#### Abstract

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

#### 1. Introduction

A mapping on a subset of a Banach space is said to be *nonexpansive* if
A point in is called a *fixed point* of if . 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 be a metric space and let . The distance from to is defined by
The set is called *proximal* if for each , there exists an element such that . Let and . We will denote the open ball centered at with radius by , the closed -*hull* of by , and the family of nonempty subsets of by . Let be the Hausdorff distance on , that is,
Let be a multivalued mapping. For each , we let
In the case of is a singleton; we will assume, without loss of generality, that is a point in . A point is called a *fixed point* of if . A point is called an *endpoint* of if is a fixed point of and . We will denote by the set of all fixed points of and by the set of all endpoints of . We see that for each mapping , and the converse is not true in general. A mapping is said to satisfy the *endpoint condition* if .

*Definition 2.1. * Let be a nonempty subset of a metric space and . Then is said to be(i)*nonexpansive* if ;(ii)*quasi-nonexpansive* if and
(iii)*weakly nonexpansive* if for each and , there exists such that
(iv)-*nonexpansive* if for each and , there exists such that
(v)*proximally nonexpansive* if the map defined by is nonexpansive;(vi)*proximally continuous* if the map is continuous;(vii)*almost lower semicontinuous* if given , for each there is an open neighborhood of such that
(viii)-*semicontinuous* if given , for each there is an open neighborhood of such that

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

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

For any pair of points in a metric space , a *geodesic path* joining these points is an isometry from a closed interval to such that and . The image of is called a *geodesic segment* joining and . If there exists exactly one geodesic joining and we denote by the geodesic joining and . For and , we denote the point such that by . The space is said to be a *geodesic space* if every two points of are joined by a geodesic, and is said to be *uniquely geodesic* if there is exactly one geodesic joining and for each . A subset of is said to be *convex* if includes every geodesic segment joining any two of its points, and is said to be *gated* if for any point there is a unique point such that for any ,
The point is called the *gate* of in . From the definition of we see that it is also the unique nearest point of in . The set is called *geodesically bounded* if there is no geodesic ray in , that is, an isometric image of . We will denote by the family of nonempty proximinal subsets of , by the family of nonempty closed convex subsets of , and by the family of nonempty compact convex subsets of .

*Definition 2.3. *An -tree (sometimes called *metric tree*) is a geodesic metric space such that:(i)there is a unique geodesic segment joining each pair of points ;(ii)if , then .

By (i) and (ii) we have(i)if , then for some .

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 is a complete -tree if and only if 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 be a complete -tree. Then the following statements hold: *(i)*[14, page 1048] the gate subsets of are precisely its closed and convex subsets; *(ii)*[11, page 176] if is a closed convex subset of , then, for each , there exists a unique point such that
*(iii)* [11, page 176] if is closed convex and if belong to , then ;*(iv)*[15, Lemma 3.1] if and are closed convex subsets of , then, for any ,
*(v)* [16, Lemma 3.2] if is closed convex, then, for any , one has either
or
*(vi)*[17, Lemma 2.5] if and , then
*(vii)* [18, Proposition 1] if is a closed convex subset of and is a quasi-nonexpansive mapping, then 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 be a nonempty closed convex subset of a complete -tree and . If for some , then .*

* Proof. *By Lemma 2.4(iii), for all . Then for , we have
This implies that is the gate of in for all . Since , then is the gate of in . By Lemma 2.4(v), for each we have
Hence as desired.

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

* Proof. *(i) () Follows from Proposition 2.2(i). (): let and . Choose . Then

(ii) () Follows from Proposition 2.2(ii). (): for each , we let . Then
This means is -nonexpansive.

(iii) We let and divide the proof to 3 cases.*Case 1*. . Then .*Case 2. * or vice versa. Let . Then by Lemma 2.4(iii), . We claim that . Let be the gate of in . Then . Since , then by Lemma 2.4(iii) we have . This implies that . Since , by Lemma 3.1 we have
By Lemma 2.4(iv),
*Case 3*. and . Let and be the gates of and in , respectively. Since and , then
Let . Then by Lemma 3.1, we have
By (3.7), we have
By Lemma 2.4(iv),

(iv) Follows from the fact that nonexpansiveness implies continuity.

(v) Given and let . Since the map is single valued continuous, then there exists such that
Let . Then is an open neighborhood of . Since
then
Therefore, is almost lower semicontinuous.

(vi) See [19, page 114].

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

Proposition 3.3. *Let be a complete -tree, a nonempty closed convex geodesically bounded subset of , and an -semicontinuous mapping. Then has a fixed point.*

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

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

#### 4. Convergence Theorems

Let be a nonempty convex subset of an -tree , a multivalued mapping and ,.

(A) The sequence of Ishikawa iterates [9] is defined by , where , and where .

Recall that a multivalued mapping is said to satisfy *Condition (I)* if there is a nondecreasing function with for such that
The mapping is called *hemicompact* if for any sequence in such that
there exists a subsequence of and such that .

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

Theorem 4.1. *Let be a complete -tree, a nonempty closed convex subset of , and a proximally nonexpansive mapping with . Let be the Ishikawa iterates defined by (A). Assume that satisfies condition (I) and . Then converges to a fixed point of .*

Theorem 4.2. *Let be a complete -tree, a nonempty closed convex subset of , and a proximally nonexpansive mapping with . Let be the Ishikawa iterates defined by (A). Assume that is hemicompact and (i) ; (ii) ; (iii) . Then converges to a fixed point of .*

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

Corollary 4.3. *Let be a complete -tree, a nonempty closed convex subset of , and a nonexpansive mapping with . Let be the Ishikawa iterates defined by (A). Assume that satisfies condition (I) and . Then converges to a fixed point of .*

Corollary 4.4. *Let be a complete -tree, a nonempty closed convex subset of , and a nonexpansive mapping with . Let be the Ishikawa iterates defined by (A). Assume that is hemicompact and (i) ; (ii) and (iii) . Then converges to a fixed point of .*

*Definition 4.5. *Let be a nonempty subset of a complete -tree and be a multivalued mapping for which . We say that is a key of if, for each is the gate of in . We say that satisfies the *gate condition* if has a key in .

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 be a nonempty convex subset of an -tree a multivalued mapping, and . Fix .

(B) The sequence of Ishikawa iterates is defined by , where is the gate of in , and where is the gate of in .

Recall that a sequence in a metric space is said to be *Fejér monotone* with respect to a subset of if

The following fact can be found in [20].

Proposition 4.6. *Let be a complete metric space, be a nonempty closed subset of , and be Fejér monotone with respect to . Then converges to some if and only if .*

Lemma 4.7. *Let be a nonempty closed convex subset of a complete -tree and be a quasi-nonexpansive mapping satisfying the gate condition. Let be a key of and let be the Ishikawa iterates defined by (B). Then is Fejér monotone with respect to and exists for each .*

* Proof. *Let . For each , we have
This shows that is Fejér monotone with respect to . Notice from (4.9) that for all . This implies that is bounded and decreasing. Hence exists.

Theorem 4.8. *Let be a nonempty closed convex subset of a complete -tree and be a quasi-nonexpansive mapping satisfying the gate condition. Let be a key of and let be the Ishikawa iterates defined by (B). Assume that satisfies condition (I) and . Then converges to a fixed point of .*

* Proof. *Let . By Lemma 2.4(vi), we have
Thus, by (4.10) we have
This implies that
and so
Thus, . Also as . Since satisfies condition (I), we have . By Lemma 4.7, is Fejér monotone with respect to . 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 be a nonempty closed convex subset of a complete -tree and be a nonexpansive mapping satisfying the gate condition. Let be a key of and let be the Ishikawa iterates defined by (B). Assume that satisfies condition (I) and . Then converges to a fixed point of .*

Theorem 4.10. *Let be a nonempty closed convex subset of a complete -tree and be a quasi-nonexpansive mapping satisfying the gate condition. Let be a key of and let be the Ishikawa iterates defined by (B). Assume that is hemicompact and continuous and (i) ; (ii) and (iii) . Then converges strongly to a fixed point of .*

* Proof. *As in the proof of Theorem 4.8, we obtain
Since is hemicompact, there is a subsequence of such that for some . Since is continuous, then
This implies that . By Lemma 4.7, exists and hence is the limit of itself.

Corollary 4.11. *Let be a nonempty closed convex subset of a complete -tree and be a nonexpansive mapping satisfying the gate condition. Let be a key of and let be the Ishikawa iterates defined by (B). Assume that is hemicompact and (i) ; (ii) ; (iii) . Then converges strongly to a fixed point of .*

#### 5. Examples

*Example 5.1 (see [7] (A nonexpansive mapping which is not -nonexpansive)). *Let be the triangle in the Euclidean plane with vertexes . Let be given by
Then for each , we have
Therefore, is nonexpansive.

For each , we denote by the point in nearest to . Thus, for with we have
This implies that is not -nonexpansive.

*Example 5.2 (see [7] (A -nonexpansive mapping which is not nonexpansive)). *Let and be defined by
Then for every . This implies that is -nonexpansive. However, we have
This shows that is not nonexpansive.

*Example 5.3. *Let and be defined by for . Then for all . This implies that is nonexpansive. We see that and is a key of . Since , then 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 be a nonempty closed convex subset of a complete -tree and be a quasi-nonexpansive mapping with . Let be the Ishikawa iterates defined by (B). Assume that satisfies condition (I) and . Does converge to a fixed point of ?

*Question 2. *Let be a nonempty closed convex subset of a complete -tree and be a quasi-nonexpansive mapping with . Let be the Ishikawa iterates defined by (B). Assume that is hemicompact and continuous and (i) ; (ii) ; (iii) . Does converge to a fixed point of ?

#### 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.