 Research
 Open Access
Strong convergence theorems for multivalued mappings in a geodesic space with curvature bounded above
 Pongsakorn Yotkaew^{1} and
 Satit Saejung^{1, 2}Email author
https://doi.org/10.1186/s136630150488x
© Yotkaew and Saejung 2015
 Received: 2 September 2015
 Accepted: 11 December 2015
 Published: 30 December 2015
Abstract
We prove two strong convergence results of the Ishikawa iteration for a multivalued quasinonexpansive mapping in a complete geodesic space with curvature bounded above. Our results improve significantly the recent results of Panyanak (Fixed Point Theory Appl. 2014:1, 2014) in the sense that many restrictions in his results are weakened. In particular, we can conclude that the convergence result of the Mann iteration which cannot be deduced from Panyanak’s results.
Keywords
 \(\operatorname{CAT}(\kappa)\) space
 fixed point
 Ishikawa iteration
 Mann iteration
 multivalued mapping
 strong convergence
1 Introduction
For a metric space X, a point \(p\in X\) is said to be a fixed point of a multivalued mapping \(T:X\to2^{X}\) if \(p \in Tp\). The set of all fixed points of T is denoted by \(\operatorname{F}(T)\). The purpose of this paper is to discuss an iterative scheme to approximate a fixed point of a multivalued mapping.
In 2009, Shahzad and Zegeye [4] proved the following result.
Theorem 1.1
([4], Theorems 2.3 and 2.5)
 (A′):

T satisfies condition (I) with respect to K, and \(\{\alpha_{n}\}, \{\beta_{n}\}\subset[a,b]\subset(0,1)\);
 (A^{∗}):

T is hemicompact and continuous, and \(\{\alpha_{n}\}, \{\beta_{n}\}\subset[0,1)\), \(\lim_{n\to\infty}\beta_{n}= 0\), and \(\sum_{n=1}^{\infty}\alpha_{n}\beta_{n}=\infty\).
Recall that the mapping T satisfies the endpoint condition if \(\mathrm{F}(T)\neq\varnothing\) and \(Tp=\{p\}\) for all \(p\in\mathrm{F}(T)\).
Recently, Panyanak [1] presented the analog result of Shahzad and Zegeye in the framework of a \(\operatorname{CAT}(1)\) space whose diameter is less than \(\pi/2\) and whose metric is convex. (See the relevant definitions in Section 2.) We quote all his main results as follows.
Theorem 1.2
([1], Theorems 3.2 and 3.3)
 (A′):

T satisfies condition (I) with respect to X, and \(\{ \alpha_{n}\}, \{\beta_{n}\}\subset[a,b]\subset(0,1)\);
 (A^{∗}):

T is hemicompact and continuous, and \(\{\alpha_{n}\}, \{\beta_{n}\}\subset[0,1)\), \(\lim_{n\to\infty}\beta_{n}= 0\), and \(\sum_{n=1}^{\infty}\alpha_{n}\beta_{n}=\infty\).
In this paper, we improve both Theorems 3.2 and 3.3 of Panyanak (see Theorem 1.2 above) in the following aspects. (1) We do not assume that the metric is convex. (2) The restrictions on the mapping T and the parameters \(\{\alpha_{n}\} \), \(\{\beta_{n}\}\) are relaxed. (3) The condition \(\operatorname{diam}(X)<\pi/2\) is replaced by a more general one and this condition is sharp. Note that the Mann algorithm is nothing but the Ishikawa algorithm (1.1) with \(\beta_{n}\equiv0\). We also show that our result can conclude the convergence for the Mann algorithm while this is not the case in Theorem 1.2.
2 Preliminaries
For a real number κ, we denote by \(M_{\kappa}\) the following: (i) \(M_{\kappa}\) is the spherical space \((1/\sqrt{\kappa})\mathbb{S}^{2}\) if \(\kappa>0\); (ii) \(M_{\kappa}\) is the Euclidean space \(\mathbb{R}^{2}\) if \(\kappa=0\); (iii) \(M_{\kappa}\) is the hyperbolic space \((1/\sqrt{\kappa})\mathbb {H}^{2}\) if \(\kappa<0\). In particular, \(D_{\kappa}:=\operatorname{diam}(M_{\kappa})=\infty\) if \(\kappa\leq 0\) and \(D_{\kappa}=\pi/\sqrt{\kappa}\) if \(\kappa>0\).

\(d(u,v)=d_{M_{\kappa}}(\overline{u}, \overline{v})\), \(d(v,w)=d_{M_{\kappa}}(\overline{v}, \overline{w})\), and \(d(u,w)=d_{M_{\kappa}}(\overline{u}, \overline{w})\);

\(d(p,q)\leq d_{M_{\kappa}}(\overline{p}, \overline{q})\) for all \(p,q\in\triangle(u,v,w)\) and \(\overline{p}, \overline{q}\in\triangle(\overline{u}, \overline{v}, \overline{w})\).
The following lemma, proved by Kimura and Satô [6], gives a very important property of \(\operatorname{CAT}(1)\) spaces.
Lemma 2.1

be quasinonexpansive if \(\mathrm{F}(T)\neq\varnothing\) and \(H(Tu,\{p\})\leq d(u,p)\) for all \(u\in C\) and \(p\in\mathrm{F}(T)\);

be closed at zero if each strongly convergent sequence \(\{ u_{n}\}\) in C satisfying \(d(u_{n},Tu_{n})\to0\) has its limit in \(\mathrm{F}(T)\);

be hemicompact if each bounded sequence \(\{u_{n}\}\) in C satisfying \(d(u_{n},Tu_{n})\to0\) has a strongly convergent subsequence;

satisfy condition (I) with respect to C [7] if \(\mathrm{F}(T)\neq\varnothing\) and there exists a nondecreasing function \(f:[0,\infty)\to[0,\infty)\) such that f vanishes only at zero and \(f(d(u,\mathrm{F}(T)))\leq d(u,Tu)\) for all \(u\in C\).
Remark 2.2
(1) \(\mathrm{F}(T)\) is a closed set if T is quasinonexpansive. (2) Every continuous mapping is closed at zero.
The following lemma is taken from the result of Senter and Dotson [7]. However, it is established for a multivalued mapping in a metric space, while the original result is for a singlevalued mapping in a Banach space.
Lemma 2.3
Let K be a bounded closed subset of a metric space X, and let \(T:K\to2^{X}\setminus\{\varnothing\}\) be a multivalued mapping with a fixed point. If T is hemicompact and closed at zero, then it satisfies condition (I) with respect to K.
Proof
3 Main results
Lemma 3.1
Let F be a nonempty closed subset of a complete metric space X and let \(\gamma>0\). Suppose that \(\{u_{n}\}\subset X\) is \((\gamma, F)\)Fejér monotone. Then \(\{u_{n}\}\) converges to an element of F if and only if \(\liminf_{n\to\infty}d(u_{n},F)=0\).
Proof
Lemma 3.2
Let K be a subset of a metric space X and let \(\{u_{n}\}\) be a bounded sequence in K. Let \(T:K\to2^{X}\setminus\{\varnothing\}\) be a multivalued mapping with a fixed point. If T satisfies condition (I) with respect to \(\{u_{n}\}\) and \(\lim_{n\to \infty}d(u_{n},Tu_{n})=0\), then \(\lim_{n\to\infty}d(u_{n},\mathrm{F}(T))=0\).
Proof
Lemma 3.3
 (i)
\(\lim_{n\to\infty} d(\alpha_{n} u_{n}\oplus(1\alpha_{n})v_{n},p)= \lambda\).
 (ii)
\(\limsup_{n\to\infty} d(u_{n},p)\leq\lambda\);
 (iii)
\(\limsup_{n\to\infty} d(v_{n},p)\leq\lambda\).
Proof
By the double extract subsequence principle, we have \(\lim_{n\to\infty }d(u_{n},v_{n})=0\). □
Lemma 3.4
Proof
Lemma 3.5
Proof
Let \(p\in\mathrm{F}(T)\) be such that \(d(x_{1},p)<\pi/2\). Since \(z_{1}\in Tx_{1}\) and T is quasinonexpansive, we get \(d(z_{1},p)\leq H(Tx_{1},\{p\})\leq d(x_{1},p)<\pi/2\). By Lemma 3.4, we see that \(y_{1}\) is well defined and \(d(y_{1},p)\leq d(x_{1},p)\). Thus \(d(z^{\prime}_{1},p)\leq H(Ty_{1},\{p\})\leq d(y_{1},p)<\pi/2\). Again, by Lemma 3.4, we see that \(x_{2}\) is well defined and \(d(x_{2},p)\leq d(x_{1},p)\). Hence, the result follows from mathematical induction. □
Lemma 3.6
 (i)
If \(\sum_{n=1}^{\infty}\alpha_{n}\beta_{n}(1\beta_{n})=\infty\), then \(\liminf_{n\to\infty}d(x_{n},v_{n})=0\).
 (ii)
If \(\sum_{n=1}^{\infty}\alpha_{n}(1\alpha_{n})=\infty\), then \(\liminf_{n\to\infty}d(x_{n},u_{n})=0\).
Proof
Now, we obtain the following convergence theorem, which improves the results of Panyanak [1].
Theorem 3.7
 (A)
T satisfies condition (I) with respect to each bounded subset of X, and \(\sum _{n=1}^{\infty}\alpha_{n}\beta_{n}(1\beta_{n})=\infty\);
 (B)
\(d(x_{1},\mathrm{F}(T))<\pi/2\).
Proof
Remark 3.8
 (1)
We do not assume convexity of the metric.
 (2)
The restrictions on the mapping T and the parameters \(\{ \alpha_{n}\}\), \(\{\beta_{n}\}\) are relaxed. In fact, it is easy to see that either the condition (A′) or (A^{∗}) implies the condition (A).
 (3)
The condition \(\operatorname{diam}(X)<\pi/2\) is weakened. It is clear that the condition \(\operatorname{diam}(X)<\pi/2\) implies the condition (B).
Remark 3.9
Our condition (B) is sharp in the sense that the condition \(d(x_{1},\mathrm{F}(T))<\pi/2\) cannot be improved to \(d(x_{1},\mathrm{F}(T))\leq\pi/2\).

\(d(x_{1},\mathrm{F}(T))=\pi/2\);

T satisfies condition (I) with respect to \(\{x_{n}\}\);

\(\{x_{n}\}\) does not converge to a fixed point of T.
Example 3.10
Based on Lemma 3.6(ii), we present a convergence theorem which can be reduced to the Mann algorithm.
Theorem 3.11

T satisfies condition (I) with respect to each bounded subset of X, and \(\sum_{n=1}^{\infty}\alpha_{n}(1\alpha_{n})=\infty\) and \(\limsup_{n\to \infty}\beta_{n}<1\);

\(d(x_{1},\mathrm{F}(T))<\pi/2\).
Proof
By Lemma 3.6(ii), we have \(\liminf_{n\to\infty} d(x_{n},u_{n})=0\). Let \(\{n_{k}\}\) be an increasing sequence of positive integers such that \(\lim_{k\to\infty} d(x_{n_{k}},u_{n_{k}})=0\).
Case 1: \(\liminf_{k\to\infty}\beta_{n_{k}}=0\). Without loss of generality, we may assume that \(\lim_{k\to\infty}\beta_{n_{k}}=0\). It follows that \(\lim_{k\to\infty}d(y_{n_{k}},x_{n_{k}})=\lim_{k\to\infty}\beta _{n_{k}}d(v_{n_{k}},x_{n_{k}})=0\). Then \(\lim_{k\to\infty} d(y_{n_{k}}, u_{n_{k}})=0\), and so \(\lim_{k\to\infty } d(y_{n_{k}}, Ty_{n_{k}})=0\). Notice that \(\mathrm{F}(T)\) is closed, \(\{y_{n}\}\) is bounded, and T satisfies condition (I) with respect to \(\{y_{n}\}\). Using Lemma 3.2 gives \(\lim_{k\to\infty} d(x_{n_{k}}, \mathrm{F}(T))=0\). Consequently, the conclusion follows from Lemma 3.1.
The next result follows as an immediate consequence of the above theorem.
Corollary 3.12

T satisfies condition (I) with respect to each bounded subset of X, and \(\sum_{n=1}^{\infty}\alpha_{n}(1\alpha_{n})=\infty\);

\(d(x_{1},\mathrm{F}(T))<\pi/2\).
The following result is a strong convergence theorem in a \(\operatorname{CAT}(\kappa)\) space where κ is any real number. It should be noted that if \(\kappa\le0\), then \(D_{\kappa}=\infty\) and the condition \(d(x_{1},\mathrm{F}(T))< D_{\kappa}/2\) is automatically satisfied.
Theorem 3.13

T satisfies condition (I) with respect to each bounded subset of X;

either (i) \(\sum_{n=1}^{\infty}\alpha_{n}\beta_{n}(1\beta_{n})=\infty\) or (ii) \(\sum_{n=1}^{\infty}\alpha_{n}(1\alpha_{n})=\infty\) and \(\limsup_{n\to \infty}\beta_{n}<1\).
Declarations
Acknowledgements
The first author is supported by the Thailand Research Fund under grant TRG5880026. He is also partially supported by the research grant from the Faculty of Science and the Research and Technology Transfer Affairs Division of Khon Kaen University. The research of the second author is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Panyanak, B: On the Ishikawa iteration processes for multivalued mappings in some \(\operatorname{CAT}(\kappa)\) spaces. Fixed Point Theory Appl. 2014, 1 (2014) MathSciNetView ArticleGoogle Scholar
 Ishikawa, S: Fixed points by a new iteration method. Proc. Amer. Math. Soc. 44, 147150 (1974) MATHMathSciNetView ArticleGoogle Scholar
 Mann, WR: Mean value methods in iteration. Proc. Amer. Math. Soc. 4, 506510 (1953) MATHMathSciNetView ArticleGoogle Scholar
 Shahzad, N, Zegeye, H: On Mann and Ishikawa iteration schemes for multivalued maps in Banach spaces. Nonlinear Anal. 71, 838844 (2009) MATHMathSciNetView ArticleGoogle Scholar
 Bridson, MR, Haefliger, A: Metric Spaces of Nonpositive Curvature. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 319. Springer, Berlin (1999) MATHGoogle Scholar
 Kimura, Y, Satô, K: Convergence of subsets of a complete geodesic space with curvature bounded above. Nonlinear Anal. 75, 50795085 (2012) MATHMathSciNetView ArticleGoogle Scholar
 Senter, HF, Dotson, WG: Approximating fixed points of nonexpansive mappings. Proc. Amer. Math. Soc. 44, 375380 (1974) MATHMathSciNetView ArticleGoogle Scholar