Pythagorean Property and Best Proximity Pair Theorems

Transcription

1 isibang/ms/2013/32 November 25th, statmath/eprints Pythagorean Property and Best Proximity Pair Theorems Rafa Espínola, G. Sankara Raju Kosuru and P. Veeramani Indian Statistical Institute, Bangalore Centre 8th Mile Mysore Road, Bangalore, India

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3 PYTHAGOREAN PROPERTY AND BEST PROXIMITY PAIR THEOREMS RAFA ESPÍNOLA, G. SANKARA RAJU KOSURU, AND P. VEERAMANI Abstract. The aim of this paper is to prove the existence and convergence theorems for cyclic contractions. We introduce a notion called proximally complete pair (A, B) on a metric space, which unify the earlier notions that are used to prove the existence of a best proximity point for a cyclic contraction. By observing geometrical properties on a Hilbert space, we introduce Pythagorean property and use this property to give sufficient conditions for a cyclic map to be cyclic contraction. 1. Introduction and Preliminaries Let A, B be nonempty closed subsets of a complete metric space X and let T be a cyclic map on A B, that is T A B and T B A. If T is a contraction, that is there is an α (0, 1) such that d(t x, T y) αd(x, y), for x A and y B, then A B and for any x 0 in A B the Picard s iteration {T n x 0 } converges to the unique fixed point of T ([14]). Motivated by this, Eldred and Veeramani in [5] introduced a notion called cyclic contraction and gave sufficient conditions (Theorem 3.10, [5]) for the existence of a unique point x A such that d(x, T x) = d(a, B) := inf{d(u, v) : u A, v B} (such a point is said to be a best proximity point) for a cyclic contraction mapping T in the settings of a uniformly convex Banach space. Many authors studied ([1, 2, 6, 12, 16, 17]) the existence and convergence of best proximity points of cyclic contractions. Since, in the literature, we do not have natural examples for such a class of cyclic contractions, in this paper we introduce a notion called Pythagorean property and thereby prove that every continuously Fréchet differentiable cyclic map T with sup x A B T x < 1 is a cyclic contraction, where T x is the Fréchet derivative of T at x. We prove that every closed convex pair (A, B) in a Hilbert space (or a CAT(0) space) has Pythagorean property. An 2000 Mathematics Subject Classification. 47H10, 46C20, 54H25. Key words and phrases. cyclic contraction, semi sharp proximinal pair, proximal complete pair, best proximity points, pythagorean property. 1

4 2 R. ESPÍNOLA, G. SANKARA RAJU KOSURU, AND P. VEERAMANI example for such a pair is given in a non-hilbert space setting. We also introduce a notion called proximally complete pair for a pair of subsets of a metric space, which unify the earlier notions, namely the property UC, cyclically completeness, that are used to prove the existence and convergence of a best proximity point of a cyclic contraction map T on A B. We prove that every pair of nonempty closed convex subsets of a uniformly convex Banach space (or boundedly compact subsets of a metric space) is proximally complete. Finally we give a type of necessary condition on (A, B) for the existence and convergence of a unique best proximity point of a cyclic contraction on A B. Let (X, d) be a metric space. We recall that a pair (A, B) of nonempty subsets of X is said to be sharp proximinal if for each x in A (respectively in B) there exists a unique y in B (respectively in A) such that d(x, y) = d(a, B). We say that a pair (A, B) of nonempty subsets of X is said to be a semi sharp proximinal if for each x in A (respectively in B) there exists at most one y in B (respectively in A) such that d(x, y) = d(a, B). We denote such a y by x, if it exists. Every closed convex pair (A, B) in a strictly convex Banach space is semi sharp proximinal ([15], Lemma 2.5). Also such examples (Example 3.13) are given, in [15], in nonstrictly convex Banach spaces. We now fix some notations and definitions, used hereafter. A 0 = {x A : d(x, y) = d(a, B), for some y in B}; B 0 = {y B : d(x, y) = d(a, B), for some x in A}. A cyclic map T on A B is said to be a cyclic contraction ([5]) if there exists α in [0, 1) such that d(t x, T y) αd(x, y) + (1 α)d(a, B), for x A and y B. Now we quote a Lemma, proved in [5], which we use to prove that every closed convex pair in a uniformly convex Banach space is proximally complete. Lemma 1.1. [5] Let A be a nonempty closed and convex subset and B be a nonempty closed subset of a uniformly convex Banach space. Let {x n } and {z n } be sequences in A and {y n } be a sequence in B satisfying: (1) z n y n d(a, B). (2) For every ϵ > 0 there exists N 0 N such that for all m > n N 0, x m y n dist(a, B) + ϵ. Then, for every ϵ > 0 there exists N 1 such that for all m > n N 1, x m z n < ϵ.

5 PYTHAGOREAN PROPERTY AND BEST PROXIMITY PAIR THEOREMS 3 Motivated by the above Lemma, Suzuki et al. in [17] introduced a notion called the Property UC and thereby obtained existence and convergence results for best proximity points. Definition 1.2. Let A and B be nonempty subsets of a metric space (X, d). Then (A, B) is said to satisfy the property UC if the following holds: (1) If {x n } and {z n } are sequences in A and {y n } is a sequence in B such that lim n d(x n, y n ) = d(a, B) and lim n d(z n, y n ) = d(a, B), then lim n d(x n, z n ) = 0 holds. It is easy to notice that a pair (A, B) in a metric space is semi sharp proximinal if it has the property UC. On the other hand the compact and convex pair (A, B), where A := {(0, x) : 0 x 1} and B := {(1, x) : 0 x 1}, in the Banach space R 2 with respect to does not have the property UC. 2. Proximally completeness For a pair (A, B) of nonempty subsets of a metric space, in this section, we introduce a notion called proximally complete pair and establish some properties of cyclically Cauchy sequences and proximally complete pairs. Definition 2.1. [13] Let X be a metric space and A, B be nonempty subsets of X. A sequence {x n } n=0 in A B, with x 2n A and x 2n+1 B for all n 0, is said to be a cyclically Cauchy sequence if for every ϵ > 0 there exists an N N such that d(x n, x m ) < d(a, B) + ϵ, when n is even, m is odd and n, m N. If d(a, B) = 0, then a sequence {x n } in A B is cyclically Cauchy if and only if {x n } is Cauchy. The following Lemma ensures the boundedness of a cyclically Cauchy sequence. Lemma 2.2. Every cyclically Cauchy sequence is bounded. Proof. Let {x n } be a cyclically Cauchy sequence in A B. Then there exists N N, such that d(x 2n, x 2N+1 ) < d(a, B) + 1 for all n N. Therefore for all n N, x 2n B(x 2N+1, r), where r = max{d(x 2, x 2N+1 ), d(x 4, x 2N+1 ),..., d(x 2N, x 2N+1 ), d(a, B) + 1}. So that {x 2n } is bounded. In a similar fashion one can prove that the sequence {x 2n+1 } is a bounded sequence and hence {x n } is bounded.

6 4 R. ESPÍNOLA, G. SANKARA RAJU KOSURU, AND P. VEERAMANI It is to be noted that for a cyclically Cauchy sequence {x n }, the sequence {x 2n } may have two different convergent subsequences. Example 2.3. Let X = (R 3, 2 ). A := {(x, y, z) X : 1 x 1, y 2 + z 2 = 1}, B := {(0, 0, 0)}. For n N define x 2n+1 = (0, 0, 0) and { ( 1, 1, 0) if n is odd, x 2n := n, 1, 0) if n is even. ( 1 n Then d(a, B) = 1 and {x n } is cyclically Cauchy. It is to noted that {x 2n } has two different convergent subsequences {( 1, 1, 0)} and {( 1, 1, 0)} which converge to (0, 1, 0) and n n (0, 1, 0) respectively. Also it is to be noted that the pair (A, B), given in Example 2.3, does not have the property UC even though A, B are compact subsets of a uniformly convex Banach space. Now we quote a result of [5], which we use in the sequel. Proposition 2.4. [5] Let A and B be nonempty sets of a metric space and let T be a cyclic contraction mapping on A B. Suppose for some x 0 in A, the sequence {T n x 0 } has a convergent subsequence {T n k x0 } that converges to x in A then (i) d(t n k 1 x 0, x) d(a, B), d(t n k x0, T x) d(a, B) and (ii) d(x, T x) = d(a, B). (iii) Further if there exists a subsequence of {T n x 0 } that converges to some y in B then d(x, y) = d(a, B). Also in [13] it is defined that, a pair (A, B) is said to be cyclically complete if for every cyclically Cauchy sequence {x n } in A B either {x 2n } or {x 2n+1 } converges. It is to be noted that, some basic properties, such as Proposition 2.6, Theorem 2.7 and Theorem 2.9 fail to hold with the above definition. To illustrates the same, let A := {(0, x) : 0 x < 1} and B := {(1, x) : 0 x 1} in the Euclidean space R 2. Then (A, B) is cyclically complete but A 0 is not a closed set. Also the sequence {x 2n } does not have any convergent subsequence in A, for the cyclically Cauchy sequence {x n } in A B, where { (0, 1 1 ) if n is even, x n := n (1, 1 1 ) if n is odd. n On the other hand the compact pair (A, B), where A := {(1 + 1 n )e i : i = 1, 3 and n N} {e 1, e 3 } and B := {(1 + 1 n )e i : i = 2, 4 and n N} {e 2, e 4 }, in the Banach space R 4

7 PYTHAGOREAN PROPERTY AND BEST PROXIMITY PAIR THEOREMS 5 with respect to p, for 1 p is not a cyclically complete pair. Motivated by these obsevations and taking into account the nature of symmetry we define: Definition 2.5. A pair (A, B) of subsets of a metric space is said to be proximally complete if for every cyclically Cauchy sequence {x n } in A B, the sequences {x 2n } and {x 2n+1 } have convergent subsequences in A and B. If A and B are closed subsets of a complete metric space with d(a, B) = 0, then (A, B) is a proximally complete pair. As a particular case, suppose A is a subset of a metric space then (A, A) is proximally compete if and only if A is complete. As a consequence of Lemma 2.2, we have every boundedly compact pair in a metric space is proximally complete. Suppose (A, B) has the property UC and A, B are complete. If {x n } is a cyclically Cauchy sequence in A B, then sequences {x 2n } and {x 2n+1 } are Cauchy and hence (A, B) is proximally complete. Also if {x 2nk } and {x 2mk +1} are convergent subsequences of {x 2n } and {x 2n+1 }, that converge to x A and y B respectively, then d(a, B) d(x, y) = lim d(x 2n k, x 2mk +1) = d(a, B) and hence we have: k Proposition 2.6. Let (A, B) be a proximally complete pair in a metric space X. Then A 0 is non empty if and only if there exists a cyclically Cauchy sequence in A B. Theorem 2.7. Let A and B be subsets of a metric space X. complete, then A 0 and B 0 are closed subsets of X. If (A, B) is proximally Proof. Let {x n } be a sequence in A 0 such that x n x in X. For n N let x n B 0 such that d(x n, x n) = d(a, B). For n N define { x m if n = 2m for some m N, y n := x m if n = 2m + 1 for some m N. Now d(y 2n, y 2m+1 ) = d(x n, x m) d(x n, x) + d(x, x m ) + d(x m, x m) d(a, B), as n, m and hence {y n } is a cyclically Cauchy sequence. Since (A, B) is proximally complete, {x n } and {x n} have convergent subsequences, which converge to x and y respectively, then d(x, y) = d(a, B) and hence A 0 is closed. In a similar fashion one can prove B 0 is also a closed set. The following theorem ensures that every closed convex pair in a uniformly convex Banach space is proximally complete.

8 6 R. ESPÍNOLA, G. SANKARA RAJU KOSURU, AND P. VEERAMANI Theorem 2.8. Any nonempty closed convex pair (A, B) in a uniformly convex Banach space is proximally complete. Further, for any cyclic Cauchy sequence {x n }, the sequences {x 2n } and {x 2n+1 } converge to some x in A and y in B respectively with d(x, y) = d(a, B). Proof. Let {x n } be a cyclically Cauchy sequence in A B. Suppose {x 2n } is not a Cauchy sequence. Then there exists ϵ 0 > 0 and subsequences {x 2nk } and {x 2mk } of {x 2n } such that d(x 2nk, x 2mk ) ϵ 0, for all k N. Also one can observe that d(x 2nk, x 2k+1 ) d(a, B) and d(x 2mk, x 2k+1 ) d(a, B), as k. By Lemma 1.1, there exists N 1 N such that d(x 2nk, x 2mk ) < ϵ 0 for all k N 1, a contradiction. Hence {x 2n } converges to some point x in A. In a similar fashion one can prove that x 2n+1 y in B. Also d(x, y) = lim n d(x 2n, x 2n+1 ) = d(a, B). The following theorem gives sufficient conditions for the convergence of {x 2n } and {x 2n+1 }, whenever {x n } is a cyclically Cauchy sequence. Theorem 2.9. Let (A, B) be a proximally complete semi sharp proximinal pair in a metric space X. If {x n } is a cyclically Cauchy sequence in A B then x 2n x, for some x in A and x 2n+1 y, for some y in B. Further d(x, y) = d(a, B). Proof. Let {x n } be a cyclically Cauchy sequence in A B. Fix a convergent subsequence {x 2nk +1} of {x 2n+1 }, that converge to y B. Let {x 2mk } and {x 2lk } be convergent subsequences of {x 2n } that converge to x 1 and x 2 in A respectively. By the semi sharp proximinality of (A, B), we have x 1 = x 2. Hence any two convergent subsequences of {x 2n } converge to a point say to x, with d(x, y) = d(a, B). Suppose {x 2n } is not Cauchy, then there exists ϵ 0 > 0 and two subsequences {x 2np }, {x 2mp } of {x 2n } such that d(x 2np, x 2mp ) ϵ 0, for all p N. Now consider the sequence {y p }, where { x 2np if p is even y p := if p is odd x p Then it is easy to see that the sequence {y p } is a cyclically Cauchy sequence and hence {x 2np } has a convergent subsequence. Similarly {x 2mp } has a convergent subsequence. Since (A, B) is a proximally complete pair, {x 2np } and {x 2mp } have convergent subsequences, that converge to x. Hence there exists P N such that d(x 2nP, x) < ϵ 0 2 and d(x 2mP, x) < ϵ 0 2. Now d(x 2nP, x 2mP ) d(x 2nP, x) + d(x 2mP, x) < ϵ ϵ 0 2 = ϵ 0, a contraction. Hence {x 2n } is Cauchy. Also {x 2n } has a convergent subsequence and hence x 2n x in A. In a similar fashion one can show x 2n+1 y in B.

9 PYTHAGOREAN PROPERTY AND BEST PROXIMITY PAIR THEOREMS 7 Now we illustrate a situation, where one can get a natural example of a proximally complete pair in a metric space. Let (X, d) be a metric space and Y be a complete subspace of X. For x y in X, if we define A := {(u, x) : u Y } and B := {(u, y) : u Y }, then it is easy to see that the pair (A, B) is a proximally complete pair in the metric space (X X, d 1 ), where d 1 ((u 1, v 1 ), (u 2, v 2 )) = d(u 1, u 2 ) + d(v 1, v 2 ). For instant X denote the set of all complex valued continuous functions on [0, 1] with the, defined as f 1 + if 2 = 1 f 0 1(t) dt + 1 f 0 2(t) dt. Let A = {sinnt : n N} and B = {sinnt + i : n N}. Note that X is not complete but A and B are complete. The pair (A, B) is proximally complete. 3. Pythagorean property and Cyclic contractions In this section we introduce a notion called Pythagorean property for a pair of subsets of a metric space and thereby we give a natural example of cyclic contractions. We also prove that every closed convex pair of subsets of a Hilbert space (or CAT(0) space) has Pythagorean property. Definition 3.1. A semisharp proximinal pair in a metric space X is said to have Pythagorean property if for each (x, y) in A 0 B 0, we have d(x, y) 2 = d(x, x ) 2 + d(x, y) 2 and d(x, y) 2 = d(y, y ) 2 + d(y, x) 2 It is easy to notice that if d(a, B) = 0, then every pair (A, B) has Pythagorean property. Let A and B be nonempty closed convex subsets in a strictly convex Banach space X with A 0 = A and B 0 = B. It was proved in [7] that, there exists h X such that B = A + h and x = x + h for all x A. That is x + h is a best approximation to x in B. In a similar way we have y is a best approximation to y h in B. Therefore, if X is a Hilbert space, then h, y (x + h) 0 and h, x + h y 0, which implies h, y (x + h) = 0. By Pythagorean theorem, we have Hence we have the following: x y 2 = h 2 + x + h y 2. Proposition 3.2. Every nonempty closed and convex pair (A, B) in a Hilbert space has Pythagorean property. We now prove that a class of spaces, namely CAT(0) space, satisfy the Pythagorean property. CAT(0) spaces can be viewed as a metric analog of Hilbert space. The idea we

10 8 R. ESPÍNOLA, G. SANKARA RAJU KOSURU, AND P. VEERAMANI use in our proof is that the classes of CAT(0) spaces are characterized by having thinner triangles then the comparisons ones in the 2-dimensional Euclidean space. Let (X, d) be a uniquely geodesic metric space. A triangle (a, b, c) in the Euclidian space is said to be a comparison triangle for a geodesic triangle (x, y, z) in X if d(x, y) = d(a, b), d(y, z) = d(b, c) and d(x, z) = d(a, c). We say that a point α in a line segment [a, b] in the Euclidian space is said to be the corresponding point of p in the geodesic [x, y] in X, tb + (d(a, b) t)a if γ is the geodesic from x to y and p = γ(t), then α =, such a point d(a, b) is denoted by p. X is said be a CAT(0) space if each geodesic triangle (x, y, z) in X has a comparison triangle ( x, ȳ, z) with d(p, q) d( p, q), for any p, q (x, y, z). Let (x, y, z) be a geodesic triangle in a CAT(0) space. The Alexandrov angle between x, z at y is denoted by y (x, z) and is defined as y (x, z) = lim sup t 1,t 2 0 ȳ( γ 1 (t 1 ), γ 2 (t 2 )), where ȳ is the angle in R 2. Where γ 1, γ 2 corresponding line segments from x to ȳ and ȳ to z respectively in the comparison triangle ( x, ȳ, z) in R 2. A subset C of a CAT(0) space is said to be convex if for every x, y C, the geodesic [x, y] C. For further details on CAT(0) space and their properties the reader can refer [3, Chapter II.1]. Let A, B be closed convex subsets of a CAT(0) space X. Suppose x A and y, z B are such that d(x, y) = d(a, B) = d(x, z). By [3, Propositin 2.4], we have a unique π(x) B such that d(x, π(x)) = d(x, B). Now d(x, B) d(x, y) = d(x, z) = d(a, B) d(x, B). Hence by the uniqueness of π(x), we have y = z. Hence we have the following Lemma. Lemma 3.3. Let (A, B) be a nonempty closed convex pair in a complete CAT(0) space X. Then (A 0, B 0 ) is a semisharp proximinal pair in X. Indeed, something more can be said on this regard. Corollary 3.4. Under the previous conditions, (A 0, B 0 ) is a sharp proximinal pair. Proof. It suffices to show that A 0 is nonempty. This follows as a simple consequence of the fact that decreasing sequences of closed bounded and convex subsets of complete CAT(0) spaces have nonempty intersection (see, for instance, [9, Proposition 3.1] for this fact). Then A 0 = ε>0 (A B(B, d(a, B) + ε)),

11 PYTHAGOREAN PROPERTY AND BEST PROXIMITY PAIR THEOREMS 9 where B(B, d(a, B) + ε) = x B B(x, d(a, B) + ε) which is convex. Now, take x A 0 and then, from Lemma 3.3, (B(x, d(a, B) + ε) B) ε>0 is a singleton {y} such that d(x, y) = d(a, B). The reader may also check [8, Appendix] for related results to this one. Theorem 3.5. Let X be a complete CAT(0) space. Then every closed convex pair (A, B) of subsets of X have Pythagorean property. Proof. By Lemma 3.3, we have (A, B) is a semisharp proximinal pair. Fix (x, y) A 0 B 0. Let (x, y ) B 0 A 0 be such that d(x, x ) = d(a, B) = d(y, y ). If, x = y then y = x and (A, B) have Pythagorean property. Suppose x y. By [3, Propositin 2.4], we have x (x, y) π 2, where x (x, y) the Alexandrov angle between x, y at x. In a similar fashion we have x (x, y ) π, 2 y(y, x ) π and 2 y (y, x) π. Now by [3, Theorem 2.11], we 2 have x (x, y) + x (x, y ) + y (y, x ) + y (y, x) = π and the convex hull of x, x, y, y is isometric to a convex quadrilateral, say (p, q, r, s), in R 2. Hence d(x, x ) = p q, d(x, y) = q r, d(y, y ) = r s, d(y, x) = s p, d(x, y) = p r, d(x, y ) = q s and (1) x (x, y) = x (x, y ) = y (y, x ) = y (y, x) = π 2. Now consider the geodesic triangle (x, x, y). If u, v are two points in the geodesic triangle (x, x, y) and ū, v be their corresponding points in the triangle (p, q, r) in R 2, then by convex isometry we have d(u, v) = d(ū, v). Hence the triangle (p, q, r) is a comparison triangle for the geodesic triangle (x, x, y). Hence q (p, r) = x(x, y) = π. 2 That is triangle (p, q, r) is a right angle triangle in R 2. Therefore, we have p r 2 = p q 2 + q r 2. Hence d(x, y) 2 = d(x, x ) 2 + d(y, x ) 2. In a similar fashion, by consider the geodesic triangle (x, y, y) and triangle (p, s, r), one can obtain d(x, y) 2 = d(x, y ) 2 + d(y, y ) 2. This completes the proof. Something very strong follows from (1). Indeed, from the Flat Quadrilateral Theorem [3, 2.11 The Flat Quadrilateral Theorem], it follows that the convex hull (see [3, p. 112]

12 10 R. ESPÍNOLA, G. SANKARA RAJU KOSURU, AND P. VEERAMANI for proper definition) of the set {x, y, x, y } is actually isometric to the Euclidean convex hull of a rectangle with sides d(x, y), d(x, x ), d(y, y ) and d(x, y ). This, however, cannot happen in any space which is CAT (κ) with κ < 0 at least the rectangle is a segment, that is, at least x = x p and y = y p which proves the next corollary. Corollary 3.6. Let X be a complete CAT (κ) space with κ < 0. Then for every closed convex pair (A, B) of subsets of X the sets A 0 and B 0 are singleton, that is, there exists one unique proximinal pair (x, y). Remark 3.7. Spaces CAT (κ) with κ < 0 are studied in [3] and some relevant examples are the hyperbolic space of constant negative curvature [3, p. 18], R-trees and the real Hilbert ball with the hyperbolic metric [10]. Now we give an example of a pair (A, B) which has Pythagorean property in a non- Hilbert space setting. Example 3.8. Consider the space X of all complex valued continuous functions on [0, 1] with supremum norm, i.e., X = (C[0, 1], ). A := {f α : α [0, 1]} and B := {g α : α [0, 1]}, where { α + t, if t [0, 1 f α (t) := ] 2 α + (1 t), if t [ 1, 1]. 2 { α + t + i(t + 1 g α (t) := ), if t [0, 1] 2 2 α + (1 t) + i( 3 t), if t [ 1, 1]. 2 2 For any fixed α [0, 1] and for any t [0, 1 2 ], g α(t) f α (t) = α + t + i(t ) α + t and for any t [ 1, 1], g 2 α(t) f α (t) = α + (1 t) + i( 3 t) α + (1 t) 1. Also 2 f α ( 1) g 2 α( 1) = 1. Therefore f 2 α g α = 1. Now for any α β [0, 1], f α g β g α ( 1) 2 f β ( 1 2 ) = (α ) + i(t + 1) (β ) > 1. Hence d(a, B) = 1 and f α = g α, for all f α A. Now for every α, β, t [0, 1], f α (t) g β (t) 2 = 1+ α β 2 = f α (t) g α (t) 2 + g α (t) g β (t) 2 and hence (A, B) has Pythagorean property. Finally we give a natural example of a cyclic contraction map using Pythagorean property. Theorem 3.9. Let A and B be nonempty subsets of a metric space X. Assume that (A, B) has Pythagorean property. Let T be a cyclic map on A B. If T is a contraction on A 0 and (T x) = T x for all x A 0 then T is a cyclic contraction on A 0 B 0.

13 PYTHAGOREAN PROPERTY AND BEST PROXIMITY PAIR THEOREMS 11 Proof. For x in A and y in B, T x T y 2 = T x T y 2 + T x T x 2 k 2 x y 2 + d(a, B) 2 = k 2 ( x y 2 x x 2 ) + d(a, B) 2 = k 2 x y 2 + (1 k) 2 d(a, B) 2 + 2k(1 k)d(a, B) 2 k 2 x y 2 + (1 k) 2 d(a, B) 2 + 2k(1 k)d(a, B) x y (k x y + (1 k)d(a, B)) 2. Hence T x T y k x y + (1 k)d(a, B). Now by using Theorem 3.9 and the Mean Value Theorem for Fréchet differentiable functions ([11]) we obtain: Theorem Let A and B be nonempty closed convex subsets of a Banach space X. Assume that (A, B) has Pythagorean property. If there exist open convex subsets G 1 and G 2 of X such that A 0 G 1, B 0 G 2 and a continuously Fréchet differentiable map T : G 1 G 2 X map satisfying: (1) sup x A0 B 0 T x < 1 where T x is the Fréchet derivative of T at x, (2) T A 0 B 0, T B 0 A 0 and (T x) = T x for all x A 0 then T is a cyclic contraction on A 0 B Existence of Best proximity points Theorem 4.1. Let (A, B) be a proximally complete pair in a metric space X. If T is a cyclic contraction on A B, then there exists (x, y) A B such that d(x, T x) = d(a, B) and d(y, T y) = d(a, B) with d(x, y) = d(a, B). Proof. For x 0 A, define x n = T n x 0 for all n N. First we prove that the sequence {x 2n } is bounded. Suppose not, for M = 2α2 d(x 1, x 2 ) 1 α 2 + d(a, B) + d(x 2, x 3 ), there exists n N,

14 12 R. ESPÍNOLA, G. SANKARA RAJU KOSURU, AND P. VEERAMANI such that d(x 3, x 2n 2 ) M and d(x 3, x 2n ) > M. Now M < d(x 3, x 2n ) α 2 d(x 1, x 2n 2 ) + (1 α 2 )d(a, B) α 2 (d(x 1, x 2 ) + d(x 2, x 2n 2 )) + (1 α 2 )d(a, B) α 2 (d(x 1, x 2 ) + d(x 2, x 3 ) + d(x 3, x 2n 2 )) + (1 α 2 )d(a, B) α 2 (d(x 1, x 2 ) + d(x 2, x 3 ) + M) + (1 α 2 )d(a, B) α 2 (d(x 1, x 2 ) + d(x 1, x 2 ) + M) + (1 α 2 )d(a, B) α 2 (2d(x 0, x 1 ) + M) + (1 α 2 )d(a, B) = α 2 M + 2α 2 d(x 0, x 1 ) + (1 α 2 )d(a, B) = α 2 M + (1 α 2 )M d(x 2, x 3 ) α 2 M + (1 α 2 )M = M a contradiction. In a similar way one can prove {x 2n+1 } is bounded and hence the sequence {x n } is bounded. For any n m in N, d(x 2n, x 2m+1 ) α m d(x 0, x 2(n m)+1 ) + (1 α m )d(a, B). Therefore the sequence {x n } is a cyclically Cauchy sequence in A B, so that the sequences {x 2n } and {x 2n+1 } have convergent subsequences and hence the conclusion follows from Proposition 2.4. For x A, define [x] = {y B : d(x, y) = d(a, B)} and in a similar way we have [y] = {u A : d(u, y) = d(a, B)}, for y B. It is easy to see that, if x i [x] for i = 1, 2 for some x A B, Then x [x 1 ] [x 2 ]. The following Proposition gives a type of necessary condition for the existence of a unique best proximity point for a cyclic contraction. Proposition 4.2. Let (A, B) be a pair in a metric space X. If there exists x A( or B such that [x] containing two different points say x 1 and x 2 with [x 1 ] [x 2 ] contains a point other then x. Then there exists a map T on A B satisfying: (i) T is a cyclic contraction with two distinct best proximity points in A. (ii) For any x 0 A, the sequences {T 2n x 0 } and {T 2n+1 x 0 } diverge.

15 PYTHAGOREAN PROPERTY AND BEST PROXIMITY PAIR THEOREMS 13 Proof. Let x A such that x 1 x 2 in [x] and [x 1 ] [x 2 ] contains an element say y x. Define T : A B A B as T (u) := x 1 x 2 if u A and u = x if u A and u x y if u B and u = x 1 x if u B and u x 1. Then T is a cyclic contraction with x, y are different best proximity points of T and for any x 0 A, the sequences {T 2n x 0 } and {T 2n+1 x 0 } diverge. The following example illustrates Proposition 4.2. Example 4.3. Let A be the line segment joining the points (0, 1, 0), ( 1, 1, 0) and B be the 2 2 line segment joining the points (0, 0, 0), ( 1, 1, 1) in 2 2 (R3, 1 ). Then it is easy to see that d(a, B) = 1. For (0, 0, 0) B, d ( (0, 0, 0), (0, 1, 0) ) = 1 = d(a, B) = d ( (0, 0, 0), ( 1 2, 1 2, 0)). Also d ( ( 1 2, 1, 1 2 ), (0, 1, 0)) = 1 = d(a, B) = d ( ( 1 2, 1, 1 2 ), ( 1 2, 1 2, 0)). That is (0, 1, 0), ( 1 2, 1 2, 0) [(0, 0, 0)] and ( 1, 1, 1) [(0, 1, 0)] [( 1, 1, 0)]. Hence one can construct a cyclic contraction T on A B, which satisfies the conclusion of Theorem 4.2. Theorem 4.4. Let (A, B) be a proximally complete semi sharp proximinal pair in a metric space X. Suppose T is a cyclic contraction on A B, then the following hold: (i) There exists a unique best proximity point x of T in A. (ii) {T 2n x 0 } and {T 2n+1 x 0 } converge to x and T x respectively, for every x 0 A. (iii) T x is the unique best proximity point of T in B. (iv) x and T x are the unique fixed points of T 2 in A and B respectively. Proof. By Theorem 4.1, there exists x A such that d(x, T x) = d(a, B). Notice that d(a, B) d(t x, T 2 x) d(x, T x) = d(a, B). Since (A, B) is a semi sharp proximinal, T 2 x = x. Suppose there exists z x A is such that d(z, T z) = d(a, B). In a similar fashion we have T z = z. d(t z, x) = d(t z, T 2 x) d(z, T x) = d(t 2 z, T x) < d(t z, x),

16 14 R. ESPÍNOLA, G. SANKARA RAJU KOSURU, AND P. VEERAMANI a contradiction, hence there exists a unique x A such that d(x, T x) = d(a, B). x 0 A. Define x n = T x n 1 for all n N. It has been proved in Theorem 4.1 that {x n } is a cyclically Cauchy sequence and hence by Theorem 2.9 {x 2n } and {x 2n+1 } are convergent sequences. Also by the claim in the proof of Theorem 4.1, {x 2n } converge to the best proximity point x in A and {x 2n+1 } converge to the best proximity point y in B, with d(x, y) = d(a, B). By semi sharp proximinality of (A, B), y = T x. Now we prove that x is a unique fixed point of T 2 in A. We have T 2 x = x. If y A satisfying T 2 y = y then T 2n y = y for all n N. We have T 2n y x and hence y = x. In a similar fashion one can show that T x is a unique fixed point of T 2 in B. As an immediate consequence of the above theorem we have, if A and B are nonempty closed convex subsets of a uniformly convex Banach space, then conclusions of Theorem 4.4 hold (Theorem 3.10,[5]). conclustions It is to be noted that Theorem 4.4 holds for the weaker class of cyclic maps satisfying (2) d(t x, T 2 x) r d(x, T x) + (1 r)d(a, B) for all x A. Suzuki et al. in [17] proved the existence and convergence of best proximity points for a type of cyclic contraction T with the contraction condition: There exists r in [0, 1) such that d(t x, T y) r max{d(x, y), d(x, T x), d(y, T y)} + (1 r)d(a, B) for all x in A and y in B. The above contraction condition implies (2) and hence as a consequence of Theorem 4.4 we have the following: Let (X, d) be a metric space and let A and B be nonempty subsets of X such that (A, B) (and (B, A)) satisfies the property UC. Assume that A and B are complete. If T is a cyclic mapping on A B that satisfies Equation 2 for some r < 1, then conclusions of Theorem 4.4 hold (Theorem 2, [17]) A cyclic map T on A B is said to be relatively nonexpansive ([4, 7]) if d(t x, T y) d(x, y) for all x A and y B. It is easy to see that for a relative nonexpansive map T on a A B, we have (T x) = T x and hence as a consequence of Proposition 3.2 and Theorem 4.4 we obtain: Theorem 4.5. Let A, B be nonempty closed convex subsets of a Hilbert space and T be a cyclic map on A B. Further if T is relatively nonexpansive and T A : A B is a contraction then conclusions of Theorem 4.4 hold. Fix

17 PYTHAGOREAN PROPERTY AND BEST PROXIMITY PAIR THEOREMS 15 Acknowledgement : Rafa Espínola was supported by DGES, Grant MTM C02-01 and Junta de Andalucía, Grant FQM-127. References [1] M. A. Al-Thagafi and N. Shahzad, Convergence and existence results for best proximity points, Nonlinear Anal. 70 (2009), no. 10, [2] C. Di Bari, T. Suzuki and C. Vetro, Best proximity points for cyclic Meir-Keeler contractions, Nonlinear Anal. 69 (2008), no. 11, [3] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer, Berlin, [4] A. A. Eldred, W. A. Kirk and P. Veeramani, proximal normal structure and relatively nonexpansive mappings, Studia Math. 171 (2005), no. 3, [5] A. A. Eldred and P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2006), no. 2, [6] A. A. Eldred and P. Veeramani, On best proximity pair solutions with applications to differential equations, J. Indian Math. Soc. (N.S.) 2007, Special volume on the occasion of the centenary year of IMS ( ), (2008). [7] R. Espínola, A new approach to relatively nonexpansive mappings, Proc. Amer. Math. Soc. 136 (2008), no. 6, [8] R. Espínola and A. Fernández-León, On best proximity points in metric and Banach spaces. Canad. J. Math. 63 (2011), no. 3, [9] R. Espínola, A. Fernández-León and B. Pi atek, Fixed points of single- and set-valued mappings in uniformly convex metric spaces with no metric convexity, Fixed Point Theory Appl. (2009), Article ID , 16 pages. [10] K. Goebel and S. Reich, Uniform convexity, hyperbolic geometry and nonexpansive mappings, Pure Appl. Math., Marcel Dekker, Inc., New York Basel, [11] C. W. Groetsch, Elements of applicable functional analysis, Monographs and Textbooks in Pure and Applied Mathematics, 55, Dekker, New York, [12] Karpagam, S and Agrawal, Sushama, Best proximity point theorems for p-cyclic Meir-Keeler contractions. Fixed Point Theory Appl. 2009, Art. ID , 9 pp, 47H10 (46B20 54H25). [13] Karpagam, S and Agrawal, Sushama, Best proximity points for cyclic contractions. (preprint). [14] W. A. Kirk, P. S. Srinivasan and P. Veeramani, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory 4 (2003), no. 1, [15] G. Sankara Raju Kosuru and P. Veeramani, On existence of best proximity pair theorems for relatively nonexpansive mappings, J. Nonlinear Convex Anal. 11 (2010), no. 1, [16] B. Pi atek, On cyclic Meir-Keeler contractions in metric spaces, Nonlinear Anal. 74 (2011), no. 1, [17] T. Suzuki, M. Kikkawa and C. Vetro, The existence of best proximity points in metric spaces with the property UC, Nonlinear Anal. 71 (2009), no. 7-8,

18 16 R. ESPÍNOLA, G. SANKARA RAJU KOSURU, AND P. VEERAMANI Departamento de Análisis Matemático, faciltad de Matemáticas, Universidad de Sevilla, Sevilla, Spain. address: Mathematics and Statistics Unit, Indian Statistical Institute Bangalore, r. v. College post, Bangalore , India. address: Department of Mathematics, Indian Institute of Technology Madras, Chennai , India. address: