Best proximity points for proximal contractions

arXiv:1207.4349v1 [math.FA] 18 Jul 2012 Noname manuscript No. (will be inserted by the editor) Best proximity points for proximal contractions Aurora Fern´andez-Le´on Abstract In this paper we improve and extend some best proximity point results concerning the socalled proximal contractions. Specifically, compactness assumptions under the sets Aand Bare removed to consider completeness conditions instead. Keywords Proximal contraction ·fixed point ·best proximity point ·approximate solution ·metric space Mathematics Subject Classification (2000) Primary 54E40 ·54E35 ·54H25 ·41A65; Secondary 47H10 A. Fern´andez-Le´on Departamento de An´alisis Matem´atico, Universidad de Sevilla, Apdo. 1160, 41080 Sevilla, Spain E-mail: [email protected] Tel: 0034649160497 2 Aurora Fern´andez-Le´on 1 Introduction A classical and very well-studied problem in Metric Fixed Point Theory is the existence of fixed point of single-valued non-self mappings Tdefined from a set Ato a set B, where Aand Bare subsets of a metric space X. Since the relation between these two sets Aand Bmay have so different natures (A∩B=∅,A∩B6=∅,A⊆B, etc...), the branch of the theory that deals with these mappings has not only studied the classical fixed point problem that tries to find solutions of the equation T x =xbut also has considered those situations where the existence of fixed points may even make no sense. In these special cases, the interest of the theory mainly focuses on the existence of approximate solutions of the equation T x =x, that is, points that are close to its image somehow. In this direction, one classical wellknown best approximation theorem due to Fan [1] is the one that states that if Ais a compact, convex and nonempty subset of a Hausdorff locally convex topological vector space Xand Tis a continuous mapping from Ato X, then there exists a point x∈Asuch that d(x, T x) = d(T x, A), where dis the semi-metric induced by a continuous semi-norm defined on X(notice that dis a metric if the semi-norm is actually a norm). Both best approximation theorems and best proximity point theorems are results in the theory that search approximate solutions, that is, points in A(the domain of T) satisfying the condition d(x, T x) = d(T x, A). The main difference between these two families of theorems is that the second one tries to go further in the approximation by searching approximate solutions which in turn are solutions of the minimization problem of the function d(x, T x) on A. In fact, best approximation results do not care whether the error d(x, T x) is the minimum possible in the context. To be precise, best proximity point theorems assure, for mappings Tdefined from Ato B, the existence of approximate solutions that minimize the value d(x, T x) in the set Aand moreover with respect to B, i.e., guarantee the existence of points x∈Asuch that d(x, T x) = d(A, B), where d(A, B) is actually the absolute infimum of d(x, T x) in this context. Points that satisfy the previous equality are known in the theory either as absolute optimal approximate solutions or as best proximity points. The study of the existence of best proximity points have been tackled in the theory from many different approaches. In this regard, generalizations of best proximity point theorems from self-maps to non-self-maps have arisen in the past two decades (see for example [2,3,4,5,6,7,8,9]). Besides, a relevant research line is the one that seek best proximity points of cyclic mappings, i.e., self-mappings Tdefined on the union of Aand Band satisfying that T(A) is contained in Band T(B) is contained in A. Many publications have arisen by assuming several additional metric conditions on these cyclic mappings, see for instance [10,11,12,13,14,15,16,17,18] and references therein. Our work developed here aims however to find best proximity points of classical non-self mappings that satisfy certain contractive conditions. The present paper is basically motivated by the papers [19] and [20]. These works include existence, uniqueness and convergence results of best proximity points for proximal contractions, thereby extending Banach’s contraction principle to the case of non-self mappings. Most of the results there considered are proved by assuming some compactness properties under the sets Aand Bbetween which a proximal contraction Tis defined. The feeling that these hypothesis are too restrictive in the context of proximal contractions has encouraged us to go into this topic. In fact, the results we prove in the sequel come directly from an attempt to improve in this direction the main results in [19] and [20]. The work is organized as follows: in Section 2 we introduce most of the definitions, notations and previous results we will need. In Section 3, we find a weaker condition for a set Athan the fact of being approximatively compact with respect to a set Bto guarantee the existence of best proximity point of proximal contractions of the first and second kind. In this regard, we consider classical completeness assumptions as the fact of being closed to improve the main results in [19] and [20]. Finally, we include an appendix where problems studied in Section 3 are considered from a different approach. Best proximity points for proximal contractions 3 2 Preliminaries In this section we introduce the main concepts and results we will need throughout this work. We begin by fixing some notations. Let (X, d) be a metric space and let Aand Bbe two nonempty subsets of X. Define d(x, A) = inf{d(x, y) : y∈A}, d(A, B) = inf{d(x, y) : x∈A, y ∈B}, A0={x∈A:d(x, y) = d(A, B) for some y∈B}, B0={y∈B:d(x, y) = d(A, B) for some x∈A}. From now on, B(a, r) will denote the closed ball in the space Xcentered at a∈Xwith radius r > 0. Next, we define a notion of compactness between sets given in [21]. This concept will play an essential role in this paper. Definition 2.1 Let Aand Bbe two nonempty subsets of a metric space X.Ais said to be approximatively compact with respect to Bif every sequence {xn}of Asatisfying the condition that d(y, xn)→d(y, A) for some y∈Bhas a convergent subsequence. It is worth mentioning that if a set Ais compact and moreover Bis approximatively compact with respect to A, then the sets A0and B0are non-empty. By the end of this paper we consider in several occasions some geodesic spaces. Next, we include some background concerning these metric spaces. A metric space (X, d) is said to be a geodesic space if every two points xand yof Xare joined by a geodesic, i.e, a map c: [0, l]⊆R→Xsuch that c(0) = x,c(l) = y, and d(c(t), c(t′)) = |t−t′|for all t, t′∈[0, l]. Moreover, Xis called uniquely geodesic if there is exactly one geodesic joining xand yfor each x, y ∈X. When there is only one geodesic between two points x and y, the image of this geodesic (called geodesic segment) is denoted by [x, y]. Thus, any Banach space is a geodesic space with usual segments as geodesic segments. A subset Aof a uniquely geodesic metric space Xis said to be convex if the geodesic segment joining each pair of points xand yof Ais contained in A. For more about geodesic spaces the reader can check [22,23,24]. The notion of strict convexity in geodesic spaces was introduced in [25] for strongly convex metric spaces [26], a wider family of metric spaces than the one of geodesic metric spaces. Notice that this notion trivially extends the strict convexity known in normed spaces to the nonlinear setting. Definition 2.2 A geodesic metric space Xis said to be strictly convex if for every r > 0, a, x and y∈X with 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. In these spaces, as it happens in strictly convex Banach spaces, if xand yare any two points on the boundary of a ball then the interior of the segment [x, y] lies strictly inside the ball. Moreover, it is easy to see that strictly convex metric spaces are in fact uniquely geodesic. Example 2.1 Spaces of nonpositive curvature in the sense of Busemann (see [24] for a detailed study on them) are strictly convex metric spaces. Note that consequently the well-known CAT(0) spaces are strictly convex metric spaces. In the next section we will also mention the so-called nonempty intersection property. Notice that this property extends the notion of reflexivity in Banach spaces to geodesic metric spaces. 4 Aurora Fern´andez-Le´on Definition 2.3 Let Xbe a uniquely geodesic metric space. Xis said to have the nonempty intersection property if for any sequence {Cn}of subsets of Xsuch that Cn+1 ⊆Cn∀n∈Nand Cnis closed convex bounded and nonempty ∀n∈N, it is the case that \ n∈N Cn6=∅. A broad family of geodesic spaces that satisfy this property is the one of uniformly convex metric spaces with either a monotone or lower semicontinuous from the right modulus of convexity (see Definition 2.1 in [27] and Proposition 2.2 in [17]). This uniform convexity in metric spaces is the result of assuming uniformity conditions on the definition of strict convexity given above and generalizes the notion of uniform convexity in Banach spaces. Thus, any uniformly convex Banach space is uniformly convex is this sense. Next we give the definitions of the mappings we will deal with in the following section. These mappings were introduced in the recent paper [20]. We begin with the definition of proximal contraction of the first kind. Definition 2.4 Let Aand Bbe two nonempty subsets of a metric space X. A map T:A→Bis a proximal contraction of the first kind if there exists a non-negative number α < 1 such that d(u, T x) = d(A, B) d(v, T y) = d(A, B)⇒d(u, v)≤αd(x, y), for all u, v, x, y ∈A. It is immediate to see that these mappings are contractions and so continuous if we restrict to the case of self-mappings. On the contrary, next example given in [21] shows that these mappings are not continuous in general. Example 2.2 Let A= [0,1] and B= [2,3] subsets of Rendowed with the Euclidean metric. Then the mapping T:A→Bdefined as f(x) = 3−x if x is rational 2 + x otherwise is a proximal contraction of the first kind. Next we give the definition of proximal contraction of the second kind. Definition 2.5 Let Aand Bbe two nonempty subsets of a metric space X. A mapping T:A→Bis a proximal contraction of the second kind if there exists a non-negative number α < 1 such that d(u, T x) = d(A, B) d(v, T y) = d(A, B)⇒d(T u, T v)≤αd(T x, T y), for all u, v, x, y ∈A. Notice that unlike the other definition, these mappings may not be even continuous when we restrict to the self case. In the sequel, we also consider the following non-self mappings. Definition 2.6 Let Aand Bbe two nonempty subsets of a metric space X. Let g:A→Abe an isometry and T:A→Ba non-self mapping. The mapping Tis said to preserve isometric distance with respect to gif d(T gx, T gy) = d(T x, T y) for every xand yin A. Best proximity points for proximal contractions 5 A counterpart notion of fixed point in the context of non-self mappings is the so-called best proximity point. Definition 2.7 Let T:A→Bbe a non-self mapping where Aand Bare two nonempty subsets of a metric space X. A point x∈Ais said to be a best proximity point for Tif d(x, T x) = d(A, B). 3 Main Results and Consequences Our first result is a generalized best proximity point theorem for proximal contractions of the first kind. This theorem extends Theorem 3.1 in [19] to more general settings by assuming weaker conditions on the sets Aand B. Theorem 3.1 Let Aand Bbe two nonempty subsets of a complete metric space X,T:A→Ba proximal contraction of the first kind and g:A→Aan isometry. Suppose that T(A0)⊆B0and A0⊆g(A0). If A0 is nonempty and closed, then there exists a unique point x∈Asuch that d(gx, T x) = d(A, B). Moreover, for every x0∈A0there exists a sequence {xn} ⊆ Asuch that d(gxn+1, T xn) = d(A, B)for every n≥0 and satisfying xn→x. Proof Let x0be a point in A0. Since T(A0)⊆B0and A0⊆g(A0), we can find a point x1∈A0such that d(gx1, T x0) = d(A, B). For this point x1∈A0, we can proceed similarly so that we get another point x2∈A0satisfying the condition d(gx2, T x1) = d(A, B). By repeating this process, once we have xn∈A0, it is possible to find xn+1 ∈A0such that d(gxn+1, T xn) = d(A, B) for every natural number n. Since Tis a proximal contraction of the first kind, d(gxn+1, gxn)≤αd(xn, xn−1) for every n∈N. The isometric character of gshows that in fact d(xn+1, xn)≤αd(xn, xn−1) for every n∈N. Thus, from this inequality we deduce that {xn}is a Cauchy sequence. By using the completeness of the space Xand the fact that A0is closed, we conclude that xn→x∈A0. In addition, the continuity of gimplies that gxn→gx. Since gxn∈A0for every n∈N,gx is also in the set A0. Since x∈A0, we have that T x ∈B0and therefore there exists a point z∈A0such that d(T x, z) = d(A, B). By applying again that Tis a proximal contraction, we obtain that d(z, gxn+1)≤αd(x, xn). Now, by letting ngo to infinity, we have that d(z, gxn+1)→0 and consequently z=gx. Thus, we conclude that d(gx, T x) = d(A, B). Now the proof follows the same patterns as the proof of [19, Theorem 3.1]. Let ˆxbe another point in A0such that d(gˆx, T ˆx) = d(A, B). Then, again by using the properties of Tand gand reasoning as above, we get that d(gx, gˆx) = d(x, ˆx)≤αd(x, ˆx). Consequently, ˆxmust be equal to xand thus xis the only point in A0such that d(gx, T x) = d(A, B), which completes the proof. 6 Aurora Fern´andez-Le´on Next, by considering the mapping gas the identity map on A, we get the following consequence. Notice also that this corollary provides existence and uniqueness of best proximity points in a general metric space for proximal contractions of the first kind. Corollary 3.1 Let Aand Bbe two nonempty subsets of a complete metric space Xand T:A→Ba proximal contraction of the first kind. Suppose that T(A0)⊆B0. If A0is nonempty and closed, then there exists a unique best proximity point x∈A. Moreover, for every x0∈A0there exists a sequence {xn} ⊆ A such that d(xn+1, T xn) = d(A, B)for every n≥0and satisfying xn→x. These two previous results notably improve those given in [19] as Theorem 3.1 and Corollary 3.1. In [19], the set Bis assumed to be approximatively compact with respect to A. This topological property, besides being very restrictive, implies that the set A0is closed, which is the property required in the present paper. Next proposition easily shows this assertion. Proposition 3.1 Let Aand Btwo nonempty subsets of a metric space X. If Aand Bare closed and A is approximatively compact with respect to B, then B0is closed. Proof Let {yn} ⊆ B0a convergent sequence and y∈Xits limit. The fact that Bis closed implies that y∈B. On the other hand, since the point ynis in B0, there exists xn∈A0such that d(yn, xn) = d(A, B) for every n∈N. As a consequence, we get that d(xn, y)→d(A, B) when ngoes to infinity. Since Ais closed and approximatively compact with respect to B, we also have that there exists a subsequence of {xn}that converges to a point z∈A. Finally, by combining the information we have, we get that d(z, y) = d(A, B), which implies that y∈B0. Next example shows that the opposite implication of the previous proposition is not true. As a consequence of this fact, we get that Theorem 3.1 and Corollary 3.1 are actually strictly more general than the counterpart results in [19]. Example 3.1 Let Xdenote the Banach space given by the set ℓ2endowed with the norm kxk=max{kxk2,√2kxk∞}. Notice that this norm is equivalent to k·k2and therefore (X, k·k) is also a reflexive Banach space. Let A={x= (xn)∈X:x1= 1 and kxk ≤ √2}and B={z= (2,0,0,...)}. It is easy to see that both Aand Bare closed and convex. Notice that e1∈A,kz−e1k=√2 and moreover, if y∈Awe have that y= (1, y2, y3,...) with Pn≥2y2 n≤1 and |yn| ≤ 1 for every n∈N. As a consequence, we have that kz−yk2≤√2 and kz−yk∞= 1 for every y∈Aand therefore d(A, B) = √2 = d(z, y) for every y∈A. Let {un}be the sequence in Agiven by un=e1+en. Since this sequence does not have any convergent subsequence, we conclude that Ais not approximatively compact with respect to B. It is worth mentioning that in general the set A0is closed whenever the metric space Xis a reflexive Banach space and the sets Aand Bare closed with Aconvex. Therefore, Example 3.1 is just a particular case in this respect where Ais not approximatively compact with respect to Band B0is closed. The next result deals with proximal contractions of the second kind. Again, a compactness assumption is removed, now from Theorem 3.1 in [20], to consider a completeness hypothesis. Also notice that here we do not impose any continuity property on T. Best proximity points for proximal contractions 7 Theorem 3.2 Let Aand Bbe two nonempty subsets of a complete metric space X,T:A→Ba proximal contraction of the second kind that preserves isometric distance with respect to an isometry g:A→A. Suppose that T(A0)⊆B0and A0⊆g(A0). If T(A0)is nonempty and closed, then there exists a point x∈Asuch that d(gx, T x) = d(A, B). Moreover, if ˆxis another point in Afor which d(gˆx, T ˆx) = d(A, B), then T x =Tˆx. Proof Following similar patterns to those considered in Theorem 3.1, we have that there exists a sequence {xn} ⊆ Asatisfying d(gxn+1, T xn) = d(A, B).(3.1) Since Tis a proximal contraction of the second kind, d(T gxn+1, T gxn)≤αd(T xn−1, T xn) for every n∈N. By using the isometric character of gand the fact that Tpreserves isometric distance with respect to g, we get d(T xn+1, T xn)≤αd(T xn, T xn−1) for every n∈N. Thus, as a consequence, we conclude that {T xn}is a Cauchy sequence. By using the completeness of the space Xand the fact that T(A0) is closed, we obtain that {T xn}converges to some point yin T(A0)⊆B0. Hence, y=T u for some u∈A0. Moreover, we also deduce that there exists z∈A0such that d(z, T u) = d(A, B). Since A0⊆g(A0), we have that z=gx for some x∈A0. Then, d(gx, T u) = d(A, B). By using (3.1) and previous equality, we get d(T gx, T gxn+1) = d(T x, T xn+1)≤αd(T u, T xn). Consequently, we obtain that T x =T u and hence d(gx, T x) = d(A, B). The uniqueness of such a point x follows similar patterns to those considered in [20, Theorem 3.1]. Let ˆxbe another point in A0such that d(gˆx, T ˆx) = d(A, B). Then, again by using the properties of Tand gand reasoning as above, we get that d(T gx, T gˆx) = d(T x, T ˆx)≤αd(T x, T ˆx). Consequently, Tˆxmust be equal to T x, which completes the proof. Remark 3.1 Notice that, in general, if a second proximal contraction Tis also continuous and Bis approximatively compact with respect to A, then T(A0) is sequentially closed in B0for sequences {T xn} ⊆ T(A0) such that d(gxn+1, T xn) = d(A, B) for every n∈N, that is what has essentially been considered in the proof of previous theorem. If we consider again the particular case of gequal to the identity map, we obtain an improvement of Corollary 3.2 of [20] as well as a best proximity point result for proximal contractions of the second kind. Corollary 3.2 Let Aand Bbe two nonempty subsets of a complete metric space Xand T:A→B a proximal contraction of the second kind. Suppose that T(A0)⊆B0. If T(A0)is nonempty and closed, then there exists a best proximity point x∈A. Moreover, if ˆxis another best proximity point in A, then T x =Tˆx. 8 Aurora Fern´andez-Le´on 4 Appendix: A Property That Involves the Pair (A, B). One interesting problem that arises when dealing with Section 3 is whether it is possible to establish or formulate its results by using a completeness property that involves both set Aand Bin such a way that we preserve the classical structure of recent theorems concerning best proximity points (see, apart from the property of being approximatively compact with respect to another set, the properties UC, WUC or HW considered when dealing with best proximity points in cyclic mappings [27,15,18]). In this regard, we introduce a useful property involving the sets Aand B. Definition 4.1 Let Aand Bbe nonempty subsets of a metric space (X, d). The pair (A, B) is said to satisfy property WAC if for every sequence {xn}in Aand every point pin Bsuch that lim n d(xn, p) = d(A, B), then it is the case that p∈B0. Next proposition shows that this property is weaker than the approximative compactness considered in Section 3. Proposition 4.1 Let Aand Bbe nonempty subsets of a metric space (X, d). If Ais approximatively compact with respect to Band Ais closed, then the pair (A, B)has property WAC. Proof We omit the proof. The following proposition establishes some general situations where we find property WAC between sets. Proposition 4.2 Let Aand Bbe two nonempty subsets of a metric space X. (1) If Xis a reflexive Banach space and the sets Aand Bare such that Ais closed and convex and Bis closed, then (A, B)has property WAC. (2) If Xis a strictly convex metric space with the nonempty intersection property (see Section 2 for definitions) and the sets Aand Bare such that Ais closed and convex and Bis closed, then (A, B) has property WAC. Proof (1) Let {xn} ⊆ Aand p∈Bsuch that limnd(xn, p) = d(A, B). Since {xn}is bounded, we have that xnweakly converges to a point x∈A. By using that fact that the norm in Xis weakly lower semicontinuous, we also get that d(x, p)≤lim infnd(xn, p) = d(A, B). Consequently, we conclude that p∈B0. (2) Let {xn} ⊆ Aand p∈Bsuch that limnd(xn, p) = d(A, B). Let us consider for every n∈N the subsets Cnof Agiven by Cn=A∩B(p, d(A, B) + 1/n). Since balls are convex in the context of strictly convex metric spaces, we may assert that Cnare closed, convex and bounded. Moreover, it is not difficult to see that for every n∈Nthere exists m0∈Nsuch that xm∈Cnfor every m≥m0. Consequently we conclude that Cnis also nonempty for every natural number n. By using the fact that Xhas the nonempty intersection property we get that TCn6=∅. Let xbe a point in this intersection. Since x∈B(p, d(A, B) + 1/n) for every n∈N, we get that d(p, x) = d(A, B) and thus p∈B0. As a consequence of the first statement of this proposition, we deduce that the pair of sets (A, B) considered in Example 3.1 also have property WAC and therefore property WAC is strictly weaker than approximative compactness. Best proximity points for proximal contractions 9 Remark 4.1 For those interested in properties UC, WUC or HW, notice that in general property WUC implies approximative compactness (and thus property UC too). Then, property WAC is more general than properties WUC and UC. However, it is possible to find pairs of sets with property HW and without property WAC. For this aim, it suffices to consider A={−1/n :n∈N}and B={1+1/n :n∈N}∪{1}. The proposition and the example given below show that property WAC is strictly stronger than the fact that A0is closed. For this reason, we have not consider property WAC in Section 3. Proposition 4.3 Let Aand Bbe two nonempty subsets of a metric space. If Ais closed and the pair (B, A)has property WAC, then A0is closed. Proof The proof follows similar patterns to those considered in Proposition 3.1. Next example shows that the inverse implication of the previous proposition is not true. Example 4.1 Let Xbe the Euclidean plane. Let A={(x, 1) : x≥1} ∪ {(x, 0) : x≥1}and B= {(0, y) : y6= 0}. Consider the sequence {yn} ⊆ Bgiven by {yn}={(0,−1/n)}n∈Nand the point q= (1,0) ∈A. Notice that A0={(1,1)}, so it is closed, but however (B, A) does not have property WAC since d(yn, q)→1 = d(A, B) but qis not in A0. Acknowledgements The author was partially supported by the Ministery of Science and Technology of Spain, Grant MTM2009-10696-C02-01 and La Junta de Antalucia, Grant P08-FQM-03543. References 1. Fan, K.: Extensions of two fixed point theorems of F. E. Browder. Math. Z. 112, 234-240 (1969) 2. Al-Thagafi, M.A., Shahzad, N.: Best proximity pairs and equilibrium pairs for Kakutani multimaps. Nonlinear Anal. 70 (30), 1209-1216 (2009) 3. Beer, G., Pai, D. V.: Proximal maps, prox maps, and coincidence points. Numer. Funct. Anal. Optim. 11, 429448 (1990) 4. Kim, W. K., Kum, S., Lee, K. H.: On general best proximity pairs and equilibrium pairs in free abstract economies. Nonlinear Anal. 68 (8), 2216-2227 (2008) 5. p Kirk, W. A., Reich, S., Veeramani, P.: Proximal retracts and best proximity pair theorems. Numer. Func. Anal. Optim. 24, 851-862 (2003) 6. Sadiq Basha, S., Veeramani, P.: Best approximations and best proximity pairs. Acta. Sci. Math. (Szeged) 63, 104-113 (1997) 7. Sadiq Basha, S., Veeramani, P.: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 103, 119-129 (2000) 8. Sehgal, V. M., Singh, S. P.: A generalization to multifunctions of Fans best approximation theorem. Proc. Amer. Math. Soc. 102, 534537 (1988) 9. Srinivasan, P. S.: Best proximity pair theorems. Acta Sci. Math. (Szeged) 67, 421-429 (2001) 10. Al-Thagafi, M.A., Shahzad, N.: Convergence and existence results for best proximity points. Nonlinear Anal. 70 (10), 3665-3671 (2009) 11. Di Bari, C., Suzuki, T., Vetro, C.: Best proximity points for cyclic Meir-Keeler contractions. Nonlinear Anal. 69 (11), 3790-3794 (2008) 12. Eldred, A. A., Veeramani, P.: Existence and convergence of best proximity points. J. Math. Anal. Appl. 323 (2), 1001-1006 (2006) 13. Eldred, A. A., Kirk, W. A., Veeramani, P.: Proximal normal structure and relatively nonexpansive mappings. Studia Math. 171 (3), 283-293 (2005) 14. Esp´ınola, R.: A new approach to relatively nonexpansive mappings. Proc. Amer. Math. Soc. 136 (6), 1987-1995 (2008) 15. Fern´andez-Le´on, A.: Existence and uniqueness of best proximity points in geodesic metric spaces. Nonlinear Anal. 73 (4), 915-921 (2010) 16. Kirk, W. A., Srinivasan, P.S., Veeramani, P.: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory 4, 79-89 (2003) 17. Kohlenbach, U., Leu¸stean, L.: Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces. J. of the European Math. Soc. 12 (1), 71-92 (2010)