# Browder’s Convergence Theorem for Multivalued Mappings in Banach Spaces without the Endpoint Condition

Authors: Thanomsak Laokul
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

We prove Browder’s convergence theorem for multivalued mappings in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm by using the notion of diametrically regular mappings. Our results are significant improvement on results of Jung (2007) and Panyanak and Suantai (2020).


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## Body

## 1. Introduction

LetE,· be a Banach space. We denote by CBE the family of nonempty closed bounded subsets of E and KE the family of nonempty compact subsets of E. For any nonempty subset C of E and x∈E, the distance from x to C is defined by
(1)distx,C≔infx−y:y∈C.Theradius of C relative to x is defined by
(2)Rx,C≔supx−y:y∈C.Thediameter of C is defined by
(3)diamC≔supx−y:x,y∈C.The setC is said to be bounded if diamC<∞. The Hausdorff distance on CBE is defined by
(4)HA,B≔maxsupa∈Adista,B,supb∈Bdistb,AforallA,B∈CBE.A multivalued mappingT:C⟶CBE is said to be contractive if there exists a constant k∈0,1 such that
(5)HTx,Ty≤kx−yforallx,y∈C.If (5) is valid when k=1, then T is said to be nonexpansive. It is clear that every contractive mapping is nonexpansive, but, in general, the converse is not true. A point x∈C is said to be a fixed point of the multivalued mapping T if x∈Tx. We denote by FixT the set of all fixed points of T. Additionally, if x=Tx, then x is called an endpoint (or a stationary point) of T. We denote by EndT the set of all endpoints of T. We can see that for each mapping T, EndT⊆FixT. We may say in other words that the concept of endpoints seems to be more complicated than the concept of fixed points. Nevertheless, both concepts are coincident when T is a single-valued mapping; in this case, EndT=FixT.LetT:C⟶KC be a multivalued nonexpansive mapping. Given u∈C and t∈0,1, let Gt:C⟶KC be defined by
(6)Gtx=tTx+1−tu,x∈C.By Nadler’s theorem [1], Gt is a contraction and admits a fixed point xt∈C, that is,
(7)xt∈tTxt+1−tu.IfT is single-valued, then
(8)xt=tTxt+1−tu.The strong convergence of the netxt for a single-valued nonexpansive mapping T, which is defined by (8), has been investigated by many authors; especially in 1967, Browder [2] established the great influence to the development of approximated fixed point theory. Some relevant definitions and concept will be given in the next section.Theorem 1 (Browder [2]).
LetC be a nonempty bounded closed convex subset of a Hilbert space H and T:C⟶C be a nonexpansive mapping. Fix u∈C and let xt be defined by (8). Then, xt converges strongly as t⟶1 to the element of FixT nearest to u.A natural question which arises is whether Browder’s convergence theorem can be extended to the multivalued case. Pietramala [3] (see also Jung [4]) gave the following example which shows that it is not true in general.Example 1.
Let C=0,1×0,1 be the square in the real plane and T:C⟶KC be defined by
(9)Ta,b≔thetrianglewithvertices0,0,a,0,0,bforalla,b∈C.It is easy to see that for anya1,b1,a2,b2∈C,
(10)HTa1,b1,Ta2,b2=maxa1−a2,b1−b2≤a1,b1−a2,b2.This implies thatT is nonexpansive. It is also easy to see that the fixed point set of T is the set a,0:0≤a≤1∪0,b:0≤b≤1. Let u=1,0; then, the mapping Gt defined by (6) has the fixed point set FixGt=a,0:1−t≤a≤1. Let
(11)xt=1/n,0,ift=1−1/n,1,0,otherwise.Then,xt satisfies (7), but it does not converge. This example also shows that the sequence FixGt does not converge as t⟶1 to FixT under the Hausdorff metric.Pietramala [3] extended Browder’s convergence theorem to the multivalued mapping case with the endpoint condition. Acedo and Xu [5] showed the strong convergence of the net xt defined by (7) in a Hilbert space under the endpoint condition. Later on, it was extended to a Banach space with a sequentially continuous duality mapping by Kim and Jung [6]. Sahu [7] proved the strong convergence of xt in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. Since then, the strong convergence of xt has been improved and many papers have appeared (see for instance [4, 8, 9]). Jung [4] provided the strong convergence of xt for a multivalued nonexpansive mapping in a uniformly convex or a reflexive Banach space having a uniformly Gâteaux differentiable norm. He also noted that the endpoint condition should be added in the main results of Sahu [7].Theorem 2 (Jung [4]).
LetE be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, C a nonempty closed convex subset of E, and T:C⟶KE a multivalued nonexpansive mapping. Suppose that C is a nonexpansive retract of E. Suppose that T satisfies the endpoint condition and that for each u∈C and t∈0,1, the contraction Gt defined by (6) has a fixed point xt∈C. Then, T has a fixed point if and only if xt remains bounded as t⟶1, and in this case, xt converges strongly as t⟶1 to a fixed point of T.Very recently, Panyanak and Suantai [10] brought the concept of diametrically regular mappings which is more general than the concept of the endpoint condition to prove Browder’s theorem for multivalued nonexpansive mappings in Hadamard spaces.Theorem 3 (Panyanak and Suantai [10]).
LetC be a nonempty closed convex subset of a Hadamard space X,ρ and T:C⟶KC be a multivalued nonexpansive mapping. Fix u∈C and let xt be defined by (7). Suppose that xt is diametrically regular for T. Then, T has an endpoint if and only if xt is bounded as t⟶1. In this case, xt converges strongly to the unique point x~ in EndT such that
(12)ρu,x~=minρu,e:e∈EndT.In this paper, we will show that the result of Jung [4] in Theorem 2 can be approached without the endpoint condition by using the notion of diametrically regular mappings. Our results extend and complement many known results including those of [2, 4, 7, 10–12].


## 2. Preliminaries and Lemmas

Throughout this paper,ℕ stands for the set of natural numbers and ℝ stands for the set of real numbers.LetE be a Banach space and let C be a nonempty subset of E. A multivalued mapping T:C⟶CBE is said to be diametrically regular if there exists the net xα in C such that limαdiamTxα=0. In this case, we call xα a diametrically regular net for T. The mapping T is said to satisfy the endpoint condition if Ty=y for any y∈FixT. Notice that if C is a nonempty closed convex subset of a uniformly convex Banach space E and T is a nonexpansive mapping, then EndT is closed and convex (see [13]).A Banach spaceE,· is said to be uniformly convex if for each ε∈0,2 there exists δ>0 such that for any x,y∈E the conditions x≤1, y≤1, and x−y≥ε imply
(13)x+y2≤1−δ.It is well known that ifE is uniformly convex, then E is reflexive and strictly convex (cf. [14]). The norm of E is said to be Gâteaux differentiable (and E is said to be smooth) if
(14)limt→0x+ty−xtexistsfor each x,y in its unit sphere SE=x∈E:x=1. It is said to be uniformly Gâteaux differentiable if for each y∈SE, this limit is attained uniformly for x∈SE. We denote by J the (normalized) duality mapping from E into 2E∗, i.e.,
(15)Jx=x∗∈E∗:x,x∗=x2=x∗2,x∈E,where ·,· denotes the generalized duality pairing.A subsetC of E is said to be a retract if there exists a continuous mapping Q:E⟶C with C=FixQ. Any such mapping Q is a retraction of E onto C. If Q is nonexpansive, then C is said to be a nonexpansive retract of E (cf. [14, 15]).Letμ be a linear continuous functional on ℓ∞ and a=a1,a2,⋯∈ℓ∞. We will sometimes write μnan in place of the value μa. A linear continuous functional μ such that μ=1=μ1 and μnan=μnan+1 for every a=a1,a2,⋯∈ℓ∞ is called a Banach limit. We know that if μ is a Banach limit, then
(16)liminfn→∞an≤μnan≤limsupn→∞anfor every a=a1,a2,⋯∈ℓ∞. Let xn be a bounded sequence in E. Then, the real-valued continuous function ϕ on E defined by
(17)ϕz=μnxn−z2,z∈E,is convex.The following lemma was given in [16].Lemma 4 (see [16]).
LetC be a nonempty closed convex subset of a Banach space E with a uniformly Ga^teaux differentiable norm and let xn be a bounded sequence in E. Let μ be a Banach limit and u∈C. Then,
(18)μnxn−u2=miny∈Cμnxn−y2if and only if
(19)μnx−u,Jxn−u≤0for all x∈C.The following result was essentially given by Reich [17] and also proved by Takahashi and Jeong [18].Lemma 5 (see [18]).
LetE be a uniformly convex Banach space, C a nonempty closed convex subset of E, and xn a bounded sequence in E. Then, the set
(20)M=u∈C:μnxn−u2=minz∈Cμnxn−z2consists of one point.From now on, we will use the notationxn⇀x (resp., xn⟶x) for a sequence xn converging weakly (resp., converging strongly) to a point x. The following two lemmas are also needed.Lemma 6 (see [19]).
LetC be a nonempty subset of a metric space E, xn be a sequence in C, and T:C⟶KE be a multivalued mapping. Then, Rxn,Txn⟶0 if and only if distxn,Txn⟶0 and diamTxn⟶0.Lemma 7 (see [20]).
LetC be a nonempty closed convex subset of a uniformly convex Banach space E and T:C⟶KE be a multivalued nonexpansive mapping. Then, the following implication holds:
(21)xn⊆C,xn⇀x,Rxn,Txn⟶0⟹x∈EndT.


## 3. Main Results

We begin this section by proving that the diametric regularity of a nonexpansive mappingT with a nonempty fixed point set is weaker than the endpoint condition. Our proof follows the ideas of proof in Panyanak and Suantai [10].Proposition 8.
LetE be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, C a nonempty closed convex subset of E, and T:C⟶KE a multivalued nonexpansive mapping with FixT≠∅. Suppose that C is a nonexpansive retract of E and that for each u∈C and t∈0,1, the contraction Gt defined by Gtx=tTx+1−tu has a fixed point xt∈C. Then, the following statement holds:
§ if T satisfies the endpoint condition, then xt is a diametrically regular net for T.Proof.
By Theorem2, limt→1xt=x~, where x~∈FixT. Since T satisfies the endpoint condition, x~∈EndT. For v,w∈Txt we have
(22)v−w≤v−x~+x~−w≤distv,Tx~+distw,Tx~≤2HTxt,Tx~≤2xt−x~.This implies thatlimt→1diamTxt=0, and hence, xt is diametrically regular for T.The following example shows that the endpoint condition in Proposition8 is necessary.Example 2.
Let C≔0,1,E≔ℝ, and T:C⟶KE be defined by
(23)Tx=0,1+xforallx∈C.Then,T is a multivalued nonexpansive mapping with FixT=0,1 and EndT=∅. Since diamTx=1+x for all x∈C, there is no net xt in C which is diametrically regular for T.The main theorem is proved as follows.Theorem 9.
LetE be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, C a nonempty closed convex subset of E, and T:C⟶KE a multivalued nonexpansive mapping. Suppose that C is a nonexpansive retract of E and that for each u∈C and t∈0,1, the contraction Gt defined by Gtx=tTx+1−tu has a fixed point xt∈C. Suppose that xt is diametrically regular for T. Then, T has an endpoint if and only if xt is bounded as t⟶1, and in this case, xt converges strongly as t⟶1 to an endpoint of T.Proof.
Lete∈EndT. By the compactness of T, for each t∈0,1 there exists yt∈Txt such that xt=tyt+1−tu. Since T is nonexpansive,
(24)yt−e=distyt,Te≤HTxt,Te≤xt−e.
This implies that(25)xt−e≤tyt−e+1−tu−e≤txt−e+1−tu−e,and hence, xt−e≤u−e. Therefore xt is bounded. Conversely, suppose that xt remains bounded as t⟶1. Let tn be a sequence in 0,1 converging to 1 and put xn≔xtn. For each n∈ℕ, let yn∈Txn be such that xn=tnyn+1−tnu. Then,
(26)distxn,Txn≤xn−yn=1−tnu−yn⟶0,asn⟶∞.
Sincexn is bounded, we define ϕ:E⟶0,∞ by
(27)ϕz=μnxn−z2.
Sinceϕ is continuous and convex, ϕz⟶∞ as z⟶∞. As E is reflexive, ϕ attains its infimum over C (cf. [21], p.79). Let x~∈C be such that
(28)μnxn−x~2=miny∈Cμnxn−y2,and let
(29)M=x∈C:μnxn−x2=miny∈Cμnxn−y2.
Then,M is nonempty because x~∈M. We also obtain that M is a bounded closed convex subset of C. Since C is a nonexpansive retract of E, we assume that Q is a nonexpansive retract of E onto C. Then, for any y∈E, we have
(30)μnxn−y2≥μnQxn−Qy2=μnxn−Qy2≥μnxn−x~2,and hence, μnxn−x~2=miny∈Eμnxn−y2. This implies that x~ is a global minimum point over all of E. Furthermore, x~ is a unique element in M by Lemma 5. Since Tx~ is compact, for each n≥1, there exists wn∈Tx~ for x~∈M such that
(31)yn−wn=distyn,Tx~≤HTxn,Tx~≤xn−x~.
Letwnk be a convergent subsequence of wn. Assume that wnk⟶w∈Tx~. For k sufficiently large, we obtain from (26) and (31) that
(32)xnk−w≤xnk−ynk+ynk−wnk+wnk−w=1−tnku−ynk+ynk−wnk+wnk−w.
Thus,μnkxnk−w2≤μnkynk−wnk2≤μnkxnk−x~2. But x~ is the unique global minimum element, w∈M. As we know from Lemma 5 that M consists of one point, it follows that x~=w∈Tx~. We now show that x~=Tx~. Take v∈Tx~ and choose unk∈Txnk so that v−unk=distv,Txnk. Thus,
(33)xnk−v≤xnk−ynk+ynk−unk+unk−v≤1−tnku−ynk+diamTxnk+HTxnk,Tx~≤1−tnku−ynk+diamTxnk+xnk−x~.
Sincexnk is diametrically regular for T and (26), we obtain that
(34)μnkxnk−v2≤μnkxnk−x~2.
Sincex~ is the global minimum point over all of E, it implies that v=x~, and so, x~=Tx~. On the other hand, for x~=Tx~, we obtain from (24) that
(35)xn−yn,Jxn−x~=xn−x~+x~−yn,Jxn−x~≥xn−x~2−yn−x~xn−x~≥xn−x~2−xn−x~2=0.
It follows that(36)0≤xn−yn,Jxn−x~=1−tnu−yn,Jxn−x~.
Hence, from (26) and (36), we obtain
(37)xn−u,Jxn−x~≤xn−yn,Jxn−x~.
Thus, forTx~=x~=M,
(38)μnxn−u,Jxn−x~≤μnxn−yn,Jxn−x~≤0.
But by Lemma4, we have μnx−x~,Jxn−x~≤0 for all x∈C. In particular, we have
(39)μnu−x~,Jxn−x~≤0.
Combining (38) and (39), we get
(40)μnxn−x~,Jxn−x~=μnxn−x~2≤0.
Therefore, there is a subsequencexnj of xn which converges strongly to x~. To complete the proof, suppose that there is another subsequence xnk of xn which converges strongly to (say) y. Then,
(41)distxnk,Txnk≤xnk−ynk=1−tnku−ynk⟶0,ask⟶∞.
Sincexnk is diametrically regular for T, we have that diamTxnk⟶0 as k⟶∞. Thus, from (41) and Lemma 6, we get
(42)Rxnk,Txnk⟶0,ask⟶∞.
Sincexnk⊆C, xnk⟶y and (42), we obtain from Lemma 7 that y∈EndT. It follows from (38) that
(43)x~−u,Jx~−y≤0,y−u,Jy−x~≤0.
Adding these two inequalities yields(44)x~−y,Jx~−y=x~−y2≤0.
Thus,y=x~. This proves the strong convergence of xt.Remark 10.
Theorem9 is an extension of Theorem 1 in [4] and Theorem 3.5 in [10].It is well known that any closed convex set in Hilbert spaces is a nonexpansive retract with the proximity map as a nonexpansive retraction. Thus, we obtain the following result.Corollary 11.
LetH be a real Hilbert space, C a nonempty closed convex subset of H, and T:C⟶KH a multivalued nonexpansive mapping. Suppose that for each u∈C and t∈0,1, the contraction Gt defined by Gtx=tTx+1−tu has a fixed point xt∈C. Suppose that xt is diametrically regular for T. Then, T has an endpoint if and only if xt is bounded as t⟶1, and in this case, xt converges strongly as t⟶1 to an endpoint of T.Remark 12.
Corollary11 is an extension of Corollary 1 in [4], and also, it is a non-self-mapping case of Theorem 3.6 in [10].The following example given by Pietramala [3] (see also [10]) shows that the diametric regularity of xt in Theorem 9 and Corollary 11 is necessary.Example 3.
LetX be the Euclidean space ℝ2 and C=0,1×0,1. Let T:C⟶KC be defined by
(45)Ta,b≔theclosedconvexhullof0,0,a,0,0,b.Then,T is a multivalued nonexpansive mapping with FixT=a,b∈C:ab=0 and EndT=0,0. Fix u∈C. For each t∈0,1, let xt=1−tu. It is easy to see that xt satisfies (7), and it is diametrically regular for T. By Corollary 11, limt→1xt=0,0. However, if u=1,0 and zt≡u, then zt satisfies (7) but it is not diametrically regular for T. Moreover, limt→1zt=u≠0,0.Recall that theinward set of C⊆E at x is defined by
(46)ICx=z∈E:z=x+λy−x:y∈C,λ≥0.It is known that ifC is convex, then the closure of ICx is ICx¯=x+TCx for any x∈C, where
(47)TCx=y∈E:liminfλ→0+dx+λy,Cλ=0.A mappingT:C⟶CBE is said to satisfy (i) weak inwardness condition if Tx⊂ICx¯ for all x∈C and (ii) inwardness condition if Tx⊂ICx for all x∈C. For details, we refer to [22].The weak inwardness condition guarantees the existence of a fixed point of the contractionGt defined by (6); the weak inwardness condition for multivalued mappings is used. It has recently been shown by Xu [23] that if C is a closed bounded convex subset of a uniformly convex Banach space E and T:C⟶KE is a multivalued nonexpansive mapping satisfying the weak inwardness condition, then T has a fixed point. Thus, we have the following results.Corollary 13.
LetE be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, C a nonempty bounded closed convex subset of E, and T:C⟶KE a multivalued nonexpansive mapping. Suppose in addition that T satisfies the weak in wardness condition. Fix u∈C and let xt be defined by (7). If xt is diametrically regular for T, then xt converges strongly as t⟶1 to the unique point x~in EndT.Proof.
Fixu∈C. For each t∈0,1, define the contraction Gt:C⟶KE by
(48)Gtx=tTx+1−tu,x∈C.As it is easily seen thatGt also satisfies the week inwardness condition: Gtx⊂I¯Cx for all x∈C, we have that Gt has a fixed point denoted by xt. Also, we have that the fixed point set of T is nonempty. Thus, the result follows from Theorem 9.Corollary 14.
LetH be a real Hilbert space, C a nonempty bounded closed convex subset of H, and T:C⟶KH a multivalued nonexpansive mapping which satisfies the weak inwardness condition. Fix u∈C and let xt be defined by (7). If xt is diametrically regular for T, then xt converges strongly as t⟶1 to an endpoint of T.Remark 15.
Corollaries13 and 14 are extensions of Corollaries 2 and 3 in [4], respectively.Remark 16.
In Theorem 2 of Jung [4], Browder’s convergence theorem for multivalued nonexpansive mappings in a reflexive Banach space with a uniformly Gâteaux differentiable norm can be also proved. In this paper, we relaxed the endpoint condition and we then applied Lemmas 6 and 7, which is a crucial tool, to prove Browder’s convergence theorem for multivalued nonexpansive mappings in uniformly convex Banach spaces with a uniformly Gâteaux differentiable norm. Therefore, the following question should be of interest.Without the endpoint condition, can the result of Jung [4] (Theorem 2) be approached by using the notion of diametrically regular mappings?



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