# On Multivalued Nonexpansive Mappings in ℝ-Trees

Authors: K. Samanmit; B. Panyanak
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

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


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## Body

## 1. Introduction

A mappingt on a subset E of a Banach space (X,∥·∥) is said to be nonexpansive if
(1.1)‖t(x)-t(y)‖≤‖x-y‖,∀x,y∈E.
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
(2.1)dist⁡(x,E)=inf⁡{d(x,y):y∈E}.
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 ɛ>0 and x0∈X. We will denote the open ball centered at x0 with radius ɛ by B(x0,ɛ), the closed ɛ-hull of E by Nɛ(E)={x∈X:dist(x,E)≤ɛ}, and the family of nonempty subsets of E by 2E. Let H(·,·) be the Hausdorff distance on 2E, that is,
(2.2)H(A,B)=max⁡{sup⁡a∈Adist⁡(a,B),sup⁡b∈Bdist⁡(b,A)},A,B∈2E.
Let T:E→2E be a multivalued mapping. For each x∈E, we let
(2.3)PT(x)(x):={u∈T(x):d(x,u)=dist⁡(x,T(x))}.
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 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.
LetE be a nonempty subset of a metric space (X,d) and T:E→2E. 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
(2.4)H(T(x),T(p))≤d(x,p)∀x∈E,p∈Fix⁡(T);(iii)
weakly nonexpansive if for each x,y∈E and ux∈T(x), there exists uy∈T(y) such that
(2.5)d(ux,uy)≤d(x,y);(iv)
*-nonexpansive if for each x,y∈E and ux∈PT(x)(x), there exists uy∈PT(y)(y) such that
(2.6)d(ux,uy)≤d(x,y);(v)
proximally nonexpansive if the map F:E→2E 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 ɛ>0, for each x∈E there is an open neighborhood U of x such that
(2.7)⋂y∈U‍Nɛ(T(y))≠∅;(viii)
ɛ-semicontinuous if given ɛ>0, for each x∈E there is an open neighborhood U of x such that
(2.8)T(y)∩Nɛ(T(x))≠∅∀y∈U.The following facts can be found in [7, 8].Proposition 2.2.
LetE be a nonempty subset of a metric space (X,d) and T:E→2E be a multivalued mapping. Then the following statements hold:(i)
ifT is weakly nonexpansive, then T is nonexpansive;(ii)
ifT is *-nonexpansive and T takes nonempty proximal values, then T is proximally nonexpansive;(iii)
the converses of (i) and (ii) hold ifT takes compact values.For any pair of pointsx,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 yx such that for any z∈E,
(2.9)d(x,z)=d(x,yx)+d(yx,z).
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.
Anℝ-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)
ifu,v,w∈X, then [u,v]∩[u,w]=[u,z] for some z∈X.Anℝ-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.
LetX 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 PE(x)∈E such that
(2.10)d(x,PE(x))=dist⁡(x,E);(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 u∈X,
(2.11)d(PA(u),PB(u))≤H(A,B);(v)
[16, Lemma 3.2]  if E is closed convex, then, for any x,y∈X, one has either
(2.12)PE(x)=PE(y)
or
(2.13)d(x,y)=d(x,PE(x))+d(PE(x),PE(y))+d(PE(y),y);(vi)
[17, Lemma 2.5]  if x,y,z∈X and α∈[0,1], then
(2.14)d2((1-α)x⊕αy,z)≤(1-α)d2(x,z)+αd2(y,z)-α(1-α)d2(x,y);(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 inℝ-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.
LetE be a nonempty closed convex subset of a complete ℝ-tree X and v∉E. If v∈[PE(v),u] for some u∈X, then PE(v)=PE(u).Proof.
By Lemma2.4(iii), PE(x)=PE(v) for all x∈[PE(v),v]. Then for z∈E, we have
(3.1)d(z,x)=d(z,PE(v))+d(PE(v),x)∀x∈[PE(v),v].
This implies that PE(v) is the gate of z in [PE(v),v] for all z∈E. Since v∈[PE(v),u], then v is the gate of u in [PE(v),v]. By Lemma 2.4(v), for each z∈E we have
(3.2)d(u,z)=d(u,v)+d(v,PE(v))+d(PE(v),z)=d(u,PE(v))+d(PE(v),z)≥d(u,PE(v)).
Hence PE(v)=PE(u) as desired.Proposition 3.2.
LetE be a nonempty subset of a complete ℝ-tree X and T:E→2E 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)
ifT is nonexpansive, then T is proximally nonexpansive;(iv)
ifT is proximally nonexpansive, then T is proximally continuous;(v)
ifT is proximally continuous, then T is almost lower semicontinuous;(vi)
ifT is almost lower semicontinuous, then T is ɛ-semicontinuous.Proof.
(i) (⇒) Follows from Proposition 2.2(i). (⇐): let x,y∈E and ux∈T(x). Choose uy=PT(y)(ux). Then
(3.3)d(ux,uy)=dist⁡(ux,T(y))≤H(T(x),T(y))≤d(x,y).
(ii) (⇒) Follows from Proposition 2.2(ii). (⇐): for each x∈E, we let ux=PT(x)(x). Then
(3.4)d(ux,uy)=d(PT(x)(x),PT(y)(y))≤d(x,y).
This means T is *-nonexpansive.
(iii) We letx,y∈E 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.
P
T
(
x
)
(
x
)
∉
[
x
,
y
]
,
P
T
(
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 v≠PT(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 v∉T(x), by Lemma 3.1 we have
(3.5)PT(x)(x)=PT(x)(v)=PT(x)(u).
By Lemma 2.4(iv),
(3.6)d(PT(x)(x),PT(y)(y))=d(PT(x)(u),PT(y)(u))≤H(T(x),T(y))≤d(x,y).
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
(3.7)PT(x)(x)=PT(x)(v),PT(y)(y)=PT(y)(w).
Let u∈[v,w]. Then by Lemma 3.1, we have
(3.8)PT(x)(v)=PT(x)(u),PT(y)(w)=PT(y)(u).
By (3.7), we have
(3.9)PT(x)(x)=PT(x)(u),PT(y)(y)=PT(y)(u).
By Lemma 2.4(iv),
(3.10)d(PT(x)(x),PT(y)(y))=d(PT(x)(u),PT(y)(u))≤H(T(x),T(y))≤d(x,y).
(iv) Follows from the fact that nonexpansiveness implies continuity.
(v) Givenɛ>0 and let x0∈E. Since the map F(x)=PT(x)(x) is single valued continuous, then there exists δ>0 such that
(3.11)d(PT(x)(x),PT(x0)(x0))<ɛ∀x∈B(x0,δ).
Let U=B(x0,δ). Then U is an open neighborhood of x0. Since
(3.12)dist⁡(PT(x0)(x0),T(x))≤d(PT(x0)(x0),PT(x)(x))<ɛ∀x∈U,
then
(3.13)PT(x0)(x0)∈⋂x∈U‍Nɛ(T(x)).
Therefore, T is almost lower semicontinuous.
(vi) See [19, page 114].The following result can be found in [19, Theorem 4].Proposition 3.3.
LetX be a complete ℝ-tree, E a nonempty closed convex geodesically bounded subset of X, and T:E→𝒞𝒞(E) an ɛ-semicontinuous mapping. Then T has a fixed point.As a consequence of Propositions3.2 and 3.3, we obtain the following.Corollary 3.4.
LetE 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

LetE be a nonempty convex subset of an ℝ-tree X,T:E→𝒫(E) a multivalued mapping and {αn},{βn}⊆[0,1].(A) The sequence of Ishikawa iterates [9] is defined by x1∈E,
(4.1)yn=βnzn⊕(1-β)xn,n≥1,
where zn∈PT(xn)(xn), and
(4.2)xn+1=αnzn′⊕(1-αn)xn,n≥1,
where zn′∈PT(yn)(yn).Recall that a multivalued mappingT: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
(4.3)dist⁡(x,T(x))≥f(dist⁡(x,Fix⁡(T)))∀x∈E.
The mapping T is called hemicompact if for any sequence {xn} in E such that
(4.4)lim⁡n→∞dist⁡(xn,T(xn))=0,
there exists a subsequence {xnk} of {xn} and q∈E such that limk→∞xnk=q.The following theorems are consequences of [10, Theorems 3.6 and 3.7].Theorem 4.1.
LetX 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 {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.
LetX 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 {xn} be the Ishikawa iterates defined by (A). Assume that T is hemicompact and (i) 0≤αn,βn<1; (ii) βn→0; (iii) ∑‍αnβn=∞. Then {xn} converges to a fixed point of T.As consequences of Proposition3.2, Theorems 4.1 and 4.2, we obtain the following.Corollary 4.3.
LetX 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 {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.
LetX 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 {xn} be the Ishikawa iterates defined by (A). Assume that T is hemicompact and (i) 0≤αn,βn<1; (ii) βn→0 and (iii) ∑‍αnβn=∞. Then {xn} converges to a fixed point of T.Definition 4.5.
LetE 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. Example5.3 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: letE be a nonempty convex subset of an ℝ-tree X,T:E→𝒞𝒞(E) a multivalued mapping, and {αn},{βn}⊆[0,1]. Fix u∈E.(B) The sequence of Ishikawa iterates is defined byx1∈E,
(4.5)yn=βnzn⊕(1-βn)xn,n≥1,
where zn is the gate of u in T(xn), and
(4.6)xn+1=αnzn′⊕(1-αn)xn,n≥1,
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
(4.7)d(xn+1,p)≤d(xn,p)∀p∈E,n≥1.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 p∈E if and only if limn→∞dist(xn,E)=0.Lemma 4.7.
LetE 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 {xn} be the Ishikawa iterates defined by (B). Then {xn} is Fejér monotone with respect to Fix(T) and limn→∞d(xn,p) exists for each p∈Fix(T).Proof.
Letp∈Fix(T). For each n, we have
(4.8)d(yn,p)=d(βnzn⊕(1-βn)xn,p)≤βnd(zn,p)+(1-βn)d(xn,p)=βnd(PT(xn)(u),PT(p)(u))+(1-βn)d(xn,p)≤βnH(T(xn),T(p))+(1-βn)d(xn,p)≤βnd(xn,p)+(1-βn)d(xn,p)≤d(xn,p),(4.9)d(xn+1,p)=d(αnzn′⊕(1-αn)xn,p)≤αnd(zn′,p)+(1-αn)d(xn,p)=αnd(PT(yn)(u),PT(p)(u))+(1-αn)d(xn,p)≤αnH(T(yn),T(p))+(1-αn)d(xn,p)≤αnd(yn,p)+(1-αn)d(xn,p)≤d(xn,p).
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 n≥1. This implies that {d(xn,p)}n=1∞ is bounded and decreasing. Hence limn→∞d(xn,p) exists.Theorem 4.8.
LetE 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 {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.
Letp∈Fix(T). By Lemma 2.4(vi), we have
(4.10)d2(xn+1,p)=d2(αnzn′⊕(1-αn)xn,p)≤(1-αn)d2(xn,p)+αnd2(zn′,p)-αn(1-αn)d2(xn,zn′)=(1-αn)d2(xn,p)+αnd2(PT(yn)(u),PT(p)(u))-αn(1-αn)d2(xn,zn′)≤(1-αn)d2(xn,p)+αnH2(T(yn),T(p))-αn(1-αn)d2(xn,zn′)≤(1-αn)d2(xn,p)+αnd2(yn,p),d2(yn,p)=d2(βnzn⊕(1-βn)xn,p)≤(1-βn)d2(xn,p)+βnd2(zn,p)-βn(1-βn)d2(xn,zn)=(1-βn)d2(xn,p)+βnd2(PT(xn)(u),PT(p)(u))-βn(1-βn)d2(xn,zn)≤(1-βn)d2(xn,p)+βnH2(T(xn),T(p))-βn(1-βn)d2(xn,zn)≤(1-βn)d2(xn,p)+βnd2(xn,p)-βn(1-βn)d2(xn,zn)≤d2(xn,p)-βn(1-βn)d2(xn,zn)≤d2(xn,p)-βn(1-βn)d2(xn,zn).
Thus, by (4.10) we have
(4.11)d2(xn+1,p)≤(1-αn)d2(xn,p)+αnd2(xn,p)-αnβn(1-βn)d2(xn,zn).
This implies that
(4.12)a2(1-b)d2(xn,zn)≤αnβn(1-βn)d2(xn,zn)≤d2(xn,p)-d2(xn+1,p)
and so
(4.13)∑n=1∞‍a2(1-b)d2(xn,zn)<∞.
Thus, limn→∞d2(xn,zn)=0. Also dist(xn,T(xn))≤d(xn,zn)→0 as n→∞. Since T satisfies condition (I), we have limn→∞d(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 Proposition3.2 and Theorem 4.8, we obtain the following.Corollary 4.9.
LetE 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 {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.
LetE 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 {xn} be the Ishikawa iterates defined by (B). Assume that T is hemicompact and continuous and (i) 0≤αn,βn<1; (ii) βn<1 and (iii) ∑‍αnβn=∞. Then {xn} converges strongly to a fixed point of T.Proof.
As in the proof of Theorem4.8, we obtain
(4.14)lim⁡n→∞dist⁡(xn,T(xn))=0.
Since T is hemicompact, there is a subsequence {xnk} of {xn} such that xnk→q for some q∈E. Since T is continuous, then
(4.15)dist⁡(q,T(q))≤d(q,xnk)+dist⁡(xnk,T(xnk))+H(T(xnk),T(q))⟶0ask⟶∞.
This implies that q∈T(q). By Lemma 4.7, limn→∞d(xn,q) exists and hence q is the limit of {xn} itself.Corollary 4.11.
LetE 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 {xn} be the Ishikawa iterates defined by (B). Assume that T is hemicompact and (i) 0≤αn,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)).
LetE 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
(5.1)T(x,y)=thesegmentjoining(0,1)and(x,0).
Then for each (x1,y1),(x2,y2)∈E, we have
(5.2)H(T(x1,y1),T(x2,y2))=|x1-x2|≤d((x1,y1),(x2,y2)).
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
(5.3)|u(x,y)-u(1,0)|>d((x,y),(1,0)).
This implies that T is not *-nonexpansive.Example 5.2 (see [7] (A *-nonexpansive mapping which is not nonexpansive)).
LetE=[0,∞) and T:E→𝒦𝒞(E) be defined by
(5.4)T(x)=[x,2x]∀x∈E.
Then ux=x for every x∈E. This implies that T is *-nonexpansive. However, we have
(5.5)H(T(x),T(y))=H([x,2x],[y,2y])=2|x-y|.
This shows that T is not nonexpansive.Example 5.3.
LetE=[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 Theorems4.8 and 4.10 can be omitted. We finish the paper with the following questions.Question 1.
LetE 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 {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.
LetE 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 {xn} be the Ishikawa iterates defined by (B). Assume that T is hemicompact and continuous and (i) 0≤αn,βn<1; (ii) βn<1; (iii) ∑‍αnβn=∞. Does {xn} converge to a fixed point of T?



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*Source: 629149-2012-07-10.xml*