Open Access

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

Fixed Point Theory and Applications20152015:5

https://doi.org/10.1186/s13663-014-0251-8

Received: 13 August 2014

Accepted: 12 December 2014

Published: 16 January 2015

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.

Keywords

fixed points \(\operatorname{CAT}(\kappa)\) spacesmultivalued quasi-nonexpansive mappings

MSC

47H1047J25

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 [35]. 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
  1. (i)
    be nonexpansive if, for all \(x,y\in D\),
    $$\begin{aligned} H(Tx,Ty)\leq d(x,y); \end{aligned}$$
     
  2. (ii)
    be quasi-nonexpansive if \(F(T)\neq\emptyset\) and
    $$\begin{aligned} H(Tx,Tp)\leq d(x,p),\quad \forall p\in F(T), x\in D; \end{aligned}$$
     
  3. (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
  1. (i)

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

     
  2. (ii)

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

     
  3. (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., [1519] 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:
  1. (i)

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

     
  2. (ii)

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

     
  3. (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 .

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 .

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 .

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 . 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].

Declarations

Acknowledgements

Supported by the General Project of Educational Department in Sichuan (No. 13ZB0182) and the National Natural Science Foundation of China (No. 11426190).

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Authors’ Affiliations

(1)
School of Science, Southwest University of Science and Technology

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