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Journal of Applied Mathematics
Volume 2012, Article ID 629149, 13 pages
http://dx.doi.org/10.1155/2012/629149
Research Article

On Multivalued Nonexpansive Mappings in ℝ-Trees

Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Received 25 April 2012; Accepted 20 June 2012

Academic Editor: Hong-Kun Xu

Copyright © 2012 K. Samanmit and B. Panyanak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The relationships between nonexpansive, weakly nonexpansive, -nonexpansive, proximally nonexpansive, proximally continuous, almost lower semicontinuous, and 𝜀-semicontinuous mappings in -trees are studied. Convergence theorems for the Ishikawa iteration processes are also discussed.

1. Introduction

A mapping 𝑡 on a subset 𝐸 of a Banach space (𝑋,) is said to be nonexpansive if 𝑡(𝑥)𝑡(𝑦)𝑥𝑦,𝑥,𝑦𝐸.(1.1) A point 𝑥 in 𝐸 is called a fixed point of 𝑡 if 𝑥=𝑡(𝑥). The existence of fixed points for nonexpansive mappings in Banach spaces was studied independently by three authors in 1965 (see Browder [1], Göhde [2], and Kirk [3]). They showed that every nonexpansive mapping defined on a bounded closed convex subset of a uniformly convex Banach space always has a fixed point. Since then many researchers generalized the concept of nonexpansive mappings in different directions and also studied the fixed point theory for various types of generalized nonexpansive mappings.

Browder-Göhde-Kirk's result was extended to multivalued nonexpansive mappings by Lim [4] in 1974. Husain and Tarafdar [5] and Husain and Latif [6] introduced the concepts of weakly nonexpansive and -nonexpansive multivalued mappings and studied the existence of fixed points for such mappings in uniformly convex Banach spaces. In 1991, Xu [7] pointed out that a weakly nonexpansive multivalued mapping must be nonexpansive and thus the main results of Husain-Tarafdar and Husain-Latif on weakly nonexpansive multivalued mappings are special cases of those of Lim [4]. Xu [7] also showed that -nonexpansiveness is different from nonexpansiveness for multivalued mappings. In 1995, Lopez Acedo and Xu [8] introduced the concept of proximally nonexpansive multivalued mappings and proved that it coincides with the concept of -nonexpansive mappings when the mappings take compact values.

In 2009, Shahzad and Zegeye [9] proved strong convergence theorems of the Ishikawa iteration for quasi-nonexpansive multivalued mappings satisfying the endpoint condition. They also constructed a modified Ishikawa iteration for proximally nonexpansive mappings and proved strong convergence theorems of the proposed iteration without the endpoint condition. Puttasontiphot [10] gave the analogous results of Shahzad and Zegeye in complete CAT(0) spaces. However, there is not any result in linear or nonlinear spaces concerning the convergence of Ishikawa iteration for quasi-nonexpansive multivalued mappings which completely removes the endpoint condition.

In this paper, motivated by the above results, we obtain the relationships between nonexpansive, weakly nonexpansive, -nonexpansive, and proximally nonexpansive mappings in a nice subclass of CAT(0) spaces, namely, -trees. We also introduce a condition on mappings which is much more general than the endpoint condition and prove strong convergence theorems of a modified Ishikawa iteration for quasi-nonexpansive multivalued mappings satisfying such condition.

2. Preliminaries

Let (𝑋,𝑑) be a metric space and let 𝐸𝑋,𝑥𝑋. The distance from 𝑥 to 𝐸 is defined by dist(𝑥,𝐸)=inf{𝑑(𝑥,𝑦)𝑦𝐸}.(2.1) The set 𝐸 is called proximal if for each 𝑥𝑋, there exists an element 𝑦𝐸 such that 𝑑(𝑥,𝑦)=dist(𝑥,𝐸). Let 𝜀>0 and 𝑥0𝑋. We will denote the open ball centered at 𝑥0 with radius 𝜀 by 𝐵(𝑥0,𝜀), the closed 𝜀-hull of 𝐸 by 𝑁𝜀(𝐸)={𝑥𝑋dist(𝑥,𝐸)𝜀}, and the family of nonempty subsets of 𝐸 by 2𝐸. Let 𝐻(,) be the Hausdorff distance on 2𝐸, that is, 𝐻(𝐴,𝐵)=maxsup𝑎𝐴dist(𝑎,𝐵),sup𝑏𝐵dist(𝑏,𝐴),𝐴,𝐵2𝐸.(2.2) Let 𝑇𝐸2𝐸 be a multivalued mapping. For each 𝑥𝐸, we let 𝑃𝑇(𝑥)(𝑥)={𝑢𝑇(𝑥)𝑑(𝑥,𝑢)=dist(𝑥,𝑇(𝑥))}.(2.3) In the case of 𝑃𝑇(𝑥)(𝑥) is a singleton; we will assume, without loss of generality, that 𝑃𝑇(𝑥)(𝑥) is a point in 𝐸. A point 𝑥𝐸 is called a fixed point of 𝑇 if 𝑥𝑇(𝑥). A point 𝑥𝐸 is called an endpoint of 𝑇 if 𝑥 is a fixed point of 𝑇 and 𝑇(𝑥)={𝑥}. We will denote by Fix(𝑇) the set of all fixed points of 𝑇 and by End(𝑇) the set of all endpoints of 𝑇. We see that for each mapping 𝑇, End(𝑇)Fix(𝑇) and the converse is not true in general. A mapping 𝑇 is said to satisfy the endpoint condition if End(𝑇)=Fix(𝑇).

Definition 2.1. Let 𝐸 be a nonempty subset of a metric space (𝑋,𝑑) and 𝑇𝐸2𝐸. Then 𝑇 is said to be(i)nonexpansive if 𝐻(𝑇(𝑥),𝑇(𝑦))𝑑(𝑥,𝑦)forall𝑥,𝑦𝐸;(ii)quasi-nonexpansive if Fix(𝑇) and 𝐻(𝑇(𝑥),𝑇(𝑝))𝑑(𝑥,𝑝)𝑥𝐸,𝑝Fix(𝑇);(2.4)(iii)weakly nonexpansive if for each 𝑥,𝑦𝐸 and 𝑢𝑥𝑇(𝑥), there exists 𝑢𝑦𝑇(𝑦) such that 𝑑𝑢𝑥,𝑢𝑦𝑑(𝑥,𝑦);(2.5)(iv)-nonexpansive if for each 𝑥,𝑦𝐸 and 𝑢𝑥𝑃𝑇(𝑥)(𝑥), there exists 𝑢𝑦𝑃𝑇(𝑦)(𝑦) such that 𝑑𝑢𝑥,𝑢𝑦𝑑(𝑥,𝑦);(2.6)(v)proximally nonexpansive if the map 𝐹𝐸2𝐸 defined by 𝐹(𝑥)=𝑃𝑇(𝑥)(𝑥) is nonexpansive;(vi)proximally continuous if the map 𝐹(𝑥)=𝑃𝑇(𝑥)(𝑥) is continuous;(vii)almost lower semicontinuous if given 𝜀>0, for each 𝑥𝐸 there is an open neighborhood 𝑈 of 𝑥 such that 𝑦𝑈𝑁𝜀(𝑇(𝑦));(2.7)(viii)𝜀-semicontinuous if given 𝜀>0, for each 𝑥𝐸 there is an open neighborhood 𝑈 of 𝑥 such that 𝑇(𝑦)𝑁𝜀(𝑇(𝑥))𝑦𝑈.(2.8)

The following facts can be found in [7, 8].

Proposition 2.2. Let 𝐸 be a nonempty subset of a metric space (𝑋,𝑑) and 𝑇𝐸2𝐸 be a multivalued mapping. Then the following statements hold:(i)if 𝑇 is weakly nonexpansive, then 𝑇 is nonexpansive;(ii)if 𝑇 is -nonexpansive and 𝑇 takes nonempty proximal values, then 𝑇 is proximally nonexpansive;(iii)the converses of (i) and (ii) hold if 𝑇 takes compact values.

For any pair of points 𝑥,𝑦 in a metric space (𝑋,𝑑), a geodesic path joining these points is an isometry 𝑐 from a closed interval [0,𝑙] to 𝑋 such that 𝑐(0)=𝑥 and 𝑐(𝑙)=𝑦. The image of 𝑐 is called a geodesic segment joining 𝑥 and 𝑦. If there exists exactly one geodesic joining 𝑥 and 𝑦 we denote by [𝑥,𝑦] the geodesic joining 𝑥 and 𝑦. For 𝑥,𝑦𝑋 and 𝛼[0,1], we denote the point 𝑧[𝑥,𝑦] such that 𝑑(𝑥,𝑧)=𝛼𝑑(𝑥,𝑦) by (1𝛼)𝑥𝛼𝑦. The space (𝑋,𝑑) is said to be a geodesic space if every two points of 𝑋 are joined by a geodesic, and 𝑋 is said to be uniquely geodesic if there is exactly one geodesic joining 𝑥 and 𝑦 for each 𝑥,𝑦𝑋. A subset 𝐸 of 𝑋 is said to be convex if 𝐸 includes every geodesic segment joining any two of its points, and 𝐸 is said to be gated if for any point 𝑥𝐸 there is a unique point 𝑦𝑥 such that for any 𝑧𝐸, 𝑑(𝑥,𝑧)=𝑑𝑥,𝑦𝑥𝑦+𝑑𝑥,𝑧.(2.9) The point 𝑦𝑥 is called the gate of 𝑥 in 𝐸. From the definition of 𝑦𝑥 we see that it is also the unique nearest point of 𝑥 in 𝐸. The set 𝐸 is called geodesically bounded if there is no geodesic ray in 𝐸, that is, an isometric image of [0,). We will denote by 𝒫(𝐸) the family of nonempty proximinal subsets of 𝐸, by 𝒞𝒞(𝐸) the family of nonempty closed convex subsets of 𝐸, and by 𝒦𝒞(𝐸) the family of nonempty compact convex subsets of 𝐸.

Definition 2.3. An -tree (sometimes called metric tree) is a geodesic metric space 𝑋 such that:(i)there is a unique geodesic segment [𝑥,𝑦] joining each pair of points 𝑥,𝑦𝑋;(ii)if [𝑦,𝑥][𝑥,𝑧]={𝑥}, then [𝑦,𝑥][𝑥,𝑧]=[𝑦,𝑧].
By (i) and (ii) we have(i)if 𝑢,𝑣,𝑤𝑋, then [𝑢,𝑣][𝑢,𝑤]=[𝑢,𝑧] for some 𝑧𝑋.

An -tree is a special case of a CAT(0) space. For a thorough discussion of these spaces and their applications, see [11]. Notice also that a metric space 𝑋 is a complete -tree if and only if 𝑋 is hyperconvex with unique metric segments, see [12]. For more about hyperconvex spaces and fixed point theorems in hyperconvex spaces, see [13]. We now collect some basic properties of -trees.

Lemma 2.4. Let 𝑋 be a complete -tree. Then the following statements hold: (i)[14, page 1048] the gate subsets of 𝑋 are precisely its closed and convex subsets; (ii)[11, page 176] if 𝐸 is a closed convex subset of 𝑋, then, for each 𝑥𝑋, there exists a unique point 𝑃𝐸(𝑥)𝐸 such that 𝑑𝑥,𝑃𝐸(𝑥)=dist(𝑥,𝐸);(2.10)(iii) [11, page 176] if 𝐸 is closed convex and if 𝑥 belong to [𝑥,𝑃𝐸(𝑥)], then 𝑃𝐸(𝑥)=𝑃𝐸(𝑥);(iv)[15, Lemma 3.1] if 𝐴 and 𝐵 are closed convex subsets of 𝑋, then, for any 𝑢𝑋, 𝑑𝑃𝐴(𝑢),𝑃𝐵(𝑢)𝐻(𝐴,𝐵);(2.11)(v) [16, Lemma 3.2] if 𝐸 is closed convex, then, for any 𝑥,𝑦𝑋, one has either 𝑃𝐸(𝑥)=𝑃𝐸(𝑦)(2.12) or 𝑑(𝑥,𝑦)=𝑑𝑥,𝑃𝐸𝑃(𝑥)+𝑑𝐸(𝑥),𝑃𝐸𝑃(𝑦)+𝑑𝐸(𝑦),𝑦;(2.13)(vi)[17, Lemma 2.5] if 𝑥,𝑦,𝑧𝑋 and 𝛼[0,1], then 𝑑2((1𝛼)𝑥𝛼𝑦,𝑧)(1𝛼)𝑑2(𝑥,𝑧)+𝛼𝑑2(𝑦,𝑧)𝛼(1𝛼)𝑑2(𝑥,𝑦);(2.14)(vii) [18, Proposition 1] if 𝐸 is a closed convex subset of 𝑋 and 𝑇𝐸𝒞𝒞(𝐸) is a quasi-nonexpansive mapping, then Fix(𝑇) is closed and convex.

3. Results in -Trees

In general metric spaces, the concepts of nonexpansive and -nonexpansive multivalued mappings are different (see Examples 5.1 and 5.2). But, if we restrict ourself to an -tree we can show that every nonexpansive mapping with nonempty closed convex values is a -nonexpansive mapping. The following lemma is very crucial.

Lemma 3.1. Let 𝐸 be a nonempty closed convex subset of a complete -tree 𝑋 and 𝑣𝐸. If 𝑣[𝑃𝐸(𝑣),𝑢] for some 𝑢𝑋, then 𝑃𝐸(𝑣)=𝑃𝐸(𝑢).

Proof. By Lemma 2.4(iii), 𝑃𝐸(𝑥)=𝑃𝐸(𝑣) for all 𝑥[𝑃𝐸(𝑣),𝑣]. Then for 𝑧𝐸, we have 𝑑(𝑧,𝑥)=𝑑𝑧,𝑃𝐸𝑃(𝑣)+𝑑𝐸𝑃(𝑣),𝑥𝑥𝐸(𝑣),𝑣.(3.1) This implies that 𝑃𝐸(𝑣) is the gate of 𝑧 in [𝑃𝐸(𝑣),𝑣] for all 𝑧𝐸. Since 𝑣[𝑃𝐸(𝑣),𝑢], then 𝑣 is the gate of 𝑢 in [𝑃𝐸(𝑣),𝑣]. By Lemma 2.4(v), for each 𝑧𝐸 we have 𝑑(𝑢,𝑧)=𝑑(𝑢,𝑣)+𝑑𝑣,𝑃𝐸𝑃(𝑣)+𝑑𝐸(𝑣),𝑧=𝑑𝑢,𝑃𝐸(𝑃𝑣)+𝑑𝐸(𝑣),𝑧𝑑𝑢,𝑃𝐸.(𝑣)(3.2) Hence 𝑃𝐸(𝑣)=𝑃𝐸(𝑢) as desired.

Proposition 3.2. Let 𝐸 be a nonempty subset of a complete -tree 𝑋 and 𝑇𝐸2𝐸 be a multivalued mapping. If 𝑇 takes closed and convex values, then the following statements hold:(i)𝑇 is weakly nonexpansive if and only if 𝑇 is nonexpansive;(ii)𝑇 is -nonexpansive if and only if 𝑇 is proximally nonexpansive;(iii)if 𝑇 is nonexpansive, then 𝑇 is proximally nonexpansive;(iv)if 𝑇 is proximally nonexpansive, then 𝑇 is proximally continuous;(v)if 𝑇 is proximally continuous, then 𝑇 is almost lower semicontinuous;(vi)if 𝑇 is almost lower semicontinuous, then 𝑇 is 𝜀-semicontinuous.

Proof. (i) () Follows from Proposition 2.2(i). (): let 𝑥,𝑦𝐸 and 𝑢𝑥𝑇(𝑥). Choose 𝑢𝑦=𝑃𝑇(𝑦)(𝑢𝑥). Then 𝑑𝑢𝑥,𝑢𝑦𝑢=dist𝑥,𝑇(𝑦)𝐻(𝑇(𝑥),𝑇(𝑦))𝑑(𝑥,𝑦).(3.3)
(ii) () Follows from Proposition 2.2(ii). (): for each 𝑥𝐸, we let 𝑢𝑥=𝑃𝑇(𝑥)(𝑥). Then 𝑑𝑢𝑥,𝑢𝑦𝑃=𝑑𝑇(𝑥)(𝑥),𝑃𝑇(𝑦)(𝑦)𝑑(𝑥,𝑦).(3.4) This means 𝑇 is -nonexpansive.
(iii) We let 𝑥,𝑦𝐸 and divide the proof to 3 cases.
Case 1. 𝑃𝑇(𝑥)(𝑥),𝑃𝑇(𝑦)(𝑦)[𝑥,𝑦]. Then 𝑑(𝑃𝑇(𝑥)(𝑥),𝑃𝑇(𝑦)(𝑦))𝑑(𝑥,𝑦).
Case 2. 𝑃𝑇(𝑥)(𝑥)[𝑥,𝑦],𝑃𝑇(𝑦)(𝑦)[𝑥,𝑦] or vice versa. Let 𝑢[𝑃𝑇(𝑦)(𝑦),𝑦]. Then by Lemma 2.4(iii), 𝑃𝑇(𝑦)(𝑦)=𝑃𝑇(𝑦)(𝑢). We claim that 𝑃𝑇(𝑥)(𝑥)=𝑃𝑇(𝑥)(𝑢). Let 𝑣 be the gate of 𝑃𝑇(𝑥)(𝑥) in [𝑥,𝑦]. Then 𝑣𝑃𝑇(𝑥)(𝑥). Since 𝑣[𝑥,𝑃𝑇(𝑥)(𝑥)], then by Lemma 2.4(iii) we have 𝑃𝑇(𝑥)(𝑣)=𝑃𝑇(𝑥)(𝑥). This implies that 𝑣[𝑃𝑇(𝑥)(𝑣),𝑢]. Since 𝑣𝑇(𝑥), by Lemma 3.1 we have 𝑃𝑇(𝑥)(𝑥)=𝑃𝑇(𝑥)(𝑣)=𝑃𝑇(𝑥)(𝑢).(3.5) By Lemma 2.4(iv), 𝑑𝑃𝑇(𝑥)(𝑥),𝑃𝑇(𝑦)𝑃(𝑦)=𝑑𝑇(𝑥)(𝑢),𝑃𝑇(𝑦)(𝑢)𝐻(𝑇(𝑥),𝑇(𝑦))𝑑(𝑥,𝑦).(3.6)
Case 3. 𝑃𝑇(𝑥)(𝑥)[𝑥,𝑦] and 𝑃𝑇(𝑦)(𝑦)[𝑥,𝑦]. Let 𝑣 and 𝑤 be the gates of 𝑃𝑇(𝑥)(𝑥) and 𝑃𝑇(𝑦)(𝑦) in [𝑥,𝑦], respectively. Since 𝑣[𝑃𝑇(𝑥)(𝑥),𝑥] and 𝑤[𝑃𝑇(𝑦)(𝑦),𝑦], then 𝑃𝑇(𝑥)(𝑥)=𝑃𝑇(𝑥)(𝑣),𝑃𝑇(𝑦)(𝑦)=𝑃𝑇(𝑦)(𝑤).(3.7) Let 𝑢[𝑣,𝑤]. Then by Lemma 3.1, we have 𝑃𝑇(𝑥)(𝑣)=𝑃𝑇(𝑥)(𝑢),𝑃𝑇(𝑦)(𝑤)=𝑃𝑇(𝑦)(𝑢).(3.8) By (3.7), we have 𝑃𝑇(𝑥)(𝑥)=𝑃𝑇(𝑥)(𝑢),𝑃𝑇(𝑦)(𝑦)=𝑃𝑇(𝑦)(𝑢).(3.9) By Lemma 2.4(iv), 𝑑𝑃𝑇(𝑥)(𝑥),𝑃𝑇(𝑦)𝑃(𝑦)=𝑑𝑇(𝑥)(𝑢),𝑃𝑇(𝑦)(𝑢)𝐻(𝑇(𝑥),𝑇(𝑦))𝑑(𝑥,𝑦).(3.10)
(iv) Follows from the fact that nonexpansiveness implies continuity.
(v) Given 𝜀>0 and let 𝑥0𝐸. Since the map 𝐹(𝑥)=𝑃𝑇(𝑥)(𝑥) is single valued continuous, then there exists 𝛿>0 such that 𝑑𝑃𝑇(𝑥)(𝑥),𝑃𝑇(𝑥0)𝑥0𝑥<𝜀𝑥𝐵0.,𝛿(3.11) Let 𝑈=𝐵(𝑥0,𝛿). Then 𝑈 is an open neighborhood of 𝑥0. Since 𝑃dist𝑇(𝑥0)𝑥0𝑃,𝑇(𝑥)𝑑𝑇(𝑥0)𝑥0,𝑃𝑇(𝑥)(𝑥)<𝜀𝑥𝑈,(3.12) then 𝑃𝑇(𝑥0)𝑥0𝑥𝑈𝑁𝜀(𝑇(𝑥)).(3.13) Therefore, 𝑇 is almost lower semicontinuous.
(vi) See [19, page 114].

The following result can be found in [19, Theorem 4].

Proposition 3.3. Let 𝑋 be a complete -tree, 𝐸 a nonempty closed convex geodesically bounded subset of 𝑋, and 𝑇𝐸𝒞𝒞(𝐸) an 𝜀-semicontinuous mapping. Then 𝑇 has a fixed point.

As a consequence of Propositions 3.2 and 3.3, we obtain the following.

Corollary 3.4. Let 𝐸 be a nonempty closed convex geodesically bounded subset of a complete -tree 𝑋 and 𝑇𝐸𝒞𝒞(𝐸) be a multivalued mapping. Then 𝑇 has a fixed point if one of the following statements holds:(i)𝑇 is weakly nonexpansive;(ii)𝑇 is nonexpansive;(iii)𝑇 is -nonexpansive;(iv)𝑇 is proximally nonexpansive;(v)𝑇 is proximally continuous;(vi)𝑇 is almost lower semicontinuous.

4. Convergence Theorems

Let 𝐸 be a nonempty convex subset of an -tree 𝑋,𝑇𝐸𝒫(𝐸) a multivalued mapping and {𝛼𝑛},{𝛽𝑛}[0,1].

(A) The sequence of Ishikawa iterates [9] is defined by 𝑥1𝐸, 𝑦𝑛=𝛽𝑛𝑧𝑛(1𝛽)𝑥𝑛,𝑛1,(4.1) where 𝑧𝑛𝑃𝑇(𝑥𝑛)(𝑥𝑛), and 𝑥𝑛+1=𝛼𝑛𝑧𝑛1𝛼𝑛𝑥𝑛,𝑛1,(4.2) where 𝑧𝑛𝑃𝑇(𝑦𝑛)(𝑦𝑛).

Recall that a multivalued mapping 𝑇𝐸𝒫(𝐸) is said to satisfy Condition (I) if there is a nondecreasing function 𝑓[0,)[0,) with 𝑓(0)=0,𝑓(𝑟)>0 for 𝑟(0,) such that dist(𝑥,𝑇(𝑥))𝑓(dist(𝑥,Fix(𝑇)))𝑥𝐸.(4.3) The mapping 𝑇 is called hemicompact if for any sequence {𝑥𝑛} in 𝐸 such that lim𝑛𝑥dist𝑛𝑥,𝑇𝑛=0,(4.4) there exists a subsequence {𝑥𝑛𝑘} of {𝑥𝑛} and 𝑞𝐸 such that lim𝑘𝑥𝑛𝑘=𝑞.

The following theorems are consequences of [10, Theorems 3.6 and 3.7].

Theorem 4.1. Let 𝑋 be a complete -tree, 𝐸 a nonempty closed convex subset of 𝑋, and 𝑇𝐸𝒫(𝐸) a proximally nonexpansive mapping with Fix(𝑇). Let {𝑥𝑛} be the Ishikawa iterates defined by (A). Assume that 𝑇 satisfies condition (I) and 𝛼𝑛,𝛽𝑛[𝑎,𝑏](0,1). Then {𝑥𝑛} converges to a fixed point of 𝑇.

Theorem 4.2. Let 𝑋 be a complete -tree, 𝐸 a nonempty closed convex subset of 𝑋, and 𝑇𝐸𝒫(𝐸) a proximally nonexpansive mapping with Fix(𝑇). Let {𝑥𝑛} be the Ishikawa iterates defined by (A). Assume that 𝑇 is hemicompact and (i) 0𝛼𝑛,𝛽𝑛<1; (ii) 𝛽𝑛0; (iii) 𝛼𝑛𝛽𝑛=. Then {𝑥𝑛} converges to a fixed point of 𝑇.

As consequences of Proposition 3.2, Theorems 4.1 and 4.2, we obtain the following.

Corollary 4.3. Let 𝑋 be a complete -tree, 𝐸 a nonempty closed convex subset of 𝑋, and 𝑇𝐸𝒞𝒞(𝐸) a nonexpansive mapping with Fix(𝑇). Let {𝑥𝑛} be the Ishikawa iterates defined by (A). Assume that 𝑇 satisfies condition (I) and 𝛼𝑛,𝛽𝑛[𝑎,𝑏](0,1). Then {𝑥𝑛} converges to a fixed point of 𝑇.

Corollary 4.4. Let 𝑋 be a complete -tree, 𝐸 a nonempty closed convex subset of 𝑋, and 𝑇𝐸𝒞𝒞(𝐸) a nonexpansive mapping with Fix(𝑇). Let {𝑥𝑛} be the Ishikawa iterates defined by (A). Assume that 𝑇 is hemicompact and (i) 0𝛼𝑛,𝛽𝑛<1; (ii) 𝛽𝑛0 and (iii) 𝛼𝑛𝛽𝑛=. Then {𝑥𝑛} converges to a fixed point of 𝑇.

Definition 4.5. Let 𝐸 be a nonempty subset of a complete -tree and 𝑇𝐸𝒞𝒞(𝐸) be a multivalued mapping for which Fix(𝑇). We say that 𝑢𝐸 is a key of 𝑇 if, for each 𝑥Fix(𝑇),𝑥 is the gate of 𝑢 in 𝑇(𝑥). We say that 𝑇 satisfies the gate condition if 𝑇 has a key in 𝐸.

It follows from the definitions that the endpoint condition implies the gate condition and the converse is not true. Example 5.3 shows that there is a nonexpansive mapping satisfying the gate condition but does not satisfy the endpoint condition.

Motivated by the above results, we introduce a modified Ishikawa iteration as follows: let 𝐸 be a nonempty convex subset of an -tree 𝑋,𝑇𝐸𝒞𝒞(𝐸) a multivalued mapping, and {𝛼𝑛},{𝛽𝑛}[0,1]. Fix 𝑢𝐸.

(B) The sequence of Ishikawa iterates is defined by 𝑥1𝐸, 𝑦𝑛=𝛽𝑛𝑧𝑛1𝛽𝑛𝑥𝑛,𝑛1,(4.5) where 𝑧𝑛 is the gate of 𝑢 in 𝑇(𝑥𝑛), and 𝑥𝑛+1=𝛼𝑛𝑧𝑛1𝛼𝑛𝑥𝑛,𝑛1,(4.6) where 𝑧𝑛 is the gate of 𝑢 in 𝑇(𝑦𝑛).

Recall that a sequence {𝑥𝑛} in a metric space (𝑋,𝑑) is said to be Fejér monotone with respect to a subset 𝐸 of 𝑋 if 𝑑𝑥𝑛+1𝑥,𝑝𝑑𝑛,𝑝𝑝𝐸,𝑛1.(4.7)

The following fact can be found in [20].

Proposition 4.6. Let (𝑋,𝑑) be a complete metric space, 𝐸 be a nonempty closed subset of 𝑋, and {𝑥𝑛} be Fejér monotone with respect to 𝐸. Then {𝑥𝑛} converges to some 𝑝𝐸 if and only if lim𝑛dist(𝑥𝑛,𝐸)=0.

Lemma 4.7. Let 𝐸 be a nonempty closed convex subset of a complete -tree 𝑋 and 𝑇𝐸𝒞𝒞(𝐸) be a quasi-nonexpansive mapping satisfying the gate condition. Let 𝑢 be a key of 𝑇 and let {𝑥𝑛} be the Ishikawa iterates defined by (B). Then {𝑥𝑛} is Fejér monotone with respect to Fix(𝑇) and lim𝑛𝑑(𝑥𝑛,𝑝) exists for each 𝑝Fix(𝑇).

Proof. Let 𝑝Fix(𝑇). For each 𝑛, we have 𝑑𝑦𝑛𝛽,𝑝=𝑑𝑛𝑧𝑛1𝛽𝑛𝑥𝑛,𝑝𝛽𝑛𝑑𝑧𝑛+,𝑝1𝛽𝑛𝑑𝑥𝑛,𝑝=𝛽𝑛𝑑𝑃𝑇(𝑥𝑛)(𝑢),𝑃𝑇(𝑝)+(𝑢)1𝛽𝑛𝑑𝑥𝑛,𝑝𝛽𝑛𝐻𝑇𝑥𝑛+,𝑇(𝑝)1𝛽𝑛𝑑𝑥𝑛,𝑝𝛽𝑛𝑑𝑥𝑛+,𝑝1𝛽𝑛𝑑𝑥𝑛𝑥,𝑝𝑑𝑛,𝑑𝑥,𝑝(4.8)𝑛+1𝛼,𝑝=𝑑𝑛𝑧𝑛1𝛼𝑛𝑥𝑛,𝑝𝛼𝑛𝑑𝑧𝑛+,𝑝1𝛼𝑛𝑑𝑥𝑛,𝑝=𝛼𝑛𝑑𝑃𝑇(𝑦𝑛)(𝑢),𝑃𝑇(𝑝)+(𝑢)1𝛼𝑛𝑑𝑥𝑛,𝑝𝛼𝑛𝐻𝑇𝑦𝑛+,𝑇(𝑝)1𝛼𝑛𝑑𝑥𝑛,𝑝𝛼𝑛𝑑𝑦𝑛+,𝑝1𝛼𝑛𝑑𝑥𝑛𝑥,𝑝𝑑𝑛.,𝑝(4.9) This shows that {𝑥𝑛} is Fejér monotone with respect to Fix(𝑇). Notice from (4.9) that 𝑑(𝑥𝑛,𝑝)𝑑(𝑥1,𝑝) for all 𝑛1. This implies that {𝑑(𝑥𝑛,𝑝)}𝑛=1 is bounded and decreasing. Hence lim𝑛𝑑(𝑥𝑛,𝑝) exists.

Theorem 4.8. Let 𝐸 be a nonempty closed convex subset of a complete -tree 𝑋 and 𝑇𝐸𝒞𝒞(𝐸) be a quasi-nonexpansive mapping satisfying the gate condition. Let 𝑢 be a key of 𝑇 and let {𝑥𝑛} be the Ishikawa iterates defined by (B). Assume that 𝑇 satisfies condition (I) and 𝛼𝑛,𝛽𝑛[𝑎,𝑏](0,1). Then {𝑥𝑛} converges to a fixed point of 𝑇.

Proof. Let 𝑝Fix(𝑇). By Lemma 2.4(vi), we have 𝑑2𝑥𝑛+1,𝑝=𝑑2𝛼𝑛𝑧𝑛1𝛼𝑛𝑥𝑛,𝑝1𝛼𝑛𝑑2𝑥𝑛,𝑝+𝛼𝑛𝑑2𝑧𝑛,𝑝𝛼𝑛1𝛼𝑛𝑑2𝑥𝑛,𝑧𝑛=1𝛼𝑛𝑑2𝑥𝑛,𝑝+𝛼𝑛𝑑2𝑃𝑇(𝑦𝑛)(𝑢),𝑃𝑇(𝑝)(𝑢)𝛼𝑛1𝛼𝑛𝑑2𝑥𝑛,𝑧𝑛1𝛼𝑛𝑑2𝑥𝑛,𝑝+𝛼𝑛𝐻2𝑇𝑦𝑛,𝑇(𝑝)𝛼𝑛1𝛼𝑛𝑑2𝑥𝑛,𝑧𝑛1𝛼𝑛𝑑2𝑥𝑛,𝑝+𝛼𝑛𝑑2𝑦𝑛,𝑑,𝑝2𝑦𝑛,𝑝=𝑑2𝛽𝑛𝑧𝑛1𝛽𝑛𝑥𝑛,𝑝1𝛽𝑛𝑑2𝑥𝑛,𝑝+𝛽𝑛𝑑2𝑧𝑛,𝑝𝛽𝑛1𝛽𝑛𝑑2𝑥𝑛,𝑧𝑛=1𝛽𝑛𝑑2𝑥𝑛,𝑝+𝛽𝑛𝑑2𝑃𝑇(𝑥𝑛)(𝑢),𝑃𝑇(𝑝)(𝑢)𝛽𝑛1𝛽𝑛𝑑2𝑥𝑛,𝑧𝑛1𝛽𝑛𝑑2𝑥𝑛,𝑝+𝛽𝑛𝐻2𝑇𝑥𝑛,𝑇(𝑝)𝛽𝑛1𝛽𝑛𝑑2𝑥𝑛,𝑧𝑛1𝛽𝑛𝑑2𝑥𝑛,𝑝+𝛽𝑛𝑑2𝑥𝑛,𝑝𝛽𝑛1𝛽𝑛𝑑2𝑥𝑛,𝑧𝑛𝑑2𝑥𝑛,𝑝𝛽𝑛1𝛽𝑛𝑑2𝑥𝑛,𝑧𝑛𝑑2𝑥𝑛,𝑝𝛽𝑛1𝛽𝑛𝑑2𝑥𝑛,𝑧𝑛.(4.10) Thus, by (4.10) we have 𝑑2𝑥𝑛+1,𝑝1𝛼𝑛𝑑2𝑥𝑛,𝑝+𝛼𝑛𝑑2𝑥𝑛,𝑝𝛼𝑛𝛽𝑛1𝛽𝑛𝑑2𝑥𝑛,𝑧𝑛.(4.11) This implies that 𝑎2(1𝑏)𝑑2𝑥𝑛,𝑧𝑛𝛼𝑛𝛽𝑛1𝛽𝑛𝑑2𝑥𝑛,𝑧𝑛𝑑2𝑥𝑛,𝑝𝑑2𝑥𝑛+1,𝑝(4.12) and so 𝑛=1𝑎2(1𝑏)𝑑2𝑥𝑛,𝑧𝑛<.(4.13) Thus, lim𝑛𝑑2(𝑥𝑛,𝑧𝑛)=0. Also dist(𝑥𝑛,𝑇(𝑥𝑛))𝑑(𝑥𝑛,𝑧𝑛)0 as 𝑛. Since 𝑇 satisfies condition (I), we have lim𝑛𝑑(𝑥𝑛,Fix(𝑇))=0. By Lemma 4.7, {𝑥𝑛} is Fejér monotone with respect to Fix(𝑇). The conclusion follows from Proposition 4.6.

As a consequence of Proposition 3.2 and Theorem 4.8, we obtain the following.

Corollary 4.9. Let 𝐸 be a nonempty closed convex subset of a complete -tree 𝑋 and 𝑇𝐸𝒞𝒞(𝐸) be a nonexpansive mapping satisfying the gate condition. Let 𝑢 be a key of 𝑇 and let {𝑥𝑛} be the Ishikawa iterates defined by (B). Assume that 𝑇 satisfies condition (I) and 𝛼𝑛,𝛽𝑛[𝑎,𝑏](0,1). Then {𝑥𝑛} converges to a fixed point of 𝑇.

Theorem 4.10. Let 𝐸 be a nonempty closed convex subset of a complete -tree 𝑋 and 𝑇𝐸𝒞𝒞(𝐸) be a quasi-nonexpansive mapping satisfying the gate condition. Let 𝑢 be a key of 𝑇 and let {𝑥𝑛} be the Ishikawa iterates defined by (B). Assume that 𝑇 is hemicompact and continuous and (i) 0𝛼𝑛,𝛽𝑛<1; (ii) 𝛽𝑛<1 and (iii) 𝛼𝑛𝛽𝑛=. Then {𝑥𝑛} converges strongly to a fixed point of 𝑇.

Proof. As in the proof of Theorem 4.8, we obtain lim𝑛𝑥dist𝑛𝑥,𝑇𝑛=0.(4.14) Since 𝑇 is hemicompact, there is a subsequence {𝑥𝑛𝑘} of {𝑥𝑛} such that 𝑥𝑛𝑘𝑞 for some 𝑞𝐸. Since 𝑇 is continuous, then dist(𝑞,𝑇(𝑞))𝑑𝑞,𝑥𝑛𝑘𝑥+dist𝑛𝑘𝑥,𝑇𝑛𝑘𝑇𝑥+𝐻𝑛𝑘,𝑇(𝑞)0as𝑘.(4.15) This implies that 𝑞𝑇(𝑞). By Lemma 4.7, lim𝑛𝑑(𝑥𝑛,𝑞) exists and hence 𝑞 is the limit of {𝑥𝑛} itself.

Corollary 4.11. Let 𝐸 be a nonempty closed convex subset of a complete -tree 𝑋 and 𝑇𝐸𝒞𝒞(𝐸) be a nonexpansive mapping satisfying the gate condition. Let 𝑢 be a key of 𝑇 and let {𝑥𝑛} be the Ishikawa iterates defined by (B). Assume that 𝑇 is hemicompact and (i) 0𝛼𝑛,𝛽𝑛<1; (ii) 𝛽𝑛<1; (iii) 𝛼𝑛𝛽𝑛=. Then {𝑥𝑛} converges strongly to a fixed point of 𝑇.

5. Examples

Example 5.1 (see [7] (A nonexpansive mapping which is not -nonexpansive)). Let 𝐸 be the triangle in the Euclidean plane with vertexes 𝑂(0,0),𝐴(1,0),𝐵(0,1). Let 𝑇𝐸𝒦𝒞(𝐸) be given by 𝑇(𝑥,𝑦)=thesegmentjoining(0,1)and(𝑥,0).(5.1) Then for each (𝑥1,𝑦1),(𝑥2,𝑦2)𝐸, we have 𝐻𝑇𝑥1,𝑦1𝑥,𝑇2,𝑦2=||𝑥1𝑥2||𝑥𝑑1,𝑦1,𝑥2,𝑦2.(5.2) Therefore, 𝑇 is nonexpansive.
For each (𝑥,𝑦)𝐸, we denote by 𝑢(𝑥,𝑦) the point in 𝑇(𝑥,𝑦) nearest to (𝑥,𝑦). Thus, for (𝑥,𝑦)𝐸 with 0<𝑥,𝑦<1 we have ||𝑢(𝑥,𝑦)𝑢(1,0)||>𝑑((𝑥,𝑦),(1,0)).(5.3) This implies that 𝑇 is not -nonexpansive.

Example 5.2 (see [7] (A -nonexpansive mapping which is not nonexpansive)). Let 𝐸=[0,) and 𝑇𝐸𝒦𝒞(𝐸) be defined by []𝑇(𝑥)=𝑥,2𝑥𝑥𝐸.(5.4) Then 𝑢𝑥=𝑥 for every 𝑥𝐸. This implies that 𝑇 is -nonexpansive. However, we have [],[]||||𝐻(𝑇(𝑥),𝑇(𝑦))=𝐻(𝑥,2𝑥𝑦,2𝑦)=2𝑥𝑦.(5.5) This shows that 𝑇 is not nonexpansive.

Example 5.3. Let 𝐸=[0,1] and 𝑇𝐸𝒞𝒞(𝐸) be defined by 𝑇(𝑥)=[0,𝑥] for 𝑥𝐸. Then 𝐻(𝑇(𝑥),𝑇(𝑦))=|𝑥𝑦| for all 𝑥,𝑦𝐸. This implies that 𝑇 is nonexpansive. We see that Fix(𝑇)=[0,1] and 𝑢=1 is a key of 𝑇. Since End(𝑇)={0}, then 𝑇 does not satisfy the endpoint condition.

6. Questions

It is not clear that the gate condition in Theorems 4.8 and 4.10 can be omitted. We finish the paper with the following questions.

Question 1. Let 𝐸 be a nonempty closed convex subset of a complete -tree 𝑋 and 𝑇𝐸𝒞𝒞(𝐸) be a quasi-nonexpansive mapping with Fix(𝑇). Let {𝑥𝑛} be the Ishikawa iterates defined by (B). Assume that 𝑇 satisfies condition (I) and 𝛼𝑛,𝛽𝑛[𝑎,𝑏](0,1). Does {𝑥𝑛} converge to a fixed point of 𝑇?

Question 2. Let 𝐸 be a nonempty closed convex subset of a complete -tree 𝑋 and 𝑇𝐸𝒞𝒞(𝐸) be a quasi-nonexpansive mapping with Fix(𝑇). Let {𝑥𝑛} be the Ishikawa iterates defined by (B). Assume that 𝑇 is hemicompact and continuous and (i) 0𝛼𝑛,𝛽𝑛<1; (ii) 𝛽𝑛<1; (iii) 𝛼𝑛𝛽𝑛=. Does {𝑥𝑛} converge to a fixed point of 𝑇?

Acknowledgments

This research was supported by the Faculty of Science, Chiang Mai University, Chiang Mai, Thailand. The first author also thanks the Graduate School of Chiang Mai University, Thailand.

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