# Best Proximity Point Theorems for Cyclic Relativelyρ-Nonexpansive Mappings in Modular Spaces

Authors: Karim Chaira; Samih Lazaiz
Journal: Abstract and Applied Analysis
 (2018)

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## Abstract

In this paper we introduce the notion of proximalρ-normal structure of pair of ρ-admissible sets in modular spaces. We prove some results of best proximity points in this setting without recourse to Zorn’s lemma. We provide some examples to support our conclusions.


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## Body

## 1. Introduction

Fixed point theory is powerful tools in different fields such as differential equations, dynamical systems, optimal control, and many other scientific branches; it treats equations of typeTx=x where T:X→X is a map of a nonempty set to itself.LetA,B⊂X and T a cyclic mapping on A∪B; that is, T:A∪B→A∪B and TA⊆B, TB⊆A; in this case, T does not necessarily possess a fixed point if, for instance, A∩B=∅. One often attempts to find a point x which is closest to Tx in the sense that the “distance” between x and Tx is equal to the distance between A and B; such a point x is said to be a best proximity point.The first result of this kind is due to Fan [1] which is stated in locally convex Hausdorff topological vector space. Afterward, many extensions and generalizations were given; see, for instance, [2–6].On the other hand, Eldred et al. in [7], after generalizing the geometric concept of normal structure for a pair of subsets A,B in Banach space introduced earlier by Brodski and Milman (see [8]), proved the existence of best proximity points for relatively nonexpansive mappings in Banach space. Recall that a map T:A∪B→A∪B is called relatively nonexpansive if Tx-Ty≤x-y for all x∈A and y∈B. This class of mapping is much larger than nonexpansive, because, it does not guarantee the continuity of T.After that, Sankar and Veeramani in [9] without using Zorn’s lemma, proved the existence of a best proximity point by using convergence theorem. Also, Espinola in [10] showed that under a suitable condition on the pair A,B the relatively nonexpansive assumption can be seen as nonexpansive one, which in fact guarantees the continuity of the map.Recently, the best proximity points results was investigated by many authors and found extension and generalization for different class of mappings and spaces; for a recent account of the theory we refer the reader to [11–18]. In this paper, we extend the notion of proximal ρ-normal structure for a pair of ρ-admissible subsets A,B which is a generalization of Khamsi and Kozlowski definition. Also, we give existence results of a best proximity point in the setting of proximal ρ-admissible subsets in modular space. Our proofs do not invoke Zorn’s lemma.


## 2. Preliminaries

We start by recalling some basic facts of modular space. For more details the reader can consult [19].LetX be an arbitrary vector space.Definition 1.
A functionρ:X→0,∞ is called a modular on X if for arbitrary x,y∈X, (1)
ρx=0 if and only if x=0(2)
ραx=ρx for every scalar α with α=1(3)
ραx+βy≤ρx+ρy if α+β=1 and α,β≥0If the following property is satisfied,(4)
ραx+βy≤αρx+βρy if α+β=1 and α,β≥0we say thatρ is a convex modular. A modular ρ defines a corresponding modular space, i.e., the vector space Xρ given by(1)Xρ=x∈X:ρλx→0asλ→0.
In general the modularρ is not subadditive and therefore does not behave as a norm or a distance.Definition 2.
LetXρ be a modular space. (1)
We say thatxn is ρ-convergent to x and write xn→xρ if and only if ρxn-x→0.(2)
A sequencexn, where xn∈Xρ, is called ρ-Cauchy if ρxn-xm→0 as n,m→∞.(3)
We say thatXρ is ρ-complete if and only if any ρ-Cauchy sequence in Xρ is ρ-convergent.(4)
A setC⊂Xρ is called ρ-closed if for any sequence xn of C; the convergence xn→x(ρ) implies that x belongs to C.(5)
A setC⊂Xρ is called ρ-bounded if supρx-y:x,y∈C<∞.(6)
A setC⊂Xρ is ρ-sequentially compact, if for any sequence xn of C, there exists a convergent subsequence xnkk of xn such that xnk→x(ρ) in C.(7)
We will sayρ satisfies Fatou property if(2)ρx≤liminfn→∞⁡ρxnwheneverxn→xρ.
We shall say that a pairA,B of sets in a modular space satisfies a property if each of the sets A and B has that property. Thus A,B is said to be ρ-closed if both A and B are ρ-closed, H,K⊆A,B⇔H⊆AandK⊆B, (A,B) is not reduced to one point which means that A and B are not singletons, etc. We shall also introduce the following notation:(3)δρA,B=sup⁡ρx-y:x∈A,y∈B;δρx,A=sup⁡ρx-y:y∈A;distρA,B=inf⁡ρx-y:x∈A,y∈B;γρA,B=max⁡inf⁡δρx,B:x∈A,inf⁡δρy,A:y∈B.
The following definitions are extensions of Definition 5.7 in [19] and are more adapted for a pair of subsets A,B.Definition 3.
LetA,B be a ρ-bounded pair.
We will say thatH,K is proximal ρ-admissible pair of A,B if(4)H=⋂i∈IBρyi,ri∩AandK=⋂i∈IBρxi,ri′∩B
wherexi,yi∈A×B, ri,ri′≥dρA,B, I is an arbitrary index set, and Bρx,r=y∈Xρ:ρx-y≤r the standard ρ-closed ball of Xρ. The family of all proximal ρ-admissible pair of A,B will be denoted by QA,B.IfD1,D2⊆A,B we write(5)coAD2D1=⋂y∈D2Bρy,δρy,D1∩AcoBD1D2=⋂x∈D1Bρx,δρx,D2∩B.Remark 4.
Note thatcoAD2D1,coBD1D2∈QA,B and is the smallest ρ-admissible pair of A,B which contains D1,D2. Indeed, let (H,K)∈QA,B such that D1,D2⊆H,K; then H=⋂y∈D2Bρy,ry∩A and for each x∈D1 and y∈D2 we have ρ(x-y)≤ry. Hence, δρ(y,D1)≤ry since D1⊆H, which implies that (6)coAD2D1=⋂y∈D2Bρy,δρy,D1∩A⊆⋂y∈D2Bρy,ry∩A.In the same manner, we obtaincoBD1D2⊆K.Definition 5.
LetA,B be a ρ-bounded pair. (1)
QA,B is said to satisfy the property R-proximal, if for any sequence(7)Ann≥1,Bmm≥1⊆QA,B,which are nonempty and decreasing, has a nonempty intersection.(2)
QA,B is said to be proximal ρ-normal, if for each proximal ρ-admissible pair H,K not reduced to one point of A,B for which distρH,K=distρA,B and δρH,K>distρH,K there exists x,y∈H×K such that(8)δρx,K<δρH,Kandδρy,H<δρH,K.(3)
We say that the pairA,B is proximal ρ-sequentially compactness provided that every sequence xnn,ynn of A,B satisfying the condition that ρxn-yn→distρA,B has a convergent subsequence in A,B.Remark 6.
Notice that theQA,A is proximal ρ-normal (resp., has the R-proximal property) if and only if QA is ρ-normal (resp., has the R-property) in the sense of Khamsi-Kozlowski (see [19, Definition 5.7]).Definition 7.
A mapT:A∪B→A∪B will be said cyclic relatively ρ-nonexpansive on A∪B if (1)
TA⊆B and TB⊆A(2)
ρTx-Ty≤ρx-y for x∈A, y∈B
We conclude this section by a modular version of Kirk’s fixed point theorem [20] which follows as a corollary from our former result Theorem 10 (see Corollary 11 below).Theorem 8 (see [19, Theorem 5.9]).
LetA be a ρ-bounded and ρ-closed nonempty subset of Xρ which satisfies R-property. Assume that QA is ρ-normal. If T:A→A is ρ-nonexpansive, then T has a fixed point.


## 3. Best Proximity Results withρ-Normal Structure

In what follows, we investigate the validity of technical lemma due to Gillespie and Williams [21] for a pair of ρ-admissible subsets in modular space. This result can be considered as the main ingredient and will play an important role in this article.Lemma 9.
LetA,B be a ρ-bounded pair of Xρ. Let T:A∪B→A∪B be a cyclic relatively ρ-nonexpansive mapping. Assume that QA,B is proximal ρ-normal. Let H,K∈QA,B be a nonempty and T-cyclic pair; i.e., TH⊆K and TK⊆H with distρH,K=distρA,B not reduced to one point. Then, there exists a nonempty T-cyclic pair H0,K0∈QA,B such that H0,K0⊆H,K and(9)δρH0,K0≤δρH,K+γρH,K2.Proof.
Setr=1/2δρH,K+γρH,K. If δρH,K=distρH,K one can choose H0,K0=H,K. We assume that δρH,K>distρH,K. Since QA,B is proximal ρ-normal, we have(10)γρH,K<δρH,Kand hence γρH,K<r. Thus, there exists x1,y1∈H×K such that(11)δx1,K<randδy1,H<r.
Let(12)DH=⋂y∈KBρy,r∩HandDK=⋂x∈HBρx,r∩K;
thenDH,DK≠∅ since x1,y1∈DH×DK.
LetF denote the set of all nonempty pairs Eα,Fαα∈Λ of QA,B which are subsets of A,B such that T is cyclic on Eα∪Fα and DH,DK⊆Eα,Fα with distρEα,Fα=distρA,B for all α∈Λ. Obviously, F is nonempty since A,B∈F. Defining L1,L2 by(13)L1=⋂αEαandL2=⋂αFα it is clear that L1,L2≠∅ since DH,DK⊂L1,L2 and T is cyclic on L1∪L2, Eα,Fα is proximal ρ-admissible for each α so it is L1,L2, and it is easy to check that distρL1,L2=distρA,B; thus L1,L2∈F.
LetM1=DH∪TL2 and M2=DK∪TL1; it is claimed that(14)coAM2M1=L1andcoBM1M2=L2.
Indeed,M1⊂L1, M2⊂L2 and the pair (L1,L2) is proximal ρ-admissible; then(15)coAM2M1,coBM1M2⊆L1,L2
sincecoAM2M1,coBM1M2 is the smallest ρ-admissible pair which contains M1,M2. Also,(16)TcoAM2M1⊆TL1andTcoBM1M2⊆TL2 which implies(17)TcoAM2M1⊆M2andTcoBM1M2⊆M1.
Note thatdistρ(T(L1),T(L2))=distρ(L1,L2) since T is relatively ρ-nonexpansive mapping. And, since M1,M2⊆coAM2M1,coBM1M2 we get (18)distρcoAM2M1,coBM1M2=distρA,B,and hence coAM2M1,coBM1M2∈F; that is, (19)coAM2M1,coBM1M2=L1,L2.
Define(20)H0=⋂y∈L2Bρy,r∩L1andK0=⋂x∈L1Bρx,r∩L2.
We claim thatH0,K0 is the desired pair. Since DH,DK⊆H0,K0 the pair H0,K0 is nonempty; also H0,K0∈QA,B.
Note that for eachx∈H0 and y∈K0 we have(21)ρx-y≤r⇒δρH0,K0≤r.
Next, we show thatT is cyclic on H0∪K0 to complete the proof. Let x∈H0; then,(22)ρTx-Ty≤ρx-y≤r∀y∈L2 since T is relatively ρ-nonexpansive. Thus, TL2⊂BρTx,r.
Recall thatDH=⋂y∈KBρy,r∩H; then if z∈DH we have for all w∈K(23)ρz-w≤r and since H,K∈F, we get L2⊂K; then L2⊂Bz,r. It is clear that Tx∈L2; that is,(24)Tx∈Bρz,r⇒z∈BρTx,r and hence DH⊂BρTx,r, which implies(25)L1=coAM2DH∪TL2⊆BρTx,r∩A; this deduces that Tx∈K0; that is, TH0⊆K0. Similarly, we can see that TK0⊆H0. Since L1,L2⊆H,K we get H0,K0⊆H,K.Theorem 10.
LetA,B be a nonempty ρ-bounded and ρ-closed pair in a modular space Xρ. Moreover, assume that QA,B has the proximal ρ-normal structure and the property R-proximal.
IfT is cyclic relatively ρ-nonexpansive on A∪B, then there exists x,y∈A×B such that(26)ρx-Tx=ρy-Ty=distρA,B.Proof.
LetF denote the set of all nonempty pairs E,F of QA,B which are subsets of A,B such that T is cyclic on E∪F and distρE,F=dρ, where dρ=distρA,B.
F is nonempty since A,B∈F. Define δ~ρ:F→0,∞ by(27)δ~ρDA,DB=inf⁡δρE,F:E,F∈FandE,F⊆DA,DB.
SetD1A,D1B=A,B; by definition of δ~ρ, there exists D2A,D2B∈F such that D2A,D2B⊆D1A,D1B, distρD2A,D2B=distρD1A,D1B=dρ, and(28)δρD2A,D2B<δ~ρD1A,D1B+1and suppose that DkA,DkBk=1,2,…,n are constructed for n≥1. Again, by definition of δ~ρ, there exists Dn+1A,Dn+1B⊆DnA,DnB such that(29)δρDn+1A,Dn+1B<δ~ρDnA,DnB+1n and distρDn+1A,Dn+1B=dρ. Since QA,B has the property R-proximal, D∞A,D∞B≠∅ where(30)D∞A=⋂n≥1DnAandD∞B=⋂n≥1DnB,
note thatTD∞A=T⋂nDnA⊆⋂nTDnA⊆⋂nDnB=D∞B; in the same manner TD∞B⊆D∞A. Also, D∞A,D∞B∈QA,B since DnA,DnB∈QA,B for all n≥1. Also, we have (31)dρA,B≤dρD∞A,D∞B=inf⁡ρx-y:x,y∈D∞A×D∞B=inf⁡ρx-y:x,y∈⋂n≥1DnA×⋂n≥1DnB=inf⁡ρx-y:x,y∈DnA×DmB∀n,m∈N∗=infn,m∈N∗⁡dρDnA,DmB≤dρDnA,DnB=dρA,Band then D∞A,D∞B∈F.
IfδρD∞A,D∞B=distρD∞A,D∞B, then for each x,y∈D∞A×D∞B we get(32)ρx-Tx=ρy-Ty=distρD∞A,D∞B.
Assume thatδρD∞A,D∞B>distρD∞A,D∞B.
First Case. If one of the pair D∞A,D∞B is reduced to one point, say, for example, D∞B=y, since T is cyclic relatively ρ-nonexpansive on D∞A∪D∞B we get for all x∈D∞A(33)ρTx-Ty=ρy-Ty≤ρx-ywhich implies that (34)ρy-Ty=ρx-Tx=distρD∞A,D∞Bfor x=Ty; note that Tx=y for each x∈D∞A.
Second Case. The pair D∞A,D∞B is not reduced to one point; by Lemma 9, there exists DA∗,DB∗⊆D∞A,D∞B(35)δρDA∗,DB∗≤δρD∞A,D∞B+γρD∞A,D∞B2which implies(36)δρDA∗,DB∗≤δρD∞A,D∞B≤δρDnA,DnB≤δ~ρDnA,DnB+1n≤δρDA∗,DB∗+1nsinceDA∗,DB∗⊆DnA,DnB for any n≥1. If we let n→∞, we get δρDA∗,DB∗=δρD∞A,D∞B. By (35) we get(37)δρD∞A,D∞B≤γρD∞A,D∞Band this is in contradiction with the assumption that QA,B is proximal ρ-normal. This completes the proof.If we setA=B, we get Theorem 8.Corollary 11.
LetA be a ρ-bounded and ρ-closed nonempty subset of Xρ. Assume that QA,A is ρ-normal and satisfies the property R-proximal. If T:A→A is ρ-nonexpansive, then T has a fixed point.We conclude by the following example.Example 12.
Let the real spaceX={x=xnn≥1∈RN∗:∑n≥1xn1/2<∞}, and define the modular functional ρ:X→[0,∞] by (38)ρx=∑n=1∞xn1/2,for  allx=xnn≥1∈X Suppose that {en} is the canonical basis of X and let(39)A=12e1∪e3+en:n∈N∖0,1,3andB=12e3∪e1+en:n∈N∖0,1.
Then,(A,B) is ρ-bounded, ρ-closed in Xρ, and not convex. Note that A (resp., B) is not ρ-sequentially compact, because the sequence e3+enn≠{0,1,3} (resp., e1+enn≠{0,1}) does not have any ρ-convergent subsequence in A (resp., in B).
We haveρ1/2e1-1/2e3=2; also, for all x∈A and n∈N∗, ρ(x-e1-en)≥2, which implies that distρ(A,B)=2.
Q A , B satisfies the property R-proximal; indeed, let Hnn≥1,Kmm≥1 be a sequence of QA,B which are nonempty and decreasing.(1)
If, for eachn∈N∗, Kn=⋂j∈JnBρ(e3+ej,n,rj,n)∩B, so, for all j∈Jn, 1/2e3∈Bρe3+ej,n,1+1/2∩B⊂⋂n≥1Kn, because ρ(e3+ej,n-1/2e3)=1+1/2, and since Kn≠∅ for each n∈N∗, we have rj,n≥1+1/2. Hence, ⋂n≥1Kn≠∅.(2)
If, for eachn∈N∗, Kn=⋂k∈Jn′Bρ(1/2e1,rk,n)∩B, so 1/2e3∈Bρ(1/2e1,2)∩B⊂⋂n≥1Kn, since Kn≠∅ for all n∈N∗. Hence ⋂n≥1Kn≠∅.(3)
If there existsn∈N∗ such that (40)Kn=⋂j∈JnBρe3+ej,n,rj,n∩⋂k∈Jn′Bρ12e1,rk,n∩B,we have1/2e3∈Bρ(e3+ej,n,1+1/2)∩Bρ(1/2e1,2)∩B⊂⋂n≥1Kn, and hence ⋂n≥1Kn≠∅.
Analogously, we obtain that⋂n≥1Hn≠∅ replacing e3+ej,n and 1/2e1 by e1+ej,n and 1/2e3, respectively.
Q A , B has the proximal ρ-normal structure. Indeed, let H,K be a proximal ρ-admissible pair of A,B not reduced to one point for which distρH,K=distρA,B=2 and δρH,K>distρH,K; then 1/2e1∈H and 1/2e3∈K. So, there exist p,q∈N∖{0,1,3}×N∖{0,1} such that (41)e3+ep∈Hande1+eq∈K.Therefore,δρ1/2e1,K=1+1/2 and δρ1/2e3,H=1+1/2; then we get (42)δρH,K≥ρe3+ep-e1-eq>max⁡δρ12e1,K,δρ12e3,H.LetT:A∪B→A∪B be a mapping defined by (43)Ty=12e1ify∈BandTx=12e3ifx=12e1e1+e2ifx∈A∖12e1.T is cyclic and for each y∈B and x=1/2e1(44)ρTx-Ty=ρ12e1-12e3=2≤ρx-y, and for each x∈A∖1/2e1,y∈B(45)ρTx-Ty=ρ12e1-e1-e2=1+12≤ρx-y. Then, T is cyclic relatively ρ-nonexpansive on A∪B. Therefore, all assumptions of Theorem 10 are satisfied, so T has a best proximity point; in particular (46)ρ12e3-T12e3=ρ12e1-T12e1=distρA,B.


## 4. Best Proximity for Proximalρ-Sequentially Compact Pair

In this section, we useA0,B0 to denote the proximal pair obtained from A,B upon setting(47)A0=x∈A:ρx-y′=distρA,Bforsomey′∈BB0=y∈B:ρx′-y=distρA,Bforsomex′∈A.Lemma 13.
LetA,B be a nonempty ρ-bounded and proximal ρ-sequentially compactness pair in a modular space Xρ for which ρ satisfies Fatou property. Then A0,B0 is a nonempty ρ-sequentially compact pair of A,B such that distρA0,B0=distρA,B.Proof.
It is clear thatdistρA0,B0=distρA,B. Let xn and yn be two sequences in A and B, respectively, such that(48)ρxn-yn→distρA,B.
SinceA,B is a proximal ρ-compactness pair, there exist subsequences xnk and ynk of xn and yn, respectively, such that xnk→x∈A and ynk→y∈B as k→∞. Since ρ has Fatou property(49)ρx-y≤liminfk→∞⁡ρxnk-ynk=distρA,B.
This implies thatA0 is nonempty, since x∈A0. Similarly, we can see that B0 is nonempty. The ρ-sequentially compact of A0 is vacuous since for each sequence xn of A0 has a convergent subsequence for which this limit is in A0 because A0 is ρ-closed in A. Indeed, let xn⊂A0 such that xn→a; then there exists a sequence yn in B0 such that(50)ρxn-yn→distρA,B and the proximal ρ-compactness of A,B implies the existence of subsequences xnk and ynk of xn and yn, respectively, such that xnk→x∈A and ynk→y∈B. Since ρ has Fatou property,(51)ρx-y≤liminfk→∞⁡ρxnk-ynk=distρA,B. Then x∈A0; the uniqueness of the limit implies that x=a. Hence A0,B0 is ρ-sequentially compact pair.Theorem 14.
LetA,B be a nonempty ρ-bounded and proximal ρ-sequentially compactness pair in a modular space Xρ for which ρ has the Fatou property. Moreover, assume that QA,B has the proximal ρ-normal structure.
IfT is cyclic relatively ρ-nonexpansive on A∪B, then there exists x,y∈A×B such that(52)ρx-Tx=ρy-Ty=distA,B.Proof.
Letx0∈A0; then there exists y0∈B such that(53)ρTx0-Ty0≤ρx0-y0=dρ That is, Tx0∈B0. Hence TA0⊆B0, similarly, TB0⊆A0 and T is cyclic relatively ρ-nonexpansive on A0∪B0.
LetF denote the set of all nonempty ρ-closed pairs E,F of QA,B which are subsets of A,B such that T is cyclic on E∪F and ρx-y=dρ for some x,y∈E×F, where dρ=distρA,B. Thus, F is nonempty since A,B∈F.
Defineδ~ρ:F→0,∞ by(54)δ~ρDA,DB=inf⁡δρE,F:E,F∈FandE,F⊆DA,DB.
SetD1A,D1B=A,B; by definition of δ~ρ, there exists D2A,D2B∈F such that D2A,D2B⊆D1A,D1B, distρD2A,D2B=dρ, and(55)δρD2A,D2B<δ~ρD1A,D1B+1Suppose that DkA,DkBk=1,2,…,n are constructed for n≥1. Again, by definition of δ~ρ, there exists Dn+1A,Dn+1B⊆DnA,DnB such that(56)δρDn+1A,Dn+1B<δ~ρDnA,DnB+1n and distρDn+1A,Dn+1B=dρ. Since A0,B0 is ρ-sequentially compact, D∞A,D∞B≠∅, where(57)D∞A=⋂n≥1DnAandD∞B=⋂n≥1DnB.
Indeed, one can choose two sequencesxn and yn such that xn,yn∈DnA∩A0×DnB∩B0 for each n≥1 and(58)ρxn-yn→dρ Using the proximal ρ-compactness of A0,B0, there exists xnk of xn and ynk of yn such that xnk→xρ and ynk→yρ; let p≥1 and define two subsets of A0 and B0 as follows:(59)CpA=xnk:k≥pandCpB=ynk:k≥p Hence x∈⋂pCpA and y∈⋂pCpB. Thus, x∈⋂n≥1DnA=⋂k≥1DnkA=D∞A and y∈⋂n≥1DnB=⋂k≥1DnkB=D∞B. Also, ρ satisfies Fatou property and we get(60)ρx-y=distρD∞A,D∞B=distρA,B.
Note that(61)TD∞A=T⋂nDnA⊆⋂nTDnA⊆⋂nDnB=D∞B In the same manner TD∞B⊆D∞A. Hence, D∞A,D∞B∈QA,B since DnA,DnB∈QA,B for all n≥1, which implies D∞A,D∞B∈F.
In this step, we can use the same argument as Theorem14 to prove that(62)δρD∞A,D∞B=distρD∞A,D∞B Hence we get for each x,y∈D∞A×D∞B(63)ρx-Tx=ρy-Ty=distρD∞A,D∞B which completes the proof.Corollary 15.
LetA be a ρ-bounded and ρ-sequentially compact nonempty subset of Xρ which satisfies Fatou property. Assume that QA,A is ρ-normal. If T:A→A is ρ-nonexpansive, then T has a fixed point.We conclude by the following example.Example 16.
Let the real spaceX={x=xnn≥1∈RN∗:∑n≥1xn1/2<∞} and define the modular functional ρ:X→[0,∞] by (64)ρx=∑n=1∞xn1/2,for  allx=xnn≥1∈X Suppose that {en} is the canonical basis of X and let(65)A=12e1∪e2+en:n∈N∖0,1,2andB=e1,e2,116e3.
Then,(A,B) is ρ-bounded and not convex. Let u=1/2e1 in A; we have ρ(u-e1)=1/2, also for each (x,y)∈A×B, ρ(x-y)≥1/2, which implies that distρ(A,B)=1/2.
Also, for all(xn)⊂A and (yn)⊂B such that limn⁡ρ(xn-yn)=1/2 there exists n0∈N such that (66)xn=12e1andyn=e1for each n≥n0, so (xn) and (yn) are ρ-convergent sequences and the pair (A,B) is proximal ρ-sequentially compactness. However, A is not ρ-sequentially compact since the sequence e2+enn≠{0,1,2} does not have any ρ-convergent subsequence in A.
Q A , B has the proximal ρ-normal structure. Indeed, let H,K be a proximal ρ-admissible pair of A,B not reduced to one point for which distρH,K=distρA,B=1/2; then 1/2e1∈H and e1∈K. Also, δρH,K>distρH,K, so there exits m∉{0,1,2} such that e2+em∈H and e2∈K or 1/16e3∈K.
IfK={e1,1/16e3} we obtain δρ1/2e1,K=1/4+1/2 and (67)δρ116e3,H=1+154ifH=12e1,e2+e32+14otherwise, and hence δρH,K≥ρ(e2+em-e1)>maxδρ1/2e1,K,δρ1/16e3,H.
Ife2∈K, then, δρ1/2e1,K=1+1/2 and δρe2,H=1+1/2. Hence, we have (68)δρH,K≥ρe2+em-e1>maxδρ12e1,K,δρe2,H.
LetT:A∪B→A∪B be a mapping defined by (69)Ty=12e1ify∈B,andTx=e1ifx=12e1116e3ifx∈A∖12e1, So, for each y∈B and x=1/2e1, we get ρ(Tx-Ty)=1/2≤ρ(x-y), and, for each y∈B and x∈A∖{1/2e1}, we obtain, ρ(Tx-Ty)=1+22/4≤ρ(x-y).
Then,T is cyclic relatively ρ-nonexpansive on A∪B. Therefore, all assumptions of Theorem 14 are satisfied, so T has a best proximity point; in particular (70)ρe1-Te1=ρ12e1-T12e1=distρA,B.



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*Source: 8084712-2018-10-02.xml*