Best Proximity Pair Theorems for Noncyclic Mappings in Banach and Metric Spaces

Fixed Point Theory, 17(2016), No. 1, 63-84 http://www.math.ubbcluj.ro/∼nodeacj/sfptcj.html BEST PROXIMITY PAIR THEOREMS FOR NONCYCLIC MAPPINGS IN BANACH AND METRIC SPACES A. FERN´ ANDEZ-LE´ ON∗AND M. GABELEH∗∗ ∗Departamento de An´alisis Matem´atico, Universidad de Sevilla St. Tarfia s/n, Sevilla Spain E-mail: [email protected] ∗∗ Department of Mathematics, Ayatollah Boroujerdi University, Boroujerd, Iran and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran E-mail: gab.mo[email protected] Abstract. Let Aand Bbe two nonempty subsets of a metric space X. A mapping T:A∪B→A∪B is said to be noncyclic if T(A)⊆Aand T(B)⊆B. For such a mapping, a pair (x, y)∈A×Bsuch that T x =x,T y =yand d(x, y) = dist(A, B) is called a best proximity pair. In this paper we give some best proximity pair results for noncyclic mappings under certain contractive conditions. Key Words and Phrases: Best proximity pair; noncyclic contraction, noncyclic contraction in the sense of Kannan; noncyclic contraction in the sense of Chatterjea, reflexive metric space. 2010 Mathematics Subject Classification: 47H10, 47H09. 1. Introduction Given a metric space (X, d), a self-mapping Ton Xis called a contraction provided that there exists α∈[0,1) such that d(Tx, Ty)≤αd(x, y), for all x, y ∈X. The well-known Banach contraction principle states that if (X, d) is a complete metric space and Tis a contraction map, then Thas a unique fixed point. The Banach contraction principle is a very important tool in nonlinear analysis. Moreover, many extensions of this principle have arisen in the literature so far (see [20] for more information). In 1968, R. Kannan [18] introduced the so-called Kannan contraction mappings in order to investigate the existence and uniqueness of fixed point for such mappings. A mapping T:X→Xis said to be a Kannan contraction if there exists α∈[0,1 2) such that d(Tx, Ty)≤α[d(x, T x) + d(y, Ty)], for each x, y ∈X. It is known that if (X, d) is a complete metric space then every Kannan contraction mapping has a unique fixed point [18]. It is interesting to note that the notions of contraction and Kannan contraction are independent, that is, there exist contraction mappings which are not Kannan contractions and Kannan contraction mappings which are not contractions. Consequently, both conditions 63 64 A. FERN´ ANDEZ-LE ´ ON AND M. GABELEH cannot be compared directly. Moreover, contractions are always continuous while Kannan contractions may not be. On the other hand, in 1972, S. K. Chatterjea [8] introduced the so-called Chatterjea contractions with a similar purpose. A mapping T:X→Xis a Chatterjea contraction if there exists α∈[0,1 2) such that d(Tx, Ty)≤α[d(x, T y) + d(y, Tx)], for each x, y ∈X. In that occasion, existence and uniqueness of fixed point was also obtained for these mappings in the setting of complete metric spaces. In the last years, specifically from 2003 with [23] and 2005 with [10], certain extensions of the previous fixed point problems have arisen. In general terms, these extensions are based on considering a smaller set of points where the metric assumption on the mapping Tis fulfilled while certain condition of cyclic nature is assumed on the mapping under discussion. In particular, in this paper, we mainly work with the so-called noncyclic mappings. Given (A, B) a nonempty pair of subsets of a metric space (X, d), a mapping T:A∪B→A∪Bis called noncyclic provided that T(A)⊆Aand T(B)⊆B. When dealing with these mappings, it is interesting to ask whether it is possible to find (x∗, y∗)∈A×Bsuch that Tx∗=x∗, Ty∗=y∗and d(x∗, y∗) = dist(A, B),(1.1) where dist(A, B) := inf{d(x, y) : x∈A, y ∈B}. Notice that such a pair of points (x∗, y∗) is actually a solution of the following minimization problem: Find (x, y)∈ A×Bsuch that min x∈Ad(x, Tx),min y∈Bd(y, T y) and min (x,y)∈A×Bd(x, y).(1.2) Thus, a point (x∗, y∗)∈A×Bis said to be a best proximity pair for the noncyclic mapping T:A∪B→A∪Bif it is an absolute optimal solution of the minimization problem (1.2), i.e., if it is a point satisfying (1.1). Notice that in [1] the authors studied sufficient conditions to ensure the existence of solutions of the nonlinear programming problem (1.2). Another condition of cyclic nature that has been considered in the literature is the one of cyclic mapping. A mapping T:A∪B→A∪Bis called cyclic if T(A)⊆B and T(B)⊆A. In this case, the fixed point equation Tx =xmay make no sense. Consequently, the interest of the theory focuses then on the study of the existence of best proximity points, that is, points x∗∈A∪Bsuch that d(x∗, Tx∗) = dist(A, B). In the current article, we first give some best proximity pair theorems in Banach spaces for various classes of noncyclic mappings such as noncyclic contractions, noncyclic contractions in the sense of Kannan, noncyclic contractions in the sense of Chatterjea and strongly noncyclic relatively nonexpansive mappings in the sense of Chatterjea. Finally, we devote Section 5 to show that the results proved in the previous sections also hold in metric settings that are more general. 2. Preliminaries Let (A, B) be a nonempty pair of subsets in a metric space (X, d). We say that the pair (A, B) satisfies a property if both Aand Bsatisfy that property. For example, (A, B) is closed if and only if both Aand Bare closed. Likewise, (A, B)⊆(C, D) BEST PROXIMITY PAIRS FOR NONCYCLIC MAPPINGS 65 if and only if A⊆Cand B⊆D. Throughout this paper we shall use the following notations and definitions: δx(A) := sup{d(x, y) : y∈A}for all x∈X, δ(A, B) := sup{δx(B) : x∈A},diam(A) := δ(A, A). From now on, B(a;r) denotes the closed ball in the space Xcentered at a∈Xwith radius r > 0. The proximal pair of a given pair (A, B) of nonempty subsets of Xis the pair (A0, B0) given by A0={x∈A:d(x, y0) = dist(A, B) for some y0∈B}, B0={y∈B:d(x0, y) = dist(A, B) for some x0∈A}. Proximal pairs may be empty but, in particular, if Aand Bare nonempty, weakly compact and convex in a Banach space X, then (A0, B0) is a nonempty weakly compact convex pair in X. A pair of sets (A, B) is said to be proximal if A=A0 and B=B0. In fact, given x∈A, we say that y∈Bis a proximal point of xif d(x, y) = dist(A, B). Similarly, we can define the proximal point of y∈B. The closed convex hull of a set Ain a Banach space Xwill be denoted by con(A). From now on in the paper, unless otherwise stated, we will use the notation din a linear setting to refer to the induced metric by the norm in a Banach space X. Definition 2.1. A Banach space Xis said to be strictly convex if the following implication holds for every x, y, p ∈Xand R > 0:      d(x, p)≤R, d(y, p)≤R, x6=y ⇒dx+y 2, p< R. In the last section, we will mainly work in the setting of geodesic metric spaces. A metric space (X, d) is said to be a (uniquely) geodesic space if every two points x and yof Xare joined by a (unique) geodesic, i.e, a map c: [0, l]⊆R→Xsuch that c(0) = x,c(l) = y, and d(c(t), c(t0)) = |t−t0|for all t, t0∈[0, l]. A subset Aof a geodesic space Xis said to be convex if the image of any geodesic that joins each pair of points xand yof A(geodesic segment [x, y]) is contained in A. A point z in Xbelongs to a geodesic segment [x, y] if and only if there exists t∈[0,1] such that d(x, z) = td(x, y) and d(y, z) = (1 −t)d(x, y) and we write z= (1 −t)x+ty for simplicity. Notice that this point may not be unique. When t=1 2, we often use the notation x+y 2to denote 1 2x+1 2y. Any Banach space is for instance a geodesic space with usual segments as geodesic segments. A geodesic metric space Xis said to be reflexive if for every decreasing chain {Cα} ⊂ Xwith α∈Isuch that Cαis closed convex bounded and nonempty for all α∈Iwe have that \ α∈I Cα6=∅. It is immediate to see that reflexive metric spaces extend the notion of reflexivity from Banach to metric spaces. It is wellknown that every complete uniformly convex metric space with either a monotone or lower semicontinuous from the right modulus of uniform convexity is reflexive (see for 66 A. FERN´ ANDEZ-LE ´ ON AND M. GABELEH instance [24, 13] for more information about uniform convexity in geodesic spaces). Other well-known examples of this type of spaces are complete CAT(0) [4] spaces or uniformly convex Banach spaces. Let (X, d) be a uniquely geodesic space. A metric d:X×X→Ris said to be convex if for any x, y, z ∈Xone has d(x, (1 −t)y+tz)≤(1 −t)d(x, y) + td(x, z) for all t∈[0,1]. A geodesic space (X, d) is Busemann convex (introduced in [7]) if given any pair of geodesics c1: [0, l1]→Xand c2: [0, l2]→Xone has d(c1(tl1), c2(tl2)) ≤(1 −t)d(c1(0), c2(0)) + td(c1(l1), c2(l2)) for all t∈[0,1]. It is well-known that Busemann convex spaces are strictly convex [14] and with convex metric. A metric space is said to be strictly convex (see [3] for more on this property) if Xis a geodesic space and for every r > 0, a, x and y∈Xwith d(x, a)≤r, d(y, a)≤rand x6=y, it is the case that d(a, p)< r, where pis any point between xand ysuch that p6=xand p6=y, i.e., pis any point in the interior of a geodesic segment that joins xand y. It is immediate that every strictly convex metric space is uniquely geodesic. Also notice that a reflexive and Busemann convex geodesic space is complete (see [15, Lemma 4.1]). 3. Noncyclic contractive type mappings In [23, 25], some fixed point results were given for certain cyclic mappings T: A∪B→A∪Bunder contractive conditions. Notice that a mapping T:A∪ B→A∪Bis cyclic if T(A)⊆Band T(B)⊆A. Many generalizations of those conditions have been considered to study the problem of finding best proximity points in absence of fixed points. For instance, the so-called cyclic contractions [12], weak cyclic Kannan contractions [26] or cyclic Meir-Keeler contractions [9], among others, have been studied. However noncyclic mappings have been much less studied in the literature. A mapping T:A∪B→A∪Bis called noncyclic if T(A)⊆A and T(B)⊆B. In the sequel we give some comments and results regarding these noncyclic natural extensions. First we point out that, for this noncyclic approach, we mainly focus on finding best proximity pairs, i.e., pairs (x, y)∈A×Bsuch that Tx =x, Ty =yand d(x, y) = dist(A, B). The first case that is natural to study in the noncyclic context is the following. Proposition 3.1. Let Aand Bbe two nonempty closed subsets of a complete metric space (X, d). Assume that T:A∪B→A∪Bis a noncyclic mapping such that for some α∈(0,1) d(Tx, Ty)≤αd(x, y),for all (x, y)∈A×B. Then dist(A, B)=0. Moreover, the mapping Thas a fixed point in A∪Bif and only if A∩B6=∅. Proof. Let {xn},{yn}be sequences in Aand Brespectively such that d(xn, yn)→ dist(A, B). Then dist(A, B)≤d(T xn, T yn)≤αd(xn, yn). Taking limit when ntends to infinity, we see that necessarily dist(A, B) = 0. BEST PROXIMITY PAIRS FOR NONCYCLIC MAPPINGS 67 Suppose first that A∩B6=∅. If we apply the Banach contraction principle in A∩B, we get that there exists a fixed point of the mapping Tthat in fact is unique in A∩B. On the other hand, suppose that Thas a fixed point y∗in A∪B. Without loss of generality, suppose that y∗∈B. Then, given a point x0∈A, if we denote xn=Tnx0, we have that d(xn, y∗)≤αd(xn−1, y∗)≤αnd(x0, y∗). Consequently, we get that {xn}converges to y∗. Since Ais closed, y∗∈A∩Band the result follows.  In 2005, the more general case of α= 1 in the previous proposition was taken under consideration [10]. In this regarding, the notion of proximal normal structure was introduced. Definition 3.2. A convex pair (K1, K2)in a Banach space Xis said to have proximal normal structure if for any bounded closed convex and proximal pair (H1, H2)⊆ (K1, K2)for which dist(H1, H2) = dist(K1, K2)and δ(H1, H2)>dist(H1, H2), there exits (x1, x2)∈H1×H2such that δx1(H2)< δ(H1, H2), δx2(H1)< δ(H1, H2). Note that if in the above definition K1=K2, then we get the notion of normal structure introduced by Brodski and Milman [5]. Also in [10], it was announced that every nonempty bounded closed convex pair of subsets of a uniformly convex Banach space Xhas proximal normal structure. The next theorem, established by Eldred, Kirk and Veeramani in [10], is the main result in the cited paper concerning the so-called noncyclic relatively nonexpansive mappings. A mapping T:A∪B→A∪Bis a noncyclic relatively nonexpansive mapping if T is noncyclic on A∪Band d(Tx, Ty)≤d(x, y) for all (x, y)∈A×B. Theorem 3.3. (Theorem 2.2 in [10]) Let (A, B)be a nonempty, weakly compact convex pair in a strictly convex Banach space Xand T:A∪B→A∪Ba noncyclic relatively nonexpansive mapping. Assume that the pair (A, B)has proximal normal structure. Then Thas a best proximity pair. One year later [12], a weaker condition of contractive type was considered, but only for cyclic mappings. The resultant cyclic mappings were called cyclic contractions and in particular satisfy d(Tx, Ty)≤αd(x, y) + (1 −α)dist(A, B), for some α∈(0,1) and for every x∈Aand y∈B. In [2], a best proximity pair result has been given for the counterpart noncyclic mappings. This result states that if A and Bare two nonempty and weakly compact convex sets in a strictly convex Banach space, then any noncyclic contraction, i.e., any noncyclic mapping T:A∪B→A∪B such that d(Tx, Ty)≤αd(x, y) + (1 −α)dist(A, B) for every x∈Aand y∈B, has a best proximity pair. In that occasion, the result was proved by using Zorn’s Lemma. Next we give a proof of this result without using this lemma. It is remarkable that the hypothesis on compactness for the sets Aand Bare removed. 68 A. FERN´ ANDEZ-LE ´ ON AND M. GABELEH First, we need the following result on the boundedness of the iterates of the mapping T. Proposition 3.4. Let Aand Bbe nonempty subsets of a metric space X. Suppose that T:A∪B→A∪Bis a noncyclic contraction. Let x0∈A,y0∈B,xn=Tnx0 and yn=Tny0for every n∈N. Then limnd(xn, yn) = dist(A, B).Moreover, {xn} and {yn}are bounded. Proof. The proof follows similar patterns to those considered in [12] in Proposition 3.3. The fact that limnd(xn, yn) = dist(A, B) is immediate. Then, it suffices to see that xn=Tnx0is bounded. Suppose that {xn}is not bounded. Then let Mbe a natural number such that M≥max α (1 −α)d(y0, Ty0) + dist(A, B), d(Ty0, x0), where αis the contraction constant. For this natural number, there exists N0∈N such that d(Ty0, TN0x0)> M and d(Ty0, TN0−1x0)≤M. Thus, M−dist(A, B) α+ dist(A, B)< d(y0, TN0−1x0) ≤d(y0, Ty0) + d(Ty0, TN0−1x0)≤d(y0, Ty0) + M, which is a contradiction and the result follows.  Next we give the following best proximity pair result for noncyclic contractions. Theorem 3.5. Let Aand Bbe two closed convex subsets of a strictly convex and reflexive Banach space. Suppose that T:A∪B→A∪Bis a noncyclic contraction. Then Thas a best proximity pair. Proof. Let d∗(x, y) := d(x, y)−dist(A, B). Then d∗(Tx, Ty)≤αd∗(x, y) for every x∈Aand y∈B. Let x0and y0be two arbitrary but fixed points in Aand B respectively. Fix m0∈Nand let m=m0+kwith k∈N. Then d∗(Tmx0, Tm0y0)≤αm0d∗(Tkx0, y0) ≤αm0sup{d(Tkx0, y0) : k∈N}=αm0M(x0, y0). Since the orbits of a noncyclic contraction map are bounded at every point z∈ A∪B, we have that, for every ε > 0, there exists m0(ε)∈Nsuch that Tmx0∈ B(Tm0(ε)y0; dist(A, B) + ε) for every m≥m0(ε). In a similar way, d∗(Tmx0, Tm0+1y0)≤αm0d∗(Tkx0, Ty0) ≤αm0sup{d(Tkx0, Ty0) : k∈N}=αm0M0(x0, y0). Let εn= 1/n. Then there is m0(εn)∈Nsuch that Tmx0∈B(Tm0(εn)y0; dist(A, B)+ εn) and Tmx0∈B(Tm0(εn)+1y0; dist(A, B)+εn) for every m≥m0(εn). Denote Bn= B(Tm0(εn)y0; dist(A, B)+εn) and B0 n=B(Tm0(εn)+1y0; dist(A, B)+εn) and consider BEST PROXIMITY PAIRS FOR NONCYCLIC MAPPINGS 69 the sequence of subsets {Cn}of Xsuch that C1=A∩B1∩B0 1and Cn=Cn−1∩Bn∩B0 n for every n≥2. By definition, it is immediate to see that, for every n∈N,Cnis closed, convex, bounded and nonempty. Then, by means of the reflexivity of the space, we conclude that \ n∈N Cn6=∅. Let p∈\ n∈N Cn⊂A. Taking m0(εn) as an increasing sequence depending on n, we have that the sequences of terms zn=Tm0(εn)y0and z0 n= Tm0(εn)+1y0for every n∈Nare subsequences of {Tny0}. Consequently, since p∈Cn for every n∈N,d(zn, p)→dist(A, B) and d(z0 n, p)→dist(A, B). Besides, we have that d(z0 n, Tp) = d(Tzn, Tp)→dist(A, B). Since {z0 n}is a bounded sequence, it has a weakly convergent subsequence to a point q∈B. Then, by using the lower semicontinuity of the norm, we have that d(q, p) = d(q, Tp) = dist(A, B), which implies in a strictly convex space that p=T p. Moreover, since d(Tp, Tq) = d(p, q) = d(Tp, q) = dist(A, B), we also get that q=Tq and the result follows.  Remark 3.6. The assumption of reflexivity for the space Xcan be dropped if Aand Bare supposed to be weakly compact and nonempty. The next example shows that the hypothesis on convexity and completeness for the sets Aand Bare necessary conditions in the previous theorem. Example 3.7. Let k∈(0,1) and let Aand Bthe subsets of the space `p,1≤p≤ ∞, defined as A={((1 + k2n)e2n) : n∈N}and B={((1 + k2m+1)e2m+1) : m∈N}. It is easy to see that dist(A, B) = 21/p in `p. Consider the mapping T:A∪B→A∪B defined by Tx =((1 + k2m+2)e2m+2 if x= (1 + k2m)e2mfor some m≥1 (1 + k2n+1)e2n+1 if x= (1 + k2n−1)e2n−1for some n≥1. Then Tis noncyclic on A∪B. Moreover, Tis a noncyclic contraction mapping. We omit the proof of this fact since it follows similar patterns to those considered in Example 3.6 in [12]. As it was pointed out in that paper, A0=B0=∅and therefore Tdoes not have any best proximity pair. In fact, notice that Tdoes not even have fixed points. In 2011, certain existence results on best proximity points were given for the socalled weak cyclic Kannan contractions [26]. It seems natural to analyse what that weak Kannan condition implies in the noncyclic case. To establish our results, we introduce the following new class of noncyclic maps. Definition 3.8. Let (A, B)be a nonempty pair of subsets of a metric space (X, d). A mapping T:A∪B→A∪Bis said to be a noncyclic contraction in the sense of Kannan if Tis noncyclic on A∪Band there exists α∈[0,1 2)such that d(Tx, Ty)≤α[d(x, T x) + d(y, Ty)] + (1 −2α)dist(A, B), for all (x, y)∈A×B. It is clear that if in the above definition A=B, then we get the notion of Kannan contraction [18]. Next we show that, unlike the cyclic case, this contractive condition seems to be very restrictive in the noncyclic case. 70 A. FERN´ ANDEZ-LE ´ ON AND M. GABELEH Proposition 3.9. Let (A, B)be a nonempty pair of subsets of a complete metric space (X, d). Let T:A∪B→A∪Bbe a noncyclic contraction in the sense of Kannan. Then Thas a best proximity pair (x, y)∈A×Bif and only if A∩B6=∅. Proof. Suppose that (x, y)∈A×Bis a best proximity pair of T, that is, d(x, y) = dist(A, B) and x=Tx,y=Ty. Since Tis a noncyclic contraction in the sense of Kannan, we have that dist(A, B) = d(x, y) = d(T x, Ty)≤ ≤α[d(x, Tx) + d(Ty, y)] + (1 −2α)dist(A, B) = (1 −2α)dist(A, B), where αis the Kannan contraction constant of the mapping T. This implies that d(A, B) = 0. Moreover, since d(x, y) = dist(A, B), the result follows. The other implication holds directly by [18].  Next we give an example of a noncyclic contraction in the sense of Kannan that is neither noncyclic relatively nonexpansive nor a noncyclic contraction. Example 3.10. Let Xbe the Banach space Rendowed with the Euclidean norm. Let A:= [−1,1] and B:= [0,2].Define the noncyclic mapping T:A∪B→A∪Bas follows: Tx =(1 5if x6= 1, 0if x= 1. Then Tis noncyclic on A∪B. We now claim that Tis a noncyclic contraction in the sense of Kannan. Let x∈Aand y∈B. If x= 1, then |Tx −Ty |=1 5and |x−Tx |+|y−Ty |= 1+ |y−1 5|. Thus, |Tx −Ty |≤ α[|x−Tx |+|y−Ty |], for each α∈[1 6,1 2). Moreover, let y0=9 10 ∈Band x0= 1 ∈A. Then d(Tx0, Ty0) = 1 5> d(x0, y0) = 1 9. Thus Tis not relatively nonexpansive and therefore neither a noncyclic contraction. Also notice that y∗=1 5is a fixed point of T. As a consequence of Proposition 3.9, we see that the problem of finding a best proximity pair for these mappings reduces to find a fixed point in the intersection of Aand B. Next we consider a metric condition on Aand Bthat guarantees the existence of best proximity pairs for T, or, in other words, that guarantees the nonempty intersection of Aand Bwhen there is a noncyclic contraction in the sense of Kannan defined on them. Definition 3.11. A pair (A, B)of subsets of a metric space Xsatisfies property (H) provided that for every nonempty closed convex bounded pair (K1, K2)⊆(A, B)we have max{diam(K1), diam(K2)} ≤ δ(K1, K2). Remark 3.12. Notice that if a pair of nonempty and convex sets (A, B)satisfies property (H), then the intersection of Aand Bcontains as most one point. BEST PROXIMITY PAIRS FOR NONCYCLIC MAPPINGS 71 Theorem 3.13. Let (A, B)be a nonempty weakly compact convex pair in a strictly convex Banach space X. Suppose that T:A∪B→A∪Bis a noncyclic contraction in the sense of Kannan. If the pair (A, B)has property (H), then Thas a unique fixed point in A∩B. Proof. Let Σ denote the collection of all nonempty weakly compact convex pairs (E, F ) which are subsets of (A, B) and such that Tis noncyclic on E∪F. Since (A, B)∈Σ, then Σ is nonempty. Note that Σ is partially ordered by the reverse inclusion, that is, (A, B)≤(C, D)⇔(C, D)⊆(A, B). By the fact that (A, B) is a weakly compact and convex pair in X, every increasing chain in Σ is bounded above. Hence, by using Zorn’s lemma we can get a maximal element, say (K1, K2)∈Σ, that is minimal respect to the set inclusion. We have (con(T(K1)), con(T(K2))) ⊆(K1, K2). Moreover, T(con(T(K1))) ⊆T(K1)⊆con(T(K1)), and also T(con(T(K2))) ⊆con(T(K2)). That is, Tis noncyclic on con(T(K1)) ∪con(T(K2)). Now, by the minimality of (K1, K2), we have con(T(K1)) = K1, con(T(K2)) = K2. Let a∈K1. Then K2⊆ B(a;δa(K2)). Now, if y∈K2, then by the fact that (A, B) has property (H) we obtain d(Ta, Ty)≤α{d(a, T a) + d(Ty, y)}+ (1 −2α)dist(A, B) ≤2αmax{diam(K1),diam(K2)}+ (1 −2α)dist(A, B) ≤2αδ(K1, K2) + (1 −2α)dist(A, B). So, for each y∈K2, we have T(K2)⊆B(Ta; 2αδ(K1, K2) + (1 −2α)dist(A, B)). Hence, K2=con(T(K2)) ⊆B(Ta; 2αδ(K1, K2) + (1 −2α)dist(A, B)). This implies that d(y, T a)≤2αδ(K1, K2) + (1 −2α)dist(A, B) for all y∈K2, and then δT a(K2)≤2αδ(K1, K2) + (1 −2α)dist(A, B).(3.1) Similarly, if b∈K2, we see that δT b(K1)≤2αδ(K1, K2) + (1 −2α)dist(A, B).(3.2) Set E1:= {x∈K1:δx(K2)≤2αδ(K1, K2) + (1 −2α)dist(A, B)}, E2:= {y∈K2:δy(K1)≤2αδ(K1, K2) + (1 −2α)dist(A, B)}. 78 A. FERN´ ANDEZ-LE ´ ON AND M. GABELEH Proof. First note that (A0, B0) is a nonempty weakly compact convex proximal pair and dist(A, B) = dist(A0, B0) (see [22] for more details). Let Σ denote the collection of all nonempty, weakly compact convex pairs (C, D)⊆(A, B) for which dist(C, D) = dist(A, B), (C, D) is a proximal pair and Tis noncyclic on C∪D. Since (A0, B0)∈Σ, Σ is nonempty. Proceeding as in [17], we can apply Zorn’s lemma to prove that Σ has a minimal element with respect to the set inclusion, say (K1, K2), that is a proximal pair and satisfies dist(K1, K2) = dist(A, B). We may observe that if δ(K1, K2) = dist(K1, K2), then the strict convexity of Ximplies that K1and K2are singleton and the conclusion trivially holds. Now, suppose δ(K1, K2)>dist(K1, K2). It now follows from the fact that (A, B) has proximal normal structure that there exist (p1, p2)∈K1×K2and r∈(0,1) such that δp1(K2)≤rδ(K1, K2), δp2(K1)≤rδ(K1, K2). Since (K1, K2) is a proximal pair, there exists a pair (q1, q2)∈K1×K2such that d(p1, q2) = d(q1, p2) = dist(K1, K2). Now, for each y∈K2we have     p1+q1 2−y    ≤kp1−yk 2+kq1−yk 2 ≤rδ(K1, K2) 2+δ(K1, K2) 2=r+ 1 2δ(K1, K2). If we set p:= p1+q1 2∈K1and h:= r+1 2, we obtain δp(K2)≤hδ(K1, K2) with h∈(0,1). Similarly, if we set q:= p2+q2 2∈K2, then δq(K1)≤hδ(K1, K2). Moreover, d(p, q) = dist(K1, K2). Let H1={x∈K1:δ(x, K2)≤αδ(K1, K2) and for its proximal point y∈K2, δ(y, K1)≤αδ(K1, K2)}, H2={y∈K2:δ(y, K1)≤αδ(K1, K2) and for its proximal point x∈K1, δ(x, K2)≤αδ(K1, K2)}. Since p∈H1and q∈H2,Hi6=∅for i= 1,2. Next we show that Hiis closed and convex for i= 1,2. We give the details for H1. For H2the proof follows similar patterns. Let {vn} ⊆ H1be a sequence that converges to a point v∈K1. Since d(vn, z)≤αδ(K1, K2) for every n∈Nand z∈K2, we get δ(v, K2)≤αδ(K1, K2). The fact that vn∈H1implies δ(wn, K1)≤αδ(K1, K2),(4.3) where wn∈K2is the proximal point of vn∈K1. Let w∈K2such that d(v, w) = dist(K1, K2). Then we get dvn+v 2,wn+w 2= dist(K1, K2). Using the notion of parallelism in Banach spaces, we may prove that [wn, vn] and [w, v] are parallel segments and therefore we have that d(wn, w) = d(vn, v) for n∈N, from where wn→w(for more on this property, see Proposition 5.7 that gives the same result for more general metric spaces). Taking limit in (4.3), we may conclude δ(w, K1)≤αδ(K1, K2). Consequently v∈H1and so H1is closed. Now, let p1, q1∈ BEST PROXIMITY PAIRS FOR NONCYCLIC MAPPINGS 79 H1. Next we see that the mid-point between p1and q1,m1=p1+q1 2, is in H1. Let p2, q2∈K2be the proximal points of p1and q1, respectively. Consider m2= p2+q2 2. Since (K1, K2) is proximal and the space Xis Busemann convex, d(m1, m2) = dist(K1, K2). Let z∈K2. Then d(z, m1)≤1 2d(p1, z) + 1 2d(q1, z)≤αδ(K1, K2). Thus, δ(m1, K2)≤αδ(K1, K2). The fact that δ(m2, K1)≤αδ(K1, K2) follows similarly since δ(p2, K1) and δ(q2, K1) are both ≤αδ(K1, K2). Then m1∈H1and so H1is convex. Notice that (H1, H2) is also a proximal pair and dist(H1, H2) = dist(K1, K2). Next we see that Tis noncyclic on H1∪H2. Let x∈H1and y∈H2such that d(x, y) = dist(H1, H2). We prove that T x ∈H1. Let z∈K2. Since d(Tx, Tz)≤ d(x, Tz)≤δx(T(K2)) ≤δx(K2)≤hδ(K1, K2), we get T(K2)⊆B(Tx;hδ(K1, K2)) ∩K2:= K0 2. Then K0 2is closed, convex and nonempty. Let K0 1⊆K1be the set K0 1={x∈K1: there exists y∈K0 2with d(x, y) = dist(K1, K2)}. Then K0 1is closed, convex and nonempty. Moreover, (K0 1, K0 2) is proximal and satisfies dist(K0 1, K0 2) = dist(K1, K2), T(K0 1)⊆K0 1and T(K0 2)⊆K0 2. Therefore, K0 1∪K0 2∈Σ and by minimality of Kit follows that K2⊆B(Tx, hδ(K1, K2)) and therefore δ(Tx, K2)≤hδ(K1, K2). Proceeding similarly, we may see that δ(Ty, K1)≤hδ(K1, K2). Since Ty ∈K2is the proximal point of Tx ∈K1, we conclude that Tx ∈H1and therefore T(H1)⊆H1. Similarly, T(H2)⊆H2. As a consequence, H1∪H2∈Σ. Since, δ(H1, H2)≤αδ(K1, K2), we get a contradiction with the minimality of K. Next we see that Tis noncyclic on H1∪H2. Let x∈H1and y∈H2such that d(x, y) = dist(H1, H2). We prove that Tx ∈H1. Let z∈K2. Since d(Tx, Tz)≤ d(x, Tz)≤δx(T(K2)) ≤δx(K2)≤hδ(K1, K2), we get T(K2)⊆B(Tx;hδ(K1, K2)) ∩K2:= K0 2. Then K0 2is closed, convex and nonempty. Let K0 1⊆K1be the set K0 1={x∈K1: there exists y∈K0 2with d(x, y) = dist(K1, K2)}. Then K0 1is closed, convex and nonempty. Moreover, (K0 1, K0 2) is proximal and satisfies dist(K0 1, K0 2) = dist(K1, K2), T(K0 1)⊆K0 1and T(K0 2)⊆K0 2. Therefore, K0 1∪K0 2∈Σ and by minimality of Kit follows that K2⊆B(Tx, hδ(K1, K2)) and therefore δ(Tx, K2)≤hδ(K1, K2). Proceeding similarly, we may see that δ(Ty, K1)≤hδ(K1, K2). Since Ty ∈K2is the proximal point of Tx ∈K1, we conclude that Tx ∈H1and therefore T(H1)⊆H1. Similarly, T(H2)⊆H2. As a consequence, H1∪H2∈Σ. Since, δ(H1, H2)≤αδ(K1, K2), we get a contradiction with the minimality of K. Remark 4.8. Aside from the cases where Tis relatively nonexpansive, there are some other natural conditions on a noncyclic mapping Tthat guarantee that T(A0)⊆A0 and T(B0)⊆B0. For instance, we cite the condition of being a “noncylic relatively 80 A. FERN´ ANDEZ-LE ´ ON AND M. GABELEH u-continuous” mapping, which derives from the concept of “relatively u-continuous mapping” introduced in [11] for cyclic mappings. Since every pair of nonempty, bounded, closed and convex subsets of a uniformly convex Banach space has proximal normal structure, we get the following corollary. Corollary 4.9. Let (A, B)be a nonempty bounded closed convex pair in a uniformly convex Banach space X. Suppose that T(A0)⊆A0and T(B0)⊆B0. If Tis strongly noncyclic relatively nonexpansive in the sense of Chatterjea, then Thas a best proximity pair. Now we raise the next problem. Question 4.10. It is interesting to ask whether Theorem 4.7 holds whenever Tis noncyclic relatively nonexpansive in the sense of Chatterjea. We finish this section by making the following reflection. In [2], the Kannan type condition d(Tx, Ty)≤1 2[d(x, Tx) + d(y, T y)] was considered in the cyclic setting. At that moment, the notion of proximal quasi-normal structure is defined to get an existence result of best proximity points. Note that the concept of proximal quasinormal structure is the analog to the notion of quasi-normal structure that is used in the literature to prove a fixed point result for mappings T:X→Xsatisfying this Kannan type condition for every x, y ∈X[20]. Similar techniques to those applied for the cyclic case in [2] seem not to be useful to solve the noncyclic case and therefore, the solution to such a problem is still an open question. 5. Several consequences in geodesic spaces When the existence of best proximity pairs and best proximity points is studied in the previous sections for certain mappings, it is natural to wonder whether similar results hold in metric settings that are more general. In this regarding, we thoroughly analyse the proofs given above and show that, in most of the cases, the underlying linear structure is not necessary to get the results. In particular, we consider the setting of reflexive metric spaces that are either Busemann convex or at least strictly convex (see the section of preliminaries for definitions). First we see that the counterpart of Theorem 3.5 holds in geodesic spaces. For this aim, we recall the following property between sets given in [16]. This property was introduced in the cited paper to give the analog result in the cyclic case. Definition 5.1. Let Aand Bbe two nonempty subsets of a metric space X. The pair (A, B)is said to satisfy property HW if for every sequence {xn} ⊂ Aand {zn} ⊂ B and every two points q∈Aand p∈Bwe have that d(xn, p)→dist(A, B) d(xn, zn)→dist(A, B) d(zn, q)→dist(A, B)   ⇒d(p, q) = dist(A, B). Theorem 5.2. Let Aand Bbe two closed convex subsets of a strictly convex and reflexive metric space and T:A∪B→A∪Ba noncyclic contraction. If (A, B)has property HW then Thas a best proximity pair. BEST PROXIMITY PAIRS FOR NONCYCLIC MAPPINGS 81 Proof. This result follows by applying similar patterns as in the proof of Theorem 3.5. However, in absence of weak convergence, several changes must be considered to get the result. As in the linear case, let d∗(x, y) := d(x, y)−dist(A, B). Then d∗(Tx, Ty)≤ αd∗(x, y) for every x∈Aand y∈B. Let x0and y0be two arbitrary but fixed points in Aand Brespectively. Fix m0∈Nand let m=m0+kwith k∈N. Then d∗(Tmx0, Tm0y0)≤αm0d∗(Tkx0, y0) ≤αm0sup{d(Tkx0, y0) : k∈N}=αm0M(x0, y0). In a similar way, we have that d∗(Tmx0, Tm0+1y0)≤αm0sup{d(Tkx0, Ty0) : k∈N}=αm0M0(x0, y0), d∗(Tmy0, Tm0x0)≤αm0sup{d(Tky0, x0) : k∈N}=αm0M00(x0, y0) and d∗(Tmy0, Tm0+1x0)≤αm0sup{d(Tky0, Tx0) : k∈N}=αm0M000(x0, y0). Since the proof of Proposition 3.4 is purely metric, we have that the orbits of T are bounded at every point z∈A∪B. Consequently, for every n∈N, there exists m0(n)∈Nsuch that for every m≥m0(n), Tmx0∈B(Tm0(n)y0; dist(A, B) + 1/n), Tmx0∈B(Tm0(n)+1y0; dist(A, B)+ 1/n), Tmy0∈B(Tm0(n)x0; dist(A, B) +1/n) and Tmy0∈B(Tm0(n)+1x0; dist(A, B)+1/n). Denote Bn=B(Tm0(n)y0; dist(A, B)+1/n), B0 n=B(Tm0(n)+1y0; dist(A, B) + 1/n), An=B(Tm0(n)x0; dist(A, B)+1/n) and A0 n=B(Tm0(n)+1x0; dist(A, B)+1/n). Consider the sequence of subsets {Cn}and {Dn}of Xsuch that C1=A∩B1∩B0 1, D1=B∩A1∩A0 1,Cn=Cn−1∩Bn∩B0 nand Dn=Dn−1∩An∩A0 nfor every n≥2. By definition, it is immediate to see that Cnand Dnare closed, bounded and nonempty for every n∈N. Since Xis strictly convex, the balls in Xare convex and so Cnand Dnare convex for every n∈N. Thus, by means of the reflexivity of X, we conclude that \ n∈N Cn6=∅and \ n∈N Dn6=∅. Let p∈\ n∈N Cn⊂Aand q∈ \ n∈N Dn⊂B. Taking m0(n) as an increasing sequence on n, we have that the sequences of terms zn=Tm0(n)y0and z0 n=Tm0(n)+1y0for every n∈Nare subsequences of {Tny0}. Similarly, wn=Tm0(n)x0and w0 n=Tm0(n)+1x0are subsequences of {Tnx0}. Consequently, since p∈Cnand q∈Dnfor every n∈N, (i)d(zn, p)→ dist(A, B), (ii)d(z0 n, p) = d(Tzn, p)→dist(A, B), (iii)d(wn, q)→dist(A, B), (iv) d(w0 n, q) = d(T wn, q)→dist(A, B), (v)d(z0 n, Tp) = d(Tzn, Tp)→dist(A, B) and (vi) d(Twn, Tq)→dist(A, B). Moreover, since Tis a noncyclic contraction, d(Tnx, Tny)→dist(A, B) for every x∈Aand y∈B. Thus, (vii)d(wn, zn) = d(Tm0(n)x0, Tm0(n)y0)→dist(A, B) and (viii)d(T wn, T zn)→dist(A, B). If we apply now property HW to (i),(iii) and (vii), we get d(q, p) = dist(A, B). Considering now the same property for (iv),(v) and (viii), we get d(p, Tp) = dist(A, B). Since Xis strictly convex, we get p=T p. In a similar way we get that q=Tq and the result follows.  82 A. FERN´ ANDEZ-LE ´ ON AND M. GABELEH Remark 5.3. Notice that the previous theorem trivially extends Theorem 3.5 since every pair (A, B)of closed, convex and nonempty subsets of a strictly convex and reflexive Banach space has property HW (see Proposition 3.9 in [16]). Next we give the counterpart to Theorem 3.13 in geodesic spaces. Specifically, we give, in terms of reflexivity in metric spaces, the counterpart to Corollary 3.16. First, notice that both the notion of convex hull of a set and Definition 3.11 may be trivially extended to metric spaces with a convexity structure. Theorem 5.4. Let (A, B)be a nonempty closed convex bounded pair in a strictly convex and reflexive metric space X. Suppose that T:A∪B→A∪Bis a noncyclic contraction in the sense of Kannan. If the pair (A, B)has property (H), then Thas a unique fixed point in A∩B. Proof. This result follows by applying similar patterns as in the proof of Theorem 3.13. Consequently, we just give a sketch of the proof and comment the steps where we find the main differences. Let Σ denotes the collection of all nonempty closed convex bounded pairs (E, F) which are subsets of (A, B) and such that Tis noncyclic on E∪F. Proceeding as in Theorem 3.13, we may use the reflexivity of Xto find a minimal element, say (K1, K2)∈Σ. By using the counterpart concept of convex hull in geodesic spaces, we get similarly that Tis noncyclic on con(T(K1)) ∪con(T(K2)). Now, by the minimality of (K1, K2), we have that con(T(K1)) = K1, con(T(K2)) = K2. Reasoning as in linear case, we define the sets E1:= {x∈K1:δx(K2)≤2αδ(K1, K2) + (1 −2α)dist(A, B)}, E2:= {y∈K2:δy(K1)≤2αδ(K1, K2) + (1 −2α)dist(A, B)}. Notice that E1and E2are nonempty and closed. Moreover, since E1=\ y∈K2 B(y; 2αδ(K1, K2) + (1 −2α)dist(A, B)) ∩K1, E2=\ x∈K1 B(x; 2αδ(K1, K2) + (1 −2α)dist(A, B)) ∩K2, the convexity of the balls in Ximplies that (E1, E2) is convex. Proceeding now as in the linear case we get that δ(K1, K2) = dist(A, B). Thus, given (x, y)∈K1×K2we have d(x, Ty) = d(T y, Tx) = dist(A, B). Then, if T x+x 2is the mid-point between xand Tx in the uniquely geodesic space X, we may use the strict convexity of Xto get dist(A, B)≤dTy, T x +x 2<dist(A, B), which is a contradiction. Then, the unique pair of points (x, y)∈K1×K2is a best proximity pair of T. Consequently, from Proposition 3.9, A∩B6=∅. The uniqueness comes from the fact that the fixed point in the intersection is unique.  BEST PROXIMITY PAIRS FOR NONCYCLIC MAPPINGS 83 In the context of noncyclic and cyclic contractions in the sense of Chatterjea, we may proceed in a similar way to get the following results. Notice first that both Propositions 3.22 and 3.23 also hold when the space under consideration is a strictly convex metric space. Theorem 5.5. Let (A, B)be a closed convex bounded pair in a strictly convex and reflexive metric space X. Suppose that T:A∪B→A∪Bis a noncyclic contraction in the sense of Chatterjea. Then Thas a best proximity pair (x, y)∈A×B. Theorem 5.6. Let (A, B)be a closed convex bounded pair in a strictly convex and reflexive metric space X. Suppose that T:A∪B→A∪Bis a cyclic contraction in the sense of Chatterjea. If (A, B)has the (H) property, then Thas a best proximity point in A∩B. Finally we point out that the counterpart to Theorem 4.7 also holds in some geodesic spaces. In this case we have to assume Xto be Busemann convex. When dealing with cyclic and noncyclic mappings in the context of Busemann convex spaces, it is very useful the following property on parallel segments by H. Busemann [6]. A proof of this result is given in [17]. Proposition 5.7. Let x, y, z, w be four points in a Busemann convex geodesic space. Suppose that d(x, y) = d(m1, m2) = d(z, w), where m1and m2are the midpoints of the geodesic segments [x, z]and [y, w]. Then d(x, z) = d(m3, m4) = d(y, w), where m3and m4are the midpoints of [x, y]and [z, w]. The notion of proximal normal structure given in Banach spaces may be equally defined in a metric space with a convexity structure. Considering this concept in geodesic spaces we get the following result. Theorem 5.8. Let (A, B)be a nonempty closed convex bounded pair in a reflexive and Busemann convex metric space and suppose that (A, B)has proximal normal structure. Let T:A∪B→A∪Bbe a strongly noncyclic relatively nonexpansive mapping in the sense of Chatterjea such that T(A0)⊆A0and T(B0)⊆B0. Then T has a best proximity pair. It is not difficult to see that the proof of Theorem 4.7 also works for the previous result. We may observe that in that theorem we have mainly used the convexity structure of the Banach space, the convexity of the metric induced by the norm and the Busemann convexity of the space. In fact, thanks to Proposition 5.7, we may even get similar conclusions on parallel segments when dealing with these objects in Busemann convex metric spaces. Acknowledgements. Aurora Fern´andez-Le´on was partially supported by DGES, Grant MTM2012-34847-C02-01, and Junta de Andaluc´ıa, Grant P08-FQM-03453. Moosa Gabeleh was in part supported by a grant from the Institute for Research in Fundamental Sciences (IPM) (No. 93470047). References [1] A. Abkar, M. Gabeleh, Global optimal solutions of noncyclic mappings in metric spaces, J. Optim. Theory Appl., 153(2012), 298-305. 84 A. FERN´ ANDEZ-LE ´ ON AND M. GABELEH [2] A. Abkar, M. Gabeleh, Proximal quasi-normal structure and a best proximity point theorem, J. Nonlinear Convex Anal. (to appear). [3] G.C. Ahuja, T.D. Narang, S. Trehan, Best Approximation on Convex Sets in a Metric Space, J. Approx. Theory, 12(1974), 94-97. [4] M.R. Bridson, A. Haefliger, Metric Spaces of Non-positive Curvature, Springer-Verlag, Berlin Heidelberg, 1999. [5] M.S. Brodskii, D.P. Milman, On the center of a convex set, Dokl. Akad. Nauk. USSR, 59(1948), 837-840. [6] H. Busemann, Spaces with non-positive curvature, Acta Math., 80(1948), 259-310. [7] H. Busemann, The Geometry of Geodesics, Academic Press, New York, 1955. [8] S.K. Chatterjea, Fixed Point Theorems, C.R. Acad. Bulgare Sci., 25(1972), 727-730. [9] C. Di Bari, T. Suzuki, C. Vetro, Best proximity points for cyclic MeirKeeler contractions, Nonlinear Anal., 69(2008), 3790-3794. [10] A.A. Eldred, W.A. Kirk, P. Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Math., 171(2005), 283-293. [11] A.A. Eldred, V. Sankar Raj, P. Veeramani, On best proximity pair theorems for relatively ucontinuous mappings, Nonlinear Anal., 74(2011), 3870-3875. [12] A.A. Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl., 323(2006) 1001-1006. [13] R. Esp´ınola, A. Fern´andez-Le´on, B. Pi¸atek, Fixed points of single- and set-valued mappings in uniformly convex metric spaces with no metric convexity, Fixed Point Theory Appl., 2010 (2010), Article ID 169837, 16 pages. [14] R. Esp´ınola, B. Pi¸atek, Fixed point property and unbounded sets in CAT(0) spaces, Preprint. [15] R. Esp´ınola, A. Nicolae, Mutually nearest and farthest points of sets and the Drop Theorem in geodesic spaces, Monatsh. Math., 165(2012), 173-197. [16] A. Fern´andez-Le´on, Existence and uniqueness of best proximity points in geodesic metric spaces, Nonlinear Anal., 73(2010), 915-921. [17] A. Fern´andez-Le´on, A. Nicolae, Relatively nonexpansive mappings in geodesic spaces, submitted for publication. [18] R. Kannan, Some results on fixed points, Bull. Calcutta. Math. Soc., 10(1968), 71-76. [19] E. Karapinar, H.K. Nashine, Fixed Point Theorem for Cyclic Chatterjea Type Contractions, J. Appl. Math., 2012, Article ID 165698, 15 pages, 2012. [20] M.A. Khamsi, W.A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics, Wiley-Interscience, New York, 2001. [21] W.A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72(1965), 1004-1006. [22] W.A. Kirk, S. Reich, P. Veeramani, Proximal retracts and best proximity pair theorems, Numer. Funct. Anal. Optim., 24(2003), 851-862. [23] W.A. Kirk, P.S. Srinivasan, P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory, 4(2003) 79-89. [24] Leu¸stean, L., Nonexpansive iterations in uniformly convex W-hyperbolic spaces, Contemp. Math., 513(2010), 193-209. [25] M.A. Petric, Some results concerning cyclical contractive mappings, Gen. Math., 18(2010), no. 4, 213-226. [26] M.A. Petric, Best proximity point theorems for weak cyclic kannan contractions, Filomat, 25(2011), no. 1, 145-154. Received: June 25, 2013; Accepted: August 2, 2013.