of a metric space $(X,d)$. Recall that an element $p\in D$ is called a fixed point of a single-valued mapping T if $p=Tp$ and of a multivalued mapping T if $p\in Tp$. The set of fixed points of T is denoted by $F(T)$. D is said to be proximinal if, for each $x\in X$, there exists an element $x^{*}\in D$ such that

$$\begin{aligned} d(x, D)=\inf\bigl\{ d(x,y):y\in D\bigr\} =d\bigl(x,x^{*}\bigr). \end{aligned}$$

It is evident that every proximinal set is closed and every compact set is proximinal (see [9]).

Let $2^{D}$ be a family of nonempty subsets of D. We denote by $\mathcal{C}(D)$, $\mathcal{P}(D)$ and $\mathcal{K}(D)$ the families of nonempty closed subsets, nonempty proximinal subsets and nonempty compact subsets of D, respectively. The Hausdorff metric on $\mathcal{K}(D)$ is defined by

$$\begin{aligned} H(A,B)=\max\Bigl\{ \sup_{x\in A}d(x,B), \sup_{y\in B}d(y,A) \Bigr\} \end{aligned}$$

for all $A,B\in\mathcal{K}(D)$, where $d(x,B)=\inf\{d(x,z): z\in B\}$.

Definition 1

A multivalued mapping $T: D\to2^{D}$ is said to

• (i) be nonexpansive if, for all $x,y\in D$,

  $$\begin{aligned} H(Tx,Ty)\leq d(x,y); \end{aligned}$$

• (ii) be quasi-nonexpansive if $F(T)\neq\emptyset$ and

  $$\begin{aligned} H(Tx,Tp)\leq d(x,p),\quad \forall p\in F(T), x\in D; \end{aligned}$$

• (iii) satisfy condition $(E_{\mu})$ provided that

  $$\begin{aligned} d(x, Ty)\leq\mu d(x, Tx)+d(x,y),\quad x,y\in D \mbox{ and } \mu\geq1. \end{aligned}$$

We say that T satisfies condition $(E)$ whenever T satisfies $(E_{\mu})$ for some $\mu\geq1$.

Remark 1

There exist multivalued quasi-nonexpansive mappings satisfying condition $(E)$. For example, define a mapping $T:[0,5]\to[0,5]$ by

$$\begin{aligned} Tx= \left \{ \begin{array}{@{}l@{\quad}l} {[}0,\frac{x}{5}],& x\neq5,\\ \{1\},& x=5. \end{array} \right . \end{aligned}$$

Let $x,y\in[0,5)$, then we get

$$\begin{aligned} H(Tx,Ty)=\biggl\vert \frac{x-y}{5}\biggr\vert \leq d(x,y). \end{aligned}$$

If $x\in[0,4]$ and $y=5$, then

$$\begin{aligned} H(Tx,Ty)=1\leq5-x= d(x,y). \end{aligned}$$

If $x\in(4,5)$ and $y=5$, we have

$$\begin{aligned} d(x,Tx)=\frac{4x}{5},\qquad d(x,y)=5-x,\qquad H(Tx, Ty)=1 \quad \mbox{and}\quad d(x,Ty)=x-1. \end{aligned}$$

Then it is easy to prove that T has the required properties.

In 1991, Xu [10] introduced the best approximation operator $P_{T}$ to find fixed points of ^∗-nonexpansive multivalued mappings. In 2013, Dehghan [11] obtained the demiclosed principle of such mappings and approximated their fixed points using $P_{T}$. Let $P_{T}: D\to2^{D}$ be a multivalued mapping defined by

$$\begin{aligned} P_{T}(x)=\bigl\{ u\in Tx: d(x,u)=d(x, Tx)\bigr\} . \end{aligned}$$

By [12] we have the following lemma.

Lemma 1

[12]

Let D be a nonempty subset of a metric space $(X,d)$ and $T: D\to \mathcal{P}(D)$ be a multivalued mapping. Then

• (i) $d(x,Tx)=d(x, P_{T}(x))$ for all $x\in D$;

• (ii) $x\in F(T)\Leftrightarrow x\in F(P_{T})\Leftrightarrow P_{T}(x)=\{x\}$;

• (iii) $F(T)=F(P_{T})$.

2 Preliminaries

The study of fixed points in $\operatorname{CAT}(\kappa)$ spaces was initiated by Kirk [13, 14]. A few recent new convergence results of classical iterations on $\operatorname{CAT}(\kappa)$ spaces have been obtained (see, e.g., [15–19] and the references therein). For example, Panyanak [19] in 2014 proved the strong convergence of two types of Ishikawa iteration processes introduced in Shahzad and Zegeye [6] for some multivalued quasi-nonexpansive mappings in $\operatorname{CAT}(1)$ spaces.

Let $(X,d)$ be a metric space and $x,y\in X$ with $l=d(x,y)$. For $x,y\in X$, a geodesic path joining x to y is an isometry $c:[0,l]\to X$ such that $c(0)=x$, $c(l)=y$. The image of a geodesic path is called a geodesic segment, and we shall denote a definite choice of this geodesic segment by $[x,y]$. A metric space X is a geodesic space (r-geodesic space) if every two points of X (every two points with distance smaller than r) are joined by a geodesic segment, and X is a uniquely geodesic space (r-uniquely geodesic space) if there is exactly one geodesic segment joining x and y for any $x, y\in X$ (for any $x, y\in X$ with $d(x,y)< r$). A subset D of X is said to be convex if D includes every geodesic segment joining any two of its points.

The n-dimensional sphere $\mathbb{S}^{n}$ is the set $\{x=(x_{1},\ldots,x_{n+1})\in\mathbb{R}^{n+1}: \langle x|x\rangle=1\}$, where $\langle\cdot|\cdot\rangle$ is the Euclidean scalar product. It is endowed with the following metric: $d_{\mathbb{S}^{n}}(x,y)=\arccos \langle x|y\rangle$, $x,y\in\mathbb{S}^{n}$.

Definition 2

Given $\kappa\in\mathbb{R}$, denote by $M^{n}_{\kappa}$ the following metric spaces:

• (i) if $\kappa=0$, then $M^{n}_{0}$ is the Euclidean space $\mathbb{R}^{n}$;

• (ii) if $\kappa>0$, then $M^{n}_{\kappa}$ is obtained from the sphere $\mathbb{S}^{n}$ by multiplying the distance function by $1/\sqrt{\kappa}$;

• (iii) if $\kappa<0$, then $M^{n}_{\kappa}$ is obtained from the hyperbolic n-space $\mathbb{H}^{n}$ by multiplying the distance function by $1/\sqrt{-\kappa}$.

A geodesic triangle $\triangle(x, y, z)$ in a geodesic space $(X,d)$ consists of three points x, y, z of X and three geodesic segments joining each pair of vertices. A comparison triangle of a geodesic triangle $\triangle(x, y, z)$ is the triangle $\overline{\triangle}(\bar{x},\bar{y},\bar{z})$ in $M^{2}_{\kappa}$ such that

$$\begin{aligned} d(x,y)=d_{M^{2}_{\kappa}}(\bar{x},\bar{y}),\qquad d(y, z)=d_{M^{2}_{\kappa}}(\bar {y}, \bar{z}),\qquad d(z,x)=d_{M^{2}_{\kappa}}(\bar{z},\bar{x}). \end{aligned}$$

If $\kappa>0$, then such a triangle $\overline{\triangle}$ always exists whenever $d(x,y)+d(y,z)+d(z,x)$ is less than $2D_{\kappa}$, where $D_{\kappa}=\pi/\sqrt{\kappa}$. A point $\bar{p}\in[\bar{x},\bar{y}]$ is called a comparison point for $p\in[x,y]$ if $d(x,p)=d_{M^{2}_{\kappa}}(\bar{x},\bar{p})$. A geodesic triangle in X is said to satisfy the $\operatorname{CAT}(\kappa)$ inequality if for any $p,q\in\triangle(x, y, z)$ and for their comparison points $\bar{p},\bar{q}\in\overline{\triangle}(\bar{x},\bar{y},\bar{z})$, we have

$$\begin{aligned} d(p,q)\leq d_{M^{2}_{\kappa}}(\bar{p},\bar{q}). \end{aligned}$$

Definition 3

Given $\kappa>0$, a metric space X is a $\operatorname{CAT}(\kappa)$ space if X is $D_{\kappa}$-geodesic and any geodesic triangle $\triangle(x, y, z)$ in X with $d(x,y)+d(y,z)+d(z,x)<2D_{\kappa}$ satisfies the $\operatorname{CAT}(\kappa)$ inequality.

In 1976, Lim [20] introduced the concept of △-convergence in a general metric space. Let $\{x_{n}\}$ be a bounded sequence in a $\operatorname{CAT}(\kappa)$ space X. For $x\in X$, we define

$$r\bigl(x,\{x_{n}\}\bigr)=\limsup_{n\to\infty}d(x,x_{n}). $$

The asymptotic radius $r(\{x_{n}\})$ of $\{x_{n}\}$ is given by

$$r\bigl(\{x_{n}\}\bigr)=\inf\bigl\{ r\bigl(x,\{x_{n}\}\bigr):x \in X\bigr\} . $$

The asymptotic center $A(\{x_{n}\})$ of $\{x_{n}\}$ is the set

$$A\bigl(\{x_{n}\}\bigr)=\bigl\{ x\in X: r\bigl(x,\{x_{n}\} \bigr)=r\bigl(\{x_{n}\}\bigr)\bigr\} . $$

A sequence $\{x_{n}\}$ in a $\operatorname{CAT}(\kappa)$ space X is said to △-converge to $x\in X$ if x is the unique asymptotic center of $\{u_{n}\}$ for every subsequence $\{u_{n}\}$ of $\{x_{n}\}$.

It follows from [21] that $\operatorname{CAT}(\kappa)$ spaces are uniquely geodesic spaces. In this paper, we mainly focus on $\operatorname{CAT}(\kappa)$ spaces with $\kappa>0$, and we now collect some elementary facts about them.

Lemma 2

[15]

Let $\kappa>0$ and $(X,d)$ be a $\operatorname{CAT}(\kappa)$ space with $\operatorname{diam}(X)=:\sup\{d(u,v): u,v\in X\}<\frac {\pi}{2\sqrt{\kappa}}$. Then $A(\{x_{n}\})$ consists of exactly one point.

Lemma 3

[15]

Let $\kappa>0$ and $(X,d)$ be a complete $\operatorname{CAT}(\kappa)$ space with $\operatorname{diam}(X)\leq\frac{\pi /2-\varepsilon}{\sqrt{\kappa}}$ for some $\varepsilon\in(0, \pi/2)$. Then every sequence in X has a △-convergent subsequence.

Lemma 4

[15]

Let $\kappa>0$ and $(X,d)$ be a complete $\operatorname{CAT}(\kappa)$ space with $\operatorname{diam}(X)\leq\frac{\pi /2-\varepsilon}{\sqrt{\kappa}}$ for some $\varepsilon\in(0, \pi/2)$. D is a closed convex subset of X. If $\{x_{n}\}\subseteq D$ and $\triangle\mbox{-}\lim_{n\to\infty}x_{n}=x$, then $x\in D$.

Since the asymptotic center is unique by Lemma 2, we can obtain the following lemma.

Lemma 5

[22]

Let $\kappa>0$ and $(X,d)$ be a complete $\operatorname{CAT}(\kappa)$ space with $\operatorname{diam}(X)\leq\frac{\pi /2-\varepsilon}{\sqrt{\kappa}}$ for some $\varepsilon\in(0, \pi/2)$. Let $\{x_{n}\}$ be a sequence in X with $A(\{x_{n}\})=\{x\}$. If $\{u_{n}\}$ is a subsequence of $\{x_{n}\}$ with $A(\{u_{n}\})=\{u\}$ and $\{ d(x_{n},u)\}$ converges, then $x=u$.

Lemma 6

[21]

Let $\kappa>0$ and $(X,d)$ be a complete $\operatorname{CAT}(\kappa)$ space with $\operatorname{diam}(X)\leq\frac{\pi /2-\varepsilon}{\sqrt{\kappa}}$ for some $\varepsilon\in(0, \pi/2)$. Then, for any $x,y,z\in X$ and $t\in[0,1]$, we have

$$\begin{aligned} d\bigl((1-t)x\oplus ty,z\bigr)\leq(1-t)d(x,z)+td(y,z). \end{aligned}$$

Lemma 7

[23]

Let $\kappa>0$ and $(X,d)$ be a $\operatorname{CAT}(\kappa)$ space with $\operatorname{diam}(X)\leq\frac{\pi/2-\varepsilon}{\sqrt {\kappa}}$ for some $\varepsilon\in(0, \pi/2)$. Then, for any $x,y,z\in X$ and $t\in[0,1]$, we have

$$\begin{aligned} d^{2}\bigl((1-t)x\oplus ty,z\bigr)\leq(1-t)d^{2}(x,z)+td^{2}(y,z)- \frac{R}{2}t(1-t)d^{2}(x,y), \end{aligned}$$

where $R=(\pi-2\varepsilon)\tan(\varepsilon)$.

3 Main results

In this section, we prove our main theorems.

Theorem 1

(Demiclosed principle)

Let $\kappa>0$ and $(X,d)$ be a complete $\operatorname{CAT}(\kappa)$ space with $\operatorname{diam}(X)\leq\frac{\pi/2-\varepsilon}{\sqrt{\kappa}}$ for some $\varepsilon\in(0, \pi/2)$. Let D be a nonempty closed convex subset of X, and let $T:D\to \mathcal{K}(D)$ be a multivalued mapping satisfying condition $(E)$. If $\{x_{n}\}$ is a sequence in D such that $\lim_{n\to\infty}d(x_{n}, Tx_{n})=0$ and $\triangle\mbox{-}\lim_{n\to\infty}x_{n}=x$, then $x\in Tx$, from which we may formally say that $I-T$ is demiclosed at zero.

Proof

Since $\triangle\mbox{-}\lim_{n\to\infty}x_{n}=x$, by Lemma 4 we have $x\in D$. For each $n\geq1$, we choose $z_{n}\in Tx$ such that

$$\begin{aligned} d(x_{n},z_{n})=d(x_{n}, Tx). \end{aligned}$$

By the compactness of Tx, there is a subsequence $\{z_{n_{k}}\}$ of $\{ z_{n}\}$ such that $\lim_{k\to\infty}z_{n_{k}}=w\in Tx$. It follows from condition $(E)$ that

$$\begin{aligned} d(x_{n_{k}},z_{n_{k}})=d(x_{n_{k}},Tx)\leq\mu d(x_{n_{k}}, Tx_{n_{k}})+d(x_{n_{k}},x) \end{aligned}$$

for some $\mu\geq1$. Note that

$$\begin{aligned} d(x_{n_{k}},w)\leq d(x_{n_{k}},z_{n_{k}})+d(z_{n_{k}},w) \leq\mu d(x_{n_{k}}, Tx_{n_{k}})+d(x_{n_{k}},x)+d(z_{n_{k}},w). \end{aligned}$$

Thus

$$\begin{aligned} \limsup_{k\to\infty}d(x_{n_{k}},w)\leq\limsup _{k\to\infty}d(x_{n_{k}},x). \end{aligned}$$

By the uniqueness of asymptotic centers, we obtain $x=w\in Tx$. The proof is completed. □

Theorem 2

Let $\kappa>0$ and $(X,d)$ be a complete $\operatorname{CAT}(\kappa)$ space with $\operatorname{diam}(X)\leq\frac{\pi/2-\varepsilon}{\sqrt{\kappa}}$ for some $\varepsilon\in(0, \pi/2)$. Let D be a nonempty closed convex subset of X, and let $T_{i}:D\to \mathcal{K}(D)$ ($i=1,\ldots,m$) be a family of multivalued quasi-nonexpansive mappings satisfying condition $(E)$. Suppose that $\mathcal{F}=\bigcap^{m}_{i=1}F(T_{i})\neq\emptyset$ and $T_{i}(p)=\{p\}$ for each $p\in\mathcal{F}$. Let $\alpha_{n,i}\in[a,b]\subset(0,1)$ ($i=1,\ldots,m$). Then $\{x_{n}\}$ defined by (1) △-converges to some point in ℱ.

Proof

We divide our proof into several steps.

Step 1. In the sequel, we shall show that $\lim_{n\to \infty}d(x_{n},p)$ exists for any $p\in\mathcal{F}$. Since $T_{1}$ is quasi-nonexpansive, by Lemma 6 we have

$$\begin{aligned} d(y_{n,1},p) =&d\bigl((1-\alpha_{n,1})x_{n}\oplus \alpha_{n,1}z_{n,1},p\bigr) \\ \leq&(1-\alpha_{n,1})d(x_{n},p)+\alpha_{n,1}d(z_{n,1},p) \\ =&(1-\alpha_{n,1})d(x_{n},p)+\alpha_{n,1}d \bigl(z_{n,1},T_{1}(p)\bigr) \\ \leq&(1-\alpha_{n,1})d(x_{n},p)+\alpha_{n,1}H \bigl(T_{1}(x_{n}),T_{1}(p)\bigr) \\ \leq&(1-\alpha_{n,1})d(x_{n},p)+\alpha_{n,1}d(x_{n},p) \\ =&d(x_{n},p) \end{aligned}$$

and

$$\begin{aligned} d(y_{n,2},p) =&d\bigl((1-\alpha_{n,2})x_{n}\oplus \alpha_{n,2}z_{n,2},p\bigr) \\ \leq&(1-\alpha_{n,2})d(x_{n},p)+\alpha_{n,2}d(z_{n,2},p) \\ =&(1-\alpha_{n,2})d(x_{n},p)+\alpha_{n,2}d \bigl(z_{n,2},T_{2}(p)\bigr) \\ \leq&(1-\alpha_{n,2})d(x_{n},p)+\alpha_{n,2}H \bigl(T_{2}(y_{n,1}),T_{2}(p)\bigr) \\ \leq&(1-\alpha_{n,2})d(x_{n},p)+\alpha_{n,2}d(y_{n,1},p) \\ \leq& d(x_{n},p). \end{aligned}$$

By continuing this process we have

$$\begin{aligned} d(x_{n+1},p) \leq& d(x_{n},p). \end{aligned}$$

It implies that $d(x_{n},p)$ is decreasing and bounded below, thus $\lim_{n\to\infty}d(x_{n},p)$ exists for any $p\in\mathcal{F}$.

Step 2. We shall show that $\lim_{n\to\infty }d(x_{n},T_{i}(x_{n}))=0$ for $i=1,\ldots,m$. In fact, by Lemma 7 we obtain

$$\begin{aligned}[b] d^{2}(y_{n,1},p)&=d^{2}\bigl((1- \alpha_{n,1})x_{n}\oplus\alpha_{n,1}z_{n,1},p \bigr) \\ &\leq(1-\alpha_{n,1})d^{2}(x_{n},p)+ \alpha_{n,1}d^{2}(z_{n,1},p)-\frac {R}{2} \alpha_{n,1}(1-\alpha_{n,1})d^{2}(x_{n},z_{n,1}) \\ &=(1-\alpha_{n,1})d^{2}(x_{n},p)+ \alpha_{n,1}d^{2}\bigl(z_{n,1},T_{1}(p) \bigr)-\frac {R}{2}\alpha_{n,1}(1-\alpha_{n,1})d^{2}(x_{n},z_{n,1}) \\ &\leq(1-\alpha_{n,1})d^{2}(x_{n},p)+ \alpha_{n,1}H^{2}\bigl(T_{1}(x_{n}),T_{1}(p) \bigr)-\frac {R}{2}\alpha_{n,1}(1-\alpha_{n,1})d^{2}(x_{n},z_{n,1}) \\ &\leq(1-\alpha_{n,1})d^{2}(x_{n},p)+ \alpha_{n,1}d^{2}(x_{n},p)-\frac {R}{2} \alpha_{n,1}(1-\alpha_{n,1})d^{2}(x_{n},z_{n,1}) \\ &=d^{2}(x_{n},p)-\frac{R}{2}\alpha_{n,1}(1- \alpha_{n,1})d^{2}(x_{n},z_{n,1}) \end{aligned} $$

and

$$\begin{aligned} d^{2}(y_{n,2},p) =&d^{2}\bigl((1- \alpha_{n,2})x_{n}\oplus\alpha_{n,2}z_{n,2},p \bigr) \\ \leq&(1-\alpha_{n,2})d^{2}(x_{n},p)+ \alpha_{n,2}d^{2}(z_{n,2},p)-\frac {R}{2} \alpha_{n,2}(1-\alpha_{n,2})d^{2}(x_{n},z_{n,2}) \\ =&(1-\alpha_{n,2})d^{2}(x_{n},p)+ \alpha_{n,2}d^{2}\bigl(z_{n,2},T_{2}(p) \bigr)-\frac {R}{2}\alpha_{n,2}(1-\alpha_{n,2})d^{2}(x_{n},z_{n,2}) \\ \leq&(1-\alpha_{n,2})d^{2}(x_{n},p)+\alpha _{n,2}H^{2}\bigl(T_{2}(y_{n,1}),T_{2}(p) \bigr)-\frac{R}{2}\alpha_{n,2}(1-\alpha _{n,2})d^{2}(x_{n},z_{n,2}) \\ \leq&(1-\alpha_{n,2})d^{2}(x_{n},p)+ \alpha_{n,2}d^{2}(y_{n,1},p)-\frac {R}{2} \alpha_{n,2}(1-\alpha_{n,2})d^{2}(x_{n},z_{n,2}) \\ \leq& d^{2}(x_{n},p)-\frac{R}{2}\alpha_{n,2} \alpha_{n,1}(1-\alpha _{n,1})d^{2}(x_{n},z_{n,1})- \frac{R}{2}\alpha_{n,2}(1-\alpha_{n,2})d^{2}(x_{n},z_{n,2}). \end{aligned}$$

Similarly, we get

$$\begin{aligned} d^{2}(x_{n+1},p) =&d^{2}\bigl((1- \alpha_{n,m})x_{n}\oplus\alpha_{n,m}z_{n,m},p \bigr) \\ \leq&(1-\alpha_{n,m})d^{2}(x_{n},p)+ \alpha_{n,m}d^{2}(z_{n,m},p)-\frac {R}{2} \alpha_{n,m}(1-\alpha_{n,m})d^{2}(x_{n},z_{n,m}) \\ =&(1-\alpha_{n,m})d^{2}(x_{n},p)+ \alpha_{n,m}d^{2}\bigl(z_{n,m},T_{2}(p) \bigr)-\frac {R}{2}\alpha_{n,m}(1-\alpha_{n,m})d^{2}(x_{n},z_{n,m}) \\ \leq&(1-\alpha_{n,m})d^{2}(x_{n},p)+\alpha _{n,m}H^{2}\bigl(T_{m}(y_{n,m-1}),T_{m}(p) \bigr)\\ &{}-\frac{R}{2}\alpha_{n,m}(1-\alpha _{n,m})d^{2}(x_{n},z_{n,m}) \\ \leq&(1-\alpha_{n,m})d^{2}(x_{n},p)+ \alpha_{n,m}d^{2}(y_{n,m-1},p)-\frac {R}{2} \alpha_{n,m}(1-\alpha_{n,m})d^{2}(x_{n},z_{n,m}) \\ \leq& d^{2}(x_{n},p)-\frac{R}{2}\alpha_{n,m}(1- \alpha _{n,m})d^{2}(x_{n},z_{n,m})\\ &{}- \frac{R}{2}\alpha_{n,m}\alpha_{n,m-1}(1-\alpha _{n,m-1})d^{2}(x_{n},z_{n,m-1})-\cdots\\ &{}-\frac{R}{2}\alpha_{n,m}\alpha_{n,m-1}\cdot\cdot \cdot \alpha_{n,1}(1-\alpha_{n,1})d^{2}(x_{n},z_{n,1}). \end{aligned}$$

Then we have

$$\begin{aligned} \frac{R}{2}a^{m}(1-b)d^{2}(x_{n},z_{n,1}) \leq& \frac{R}{2}\alpha_{n,m}\alpha _{n,m-1}\cdot\cdot\cdot \alpha_{n,1}(1-\alpha_{n,1})d^{2}(x_{n},z_{n,1}) \\ \leq& d^{2}(x_{n},p)-d^{2}(x_{n+1},p), \end{aligned}$$

which yields that

$$\begin{aligned} \sum_{n=1}^{\infty}\frac {R}{2}a^{m}(1-b)d^{2}(x_{n},z_{n,1}) \leq d^{2}(x_{1},p)<\infty, \end{aligned}$$

and hence

$$\begin{aligned} \lim_{n\to\infty}d(x_{n},z_{n,1})=0. \end{aligned}$$

Similarly, we can also have

$$\begin{aligned} \lim_{n\to\infty}d(x_{n},z_{n,k})=0\quad (k=2, \ldots,m). \end{aligned}$$

Thus we obtain

$$\begin{aligned}& \lim_{n\to\infty}d\bigl(x_{n}, T_{1}(x_{n})\bigr)\leq\lim_{n\to\infty }d(x_{n},z_{n,1})=0, \end{aligned}$$

(2)

$$\begin{aligned}& \lim_{n\to\infty}d\bigl(x_{n}, T_{k}(y_{n,k-1})\bigr)\leq\lim_{n\to \infty} d(x_{n},z_{n,k})=0 \end{aligned}$$

(3)

and

$$\begin{aligned} \lim_{n\to\infty}d(x_{n}, y_{n,k-1})= \alpha_{n,k-1}\lim_{n\to\infty}d(x_{n},z_{n,k-1})=0 \end{aligned}$$

(4)

for $k=2,\ldots,m$. Now, by condition $(E)$, (3) and (4), we have, for some $\mu\geq1$,

$$\begin{aligned} d\bigl(x_{n},T_{k}(x_{n})\bigr) \leq& d(x_{n}, y_{n,k-1})+d\bigl(y_{n,k-1}, T_{k}(x_{n})\bigr) \\ \leq& d(x_{n}, y_{n,k-1})+\mu d\bigl(y_{n,k-1},T_{k}(y_{n,k-1}) \bigr)+d(x_{n},y_{n,k-1}) \\ \leq& d(x_{n}, y_{n,k-1})+\mu d(y_{n,k-1},x_{n})+ \mu d\bigl(x_{n},T_{k}(y_{n,k-1})\bigr) \\ &{}+d(x_{n},y_{n,k-1}) \to0 \end{aligned}$$

(5)

as $n\to\infty$ (for $k=2,\ldots,m$). By (2) and (5) we have

$$\begin{aligned} \lim_{n\to\infty}d\bigl(x_{n},T_{i}(x_{n}) \bigr)=0 \end{aligned}$$

for $i=1,\ldots,m$.

Step 3. Now we are in a position to prove the △-convergence of $\{x_{n}\}$. In fact, let $W_{\omega}(x_{n}):=\cup A (\{u_{n}\})$ for all subsequences $\{ u_{n}\}$ of $\{x_{n}\}$. We claim that $W_{\omega}(x_{n})\subset\mathcal{F}$. Let $u\in W_{\omega}(x_{n})$, then there exists a subsequence $\{u_{n}\}$ of $\{x_{n}\}$ such that $A (\{u_{n}\})=\{u\}$. By Lemma 3 and Lemma 4, there exists a subsequence $\{v_{n}\}$ of $\{u_{n}\}$ such that $\triangle\mbox{-}\lim_{n\to\infty}v_{n}=v\in D$. Since $\lim_{n\to\infty}d(v_{n}, T_{i}v_{n})=0$ ($i=1,\ldots,m$), it follows from Theorem 1 that $v\in\mathcal{F}$ and thus $\lim_{n\to\infty}d(x_{n},v)$ exists by Step 1. By Lemma 5, $u=v\in \mathcal{F}$, which implies that $W_{\omega}(x_{n})\subset\mathcal{F}$. Let $\{u_{n}\}$ be a subsequence of $\{x_{n}\}$ with $A(\{u_{n}\})=\{u\}$, and let $A(\{x_{n}\})=\{x\}$. Since $u\in W_{\omega}(x_{n})\subset\mathcal{F}$ and $\lim_{n\to\infty }d(x_{n},u)$ converges, we get $x=u$ by Lemma 5. It implies that $W_{\omega}(x_{n})$ consists of exactly one point. The proof is completed. □

Remark 2

Theorem 2 improves and extends the corresponding results in Abkar and Eslamian [8, Theorem 3.6].

In the sequel, we make use of condition $(A)$ introduced by Senter and Dotson [24]. A mapping $T: D\to D$, where D is a subset of a normed space E, is said to satisfy condition $(A)$ if there exists a nondecreasing function $f:[0,\infty)\to[0,\infty)$ with $f(0)=0$, $f(r)>0$ for all $r>0$ such that

$$\begin{aligned} \|x-Tx\|\geq f\bigl(d\bigl(x, F(T)\bigr)\bigr) \quad\mbox{for all } x\in D. \end{aligned}$$

Theorem 3

Let $\kappa>0$ and $(X,d)$ be a complete $\operatorname{CAT}(\kappa)$ space with $\operatorname{diam}(X)\leq\frac{\pi/2-\varepsilon}{\sqrt{\kappa}}$ for some $\varepsilon\in(0, \pi/2)$. Let D be a nonempty closed convex subset of X, and let $T_{i}:D\to \mathcal{C}(D)$ ($i=1,\ldots,m$) be a family of multivalued quasi-nonexpansive mappings satisfying condition $(E)$. Suppose that $\mathcal{F}=\bigcap^{m}_{i=1}F(T_{i})\neq\emptyset$ and $T_{i}p=\{p\}$ for each $p\in\mathcal{F}$. Let $\alpha_{n,i}\in[a,b]\subset(0,1)$ ($i=1,\ldots,m$). Assume that there is a nondecreasing function $f:[0,\infty )\to[0,\infty)$ with $f(0)=0$, $f(r)>0$ for all $r>0$ such that for some $i=1,\ldots,m$,

$$\begin{aligned} d\bigl(x_{n},T_{i}(x_{n})\bigr)\geq f\bigl(d(x_{n},\mathcal{F})\bigr). \end{aligned}$$

(6)

Then $\{x_{n}\}$ defined by (1) converges strongly to some point in ℱ.

Proof

As in the proof of Theorem 2, for $i=1,\ldots,m$, we have $\lim_{n\to\infty}d(x_{n}, T_{i}(x_{n}))=0$. Hence by assumption (6) we obtain $\lim_{n\to\infty}d(x_{n},\mathcal{F})=0$. Now we can choose a subsequence $\{x_{n_{k}}\}\subset\{x_{n}\}$ and a subsequence $\{p_{k}\}\subset\mathcal {F}$ such that for all positive integer $k\geq1$,

$$\begin{aligned} d(x_{n_{k}},p_{k})<\frac{1}{2^{k}}. \end{aligned}$$

Since for each $p\in\mathcal{F}$ the sequence $\{d(x_{n},p)\}$ is decreasing, we get

$$\begin{aligned} d(x_{n_{k+1}},p_{k})\leq d(x_{n_{k}},p_{k})< \frac{1}{2^{k}}. \end{aligned}$$

Hence

$$\begin{aligned} d(p_{k+1},p_{k})\leq d(x_{n_{k+1}},p_{k+1})+d(x_{n_{k+1}},p_{k})< \frac {1}{2^{k+1}}+\frac{1}{2^{k}}<\frac{1}{2^{k-1}}. \end{aligned}$$

Then $\{p_{k}\}$ is a Cauchy sequence in D. Without loss of generality, we can assume that $p_{k}\to p^{*}\in D$. Since for each $i=1,\ldots,m$

$$\begin{aligned} d\bigl(p^{*},T_{i}\bigl(p^{*}\bigr)\bigr)=\lim_{n\to\infty}d \bigl(p_{k},T_{i}\bigl(p^{*}\bigr)\bigr)\leq\lim _{n\to\infty }H\bigl(T_{i}(p_{k}),T_{i} \bigl(p^{*}\bigr)\bigr)\leq\lim_{k\to\infty}d\bigl(p_{k},p^{*} \bigr)=0, \end{aligned}$$

then $p^{*}\in\mathcal{F}$ and $\{x_{n_{k}}\}$ converges strongly to $p^{*}$. Since $\lim_{n\to\infty}d(x_{n},p^{*})$ exists, it follows that $\{ x_{n}\}$ converges strongly to $p^{*}$. The proof is completed. □

Remark 3

Theorem 3 improves and extends the corresponding results in Abkar and Eslamian [8, Theorem 3.9] and Panyanak [19, Theorem 3.2].

Theorem 4

Let $\kappa>0$ and $(X,d)$ be a complete $\operatorname{CAT}(\kappa)$ space with $\operatorname{diam}(X)\leq\frac{\pi/2-\varepsilon}{\sqrt{\kappa}}$ for some $\varepsilon\in(0, \pi/2)$. D is a nonempty closed convex subset of X. Let $T_{i}:D\to\mathcal{P}(D)$ ($i=1,\ldots,m$) be a family of multivalued mappings with $\mathcal{F}=\bigcap^{m}_{i=1}F(T_{i})\neq\emptyset$ such that $P_{T_{i}}$ is quasi-nonexpansive satisfying condition $(E)$. For $x_{1}\in D$, define the sequence $\{x_{n}\}\subset D$ as follows:

$$\begin{aligned} \begin{cases} y_{n,1}=(1-\beta_{n,1})x_{n}\oplus\beta_{n,1}z_{n,1},\\ y_{n,2}=(1-\beta_{n,2})x_{n}\oplus\beta_{n,2}z_{n,2},\\ \cdots\\ y_{n,m-1}=(1-\beta_{n,m-1})x_{n}\oplus\beta_{n,m-1}z_{n,m-1},\\ x_{n+1}=(1-\beta_{n,m})x_{n}\oplus\beta_{n,m}z_{n,m},\quad n\geq1, \end{cases} \end{aligned}$$

(7)

where $z_{n,1}\in P_{T_{1}}(x_{n})$, $z_{n,k}\in P_{T_{k}}(y_{n,k-1})$ ($k=2,\ldots,m$) and $\beta_{n,i}\in[a,b]\subset(0,1)$ ($i=1,\ldots,m$). Assume that there is a nondecreasing function $f:[0,\infty)\to[0,\infty )$ with $f(0)=0$, $f(r)>0$ for all $r>0$ such that for some $i=1,\ldots,m$,

$$\begin{aligned} d\bigl(x_{n},T_{i}(x_{n})\bigr)\geq f\bigl(d(x_{n},\mathcal{F})\bigr). \end{aligned}$$

(8)

Then $\{x_{n}\}$ defined by (7) converges strongly to some point in ℱ.

Proof

It follows from Lemma 1 and (8) that

$$\begin{aligned} d\bigl(x_{n},P_{T_{i}}(x_{n})\bigr)=d \bigl(x_{n},T_{i}(x_{n})\bigr)\geq f\bigl(d(x_{n},\mathcal{F})\bigr)=f\Biggl(d\Biggl(x_{n},\bigcap ^{m}_{i=1} F(P_{T_{i}})\Biggr)\Biggr) \end{aligned}$$

for some $i=1,\ldots,m$. Next we show that $P_{T_{i}}(x)$ is closed for any $i=1,\ldots,m$ and $x\in D$. In fact, let $\{y_{n}\}\subset P_{T_{i}}(x)$ and $\lim_{n\to\infty}y_{n}=y$ for some $y\in D$. Then

$$\begin{aligned} d(x,y_{n})=d\bigl(x, T_{i}(x)\bigr) \quad\mbox{and}\quad \lim _{n\to\infty }d(x,y_{n})=d(x,y). \end{aligned}$$

It follows that $d(x,y)=d(x,T_{i}(x))$ and hence $y\in P_{T_{i}}(x)$. Now applying Theorem 3 to the mappings $P_{T_{i}}$, we conclude that the sequence $\{x_{n}\}$ defined by (7) converges strongly to some point in ℱ. The proof is completed. □

Remark 4

Theorem 4 improves and extends the corresponding results in Abkar and Eslamian [8, Theorem 3.12] and Panyanak [19, Theorem 3.4].