# Cyclic Relatively Nonexpansive Mappings with Respect to Orbits and Best Proximity Point Theorems

Authors: Laishram Shanjit; Yumnam Rohen; K. Anthony Singh
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this article, we introduce cyclic relatively nonexpansive mappings with respect to orbits and prove that every cyclic relatively nonexpansive mapping with respect to orbitsT satisfying TA⊆B,TB⊆A has a best proximity point. We also prove that Mann’s iteration process for a noncyclic relatively nonexpansive mapping with respect to orbits converges to a fixed point. These relatively nonexpansive mappings with respect to orbits need not be continuous. Some illustrations are given in support of our results.


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## Body

## 1. Introduction and Preliminaries

In the year 1948, Brodskiĭ et al. [1] introduced a geometrical property known as the normal structure to study the existence of a common fixed point for a family of isometry mappings. In [2], Kirk used a geometrical notion, namely, normal structure for a nonempty bounded closed and convex subset K of a reflexive Banach space X, thereby getting the existence of a fixed point for a mapping that does not increase distance (a nonexpansive mapping). Let G,H be a pair of subsets in a Banach space X. A mapping T:G∪H⟶G∪H is said to be a relatively nonexpansive mapping if TG⊆H,TH⊆G (or TG⊆G,TH⊆H) and Tu−Tv≤u−v, for u∈G,v∈H. If G=H, then T is a nonexpansive self-mapping. In the year 2005, Eldred et al. [3] introduced a geometrical notion called proximal normal structure, for a nonempty, convex weakly compact pair G,H, in order to study the existence of a fixed point (the best proximity point) for relatively nonexpansive mappings under suitable assumptions. Motivated by Harandi et al. [4], in the year 2020, Gabeleh et al. [5] introduced a notion called noncyclic relatively nonexpansive mapping with respect to orbits T satisfying TG⊆G,TH⊆H and showed the existence of a fixed point (best proximity point) under certain assumptions. In the setting of Banach spaces, the existence of fixed point and best proximity point theorems, respectively, for nonexpansive and relatively nonexpansive mappings has been extensively studied by many authors (see [3, 5–8]). For more exciting results about the best proximity point for contraction and multivalued mappings studied by many authors, see [9–12]. For any pair of subsets C,D of X, the following notations will be used in the sequel:(1)Ru,D≔supu−v:v∈D,u∈C;δC,D≔supRu,D:u∈C;C0≔u∈C:u−v=distC,D,v∈D;D0≔v∈D:u−v=distC,D,u∈C,where distC,D≔infu−v:u∈C,v∈D.Definition 1.
(see [3]). A nonempty pair C,D of subsets in a normed linear space X is known as a proximal pair if for every u1∈C,v1∈D there exist u2∈C,v2∈D such that u1−v2=distC,D=u2−v1.Note: the pair C0,D0 is a proximal pair and (C0, D0) ⊆ (C, D), that is C0 ⊆ C, D0 ⊆ D.Definition 2.
(see [13]). A nonempty pair C,D of subsets in a Banach space X is said to be a proximal parallel pair if(i)
C,D is a sharp proximal pair, that is, for each u1,v1∈C×D, there is unique element u2,v2∈C×D such that u1−v2=distC,D=u2−v1.(ii)
D = C + h, for a certain h∈X.By using the above definition, Rajesh et al. [14] introduced the rectangle property as follows.Definition 3.
LetX be a Banach space and let C,D be a nonempty, convex proximal parallel pair. The nonempty proximal parallel pair C,D is assumed to have the rectangle property iff u+h−v=v+h−u, for u,v∈C, where D=C+h, h∈X.Definition 4.
(see [3]). Let X be a Banach space. A nonempty convex pair G,H of subsets in X is assumed to have proximal normal structure if for every nonempty, closed convex bounded proximal pair C,D⊆G,H for which distC,D=distG,H and δC,D>distC,D, there is u,v∈C,D such that(2)maxRu,D,Rv,C<δC,D.The following two theorems are obtained from Eldred et al. [3].Theorem 1.
LetX be a Banach space and let G,H be a nonempty, convex weakly compact proximal pair in X. Let T:G∪H⟶G∪H with TG⊆H,TH⊆G and Tu−Tv≤u−v, for u∈G,v∈H. Suppose G,H has proximal normal structure. Then, there exist u0∈G,v0∈H such that u0−Tu0=distG,H=v0−Tv0.Theorem 2.
LetX be a strictly convex Banach space and let G,H be a nonempty, convex weakly compact proximal pair in X. Let T:G∪H⟶G∪H with TG⊆G,TH⊆H and Tu−Tv≤u−v, for u∈G,v∈H. Suppose G,H has proximal normal structure. Then, there exist u0∈G,v0∈H such that Tu0=u0,Tv0=v0 and u0−v0=distG,H.Remark 1.
In the above two theorems, ifG=H, then the main result of Kirk [2] is obtained.Definition 5.
(see [15]). A uniformly convex space X is a normed linear space such that for each ε>0,(3)inf1−u+v2:u≤1,v≤1,u−v≥ε>0.Moreover, ifu,v,w∈X,R>0,0≤ε≤2R, such that u−w≤R,v−w≤R, u−v≥ε which implies(4)u+v2−w≤1−δεRR.The following result based on Mann’s iteration process is well known.Proposition 1.
(see [16]). Let X be a uniformly convex space, ε>0, and 0<ξ<1. For any R>0, if u,v∈X, u≤R,v≤R, u−v≥ε, then there is δ=δε/R>0, so that(5)ξu+1−ξv≤1−2δεRminξ,1−ξR.Espinola proved the following result.Lemma 1.
(see [13]). Let X be a strictly convex Banach space. If C,D is a proximal pair in X, then C,D is a proximal parallel pair.Definition 6.
(see [5]). Let X be a Banach space and let G and H be nonempty subsets of X. A mapping T:G∪H⟶G∪H satisfying TG⊆G,TH⊆H is known as noncyclic relatively nonexpansive mapping with respect to orbits provided that for every u,v∈G×H if u−v=distG,H, then Tu−Tv=distG,H; otherwise,(6)Tu−Tv≤Ru,Ov,Tu−Tv≤Rv,Ou,where Ov≔v,Tv,…,Tnv,…⊆H and Ou⊆G.In the above definition, ifG=H, we obtain the definition of Harandi et al. [4] as follows.Definition 7.
LetX be a Banach space and H be a nonempty subset of X. A mapping T:H⟶H is known as a nonexpansive mapping with respect to orbits if(7)Tu−Tv≤Ru,Ov,∀u,v∈H.Remark 2.
Every nonexpansive mapping is a nonexpansive mapping with respect to orbits, whereas the converse may not be true.


## 2. Results for Cyclic Relatively Nonexpansive Mappings with respect to Orbits

Motivated by Harandi et al. [4] and Gabeleh et al. [5], we define the following definition which will be extensively used in the sequel.Definition 8.
LetG and H be nonempty subsets of a Banach space X. A mapping T:G∪H⟶G∪H satisfying TG⊆H,TH⊆G is known as cyclic relatively nonexpansive mapping with respect to orbits provided that for every u,v∈G×H if u−v=distG,H, then Tu−Tv=distG,H; otherwise,(8)Tu−Tv≤Ru,Ov,Tv−Tu≤Rv,Ou,where Ov≔v,Tv,…,Tnv,…, T2n+1v∈G, T2nv∈H, and Ou⊆G∪H for n=0,1,2,…,.Example 1.
LetX=ℓ∞=ℓ∞ℕ and let em=δmk be the standard basis for ℓ∞ (where em denotes the element with 1 in the mth place and 0 elsewhere). Consider the sets(9)A=aei:0<a≤1andB=bej:0<b≤1,wherei≠j.
LetT:A∪B⟶A∪B be a mapping defined by(10)Taei=aejandTbej=bei.
Clearly,T is not a cyclic relatively nonexpansive mapping. Let u=aei and v=bej. Then,(11)Tu−Tv∞=aej−bei∞=maxa,b,Ru,Ov=supnu−Tnv∞=supnmaxa,b2n=1,Rv,Ou=supnv−Tnu∞=supnmaxb,a2n=1.
It shows thatT is a cyclic relatively nonexpansive mapping with respect to orbits.Now, we prove the following proposition.Proposition 2.
IfT:G∪H⟶G∪H is a cyclic relatively nonexpansive mapping, then T is a cyclic relatively nonexpansive mapping with respect to orbits.Proof.
LetT be a cyclic relatively nonexpansive mapping. For u∈G,v∈H, Tu−Tv≤u−v≤Ru,Ov. Similarly, Tv−Tu≤Rv,Ou. This shows that T is a cyclic relatively nonexpansive mapping with respect to orbits.Lemma 2.
LetX be a strictly convex Banach space and let C,D be a nonempty, convex weakly compact proximal pair of subsets in X. Suppose C,D has the rectangle property. Let T:C∪D⟶C∪D be a cyclic relatively nonexpansive mapping with respect to orbits. Further, assume that proximal pair C,D does not contain any proximal pair G,H of subsets in X such that TG⊆H,TH⊆G. Then, C,D⊆conv¯TD,conv¯TC.Proof.
LetG=conv¯TD∩C and H=conv¯TC∩D. Then, G and H are, respectively, convex weakly compact subsets of C and D. Suppose u,v∈C×D such that u−v=distC,D. Then, Tv,Tu∈TD×TCandTD,TC⊆C,D which implies Tv,Tu∈G×H. Since u−v=distC,D, by Definition 8, we get Tv−Tu=distC,D. Hence, distG,H=distC,D. It suffices to prove that G,H is a proximal pair if for every u∈G, there is v∈H such that(12)distC,D=u−v.
Letu∈convTD∩C. Then, u=∑i=1nciTvi, where vi∈D,ci≥0 and ∑i=1nci=1. Since C,D is a proximal pair, there is vi′∈C such that(13)distC,D=vi′−vi,i=1,2,…,n.
Then,u′=∑i=1nciTvi′∈convTC such that u−u′=distC,D and u′∈H. Thus, G,H⊆C,D is a proximal pair (and hence proximal parallel pair). It is easy to see that G,H is invariant under T, that is, TG⊆H,TH⊆G. This proves that C,D=G,H. Hence, C,D⊆conv¯TD,conv¯TC.Theorem 3.
LetX be a strictly convex Banach space and let A,B be a nonempty, convex weakly compact proximal pair in X. Let T:A∪B⟶A∪B be a cyclic relatively nonexpansive mapping with respect to orbits. Suppose A,B has proximal normal structure and the rectangle property. Then, there is u,v∈A×B such that u−Tu=distA,B=v−Tv.Proof.
LetF be the set of all nonempty, convex weakly closed proximal pair of subsets (C, D) of (A, B) satisfying T(C) ⊆ D, T(D) ⊆ C and distA,B=distC,D. Let ″≺¯″ be a relation defined on F by C1,D1≺¯C2,D2 if and only if C1,D1⊆C2,D2, where C1,D1,C2,D2∈F. It is easy to see that F is a partially ordered set. Also, if T is a totally ordered set, then T⊆F has the finite intersection property (FIP). Hence,(14)E,F≔∩C,D∈TC,Dis a nonempty, convex weakly compact proximal pair and E,F∈F. By Zorn’s lemma, partially ordered set F has a minimal element, say C,D. Therefore, from Lemma 2, we conclude that(15)C,D⊆conv¯TD,conv¯TC.
IfδC,D=distA,B, we get our result and the theorem is complete. Suppose(16)δC,D>distA,B.
By proximal normal structure, there isu1,v1∈C×D and η∈0,1 such that(17)maxRu1,D,Rv1,C≤ηδC,D.
SinceC,D is a proximal pair, there is u1′,v1′∈C×D such that(18)u1−v1′=u1′−v1=distA,B.
So, for anyw∈D, we have(19)u1+u1′2−w≤u1−w2+u1′−w2≤ηδC,D2+δC,D2≤βδC,D,where,β=1+η2∈0,1.
Letu2=u1+u1′/2 and v2=v1+v1′/2. Then,(20)Ru2,D≤βδC,D,Rv2,C≤βδC,Dandu2−v2=distC,D.
Define(21)G=u∈C:Ru,D≤βδC,DH=v∈D:Rv,C≤βδC,D.
Then,G,H is a nonempty, convex weakly compact proximal pair of subset of C,D. By Lemma 1, G,H is also a proximal parallel pair of subset of C,D. Now, we need to claim that TG⊆H,TH⊆G. Since C,D⊆conv¯TD,conv¯TC and A,B has rectangle property, for u∈G,(22)RTu,C=supTu−w:w∈C≤supTu−w:w∈conv¯TD=supTu−Tv:Tv∈TD≤supRu,Ov:v∈D≤Ru,D≤βδC,D.
This shows thatTG⊆H. Similarly, for v∈H,(23)RTv,D=supTv−z:z∈D≤supTv−z:z∈conv¯TC=supTv−Tu:Tu∈TC≤supRv,Ou:u∈C≤Rv,C≤βδC,D.
This shows thatTH⊆G. Therefore, G,H∈F. But δC,D=supu∈GRu,D≤βδC,D<δC,D, which is a contradiction. This shows that C and D are singleton sets. Therefore, there is u,v∈A,B such that u−Tu=distA,B=v−Tv.In Theorem3, if A=B, we obtain the result of Harandi et al. (see Theorem 2.2 in [4]). The following example illustrates that the rectangle property in Theorem 3 cannot be dropped.Example 2.
LetX=ℝ2,., an Euclidean space, and let(24)A=u,5−5u:u∈0,1B=u+1,5−5u:u∈0,1.
Then, the pairA,B with B=A+h, h=1,0 satisfies all the assumptions of Theorem 3 except the rectangle property. It is easy to see that the pair A,B does not have the rectangle property by taking particular points 1,0,0,5∈A and 2,0,1,5∈B. Let T:A∪B⟶A∪B be a mapping as in Definition 8. We need to claim that(25)Tu−Tv≤Ru,Ov≤Ru,B,u,v∈A×B,Ov≔v,Tv,…,Tnvdots,T2n+1v∈A,T2nv∈B.
Now, in particular, letu=1,0∈A and v∈B. If T2n+1v=0,5∈Ov for some v∈B, then Ru,B<Ru,Ov, which is absurd. It happened because A,B does not have the rectangle property.The next example will illustrate our result, Theorem3.Example 3.
LetX=ℝ2,., an Euclidean space, and let(26)A=−12,v:−12≤v≤12B=12,v:−12≤v≤12.
Then, the pairA,B satisfies all the assumptions of Theorem 3, and B=A+h, where h=1,0. Let T:A∪B⟶A∪B be a mapping defined by(27)Tu1,v1=u1,−v12+1,0,Tu2,v2=u2,−v24−1,0,where u1,v1∈A, u2,v2∈B. Clearly, T is cyclic relatively nonexpansive with respect to orbits but not cyclic relatively nonexpansive. Then, ∃−1/2,0,1/2,0∈A×B such that(28)−12,0−T−12,0=1=12,0−T12,0.


## 3. Results for Noncyclic Relatively Nonexpansive Mappings with respect to Orbits

Gabeleh et al. [5] proved the following interesting theorem.Theorem 4.
LetX be a strictly convex Banach space and let C,D be a nonempty, convex weakly compact pair in X. Then, the pair C,D has proximal normal structure iff for every noncyclic relatively nonexpansive mapping T with respect to orbits, there is u,v∈C×D such that Tu=u, Tv=v and u−v=distC,D.In Theorem4, if A=B, the result of Harandi et al. (see Theorem 2.2 in [4]) is obtained. Next, we show that Mann’s iteration process (see [7, 17, 18]) converges to a fixed point if X is a uniformly convex Banach space.Theorem 5.
LetX be a uniformly convex Banach space and let A and B be nonempty, convex closed bounded subsets of X. Let T:A∪B⟶A∪B be a noncyclic relatively nonexpansive mapping with respect to orbits. Let an initial point u0∈A, and define a sequence(29)un+1=1−ξnun+ξnTun,whereθ<ξn<1−θ,θ∈0,1/2,n=0,1,2,3,…Then,limn⟶+∞un−Tun=0. Moreover, if dun,A0⟶0, T is continuous, and TA is subset of a compact set, then the limit point of the sequence un is a fixed point for T.Proof.
SupposedistA,B>0. From Theorem 4, there is v∈B such that Tv=v. Since(30)un+1−v≤1−ξnun−v+ξnTun−Tv≤1−ξnun−v+ξnRun,Ov=un−v,and un−v is nonincreasing sequence, there is a positive real number k>0 so that limn⟶+∞un−v=k. Since X is uniformly convex, then δε>0 for ε>0 is a strictly increasing (and continuous) function. So, it is possible to choose ε1>0 so small that(31)1−aδεk+ε1k+ε1<k,where,a>0.
Let us assume that there exists a subsequenceuni of un with uni−Tuni≥ε>0. Choose i, such that uni−v≤k+ε1. From Definition 6, we have Tuni−Tv≤Runi,Ov=uni−v≤k+ε1. Now,(32)v−uni+1=1−ξnv−uni+ξnTv−Tuni≤1−2δεk+ε1minξn,1−ξnk+ε1≤1−bδεk+ε1k+ε1,where,0<b≤2minξn,1−ξn.
By choosingε1>0 so small, we get(33)1−bδεk+ε1k+ε1<k,which is a contradiction. Hence, limn⟶+∞un−Tun=limn⟶+∞un−un+1=0. Since TA is a subset of a compact set, the sequence un has a subsequence uni such that limi⟶+∞uni=u∈A. As dun,A0⟶0, there is wn∈A0 such that(34)limn⟶+∞un−wn=0.
Hence, there exists a subsequencewni of wn in A0 such that limi⟶+∞wni=u∈A0. Thus, there is w∈B0 such that u−w=distA,B. Now,(35)uni+1−Tw=1−ξnuni+ξnTuni−1−ξnTw+ξnTw≤1−ξnuni−Tw+ξnTuni−Tw.
Sinceu−w=distA,B, by Definition 6, we get Tu−Tw=distA,B. Therefore, from equation (35), when i⟶+∞, we have u−Tw≤distA,B which implies u−Tw=distA,B. By strict convexity of the norm, Tw=w which implies Tu=u because u is the unique point of A nearest to w.From the above theorem, ifdistA,B=0, we get the following corollary.Corollary 1.
LetX be a uniformly convex Banach space and let A be nonempty, closed bounded convex subset of X. Let T:A⟶A be a nonexpansive mapping with respect to orbits. Let the initial point u0∈A, and define a sequence(36)un+1=1−ξnun+ξnTun,where,θ<ξn<1−θ,θ∈0,1/2,n=0,1,2,3,….Then,limn⟶+∞un−Tun=0. Moreover, if T is continuous and TA is subset of a compact set, then the limit point of the sequence un is a fixed point for T.The following example illustrates Theorem5.Example 4.
LetX=ℝ2,., an Euclidean space, and let(37)A=u,v:−2≤u≤−1,−12≤v≤12B=u,v:1≤u≤2,−12≤v≤12.
DefineT:A⟶A by(38)Tu,v=T1u,T2v=u−67,−v2,foru,v∈A,and T:B⟶B by(39)Tu,v=u+67,−v4,foru,v∈B,where T1:−2,−1⟶−2,−1 and T2:−1/2,1/2⟶−1/2,1/2. It clearly shows that T:A∪B⟶A∪B is noncyclic relatively nonexpansive with respect to orbits but not noncyclic relatively nonexpansive. Let the initial point u,v∈A, and set u1=1−ξ1u+ξ1T1u. We have T1u=u−6/7. Therefore,(40)u1=u7−6ξ1−6ξ17.
Following the same line of Example 2.2 in [7], we get(41)un=17nu7−6ξ1⋯7−6ξn−6ξ17−6ξ2⋯7−6ξn−6.7ξ27−6ξ3⋯7−6ξn−⋯−6.7n−2ξn−17−6ξn−6.7n−1ξn,and limn⟶∞un=−1. Again, we have T2v=−v/2. Let(42)v1=1−ξ1v+ξ1T2v=v2−3ξ12.
Now,T2v1=−v1/2=−v2−3ξ1/22. Therefore,(43)v2=1−ξ2v1+ξ2T2v1=v2−3ξ22−3ξ122,and so on.
In general,(44)vn=v2−3ξn2−3ξn−1…2−3ξ12n,and limn⟶+∞vn=0. Hence, limn⟶+∞un,vn=−1,0∈A. Since −1,0∈A0, there exists 1,0∈B0 such that −1,0−1,0=2=distA,B. Again, T−1,0−T1,0=2=distA,B. This shows that sequence un,vn⟶−1,0∈A as n⟶+∞, a fixed point for T. In a similar way, we can show that if u′,v′∈B, then limn⟶+∞un′,vn′=1,0∈B, a fixed point for T.


## 4. Conclusions

From the results obtained in this article, we draw the following conclusions.In this research article, we study the existence of the best proximity point for cyclic relatively nonexpansive mappings with respect to orbits. Additionally, this research article establishes the convergence of Mann’s iteration process for a noncyclic relatively nonexpansive mapping with respect to orbits and nonexpansive mapping with respect to orbits to a fixed point. There may be scope for extension of results obtained in Section3 by using other well-known iteration processes.



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