A characterization of weak proximal normal structure and best proximity pairs

Noname manuscript No. (will be inserted by the editor) A Characterization of Weak Proximal Normal Structure and Best Proximity Pairs Abhik Digar ·Rafael Esp´ınola Garc´ıa ·G. Sankara Raju Kosuru Received: date / Accepted: date Abstract The aim of this paper is to address an open problem given in [Kirk, W. A., Shahzad, Naseer, Normal structure and orbital fixed point conditions, J. Math. Anal. Appl. vol 463(2), (2018) 461–476]. We give a characterization of weak proximal normal structure using best proximity pair property. We also introduce a notion of pointwise cyclic contraction wrt orbits and therein prove the existence of a best proximity pair in the setting of reflexive Banach spaces. Keywords Best proximity pairs ·Proximal normal structure ·Relatively nonexpansive mappings. Mathematics Subject Classification (2010) 46E15 ·47H10 ·54H25 1 Introduction and Preliminaries Let A, B be two non-empty subsets of a Banach space and Tbe a cyclic mapping on A∪B(T(A)⊆B, T(B)⊆A). A pair (x, y)∈A×Bis said to be a best proximity pair for Tif kx−Txk=ky−T yk=d(A, B) = inf{kx−yk:x∈A, y ∈B}. The geometry of Banach spaces plays a crucial role for the existence of best proximity pairs. The analysis of proximal normal structure and weak or semi-normal structure, the property UC, the projectional property due to Eldred et al. ([2]), Moosa ([4]), Suzuki et al. ([9]), G. S. Raju et al. ([7]) etc., respectively are widely used to prove the existence of a best proximity pair for cyclic maps. We denote sup{kx−yk:y∈B}for x∈Aby δ(x, B).We shall say that the pair (A, B) is proximal pair if for every xin A(resp. in B), there exists yin B(resp. in A) such Abhik Digar Department of Mathematics, Indian Institute of Technology Ropar, Punjab-140 001, India. E-mail: [email protected] Rafael Esp´ınola Garc´ıa Departamento de An´alisis Matem´atico - IMUS, Universidad de Sevilla, Sevilla, Spain. E-mail: [email protected] G. Sankara Raju Kosuru Department of Mathematics, Indian Institute of Technology Ropar, Punjab-140 001, India. E-mail: ra[email protected] 2 Abhik Digar et al. that kx−yk=d(A, B).Further, if such a yis unique, then (A, B) is said to be a sharp proximal pair ([7]). In this case we denote yby x0.Also, (A, B) is said to be a proximal parallel pair if (A, B) is sharp proximal and B=A+hfor some h∈X ([3]). It is shown in [3] that if Xis strictly convex and A, B are weakly compact convex subsets of X, then every (A0, B0) is a non-empty proximal parallel pair. Here A0={x∈A: there exists y∈Bsuch that kx−yk=d(A, B)}and B0={x∈B: there exists y∈Asuch that kx−yk=d(A, B)}.Also, in [7], the authors have given example(s) of sharp proximal pair which are not parallel. In [2], the authors introduced a geometrical notion called proximal normal structure to prove the existence of a best proximity pair of a relatively nonexpansive mapping (kTx−T yk ≤ kx−yk for all x∈A, y ∈B). We say (A, B) has proximal normal structure ([2]) [respectively, weak proximal normal structure ([5])] if (A, B) is convex and for any closed bounded [respectively, weakly compact] convex proximal pair (H1, H2) of subsets of (A, B) for which d(H1, H2) = d(A, B) and δ(H1, H2)> d(H1, H2),there exists (x, y)∈(H1, H2) such that δ(x, H2)< δ(H1, H2) and δ(y, H1)< δ(H1, H2).It is well known that every non-empty closed bounded convex pair (A, B) of a uniformly convex Banach space has proximal normal structure. In fact, every non-empty compact convex pair (A, B) of a Banach space has proximal normal structure. It is proved (Proposition 3.2 in [5]) that a bounded convex pair has proximal normal structure if and only if it doesn’t contain any proximal diametral sequence. A pair ({xn},{yn}) of sequences in (A, B) with kxn−ynk=d(A, B), n ≥1 is said to be a proximal diametral sequence ([5]) if d(A, B)< δ({xn},{yn}) and max{lim n→∞ d(xn+1,co ({y1, y2, ..., yn})) ,lim n→∞ d(yn+1,co ({x1, x2, ..., xn}))}= δ({xn},{yn}).It is easy to see that proximal normal structure coincides with weak proximal normal structure in reflexive Banach spaces ([5]). Moreover, therein the author proved the existence of a best proximity pair in the settings of a reflexive Banach space. Recently, in [6], the authors posed an open problem for the existence of a best proximity pair for a more general class of relatively nonexpansive mappings wrt orbits. Also therein the authors indicated that an affirmative answer may provide a characterization of proximal normal structure. Motivated by this, we aim to give a partial affirmative answer for the same. We also provide a characterization of weak proximal normal structure by using the existence of a best proximity pair for relatively orbital nonexpansive mappings. Finally, we introduce the notion of pointwise cyclic contraction wrt orbits and prove the minimal invariant subsets of such a map have nondiametral points. This guarantees the existence of a best proximity pair for such a class in the setting of a reflexive Banach space. Finally, we prove the existence of a best proximity pair for the class of pointwise cyclic contraction wrt orbits. 2 Existence of Best Proximity Pairs Let A, B be two closed convex subsets of a Banach space X. Let T:A∪B→ A∪Bbe a cyclic map. If Tadmits a best proximity pair, then A06=∅ 6=B0. Also, if Tis relatively nonexpansive, then A0∪B0is cyclically invariant under T (TA0⊆B0, TB0⊆A0). The following theorem is due to Eldred et al.([2]). A Characterization of Weak Proximal Normal Structure and Best Proximity Pairs 3 Theorem 1 Let (K1, K2)be a non-empty weakly compact convex pair in a Banach space and suppose (K1, K2)has proximal normal structure. Then every relatively nonexpansive mapping Ton A∪Bhas a best proximity pair in (K1, K2). The main tool to prove the same is to use the geometrical notion called “proximal normal structure” on A0∪B0. Later many authors established the existence of a best proximity pair for relatively nonexpansive mappings in different settings using variants of geometry ([3],[4],[8],[9]). In [5], Moosa introduced pointwise relatively nonexpansive mappings involving orbits and therein proved the existence of a best proximity pair for such a class of mappings. Recently, in 2018, Kirk and Shahzad discussed the existence of a best proximity pair for relatively nonexpansive mappings wrt orbits and therein they raised the question “can the assumption that Tis relatively nonexpansive in Theorem 1 be replaced by the assumption that Tis relatively nonexpansive wrt orbits?” Tis said to be a relatively nonexpansive mapping wrt orbits if kTx −T yk ≤ rx(O(y)) = δ(x, {y, Ty, T 2y, ...}) for all (x, y)∈(A, B).Using the following example, we can conclude that the answer is negative for the above open problem. Example 1 Let A={x∈R:−2≤x≤ −1}, B ={x∈R: 1 ≤x≤2}.Define T(x) = (−x, if x∈A; −1−x 2,if x∈B. Let y∈B. For any n, T2ny= 1 + 2n−1−1 2n−1+y 2n= 2 −1 2n−1+y 2nand T2n+1y= −1−T2ny 2=−2 + 1 2n−y 2n+1 .Now, for any x∈A, y ∈B, kTx −Tyk= (−x)−−1−y 2≤2−x=rx(O(y)) .Similarly, for x∈B, y ∈Aone can have kT x −T yk ≤ rx(O(y)) .Thus Tis relatively nonexpansive wrt orbits, but T doesn’t have any best proximity pair. ut It has to be observed that a cyclic map Ton A∪Bthat satisfies kTx −T yk ≤ rx(O(y)) does not guarantee A0∪B0is cyclically invariant under T. Hence, it is not reasonable to expect the existence of a best proximity pair for such a map T. To overcome this, we redefine the relatively nonexpansive mappings wrt orbits. For x∈A∪B, we denote {T2nx:n∈N∪ {0}} by O2(x). Definition 1 Let A, B be two non-empty subsets of a Banach space X. A cyclic map T:A∪B→A∪Bis said to be a relatively orbital nonexpansive mapping if (i) kTx −Tyk=d(A, B) if kx−yk=d(A, B) for x∈A, y ∈B; (ii) for all x∈A, y ∈B, kTx −Tyk ≤ min{rxO2(y), ryO2(x)}. It is worth mentioning that a relatively orbital nonexpanive mapping is not necessarily relatively nonexpansive. Example 2 Let A={(0, x)∈R2: 0 ≤x≤1}, B ={(1, y)∈R2: 0 ≤y≤1}and T:A∪B→A∪Bbe defined by x∈A, T(x) = ((1,x 4) if x≥1 2; (1,x 2) if x < 1 2. y∈B, T(y) = ((0,y 4) if y≥1 2; (0,y 2) if y < 1 2. 4 Abhik Digar et al. We see that Tis not relatively nonexpansive but it is a relatively orbital nonexpansive mapping. ut Let (A, B) be a non-empty sharp proximal pair in a Banach space Xand let T be a relatively orbital nonexpansive mapping on A∪B. Then it is easy to see that (A0, B0) is cyclically invariant under Tand Tx0= (T x)0.The pair (A, B) is said to satisfy the weak best proximity pair property (WBPP) if every relatively orbital nonexpansive mapping on A∪Bhas a best proximity pair. The following theorem ensures that every non-empty weakly compact convex pair of subsets of a strictly convex Banach space satisfies the WBPP. The following theorem is in a way different than Theorem 2.6 of [5]. For the sake the completeness, we prove the same here. Theorem 2 Let A, B be two non-empty weakly compact convex subsets of a strictly convex Banach space X. If (A, B)is having weak proximal normal structure, then (A, B)has WBPP. Proof Without loss of generality we may assume that A0=Aand B0=B. Let Tbe a relatively orbital nonexpansive mapping of A∪B. Let Fdenote the collection of non-empty closed bounded convex proximal pair (E1, E2) of subsets of (A0, B0) with (E1, E2) cyclically invariant under Tand d(E1, E2) = d(A, B). F6=∅,since (A0, B0)∈F.By Zorn’s Lemma Fhas a minimal element under the set inclusion order “ ⊆”, say, (F1, F2).If (F1, F2) is a singleton pair, we have δ(F1, F2) = d(A, B),i.e., Thas a best proximity pair. Suppose (F1, F2) is not singleton. By weak proximal normal structure, there exists (x1, y1)∈(F1, F2) such that m1=δ(x1, F2)< δ(F1, F2) and m2=δ(y1, F1)< δ(F1, F2).Set m= max{m1, m2}. Define L1={x∈F1:δ(x, F2)≤m} L2={y∈F2:δ(y, F1)≤m}. L16=∅, L26=∅,since x1∈L1, y1∈L2.Being closed subset of a weakly compact subset, L1, L2are weakly compact. To see L1is convex, let a, b ∈L1.For any λ∈[0,1], δ(λa + (1 −λ)b, F2)≤λδ(a, F2) + (1 −λ)δ(b, F2)≤λm + (1 −λ)m=m. Hence we can conclude that (L1, L2) is a convex pair. Let v∈F2.Suppose the unique best approximation of an element z∈A∪Bis denoted by z0. Then     x1+y0 1 2−v    ≤1 2kx1−vk+ y0 1−v  =1 2kx1−vk+ y1−v0  ≤1 2[δ(x1, F2) + δ(y1, F1)] ≤m. Since v∈F2is arbitrary, δx1+y0 1 2, F2≤m. Hence, x1+y0 1 2∈L1.Similarly, x0 1+y1 2∈L2.Moreover,    x1+y0 1 2−x0 1+y1 2  =d(A, B).Hence, d(L1, L2) = d(A, B). To see (L1, L2) is a proximal pair, let x∈L1.Then x∈F1and hence x0∈F2. A Characterization of Weak Proximal Normal Structure and Best Proximity Pairs 5 Therefore δ(x0, F1) = δ(x, F2)≤m. Thus x0∈L2.It infers (L1, L2) is a proximal pair. In fact, L2={x0∈F2:x∈L1}. Next, let x∈L1, v ∈F2.Then, kTx −Tvk ≤ rxO2(v)=δx, O2(v)≤ δ(x, F2)≤m. It follows that T(F2)⊆B(Tx;m)∩F1=: F0 1.Similarly, T(F1)⊆ BTx0;m∩F2=: F0 2.Clearly, (F0 1, F0 2)∈F.By minimality, F0 1=F1, F0 2=F2. Then F1⊆B(Tx;m) and F2⊆B(Tx0;m).For any u∈F1,ku−Txk ≤ m, hence, δ(Tx, F1)≤m. Therefore, Tx ∈L2.Hence, T(L1)⊆L2.Further, if y∈L2,then y0∈L1.This implies (Ty)0=T y0∈L2.Thus T y ∈L1.As y∈L2is arbitrary, we have T(L2)⊆L1.Hence, (L1, L2)∈F.For z∈L1, w ∈L2,kz−wk ≤ δ(z, F2)≤ m < δ(F1, F2).This infers that δ(L1, L2)< δ(F1, F2).This contradicts the minimality of (F1, F2).ut Let Tbe a cyclic map on A∪B. We say that the pair (A, B) has a proximal nondiametral pair if there exists (x, y)∈A×Bsuch that max{δ(x, B), δ(y, A)}< δ(A, B) whenever d(A, B)< δ(A, B).A similar technique can be used to obtain the following: Theorem 3 Let (A, B)be a non-empty closed bounded convex proximal pair of subsets of a Banach space and let Tbe a relatively orbital nonexpansive mapping on A∪B. If Thas a non-empty closed bounded convex proximal minimal cyclically invariant pair (A, B)having a nondiametral pair then Thas a best proximity pair. Example 3 Let A, B and Tas in the Example 2. It is easy to see that ((0,0),(1,0)) is a best proximity pair. ut 3 Characterization of weak proximal normal structure Let (A, B) be a non-empty bounded convex proximal pair of a Banach space X. A non-constant pair of sequences ({xn},{yn}) of (A, B) is said to be a proximal diametral sequence ([5]) if kxn−ynk=d(A, B) for every n∈Nand δ({xn},{yn}) = lim n→∞ d(xn+1,co ({y1, y2, ..., yn})) = lim n→∞ d(yn+1,co ({x1, x2, ..., xn})) such that d(A, B)< δ({xn},{yn}).It has to be observed that if d(A, B) = 0,then the proximal diametral sequence turns out to be a diametral sequence in A∩Bin the sense of Brodski˘ı and Mil0man ([1]). The following result is discussed in [5]. Theorem 4 A bounded convex pair (A, B)of a Banach space Xhas proximal normal structure if and only if it does not contain a proximal diametral sequence. Let (A, B) be a non-empty weakly compact convex sharp proximal pair of subsets of a Banach space having WBPP. Suppose (A, B) does not have weak proximal normal structure. Then by Theorem 4, (A, B) has a proximal diametral sequence, say, ({xn},{yn}).Consequently, lim n→∞ d(xn+1,co ({y1, y2, ..., yn})) = δ({xn},{yn}) = lim n→∞ d(yn+1,co ({x1, x2, ..., xn})) .Since, (A, B) is weakly compact, there exists a subsequence ({xnk},{ynk}) of ({xn},{yn}) which is weakly convergent. It is easy to see that the sequence ({xnk},{ynk}) is a proximal diametral subsequence. Hence, without loss of any generality, we may assume that the sequence ({xn},{yn}) is proximal diametral and weakly convergent. Now, H= co ({x1, x2, ...}), K = co ({y1, y2, ...}) are weakly compact convex subsets of A, B respectively. Define T:H∪K→H∪Kby 6 Abhik Digar et al. T(x) = (y1,if x /∈ {xn:n∈N} yn+1,if x=xnfor some n∈N; T(y) = (x1,if y /∈ {yn:n∈N} xn+1,if y=ynfor some n∈N. Clearly, δ(H, K) = δ({xn},{yn}) and lim n→∞ kxn−zk=δ(H, K) = lim n→∞ kyn−vk for any z∈K, v ∈H. Hence, rxO2(y)=δ(H, K) for each x∈H, y ∈K. Now, kTx −Tyk ≤ δ(H, K) = rxO2(y)for each x∈H, y ∈K. Also, if (x, y)∈H×Kwith kx−yk=d(H, K),then kTx −Tyk=d(H, K). Therefore Tis a relatively orbital nonexpansive mapping. As (A, B) is a sharp proximal pair, then so is (H, K) and Tdoes not have any best proximity pair. Thus we have the following: Proposition 1 Let A, B be two non-empty weakly compact convex subsets of a Banach space X. If (A, B)is a sharp proximal pair and (A, B)has WBPP, then (A, B)has weak proximal normal structure. By Theorem 2 and Proposition 1 we have the following characterization: Theorem 5 Let A, B be two non-empty weakly compact convex subsets of a strictly convex Banach space X. Then (A, B)has weak proximal normal structure if and only if every relatively orbital nonexpansive mapping T:A∪B→A∪Bhas a best proximity pair. 4 Pointwise Cyclic Contraction wrt Orbits Let (A, B) be a pair of subsets of a normed linear space. A cyclic map Ton A∪Bis said to be a proximal pointwise contraction if for any x∈A, there exists α(x)∈[0,1) such that kTx −T yk ≤ α(x)kx−yk([10]). Later many authors obtained the existence of a best proximity pair for certain types of pointwise cyclic contractions ([8], [11], [12]). Now we introduce the notion of pointwise cyclic contraction wrt orbits and prove the existence of a best proximity pair for such a map. Our result is a generalization of the main results given in the aforementioned articles. Definition 2 A cyclic map Ton a non-empty pair (A, B) of subsets of a Banach space is said to be pointwise cyclic contraction wrt orbits if it satisfies (i) kTx −Tyk=d(A, B) whenever kx−yk=d(A, B) for (x, y)∈A×B; (ii) for each (x, w)∈(A, B) there exists α(x), α(w)∈(0,1) such that kTx −Tyk ≤ α(x)rxO2(y)+ (1 −α(x)) d(A, B) for all y∈B, and kTw −Tuk ≤ α(w)rwO2(u)+ (1 −α(w)) d(A, B) for all u∈A. It is easy to see that every pointwise cyclic contraction mapping wrt orbits is relatively orbital nonexpansive. A Characterization of Weak Proximal Normal Structure and Best Proximity Pairs 7 Theorem 6 Suppose (A, B)is a weakly compact, convex pair of a strictly convex Banach space Xand T:A∪B→A∪Bis a pointwise cyclic contraction wrt orbits. Then Thas a best proximity pair. Proof Let Fdenote the collection of all non-empty proximal closed convex subsets (H1, H2) of (A0, B0) such that TH1⊆H2, T H2⊆H1and d(H1, H2) = d(A, B). Since A0∪B0∈F,we have F6=∅. By Zorn’s lemma, Fhas a minimal, say, (K1, K2) with respect to the partial order “ ⊆”.Let (x, y)∈(K1, K2) be such that kx−yk=d(K1, K2) = d(A, B).If δ(x, K2) = d(A, B),then d(A, B) = d(K1, K2)≤ kx−Txk ≤ δ(x, K2) = d(A, B).This infers kx−Txk=d(A, B). Since, Tis pointwise cyclic contraction wrt orbits, we have kT x−T2xk=d(A, B). Therefore, (x, Tx) is a best proximity pair. Similarly, if δ(y, K1) = d(A, B),then (y, T y) is a best proximity pair. Hence, we may assume that δ(x, K2)> d(A, B) and δ(y, K1)> d(A, B). Define Kx={z∈K1:kz−Txk ≤ α(x)δ(x, K2) + (1 −α(x)) d(A, B)}; Ky={w∈K2:kw−Tyk ≤ α(y)δ(y, K1) + (1 −α(x)) d(A, B)}. Here we have kTx −Tyk=d(A, B) = α(x)d(A, B) + (1 −α(x))d(A, B) < α(x)δ(x, K2) + (1 −α(x))d(A, B). Then (Ty, T x)∈(Kx, Ky) and hence Kx6=∅ 6=Ky.It is easy to see that (Kx, Ky) is convex. If {un}∞ n=1 ⊆Kxis a sequence converges to u∈Xweakly, then u∈K1. Now, ku−Txk ≤ lim inf{kun−Txk:n∈N} ≤ α(x)δ(x, K2) + (1 −α(x))d(A, B). Then u∈Kxand hence Kxis closed. Further, for any z∈Kx,kTz −T yk ≤ α(y)ryO2(z)+(1−α(y))d(A, B)≤α(y)δ(y, K1)+(1−α(y))d(A, B). This implies that Tz ∈Ky.Hence, TKx⊆Ky. Similarly, T Ky⊆Kx. Therefore, (Kx, Ky)∈ F.By minimality, Kx=K1, Ky=K2. Now, for any w∈K2,kw−T yk ≤ α(y)δ(y, K1) + (1 −α(y))d(A, B)< δ(y, K1)≤δ(K1, K2).Hence, δ(Ty, K2)< δ(K1, K2).Similarly, δ(Tx, K1)< δ(K1, K2).Thus (K1, K2) has a proximal nondiametral pair. By Theorem 3, T has a best proximity pair. ut Acknowledgements The authors would like to thank the referee for the valuable comments and suggestions. The research of Rafael Esp´ınola Garc´ıa was supported by Ministerio de Ciencia, Innovaci´on y Universidades del Gobierno de Espa˜na through the grant PGC2018-098474- B-C21, Plan Estatal 2017-2020 Generaci´on Conocimiento - Proyectos I+D+i and Proyecto de Excelencia de la Junta de Andaluci´on FQM-127. References 1. Brodski˘ı, M. S., Mil0man, D. P., On the center of a convex set, Doklady Akad. Nauk SSSR (N.S.), vol 59, (1948) 837–840. 2. Eldred, A. Anthony, Kirk, W. A., Veeramani, P., proximal normal structure and relatively nonexpansive mappings, Studia Math., vol 171(3), (2005) 283–293. 3. Esp´ınola, Rafa, A new approach to relatively nonexpansive mappings, Proc. Amer. Math. Soc., vol 136(6), (2008) 1987–1995. 4. Gabeleh, Moosa, Shahzad, Naseer, Seminormal structure and fixed points of cyclic relatively nonexpansive mappings, Abstr. Appl. Anal., (2014) 1085-3375. 5. Gabeleh, Moosa, A characterization of proximal normal structure via proximal diametral sequences, J. Fixed Point Theory Appl. vol 19(4), (2017) 2909–2925. 8 Abhik Digar et al. 6. Kirk, W. A., Shahzad, Naseer, Normal structure and orbital fixed point conditions, J. Math. Anal. Appl. vol 463(2), (2018) 461–476. 7. Raju Kosuru, G. Sankara, Veeramani, P., On existence of best proximity pair theorems for relatively nonexpansive mappings, J. Nonlinear Convex Anal., vol 11(1), (2010) 71–77. 8. Kosuru, G. Sankara Raju, Veeramani, P., A note on existence and convergence of best proximity points for pointwise cyclic contractions, Numer. Funct. Anal. Optim., vol 32(7), (2011) 821–830. 9. Suzuki, Tomonari, Kikkawa, Misako, Vetro, Calogero, The existence of best proximity points in metric spaces with the property UC, Nonlinear Anal., vol 71(7-8), (2009) 2918–2926. 10. Anuradha, J. and Veeramani, P., Proximal pointwise contraction, Topology Appl., vol 156(18), (2009) 2942–2948. 11. Mongkolkeha, Chirasak and Kumam, Poom, Best proximity points for asymptotic proximal pointwise weaker Meir-Keeler-type ψ-contraction mappings, J. Egyptian Math. Soc., vol 21(2), (2013) 87–90. 12. Gabeleh, Moosa, On generalized pointwise noncyclic contractions without proximal normal structure, Ann. Funct. Anal., vol 9(2), (2018) 220–232.