isibang/ms/2013/32
November 25th, 2013
http://www.isibang.ac.in/e statmath/eprints
Pythagorean Property and Best Proximity
Pair Theorems
Rafa Espı́nola, G. Sankara Raju Kosuru and P. Veeramani
Indian Statistical Institute, Bangalore Centre
8th Mile Mysore Road, Bangalore, 560059 India
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 x0 in A∩B the Picard’s iteration {T n x0 } 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 supx∈A∪B ∥Tx′ ∥ < 1 is a cyclic
contraction, where Tx′ 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
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.
A0 = {x ∈ A : d(x, y) = d(A, B), for some y in B};
B0 = {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 {xn } and {zn } be sequences in A and {yn }
be a sequence in B satisfying:
(1) ∥zn − yn ∥ → d(A, B).
(2) For every ϵ > 0 there exists N0 ∈ N such that for all m > n ≥ N0 , ∥xm − yn ∥ ≤
dist(A, B) + ϵ.
Then, for every ϵ > 0 there exists N1 such that for all m > n ≥ N1 , ∥xm − zn ∥ < ϵ.
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 {xn } and {zn } are sequences in A and {yn } is a sequence in B such that limn d(xn , yn ) =
d(A, B) and limn d(zn , yn ) = d(A, B), then limn d(xn , zn ) = 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 R2 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 {xn }∞
n=0 in A ∪ B, with x2n ∈ A and x2n+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(xn , xm ) < d(A, B) + ϵ, when n is even, m is odd and n, m ≥ N.
If d(A, B) = 0, then a sequence {xn } in A ∪ B is cyclically Cauchy if and only if {xn } is
Cauchy. The following Lemma ensures the boundedness of a cyclically Cauchy sequence.
Lemma 2.2. Every cyclically Cauchy sequence is bounded.
Proof. Let {xn } be a cyclically Cauchy sequence in A ∪ B. Then there exists N ∈ N, such
that d(x2n , x2N +1 ) < d(A, B) + 1 for all n ≥ N . Therefore for all n ∈ N, x2n ∈ B(x2N +1 , r),
where
r = max{d(x2 , x2N +1 ), d(x4 , x2N +1 ), . . . , d(x2N , x2N +1 ), d(A, B) + 1}.
So that {x2n } is bounded. In a similar fashion one can prove that the sequence {x2n+1 } is
a bounded sequence and hence {xn } is bounded.
4
R. ESPÍNOLA, G. SANKARA RAJU KOSURU, AND P. VEERAMANI
It is to be noted that for a cyclically Cauchy sequence {xn }, the sequence {x2n } may
have two different convergent subsequences.
Example 2.3. Let X = (R3 , ∥ · ∥2 ). A := {(x, y, z) ∈ X : −1 ≤ x ≤ 1, y 2 + z 2 = 1}, B :=
{(0, 0, 0)}. For n ∈ N define x2n+1 = (0, 0, 0) and
{
, 1, 0) if n is odd,
( −1
n
x2n :=
1
( n , −1, 0) if n is even.
Then d(A, B) = 1 and {xn } is cyclically Cauchy. It is to noted that {x2n } has two different convergent subsequences {( −1
, 1, 0)} and {( n1 , −1, 0)} which converge to (0, 1, 0) and
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 x0 in A, the sequence {T n x0 } has
a convergent subsequence {T nk x0 } that converges to x in A then
(i) d(T nk −1 x0 , x) → d(A, B), d(T nk 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 x0 } 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 {xn } in A ∪ B either {x2n } or {x2n+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 R2 . Then (A, B) is cyclically
complete but A0 is not a closed set. Also the sequence {x2n } does not have any convergent
subsequence in A, for the cyclically Cauchy sequence {xn } in A ∪ B, where
{
(0, 1 − n1 ) if n is even,
xn :=
(1, 1 − n1 ) if n is odd.
On the other hand the compact pair (A, B), where A := {(1 + n1 )ei : i = 1, 3 and n ∈
N} ∪ {e1 , e3 } and B := {(1 + n1 )ei : i = 2, 4 and n ∈ N} ∪ {e2 , e4 }, in the Banach space R4
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 {xn } in A ∪ B, the sequences {x2n } and {x2n+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 {xn } is a cyclically Cauchy
sequence in A ∪ B, then sequences {x2n } and {x2n+1 } are Cauchy and hence (A, B) is
proximally complete. Also if {x2nk } and {x2mk +1 } are convergent subsequences of {x2n }
and {x2n+1 }, that converge to x ∈ A and y ∈ B respectively, then d(A, B) ≤ d(x, y) =
lim d(x2nk , x2mk +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 A0
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. If (A, B) is proximally
complete, then A0 and B0 are closed subsets of X.
Proof. Let {xn } be a sequence in A0 such that xn → x in X. For n ∈ N let x′n ∈ B0 such
that d(xn , x′n ) = d(A, B). For n ∈ N define
{
xm if n = 2m for some m ∈ N,
yn :=
x′m if n = 2m + 1 for some m ∈ N.
Now d(y2n , y2m+1 ) = d(xn , x′m ) ≤ d(xn , x) + d(x, xm ) + d(xm , x′m ) → d(A, B), as n, m → ∞
and hence {yn } is a cyclically Cauchy sequence. Since (A, B) is proximally complete, {xn }
and {x′n } have convergent subsequences, which converge to x and y respectively, then
d(x, y) = d(A, B) and hence A0 is closed. In a similar fashion one can prove B0 is also a
closed set.
The following theorem ensures that every closed convex pair in a uniformly convex Banach space is proximally complete.
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 {xn }, the sequences
{x2n } and {x2n+1 } converge to some x in A and y in B respectively with d(x, y) = d(A, B).
Proof. Let {xn } be a cyclically Cauchy sequence in A ∪ B. Suppose {x2n } is not a Cauchy
sequence. Then there exists ϵ0 > 0 and subsequences {x2nk } and {x2mk } of {x2n } such
that d(x2nk , x2mk ) ≥ ϵ0 , for all k ∈ N. Also one can observe that d(x2nk , x2k+1 ) → d(A, B)
and d(x2mk , x2k+1 ) → d(A, B), as k → ∞. By Lemma 1.1, there exists N1 ∈ N such
that d(x2nk , x2mk ) < ϵ0 for all k ≥ N1 , a contradiction. Hence {x2n } converges to some
point x in A. In a similar fashion one can prove that x2n+1 → y in B. Also d(x, y) =
limn d(x2n , x2n+1 ) = d(A, B).
The following theorem gives sufficient conditions for the convergence of {x2n } and {x2n+1 },
whenever {xn } 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 {xn } is a cyclically Cauchy sequence in A ∪ B then x2n → x, for some x in A
and x2n+1 → y, for some y in B. Further d(x, y) = d(A, B).
Proof. Let {xn } be a cyclically Cauchy sequence in A ∪ B.
Fix a convergent subse-
quence {x2nk +1 } of {x2n+1 }, that converge to y ∈ B. Let {x2mk } and {x2lk } be convergent subsequences of {x2n } that converge to x1 and x2 in A respectively. By the semi
sharp proximinality of (A, B), we have x1 = x2 . Hence any two convergent subsequences
of {x2n } converge to a point say to x, with d(x, y) = d(A, B). Suppose {x2n } is not
Cauchy, then there exists ϵ0 > 0 and two subsequences {x2np }, {x2mp } of {x2n } such that
d(x2np , x2mp ) ≥ ϵ0 , for all p ∈ N. Now consider the sequence {yp }, where
{
x2np if p is even
yp :=
xp
if p is odd
Then it is easy to see that the sequence {yp } is a cyclically Cauchy sequence and hence
{x2np } has a convergent subsequence. Similarly {x2mp } has a convergent subsequence. Since
(A, B) is a proximally complete pair, {x2np } and {x2mp } have convergent subsequences, that
converge to x. Hence there exists P ∈ N such that d(x2nP , x) <
Now d(x2nP , x2mP ) ≤ d(x2nP , x) + d(x2mP , x) <
ϵ0
2
+
ϵ0
2
ϵ0
2
and d(x2mP , x) <
ϵ0
.
2
= ϵ0 , a contraction. Hence {x2n } is
Cauchy. Also {x2n } has a convergent subsequence and hence x2n → x in A. In a similar
fashion one can show x2n+1 → y in B.
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, d1 ), where d1 ((u1 , v1 ), (u2 , v2 )) = d(u1 , u2 ) + d(v1 , v2 ). For instant X denote the set
of all complex valued continuous functions on [0, 1] with the ∥ · ∥, defined as ∥f1 + if2 ∥ =
∫1
∫1
|f1 (t)|dt + 0 |f2 (t)|dt. Let A = {sinnt : n ∈ N} and B = {sinnt + i : n ∈ N}. Note that
0
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 A0 × B0 , 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
A0 = A and B0 = 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
∥x − y∥2 = ∥h∥2 + ∥x + h − y∥2 .
Hence we have the following:
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
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 p̄. X is said be a CAT(0) space if each geodesic triangle △(x, y, z) in X
has a comparison triangle △(x̄, ȳ, z̄) with d(p, q) ≤ d(p̄, q̄), 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
¯ ȳ (γ̄1 (t1 ), γ̄2 (t2 )), where ∠
¯ ȳ is the angle in R2 .
∠y (x, z) = lim sup ∠
t1 ,t2 →0
Where γ̄1 , γ̄2 corresponding line segments from x̄ to ȳ and ȳ to z̄ respectively in the comparison triangle △(x̄, ȳ, z̄) in R2 . 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 (A0 , B0 ) is a semisharp proximinal pair in X.
Indeed, something more can be said on this regard.
Corollary 3.4. Under the previous conditions, (A0 , B0 ) is a sharp proximinal pair.
Proof. It suffices to show that A0 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
A0 =
∩
ε>0
(A ∩ B(B, d(A, B) + ε)) ,
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 ∈ A0 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) ∈ A0 × B0 .
Let (x′ , y ′ ) ∈ B0 × A0 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) ≥ π2 , where ∠x′ (x, y) the Alexandrov angle between x, y at x′ . In a similar fashion
we have ∠x (x′ , y ′ ) ≥ π2 , ∠y (y ′ , x′ ) ≥
π
2
and ∠y′ (y, x) ≥ π2 . Now by [3, Theorem 2.11], we
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 R2 . 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, v̄ be their corresponding points in the triangle △(p, q, r) in R2 ,
then by convex isometry we have d(u, v) = d(ū, v̄). 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 R2 . 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]
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 = xp 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 A0 and B0 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 nonHilbert 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 := {fα : α ∈ [0, 1]} and B := {gα : α ∈ [0, 1]}, where
{
α + t, if t ∈ [0, 21 ]
fα (t) :=
α + (1 − t), if t ∈ [ 21 , 1].
{
α + t + i(t + 21 ), if t ∈ [0, 21 ]
gα (t) :=
α + (1 − t) + i( 32 − t), if t ∈ [ 21 , 1].
For any fixed α ∈ [0, 1] and for any t ∈ [0, 12 ], |gα (t) − fα (t)| = |α + t + i(t + 21 ) − α + t| ≤ |
and for any t ∈ [ 12 , 1], |gα (t) − fα (t)| = |α + (1 − t) + i( 23 − t) − α + (1 − t)| ≤ 1. Also
|fα ( 21 )−gα ( 12 )| = 1. Therefore ∥fα −gα ∥ = 1. Now for any α ̸= β ∈ [0, 1], ∥fα −gβ ∥ ≥ |gα ( 21 )−
fβ ( 21 )| = |(α + 12 ) + i(t + 1) − (β + 21 )| > 1. Hence d(A, B) = 1 and fα′ = gα , for all fα ∈ A.
Now for every α, β, t ∈ [0, 1], |fα (t)−|gβ (t)|2 = 1+|α−β|2 = |fα (t)−gα (t)|2 +|gα (t)−gβ (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 A0
and (T x)′ = T x′ for all x ∈ A0 then T is a cyclic contraction on A0 ∪ B0 .
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 3.10. 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 G1 and
G2 of X such that A0 ⊂ G1 , B0 ⊂ G2 and a continuously Fréchet differentiable map
T : G1 ∪ G2 → X map satisfying:
(1) supx∈A0 ∪B0 ∥Tx′ ∥ < 1 where Tx′ is the Fréchet derivative of T at x,
(2) T A0 ⊆ B0 , T B0 ⊆ A0 and (T x)′ = T x′ for all x ∈ A0
then T is a cyclic contraction on A0 ∪ B0 .
4. 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 x0 ∈ A, define xn = T n x0 for all n ∈ N. First we prove that the sequence {x2n }
2α2 d(x1 , x2 )
+ d(A, B) + d(x2 , x3 ), there exists n ∈ N,
is bounded. Suppose not, for M =
1 − α2
12
R. ESPÍNOLA, G. SANKARA RAJU KOSURU, AND P. VEERAMANI
such that d(x3 , x2n−2 ) ≤ M and d(x3 , x2n ) > M.
Now M < d(x3 , x2n ) ≤ α2 d(x1 , x2n−2 ) + (1 − α2 )d(A, B)
≤ α2 (d(x1 , x2 ) + d(x2 , x2n−2 )) + (1 − α2 )d(A, B)
≤ α2 (d(x1 , x2 ) + d(x2 , x3 ) + d(x3 , x2n−2 )) + (1 − α2 )d(A, B)
≤ α2 (d(x1 , x2 ) + d(x2 , x3 ) + M ) + (1 − α2 )d(A, B)
≤ α2 (d(x1 , x2 ) + d(x1 , x2 ) + M ) + (1 − α2 )d(A, B)
≤ α2 (2d(x0 , x1 ) + M ) + (1 − α2 )d(A, B)
= α2 M + 2α2 d(x0 , x1 ) + (1 − α2 )d(A, B)
= α2 M + (1 − α2 )M − d(x2 , x3 )
≤ α2 M + (1 − α2 )M = M
a contradiction. In a similar way one can prove {x2n+1 } is bounded and hence the sequence {xn } is bounded. For any n ≥ m in N, d(x2n , x2m+1 ) ≤ αm d(x0 , x2(n−m)+1 ) + (1 −
αm )d(A, B). Therefore the sequence {xn } is a cyclically Cauchy sequence in A ∪ B, so that
the sequences {x2n } and {x2n+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 xi ∈ [x] for i = 1, 2
for some x ∈ A∪B, Then x ∈ [x1 ]∩[x2 ]. 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 x1 and x2 with [x1 ] ∩ [x2 ] 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 x0 ∈ A, the sequences {T 2n x0 } and {T 2n+1 x0 } diverge.
PYTHAGOREAN PROPERTY AND BEST PROXIMITY PAIR THEOREMS
13
Proof. Let x ∈ A such that x1 ̸= x2 in [x] and [x1 ] ∩ [x2 ] contains an element say y ̸= x.
Define T : A ∪ B → A ∪ B as

x1



 x
2
T (u) :=

y



x
if
if
if
if
u∈A
u∈A
u∈B
u∈B
and u = x
and u ̸= x
and u = x1
and u ̸= x1 .
Then T is a cyclic contraction with x, y are different best proximity points of T and for
any x0 ∈ A, the sequences {T 2n x0 } and {T 2n+1 x0 } diverge.
The following example illustrates Proposition 4.2.
Example 4.3. Let A be the line segment joining the points (0, 1, 0), ( 21 , 12 , 0) and B be the
line segment joining the points (0, 0, 0), ( 21 , 1, 12 ) in (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), ( 21 , 12 , 0) .
(
)
(
)
Also d ( 12 , 1, 12 ), (0, 1, 0) = 1 = d(A, B) = d ( 21 , 1, 21 ), ( 12 , 12 , 0) . That is (0, 1, 0), ( 12 , 12 , 0) ∈
∩
[(0, 0, 0)] and ( 12 , 1, 12 ) ∈ [(0, 1, 0)] [( 12 , 21 , 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 x0 } and {T 2n+1 x0 } converge to x and T x respectively, for every x0 ∈ 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),
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). Fix
x0 ∈ A. Define xn = T xn−1 for all n ∈ N. It has been proved in Theorem 4.1 that {xn } is
a cyclically Cauchy sequence and hence by Theorem 2.9 {x2n } and {x2n+1 } are convergent
sequences. Also by the claim in the proof of Theorem 4.1, {x2n } converge to the best
proximity point x in A and {x2n+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
d(T x, T 2 x) ≤ r d(x, T x) + (1 − r)d(A, B) for all x ∈ A.
(2)
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.
PYTHAGOREAN PROPERTY AND BEST PROXIMITY PAIR THEOREMS
15
Acknowledgement : Rafa Espı́nola was supported by DGES, Grant MTM2012-34847C0201 and Junta de Andalucı́a, Grant FQM-127.
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16
R. ESPÍNOLA, G. SANKARA RAJU KOSURU, AND P. VEERAMANI
Departamento de Análisis Matemático, faciltad de Matemáticas, Universidad de Sevilla,
41010-Sevilla, Spain.
E-mail address: [email protected]
Mathematics and Statistics Unit, Indian Statistical Institute Bangalore, r. v. College
post, Bangalore-560 059, India.
E-mail address: [email protected]
Department of Mathematics, Indian Institute of Technology Madras, Chennai-600 036,
India.
E-mail address: [email protected]