Remarks on endpoints of multivalued mappings in geodesic spaces
 Satit Saejung^{1, 2}Email author
https://doi.org/10.1186/s1366301605414
© Saejung 2016
Received: 29 September 2015
Accepted: 8 April 2016
Published: 16 April 2016
Abstract
We discuss Panayanak’s results on the existence of an endpoint of a multivalued nonexpansive mapping. We show that all of his results can be extended and some can be established in a wider class of mappings. Out of his three open questions, two of them are solved in affirmative.
1 Introduction
An element \(x\in E\) is a fixed point of a multivalued mapping \(T:E\to\mathcal{BC}(X)\) if x belongs to the set Tx. Moreover, if \(\{ x\}=Tx\), then x is called an endpoint of T (or a stationary point of T). The set of all endpoints (all fixed points, respectively) of T is denoted by \(\operatorname{End}(T)\) (\(\operatorname{Fix}(T)\), respectively). It is clear that \(\operatorname {End}(T)\subset\operatorname{Fix}(T)\) and the inclusion may be proper (see Example 1). In the case of singlevalued mappings, both notions coincide.
The existence of an endpoint and of a fixed point of a multivalued mapping has been widely investigated by many researchers (see, e.g., [1–8]). Corley [1] proved that a maximization with respect to a cone, which subsumes ordinary and Pareto optimization, is equivalent to the problem of finding an endpoint of certain multivalued mapping. Note that the results in multivalued case are suggested but do not follow directly from the one in the singlevalued case. In spite of the Michael selection theorem, which gives a continuous selection for multivalued upper semicontinuos mappings, almost nothing is known about obtaining a nonexpansive selection. In our problem studied below, we do not know how they can be proved via the classical results for singlevalued mappings.
First, let us recall the following simple example of Ko [9]. It suggests that fixed point results in the singlevalued case should be extended to endpoint results in the corresponding multivalued one.
Example 1

It is clear that \(\operatorname{Fix}(T)=\{(a,b)\in E:ab=0\}\) and \(\operatorname{End}(T)=\{(0,0)\}\). Hence, \(\operatorname{End}(T)\) is convex, but \(\operatorname{Fix}(T)\) is not.

This example shows that even in a Hilbert space the class of Γtype mappings, which is given analogously to Bruck’s theorem does not include all nonexpansive mappings (see [5], Example 5.1, for more details).

For a fixed element \((a_{0},b_{0})\in E\) and \(t\in(0,1)\), define \(S_{t}:E\to\mathcal{K}(E)\) byIt follows that, as \(t\uparrow1\), the net \(\{\operatorname{Fix}(S_{t})\} _{t\in(0,1)}\) does not converge to the fixed point of T nearest to \((a_{0},b_{0})\) even in the weaker convergence of sets (see [6]).$$S_{t}(a,b):=(1t) (a_{0},b_{0})+tT(a,b). $$
Recently, Panyanak [15] showed that (A) holds in some situations. We first quote all main results recently proved in his paper. Some relevant definitions and concept will be given in the next section.
Theorem P1
Suppose that E is a nonempty bounded closed convex subset of a uniformly convex Banach space X. If \(T:E\to\mathcal{K}(E)\) is a nonexpansive mapping, then (A) holds.
Theorem P2
Suppose that E is a nonempty bounded closed convex subset of a reflexive Banach space X with the Opial property. If \(T:E\to\mathcal {K}(X)\) is a nonexpansive mapping, then (A) holds.
Theorem P3
Suppose that E is a nonempty bounded closed convex subset of a complete \(\operatorname {CAT}(0)\) space X. If \(T:E\to\mathcal{K}(X)\) is a nonexpansive mapping, then (A) holds.
Note that the proofs of Theorems P1, P2, and P3 are based on the technique of asymptotic center introduced by Lim [16].
Theorem P4
Suppose that E is a nonempty convex subset of a complete \(\operatorname {CAT}(0)\) space X. If \(T:E\to\mathcal{K}(X)\) is a nonexpansive mapping, then \(\operatorname{End}(T)\) is convex.
We also quote some of his questions.
Question P1
Let E be a nonempty bounded closed convex subset of a uniformly convex Banach space X, and \(T:E\to\mathcal{K}(X)\) be a nonexpansive mapping. Does (A) hold?
Question P2
Let E be a nonempty closed convex subset of a uniformly convex Banach space X, and \(T:E\to\mathcal{BC}(X)\) be a nonexpansive mapping. Is \(\operatorname{End}(T)\) convex?
The purpose of this paper is to give significant extensions of all theorems above. We also give affirmative answers to the preceding two questions. It should be noted that our proofs are different from those in Panyanak’s paper. Moreover, many results are established under weaker assumptions.
2 Main results

T has an end point if and only if there exists an element \(x\in E\) such that \(D(x,Tx)=0\).

T has the approximate endpoint property if and only if there exists a sequence \(\{x_{n}\}\) in E such that \(D(x_{n},Tx_{n})\to0\).
Let us first start with the following easy observation.
Proposition 2

If \(y'\in Ty\), then \(d(x,y')\le D(x,Tx)+H(Tx,Ty)\).

If \(y'\in Ty\) and T is nonexpansive, then \(d(x,y')\le D(x,Tx)+d(x,y)\).
Proof
The following result follows easily from the preceding proposition.
Lemma 3
Let E be a nonempty subset of a metric space X, and \(T:E\to\mathcal {BC}(X)\) be a continuous mapping. If \(\{u_{n}\}\) is a sequence in E such that \(\lim_{n}D(u_{n},Tu_{n})=0\) and \(\{u_{n}\}\) converges to some element \(u\in E\), then \(u\in\operatorname{End}(T)\).
Proof
Let \(u'\in Tu\). It follows from Proposition 2 that \(d(u_{n},u')\le D(u_{n},Tu_{n})+H(Tu,Tu_{n})\). Since T is continuous, we have \(\lim_{n} H(Tu,Tu_{n})=0\). Then the sequence \(\{u_{n}\}\) converges to \(u'\), and hence \(u=u'\in Tu\). This finishes the proof. □
2.1 Endpoint results in strictly convex spaces and uniformly convex spaces
First, we give an affirmative answer to Question P2. Moreover, we show that the uniform convexity can be weaken to the strict convexity. The following lemma seems to be known, but we give a proof for completeness.
Lemma 4
If u and v are two elements in a strictly convex space such that \(\Vert u+v\Vert =\Vert u\Vert +\Vert v\Vert \), then \(\Vert v\Vert u=\Vert u\Vert v\).
Proof
Theorem 5
Let E be a nonempty closed convex subset of a strictly convex Banach space X. If \(T:E\to\mathcal{BC}(X)\) is a nonexpansive mapping, then \(\operatorname{End}(T)\) is convex.
Proof
We simultaneously extend Theorem P1 and give an affirmative answer to Question P1. We also introduce the following concept, which is a multivalued version of [19].
Definition 6
Let E be a nonempty convex subset of a Banach space X. A multivalued mapping \(T:E\to\mathcal{BC}(X)\) is of convex type if \(\lim_{n}D(z_{n},Tz_{n})=0\) whenever \(\{u_{n}\}\) and \(\{v_{n}\}\) are sequences in E such that \(\lim_{n}D(u_{n},Tu_{n})=\lim_{n}D(v_{n},Tv_{n})=0\) and \(z_{n}=\frac{1}{2}u_{n}+\frac{1}{2}v_{n}\) for all \(n\ge1\).
In the presence of uniform convexity, this class of mappings includes all nonexpansive ones. Note that X is uniformly convex if and only if \(\lim_{n}\Vert x_{n}y_{n}\Vert =0\) whenever \(\{x_{n}\}\) and \(\{y_{n}\}\) are sequences in X satisfying \(\lim_{n}\Vert x_{n}\Vert =\lim_{n}\Vert y_{n}\Vert =\lim_{n}\frac{1}{2}\Vert x_{n}+y_{n}\Vert =1\) (see [18], Proposition 5.2.8).
Lemma 7
Let E be a nonempty bounded closed convex subset of a uniformly convex Banach space X. If \(T:E\to\mathcal{BC}(X)\) is a nonexpansive mapping, then it is of convex type.
Proof
Theorem 8
Suppose that E is a nonempty bounded closed convex subset of a uniformly convex Banach space X. If \(T:E\to\mathcal{BC}(X)\) is a continuous multivalued mapping of convex type, then (A) holds.
Proof
2.2 Endpoint results in reflexive spaces with Opial property
Proposition 9
Let E be a nonempty subset of a Banach space X with the Opial property, and \(T:E\to\mathcal{BC}(X)\) be a nonexpansive mapping. If \(\{ u_{n}\}\) is a sequence in E such that \(D(u_{n},Tu_{n})\to0\) and \(\{u_{n}\}\) converges weakly to \(u\in E\), then \(u\in\operatorname{End}(T)\).
Proof
The following result extends Theorem P2 from \(T:E\to\mathcal{K}(X)\) to \(T:E\to\mathcal{BC}(X)\).
Theorem 10
Suppose that E is a nonempty bounded closed convex subset of a reflexive Banach space X with the Opial property. If \(T:E\to\mathcal {BC}(X)\) is a nonexpansive mapping, then (A) holds.
Proof
Assume that T has an approximate endpoint property. Let \(\{ u_{n}\}\) be a sequence in E such that \(D(u_{n},Tu_{n})\to0\). Since \(\{u_{n}\} \) is bounded, there exists a subsequence \(\{u_{n_{k}}\}\) of \(\{u_{n}\}\) such that \(\{u_{n_{k}}\}\) converges weakly to some element \(u\in E\). It follows from the preceding proposition that \(u\in\operatorname{End}(T)\). □
2.3 Endpoint results in geodesic spaces whose curvature is bounded above
In this section, we extend both Theorems P3 and P4 in a more general setting, that is, we consider geodesic spaces whose curvature is bounded above. Let us recall relevant definitions and a concept as follows. For more details on the subject, we refer to [21].

\(d(x,y)=d_{M_{\kappa}}(\overline{x},\overline{y})\), \(d(y,z)=d_{M_{\kappa}}(\overline{y},\overline{z})\), and \(d(x,z)=d_{M_{\kappa}}(\overline{x},\overline{z})\);

\(d(p,q)\le d_{M_{\kappa}}(\overline{p},\overline{q})\) for all \(p,q\in\triangle(x,y,z)\) and \(\overline{p},\overline{q}\in\triangle (\overline{x},\overline{y},\overline{z})\).
Note that in this study we can consider only \(\operatorname {CAT}(1)\) spaces because all the results can be easily extended to \(\operatorname {CAT}(\kappa)\) spaces with \(\kappa>0\) by resizing the space. Moreover, every \(\operatorname {CAT}(\kappa_{1})\) space is a \(\operatorname {CAT}(\kappa_{2})\) space whenever \(\kappa_{1}\le\kappa_{2}\).
Lemma 11
([22], Lemma 3.1)
Theorem 12
Let E be a nonempty convex subset of a complete \(\operatorname {CAT}(1)\) space X. If \(T:E\to\mathcal{BC}(X)\) is a nonexpansive mapping, then \(\operatorname {End}(T)\) is πconvex.
Proof
Let E be a nonempty subset of a \(\operatorname {CAT}(1)\) space X such that \(d(u,v)<\pi\) for all \(u,v\in E\). Analogously to Definition 6, we introduce the following concept: \(T:E\to\mathcal {BC}(X)\) is of convex type if \(\lim_{n}D(z_{n},Tz_{n})=0\) whenever \(\{ u_{n}\}\) and \(\{v_{n}\}\) are sequences in E such that \(\lim_{n}D(u_{n},Tu_{n})=\lim_{n}D(v_{n},Tv_{n})=0\) and \(z_{n}=\frac{1}{2}u_{n}\oplus\frac{1}{2}v_{n}\) for all \(n\ge1\). It is shown in the following theorem that this class of mappings includes all nonexpansive ones.
Theorem 13
Let E be a nonempty convex subset of a complete \(\operatorname {CAT}(1)\) space X such that \(\operatorname{diam}(E)<\pi\). If \(T:E\to\mathcal{BC}(X)\) is nonexpansive, then it is of convex type.
Proof
The proof of the following result is very similar to that of Theorem 8.
Theorem 14
Let E be a nonempty closed convex subset of a complete \(\operatorname {CAT}(1)\) space X such that \(\operatorname{diam}(E)<\pi\) and \(d(x,y)+d(y,z)+d(z,x)<2\pi\) for all \(x,y,z\in E\). If \(T:E\to\mathcal {BC}(X)\) is of convex type and continuous, then (A) holds.
Proof
The following result improves that of Espínola and FernándezLeón [23]. It is clear that the condition \(\operatorname{diam}(E)<\pi/2\) implies \(\operatorname{diam}(E)<\pi\) and \(d(x,y)+d(y,z)+d(z,x)<2\pi\) for all \(x,y,z\in E\). In the proof above, we do not use the modulus of convexity of \(\mathbb{S}^{2}\) endowed with the spherical distance as was the case in [23].
Corollary 15
Let E be a nonempty closed convex subset of a complete \(\operatorname {CAT}(1)\) space X such that \(\operatorname{diam}(E)<\pi\) and \(d(x,y)+d(y,z)+d(z,x)<2\pi\) for all \(x,y,z\in E\). Suppose that \(T:E\to X\) is of convex type and is continuous. Then \(\operatorname{Fix}(T)\neq \varnothing\) if and only if \(\inf\{d(x,Tx):x\in E\}=0\).
Declarations
Acknowledgements
The author would like to thank the two referees for their comments and suggestions. This work is partially supported by the Research Center for Environmental and Hazardous Substance Management, Khon Kaen University.
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
 Corley, HW: Some hybrid fixed point theorems related to optimization. J. Math. Anal. Appl. 120(2), 528532 (1986) MathSciNetView ArticleMATHGoogle Scholar
 Itoh, S, Takahashi, W: Singlevalued mappings, multivalued mappings and fixedpoint theorems. J. Math. Anal. Appl. 59(3), 514521 (1977) MathSciNetView ArticleMATHGoogle Scholar
 Klein, E, Thompson, AC: Theory of Correspondences. Including Applications to Mathematical Economics. Canadian Mathematical Society Series of Monographs and Advanced Texts. A WileyInterscience Publication. Wiley, New York (1984) MATHGoogle Scholar
 Kojima, M, Yamamoto, Y: A unified approach to the implementation of several restart fixed point algorithms and a new variable dimension algorithm. Math. Program. 28(3), 288328 (1984) MathSciNetView ArticleMATHGoogle Scholar
 LlorensFuster, E: Multivalued nonexpansive mappings with an almost convex displacement function. J. Nonlinear Convex Anal. 16(9), 18351845 (2015) MathSciNetMATHGoogle Scholar
 Pietramala, P: Convergence of approximating fixed points sets for multivalued nonexpansive mappings. Comment. Math. Univ. Carol. 32(4), 697701 (1991) MathSciNetMATHGoogle Scholar
 Reich, S: Fixed point theorems for setvalued mappings. J. Math. Anal. Appl. 69(2), 353358 (1979) MathSciNetView ArticleMATHGoogle Scholar
 Todd, MJ: The Computation of Fixed Points and Applications. Lecture Notes in Economics and Mathematical Systems, vol. 124. Springer, Berlin (1976) MATHGoogle Scholar
 Ko, HM: Fixed point theorems for pointtoset mappings and the set of fixed points. Pac. J. Math. 42, 369379 (1972) MathSciNetView ArticleMATHGoogle Scholar
 Belluce, LP, Kirk, WA: Some fixed point theorems in metric and Banach spaces. Can. Math. Bull. 12, 481491 (1969) MathSciNetView ArticleMATHGoogle Scholar
 Montagnana, M, Vignoli, A: On quasiconvex mappings and fixed point theorems. Boll. Unione Mat. Ital. (4) 4, 870878 (1971) MathSciNetMATHGoogle Scholar
 Danes̆, J: Fixed point theorems, Nemyckii and Uryson operators, and continuity of nonlinear mappings. Comment. Math. Univ. Carol. 11, 481500 (1970) MathSciNetGoogle Scholar
 GarcíaFalset, J, LlorensFuster, E, Sims, B: Fixed point theory for almost convex functions. Nonlinear Anal. 32(5), 601608 (1998) MathSciNetView ArticleMATHGoogle Scholar
 LlorensFuster, E: Setvalued αalmost convex mappings. J. Math. Anal. Appl. 233(2), 698712 (1999) MathSciNetView ArticleMATHGoogle Scholar
 Panyanak, B: Endpoints of multivalued nonexpansive mappings in geodesic spaces. Fixed Point Theory Appl. 2015, 147 (2015) MathSciNetView ArticleGoogle Scholar
 Lim, TC: Remarks on some fixed point theorems. Proc. Am. Math. Soc. 60, 179182 (1976) MathSciNetView ArticleMATHGoogle Scholar
 Clarkson, JA: Uniformly convex spaces. Trans. Am. Math. Soc. 40(3), 396414 (1936) MathSciNetView ArticleMATHGoogle Scholar
 Megginson, RE: An Introduction to Banach Space Theory. Graduate Texts in Mathematics, vol. 183. Springer, New York (1998) MATHGoogle Scholar
 Kirk, WA: Nonexpansive mappings in metric and Banach spaces. Rend. Semin. Mat. Fis. Milano 51, 133144 (1981) MathSciNetView ArticleMATHGoogle Scholar
 Opial, Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591597 (1967) MathSciNetView ArticleMATHGoogle 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: Halpern iteration for strongly quasinonexpansive mappings on a geodesic space with curvature bounded above by one. Fixed Point Theory Appl. 2013, 7 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Espínola, R, FernándezLeón, A: \(\operatorname {CAT}(k)\)Spaces, weak convergence and fixed points. J. Math. Anal. Appl. 353(1), 410427 (2009) MathSciNetView ArticleMATHGoogle Scholar