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.
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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:
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:
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
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
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
Let \(x,y\in[0,5)\), then we get
If \(x\in[0,4]\) and \(y=5\), then
If \(x\in(4,5)\) and \(y=5\), we have
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
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
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
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
The asymptotic radius \(r(\{x_{n}\})\) of \(\{x_{n}\}\) is given by
The asymptotic center \(A(\{x_{n}\})\) of \(\{x_{n}\}\) is the set
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
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
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
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
for some \(\mu\geq1\). Note that
Thus
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
and
By continuing this process we have
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
and
Similarly, we get
Then we have
which yields that
and hence
Similarly, we can also have
Thus we obtain
and
for \(k=2,\ldots,m\). Now, by condition \((E)\), (3) and (4), we have, for some \(\mu\geq1\),
as \(n\to\infty\) (for \(k=2,\ldots,m\)). By (2) and (5) we have
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
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 \(i=1,\ldots,m\),
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 \(k\geq1\),
Since for each \(p\in\mathcal{F}\) the sequence \(\{d(x_{n},p)\}\) is decreasing, we get
Hence
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 \(i=1,\ldots,m\)
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:
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 \(i=1,\ldots,m\),
Then \(\{x_{n}\}\) defined by (7) converges strongly to some point in ℱ.
Proof
It follows from Lemma 1 and (8) that
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
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].
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Supported by the General Project of Educational Department in Sichuan (No. 13ZB0182) and the National Natural Science Foundation of China (No. 11426190).
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Wan, LL. Some convergence results for multivalued quasi-nonexpansive mappings in \(\operatorname{CAT}(\kappa)\) spaces. Fixed Point Theory Appl 2015, 5 (2015). https://doi.org/10.1186/s13663-014-0251-8
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DOI: https://doi.org/10.1186/s13663-014-0251-8