On Multivalued Nonexpansive Mappings in ℝ-Trees

Abstract

The relationships between nonexpansive, weakly nonexpansive, ∗-nonexpansive, proximally nonexpansive, proximally continuous, almost lower semicontinuous, and 𝜀-semicontinuous mappings in ℝ-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)‖≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐸.(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, ℝ-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 ∅≠𝐸⊆𝑋,𝑥∈𝑋. The distance from 𝑥 to 𝐸 is defined by dist(𝑥,𝐸)=inf{𝑑(𝑥,𝑦)∶𝑦∈𝐸}.(2.1) The set 𝐸 is called proximal if for each 𝑥∈𝑋, there exists an element 𝑦∈𝐸 such that 𝑑(𝑥,𝑦)=dist(𝑥,𝐸). Let 𝜀>0 and 𝑥₀∈𝑋. We will denote the open ball centered at 𝑥₀ with radius 𝜀 by 𝐵(𝑥₀,𝜀), the closed 𝜀-hull of 𝐸 by 𝑁_𝜀(𝐸)={𝑥∈𝑋∶dist(𝑥,𝐸)≤𝜀}, and the family of nonempty subsets of 𝐸 by 2^𝐸. Let 𝐻(⋅,⋅) be the Hausdorff distance on 2^𝐸, that is, 𝐻(𝐴,𝐵)=maxsup_𝑎∈𝐴dist(𝑎,𝐵),sup_𝑏∈𝐵dist(𝑏,𝐴),𝐴,𝐵∈2^𝐸.(2.2) Let 𝑇∶𝐸→2^𝐸 be a multivalued mapping. For each 𝑥∈𝐸, we let 𝑃_𝑇(𝑥)(𝑥)∶={𝑢∈𝑇(𝑥)∶𝑑(𝑥,𝑢)=dist(𝑥,𝑇(𝑥))}.(2.3) In the case of 𝑃_𝑇(𝑥)(𝑥) is a singleton; we will assume, without loss of generality, that 𝑃_𝑇(𝑥)(𝑥) is a point in 𝐸. A point 𝑥∈𝐸 is called a fixed point of 𝑇 if 𝑥∈𝑇(𝑥). A point 𝑥∈𝐸 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 𝑇∶𝐸→2^𝐸. Then 𝑇 is said to be(i)nonexpansive if 𝐻(𝑇(𝑥),𝑇(𝑦))≤𝑑(𝑥,𝑦)forall𝑥,𝑦∈𝐸;(ii)quasi-nonexpansive if Fix(𝑇)≠∅ and 𝐻(𝑇(𝑥),𝑇(𝑝))≤𝑑(𝑥,𝑝)∀𝑥∈𝐸,𝑝∈Fix(𝑇);(2.4)(iii)weakly nonexpansive if for each 𝑥,𝑦∈𝐸 and 𝑢_𝑥∈𝑇(𝑥), there exists 𝑢_𝑦∈𝑇(𝑦) such that 𝑑𝑢_𝑥,𝑢_𝑦≤𝑑(𝑥,𝑦);(2.5)(iv)∗-nonexpansive if for each 𝑥,𝑦∈𝐸 and 𝑢_𝑥∈𝑃_𝑇(𝑥)(𝑥), there exists 𝑢_𝑦∈𝑃_𝑇(𝑦)(𝑦) such that 𝑑𝑢_𝑥,𝑢_𝑦≤𝑑(𝑥,𝑦);(2.6)(v)proximally nonexpansive if the map 𝐹∶𝐸→2^𝐸 defined by 𝐹(𝑥)∶=𝑃_𝑇(𝑥)(𝑥) is nonexpansive;(vi)proximally continuous if the map 𝐹(𝑥)∶=𝑃_𝑇(𝑥)(𝑥) is continuous;(vii)almost lower semicontinuous if given 𝜀>0, for each 𝑥∈𝐸 there is an open neighborhood 𝑈 of 𝑥 such that _𝑦∈𝑈𝑁_𝜀(𝑇(𝑦))≠∅;(2.7)(viii)𝜀-semicontinuous if given 𝜀>0, for each 𝑥∈𝐸 there is an open neighborhood 𝑈 of 𝑥 such that 𝑇(𝑦)∩𝑁_𝜀(𝑇(𝑥))≠∅∀𝑦∈𝑈.(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 𝑇∶𝐸→2^𝐸 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,𝑦∈𝑋 and 𝛼∈[0,1], we denote the point 𝑧∈[𝑥,𝑦] such that 𝑑(𝑥,𝑧)=𝛼𝑑(𝑥,𝑦) by (1−𝛼)𝑥⊕𝛼𝑦. 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,𝑦∈𝑋. 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 𝑥∉𝐸 there is a unique point 𝑦_𝑥 such that for any 𝑧∈𝐸, 𝑑(𝑥,𝑧)=𝑑𝑥,𝑦_𝑥+𝑑𝑦_𝑥,𝑧.(2.9) The point 𝑦_𝑥 is called the gate of 𝑥 in 𝐸. From the definition of 𝑦_𝑥 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 ℝ-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,𝑦∈𝑋;(ii)if [𝑦,𝑥]∩[𝑥,𝑧]={𝑥}, then [𝑦,𝑥]∪[𝑥,𝑧]=[𝑦,𝑧].
By (i) and (ii) we have(i)if 𝑢,𝑣,𝑤∈𝑋, then [𝑢,𝑣]∩[𝑢,𝑤]=[𝑢,𝑧] for some 𝑧∈𝑋.

An ℝ-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 ℝ-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 ℝ-trees.

Lemma 2.4. Let 𝑋 be a complete ℝ-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 𝑥∈𝑋, there exists a unique point 𝑃_𝐸(𝑥)∈𝐸 such that 𝑑𝑥,𝑃_𝐸(𝑥)=dist(𝑥,𝐸);(2.10)(iii) [11, page 176] if 𝐸 is closed convex and if 𝑥^ belong to [𝑥,𝑃_𝐸(𝑥)], then 𝑃_𝐸(𝑥^)=𝑃_𝐸(𝑥);(iv)[15, Lemma 3.1] if 𝐴 and 𝐵 are closed convex subsets of 𝑋, then, for any 𝑢∈𝑋, 𝑑𝑃_𝐴(𝑢),𝑃_𝐵(𝑢)≤𝐻(𝐴,𝐵);(2.11)(v) [16, Lemma 3.2] if 𝐸 is closed convex, then, for any 𝑥,𝑦∈𝑋, one has either 𝑃_𝐸(𝑥)=𝑃_𝐸(𝑦)(2.12) or 𝑑(𝑥,𝑦)=𝑑𝑥,𝑃_𝐸(𝑥)+𝑑𝑃_𝐸(𝑥),𝑃_𝐸(𝑦)+𝑑𝑃_𝐸(𝑦),𝑦;(2.13)(vi)[17, Lemma 2.5] if 𝑥,𝑦,𝑧∈𝑋 and 𝛼∈[0,1], then 𝑑²((1−𝛼)𝑥⊕𝛼𝑦,𝑧)≤(1−𝛼)𝑑²(𝑥,𝑧)+𝛼𝑑²(𝑦,𝑧)−𝛼(1−𝛼)𝑑²(𝑥,𝑦);(2.14)(vii) [18, Proposition 1] if 𝐸 is a closed convex subset of 𝑋 and 𝑇∶𝐸→𝒞𝒞(𝐸) is a quasi-nonexpansive mapping, then Fix(𝑇) is closed and convex.

3. Results in ℝ-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 ℝ-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 ℝ-tree 𝑋 and 𝑣∉𝐸. If 𝑣∈[𝑃_𝐸(𝑣),𝑢] for some 𝑢∈𝑋, then 𝑃_𝐸(𝑣)=𝑃_𝐸(𝑢).

Proof. By Lemma 2.4(iii), 𝑃_𝐸(𝑥)=𝑃_𝐸(𝑣) for all 𝑥∈[𝑃_𝐸(𝑣),𝑣]. Then for 𝑧∈𝐸, we have 𝑑(𝑧,𝑥)=𝑑𝑧,𝑃_𝐸(𝑣)+𝑑𝑃_𝐸(𝑣),𝑥∀𝑥∈𝑃_𝐸(𝑣),𝑣.(3.1) This implies that 𝑃_𝐸(𝑣) is the gate of 𝑧 in [𝑃_𝐸(𝑣),𝑣] for all 𝑧∈𝐸. Since 𝑣∈[𝑃_𝐸(𝑣),𝑢], then 𝑣 is the gate of 𝑢 in [𝑃_𝐸(𝑣),𝑣]. By Lemma 2.4(v), for each 𝑧∈𝐸 we have 𝑑(𝑢,𝑧)==𝑑𝑑(𝑢𝑢,,𝑣𝑃)_𝐸+𝑑𝑣,𝑃_𝐸(𝑣)+𝑑𝑃_𝐸(𝑣),𝑧≥𝑑𝑢,𝑃_𝐸(𝑣)+𝑑𝑃_𝐸(𝑣),𝑧(𝑣).(3.2) Hence 𝑃_𝐸(𝑣)=𝑃_𝐸(𝑢) as desired.

Proposition 3.2. Let 𝐸 be a nonempty subset of a complete ℝ-tree 𝑋 and 𝑇∶𝐸→2^𝐸 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,𝑦∈𝐸 and 𝑢_𝑥∈𝑇(𝑥). Choose 𝑢_𝑦=𝑃_𝑇(𝑦)(𝑢_𝑥). Then 𝑑𝑢_𝑥,𝑢_𝑦≤≤=𝐻𝑑di(s𝑥(t𝑇,𝑦(𝑢)𝑥_𝑥.),,𝑇𝑇(𝑦(𝑦)))(3.3)
(ii) (⇒) Follows from Proposition 2.2(ii). (⇐): for each 𝑥∈𝐸, we let 𝑢_𝑥=𝑃_𝑇(𝑥)(𝑥). Then 𝑑𝑢_𝑥,𝑢_𝑦=𝑑𝑃_𝑇(𝑥)(𝑥),𝑃_𝑇(𝑦)(𝑦)≤𝑑(𝑥,𝑦).(3.4) This means 𝑇 is ∗-nonexpansive.
(iii) We let 𝑥,𝑦∈𝐸 and divide the proof to 3 cases.
Case 1. 𝑃_𝑇(𝑥)(𝑥),𝑃_𝑇(𝑦)(𝑦)∈[𝑥,𝑦]. Then 𝑑(𝑃_𝑇(𝑥)(𝑥),𝑃_𝑇(𝑦)(𝑦))≤𝑑(𝑥,𝑦).
Case 2. 𝑃_𝑇(𝑥)(𝑥)∉[𝑥,𝑦],𝑃_𝑇(𝑦)(𝑦)∈[𝑥,𝑦] or vice versa. Let 𝑢∈[𝑃_𝑇(𝑦)(𝑦),𝑦]. Then by Lemma 2.4(iii), 𝑃_𝑇(𝑦)(𝑦)=𝑃_𝑇(𝑦)(𝑢). We claim that 𝑃_𝑇(𝑥)(𝑥)=𝑃_𝑇(𝑥)(𝑢). Let 𝑣 be the gate of 𝑃_𝑇(𝑥)(𝑥) in [𝑥,𝑦]. Then 𝑣≠𝑃_𝑇(𝑥)(𝑥). Since 𝑣∈[𝑥,𝑃_𝑇(𝑥)(𝑥)], then by Lemma 2.4(iii) we have 𝑃_𝑇(𝑥)(𝑣)=𝑃_𝑇(𝑥)(𝑥). This implies that 𝑣∈[𝑃_𝑇(𝑥)(𝑣),𝑢]. Since 𝑣∉𝑇(𝑥), by Lemma 3.1 we have 𝑃_𝑇(𝑥)(𝑥)=𝑃_𝑇(𝑥)(𝑣)=𝑃_𝑇(𝑥)(𝑢).(3.5) By Lemma 2.4(iv), 𝑑𝑃_𝑇(𝑥)(𝑥),𝑃_𝑇(𝑦)(𝑦)≤≤=𝐻𝑑𝑑(𝑥(𝑃𝑇,𝑦()𝑥_𝑇(𝑥).)(𝑢,)𝑇,𝑃(𝑦))_𝑇(𝑦)(𝑢)(3.6)
Case 3. 𝑃_𝑇(𝑥)(𝑥)∉[𝑥,𝑦] and 𝑃_𝑇(𝑦)(𝑦)∉[𝑥,𝑦]. Let 𝑣 and 𝑤 be the gates of 𝑃_𝑇(𝑥)(𝑥) and 𝑃_𝑇(𝑦)(𝑦) in [𝑥,𝑦], respectively. Since 𝑣∈[𝑃_𝑇(𝑥)(𝑥),𝑥] and 𝑤∈[𝑃_𝑇(𝑦)(𝑦),𝑦], then 𝑃_𝑇(𝑥)(𝑥)=𝑃_𝑇(𝑥)(𝑣),𝑃_𝑇(𝑦)(𝑦)=𝑃_𝑇(𝑦)(𝑤).(3.7) Let 𝑢∈[𝑣,𝑤]. Then by Lemma 3.1, we have 𝑃_𝑇(𝑥)(𝑣)=𝑃_𝑇(𝑥)(𝑢),𝑃_𝑇(𝑦)(𝑤)=𝑃_𝑇(𝑦)(𝑢).(3.8) By (3.7), we have 𝑃_𝑇(𝑥)(𝑥)=𝑃_𝑇(𝑥)(𝑢),𝑃_𝑇(𝑦)(𝑦)=𝑃_𝑇(𝑦)(𝑢).(3.9) By Lemma 2.4(iv), 𝑑𝑃_𝑇(𝑥)(𝑥),𝑃_𝑇(𝑦)(𝑦)≤≤=𝐻𝑑𝑑(𝑥(𝑃𝑇,𝑦()𝑥_𝑇(𝑥).)(𝑢,)𝑇,𝑃(𝑦))_𝑇(𝑦)(𝑢)(3.10)
(iv) Follows from the fact that nonexpansiveness implies continuity.
(v) Given 𝜀>0 and let 𝑥₀∈𝐸. Since the map 𝐹(𝑥)=𝑃_𝑇(𝑥)(𝑥) is single valued continuous, then there exists 𝛿>0 such that 𝑑𝑃_𝑇(𝑥)(𝑥),𝑃_𝑇(𝑥0)𝑥₀<𝜀∀𝑥∈𝐵𝑥₀,𝛿.(3.11) Let 𝑈=𝐵(𝑥₀,𝛿). Then 𝑈 is an open neighborhood of 𝑥₀. Since dist𝑃_𝑇(𝑥0)𝑥₀,𝑇(𝑥)≤𝑑𝑃_𝑇(𝑥0)𝑥₀,𝑃_𝑇(𝑥)(𝑥)<𝜀∀𝑥∈𝑈,(3.12) then 𝑃_𝑇(𝑥0)𝑥₀∈_𝑥∈𝑈𝑁_𝜀(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 ℝ-tree, 𝐸 a nonempty closed convex geodesically bounded subset of 𝑋, and 𝑇∶𝐸→𝒞𝒞(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 ℝ-tree 𝑋 and 𝑇∶𝐸→𝒞𝒞(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 ℝ-tree 𝑋,𝑇∶𝐸→𝒫(𝐸) a multivalued mapping and {𝛼_𝑛},{𝛽_𝑛}⊆[0,1].

(A) The sequence of Ishikawa iterates [9] is defined by 𝑥₁∈𝐸, 𝑦_𝑛=𝛽_𝑛𝑧_𝑛⊕(1−𝛽)𝑥_𝑛,𝑛≥1,(4.1) where 𝑧_𝑛∈𝑃_{𝑇(𝑥𝑛)}(𝑥_𝑛), and 𝑥_𝑛+1=𝛼_𝑛𝑧^𝑛⊕1−𝛼_𝑛𝑥_𝑛,𝑛≥1,(4.2) where 𝑧^𝑛∈𝑃_{𝑇(𝑦𝑛)}(𝑦_𝑛).

Recall that a multivalued mapping 𝑇∶𝐸→𝒫(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)))∀𝑥∈𝐸.(4.3) The mapping 𝑇 is called hemicompact if for any sequence {𝑥_𝑛} in 𝐸 such that lim_𝑛→∞dist𝑥_𝑛,𝑇𝑥_𝑛=0,(4.4) there exists a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} and 𝑞∈𝐸 such that lim_𝑘→∞𝑥_𝑛𝑘=𝑞.

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

Theorem 4.1. Let 𝑋 be a complete ℝ-tree, 𝐸 a nonempty closed convex subset of 𝑋, and 𝑇∶𝐸→𝒫(𝐸) a proximally nonexpansive mapping with Fix(𝑇)≠∅. Let {𝑥_𝑛} be the Ishikawa iterates defined by (A). Assume that 𝑇 satisfies condition (I) and 𝛼_𝑛,𝛽_𝑛∈[𝑎,𝑏]⊂(0,1). Then {𝑥_𝑛} converges to a fixed point of 𝑇.

Theorem 4.2. Let 𝑋 be a complete ℝ-tree, 𝐸 a nonempty closed convex subset of 𝑋, and 𝑇∶𝐸→𝒫(𝐸) a proximally nonexpansive mapping with Fix(𝑇)≠∅. Let {𝑥_𝑛} be the Ishikawa iterates defined by (A). Assume that 𝑇 is hemicompact and (i) 0≤𝛼_𝑛,𝛽_𝑛<1; (ii) 𝛽_𝑛→0; (iii) ∑𝛼_𝑛𝛽_𝑛=∞. Then {𝑥_𝑛} 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 ℝ-tree, 𝐸 a nonempty closed convex subset of 𝑋, and 𝑇∶𝐸→𝒞𝒞(𝐸) a nonexpansive mapping with Fix(𝑇)≠∅. Let {𝑥_𝑛} be the Ishikawa iterates defined by (A). Assume that 𝑇 satisfies condition (I) and 𝛼_𝑛,𝛽_𝑛∈[𝑎,𝑏]⊂(0,1). Then {𝑥_𝑛} converges to a fixed point of 𝑇.

Corollary 4.4. Let 𝑋 be a complete ℝ-tree, 𝐸 a nonempty closed convex subset of 𝑋, and 𝑇∶𝐸→𝒞𝒞(𝐸) a nonexpansive mapping with Fix(𝑇)≠∅. Let {𝑥_𝑛} be the Ishikawa iterates defined by (A). Assume that 𝑇 is hemicompact and (i) 0≤𝛼_𝑛,𝛽_𝑛<1; (ii) 𝛽_𝑛→0 and (iii) ∑𝛼_𝑛𝛽_𝑛=∞. Then {𝑥_𝑛} converges to a fixed point of 𝑇.

Definition 4.5. Let 𝐸 be a nonempty subset of a complete ℝ-tree and 𝑇∶𝐸→𝒞𝒞(𝐸) be a multivalued mapping for which Fix(𝑇)≠∅. We say that 𝑢∈𝐸 is a key of 𝑇 if, for each 𝑥∈Fix(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 ℝ-tree 𝑋,𝑇∶𝐸→𝒞𝒞(𝐸) a multivalued mapping, and {𝛼_𝑛},{𝛽_𝑛}⊆[0,1]. Fix 𝑢∈𝐸.

(B) The sequence of Ishikawa iterates is defined by 𝑥₁∈𝐸, 𝑦_𝑛=𝛽_𝑛𝑧_𝑛⊕1−𝛽_𝑛𝑥_𝑛,𝑛≥1,(4.5) where 𝑧_𝑛 is the gate of 𝑢 in 𝑇(𝑥_𝑛), and 𝑥_𝑛+1=𝛼_𝑛𝑧^𝑛⊕1−𝛼_𝑛𝑥_𝑛,𝑛≥1,(4.6) where 𝑧^𝑛 is the gate of 𝑢 in 𝑇(𝑦_𝑛).

Recall that a sequence {𝑥_𝑛} in a metric space (𝑋,𝑑) is said to be Fejér monotone with respect to a subset 𝐸 of 𝑋 if 𝑑𝑥_𝑛+1,𝑝≤𝑑𝑥_𝑛,𝑝∀𝑝∈𝐸,𝑛≥1.(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 {𝑥_𝑛} be Fejér monotone with respect to 𝐸. Then {𝑥_𝑛} converges to some 𝑝∈𝐸 if and only if lim_𝑛→∞dist(𝑥_𝑛,𝐸)=0.

Lemma 4.7. Let 𝐸 be a nonempty closed convex subset of a complete ℝ-tree 𝑋 and 𝑇∶𝐸→𝒞𝒞(𝐸) be a quasi-nonexpansive mapping satisfying the gate condition. Let 𝑢 be a key of 𝑇 and let {𝑥_𝑛} be the Ishikawa iterates defined by (B). Then {𝑥_𝑛} is Fejér monotone with respect to Fix(𝑇) and lim_𝑛→∞𝑑(𝑥_𝑛,𝑝) exists for each 𝑝∈Fix(𝑇).

Proof. Let 𝑝∈Fix(𝑇). For each 𝑛, we have 𝑑𝑦_𝑛,𝑝≤=𝑑𝛽_𝑛𝛽_𝑛𝑧_𝑛⊕1−𝛽_𝑛𝑥_𝑛,𝑝=𝛽_𝑛𝑑𝑧_𝑛,𝑝+1−𝛽_𝑛𝑑𝑥_𝑛,𝑝≤𝛽_𝑛𝑑𝑃_{𝑇(𝑥𝑛)}(𝑢),𝑃_𝑇(𝑝)(𝑢)+1−𝛽_𝑛𝑑𝑥_𝑛,𝑝≤𝛽_𝑛𝐻𝑇𝑥_𝑛,𝑇(𝑝)+1−𝛽_𝑛𝑑𝑥_𝑛,𝑝≤𝑑𝑑𝑥_𝑛𝑥_𝑛,𝑝+1−𝛽_𝑛𝑑𝑥_𝑛,𝑝𝑑𝑥,𝑝,(4.8)_𝑛+1,𝑝≤=𝑑𝛼_𝑛𝛼_𝑛𝑧^𝑛⊕1−𝛼_𝑛𝑥_𝑛,𝑝=𝛼_𝑛𝑑𝑧^𝑛,𝑝+1−𝛼_𝑛𝑑𝑥_𝑛,𝑝≤𝛼_𝑛𝑑𝑃_{𝑇(𝑦𝑛)}(𝑢),𝑃_𝑇(𝑝)(𝑢)+1−𝛼_𝑛𝑑𝑥_𝑛,𝑝≤𝛼_𝑛𝐻𝑇𝑦_𝑛,𝑇(𝑝)+1−𝛼_𝑛𝑑𝑥_𝑛,𝑝≤𝑑𝑑𝑥_𝑛𝑦_𝑛,𝑝+1−𝛼_𝑛𝑑𝑥_𝑛,𝑝,𝑝.(4.9) This shows that {𝑥_𝑛} is Fejér monotone with respect to Fix(𝑇). Notice from (4.9) that 𝑑(𝑥_𝑛,𝑝)≤𝑑(𝑥₁,𝑝) for all 𝑛≥1. This implies that {𝑑(𝑥_𝑛,𝑝)}^∞𝑛=1 is bounded and decreasing. Hence lim_𝑛→∞𝑑(𝑥_𝑛,𝑝) exists.

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

Proof. Let 𝑝∈Fix(𝑇). By Lemma 2.4(vi), we have 𝑑²𝑥_𝑛+1,𝑝≤=𝑑²1−𝛼_𝑛𝛼𝑧^𝑛_𝑛⊕1−𝛼_𝑛𝑥_𝑛,𝑝=1−𝛼_𝑛𝑑²𝑥_𝑛,𝑝+𝛼_𝑛𝑑²𝑧^𝑛,𝑝−𝛼_𝑛1−𝛼_𝑛𝑑²𝑥_𝑛,𝑧^𝑛≤1−𝛼_𝑛𝑑²𝑥_𝑛,𝑝+𝛼_𝑛𝑑²𝑃_{𝑇(𝑦𝑛)}(𝑢),𝑃_𝑇(𝑝)(𝑢)−𝛼_𝑛1−𝛼_𝑛𝑑²𝑥_𝑛,𝑧^𝑛≤1−𝛼_𝑛𝑑²𝑥_𝑛,𝑝+𝛼_𝑛𝐻²𝑇𝑦_𝑛,𝑇(𝑝)−𝛼_𝑛1−𝛼_𝑛𝑑²𝑥_𝑛,𝑧^𝑛𝑑𝑑²𝑥_𝑛,𝑝+𝛼_𝑛𝑑²𝑦_𝑛,𝑝₂,𝑦_𝑛,𝑝≤=𝑑²1−𝛽_𝑛𝛽𝑧_𝑛𝑛⊕1−𝛽_𝑛𝑥_𝑛,𝑝=1−𝛽_𝑛𝑑²𝑥_𝑛,𝑝+𝛽_𝑛𝑑²𝑧_𝑛,𝑝−𝛽_𝑛1−𝛽_𝑛𝑑²𝑥_𝑛,𝑧_𝑛≤1−𝛽_𝑛𝑑²𝑥_𝑛,𝑝+𝛽_𝑛𝑑²𝑃_{𝑇(𝑥𝑛)}(𝑢),𝑃_𝑇(𝑝)(𝑢)−𝛽_𝑛1−𝛽_𝑛𝑑²𝑥_𝑛,𝑧_𝑛≤1−𝛽_𝑛𝑑²𝑥_𝑛,𝑝+𝛽_𝑛𝐻²𝑇𝑥_𝑛,𝑇(𝑝)−𝛽_𝑛1−𝛽_𝑛𝑑²𝑥_𝑛,𝑧_𝑛≤𝑑₂𝑑²𝑥_𝑛,𝑝+𝛽_𝑛𝑑²𝑥_𝑛,𝑝−𝛽_𝑛1−𝛽_𝑛𝑑²𝑥_𝑛,𝑧_𝑛≤𝑑₂𝑥_𝑛,𝑝−𝛽_𝑛1−𝛽_𝑛𝑑²𝑥_𝑛,𝑧_𝑛𝑥_𝑛,𝑝−𝛽_𝑛1−𝛽_𝑛𝑑²𝑥_𝑛,𝑧_𝑛.(4.10) Thus, by (4.10) we have 𝑑²𝑥_𝑛+1,𝑝≤1−𝛼_𝑛𝑑²𝑥_𝑛,𝑝+𝛼_𝑛𝑑²𝑥_𝑛,𝑝−𝛼_𝑛𝛽_𝑛1−𝛽_𝑛𝑑²𝑥_𝑛,𝑧_𝑛.(4.11) This implies that 𝑎²(1−𝑏)𝑑²𝑥_𝑛,𝑧_𝑛≤𝛼_𝑛𝛽_𝑛1−𝛽_𝑛𝑑²𝑥_𝑛,𝑧_𝑛≤𝑑²𝑥_𝑛,𝑝−𝑑²𝑥_𝑛+1,𝑝(4.12) and so ^∞_𝑛=1𝑎²(1−𝑏)𝑑²𝑥_𝑛,𝑧_𝑛<∞.(4.13) Thus, lim_𝑛→∞𝑑²(𝑥_𝑛,𝑧_𝑛)=0. Also dist(𝑥_𝑛,𝑇(𝑥_𝑛))≤𝑑(𝑥_𝑛,𝑧_𝑛)→0 as 𝑛→∞. Since 𝑇 satisfies condition (I), we have lim_𝑛→∞𝑑(𝑥_𝑛,Fix(𝑇))=0. By Lemma 4.7, {𝑥_𝑛} 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 ℝ-tree 𝑋 and 𝑇∶𝐸→𝒞𝒞(𝐸) be a nonexpansive mapping satisfying the gate condition. Let 𝑢 be a key of 𝑇 and let {𝑥_𝑛} be the Ishikawa iterates defined by (B). Assume that 𝑇 satisfies condition (I) and 𝛼_𝑛,𝛽_𝑛∈[𝑎,𝑏]⊂(0,1). Then {𝑥_𝑛} converges to a fixed point of 𝑇.

Theorem 4.10. Let 𝐸 be a nonempty closed convex subset of a complete ℝ-tree 𝑋 and 𝑇∶𝐸→𝒞𝒞(𝐸) be a quasi-nonexpansive mapping satisfying the gate condition. Let 𝑢 be a key of 𝑇 and let {𝑥_𝑛} be the Ishikawa iterates defined by (B). Assume that 𝑇 is hemicompact and continuous and (i) 0≤𝛼_𝑛,𝛽_𝑛<1; (ii) 𝛽_𝑛<1 and (iii) ∑𝛼_𝑛𝛽_𝑛=∞. Then {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Proof. As in the proof of Theorem 4.8, we obtain lim_𝑛→∞dist𝑥_𝑛,𝑇𝑥_𝑛=0.(4.14) Since 𝑇 is hemicompact, there is a subsequence {𝑥_𝑛𝑘} of {𝑥_𝑛} such that 𝑥_𝑛𝑘→𝑞 for some 𝑞∈𝐸. Since 𝑇 is continuous, then dist(𝑞,𝑇(𝑞))≤𝑑𝑞,𝑥_𝑛𝑘+dist𝑥_𝑛𝑘,𝑇𝑥_𝑛𝑘+𝐻𝑇𝑥_𝑛𝑘,𝑇(𝑞)⟶0as𝑘⟶∞.(4.15) This implies that 𝑞∈𝑇(𝑞). By Lemma 4.7, lim_𝑛→∞𝑑(𝑥_𝑛,𝑞) exists and hence 𝑞 is the limit of {𝑥_𝑛} itself.

Corollary 4.11. Let 𝐸 be a nonempty closed convex subset of a complete ℝ-tree 𝑋 and 𝑇∶𝐸→𝒞𝒞(𝐸) be a nonexpansive mapping satisfying the gate condition. Let 𝑢 be a key of 𝑇 and let {𝑥_𝑛} be the Ishikawa iterates defined by (B). Assume that 𝑇 is hemicompact and (i) 0≤𝛼_𝑛,𝛽_𝑛<1; (ii) 𝛽_𝑛<1; (iii) ∑𝛼_𝑛𝛽_𝑛=∞. Then {𝑥_𝑛} 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 𝑇∶𝐸→𝒦𝒞(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 𝐻𝑇𝑥₁,𝑦₁,𝑇𝑥₂,𝑦₂=||𝑥₁−𝑥₂||≤𝑑𝑥₁,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 𝑇∶𝐸→𝒦𝒞(𝐸) be defined by 𝑇(𝑥)=[𝑥,2𝑥]∀𝑥∈𝐸.(5.4) Then 𝑢_𝑥=𝑥 for every 𝑥∈𝐸. This implies that 𝑇 is ∗-nonexpansive. However, we have 𝐻(𝑇(𝑥),𝑇(𝑦))=𝐻([𝑥,2𝑥],[𝑦,2𝑦])=2||𝑥−𝑦||.(5.5) This shows that 𝑇 is not nonexpansive.

Example 5.3. Let 𝐸=[0,1] and 𝑇∶𝐸→𝒞𝒞(𝐸) be defined by 𝑇(𝑥)=[0,𝑥] for 𝑥∈𝐸. Then 𝐻(𝑇(𝑥),𝑇(𝑦))=|𝑥−𝑦| for all 𝑥,𝑦∈𝐸. 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 ℝ-tree 𝑋 and 𝑇∶𝐸→𝒞𝒞(𝐸) be a quasi-nonexpansive mapping with Fix(𝑇)≠∅. Let {𝑥_𝑛} be the Ishikawa iterates defined by (B). Assume that 𝑇 satisfies condition (I) and 𝛼_𝑛,𝛽_𝑛∈[𝑎,𝑏]⊂(0,1). Does {𝑥_𝑛} converge to a fixed point of 𝑇?

Question 2. Let 𝐸 be a nonempty closed convex subset of a complete ℝ-tree 𝑋 and 𝑇∶𝐸→𝒞𝒞(𝐸) be a quasi-nonexpansive mapping with Fix(𝑇)≠∅. Let {𝑥_𝑛} be the Ishikawa iterates defined by (B). Assume that 𝑇 is hemicompact and continuous and (i) 0≤𝛼_𝑛,𝛽_𝑛<1; (ii) 𝛽_𝑛<1; (iii) ∑𝛼_𝑛𝛽_𝑛=∞. Does {𝑥_𝑛} 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.