@
Appl. Gen. Topol. 18, no. 1 (2017), 13-21

doi:10.4995/agt.2017.3424

c© AGT, UPV, 2017

Best proximity points of contractive mappings

on a metric space with a graph and applications

Asrifa Sultana and V. Vetrivel∗

Department of Mathematics, Indian Institute of Technology Madras, Chennai-600036, India.

(asrifa.iitg@gmail.com, vetri@iitm.ac.in)

Communicated by M. Abbas

Abstract

We establish an existence and uniqueness theorem on best proximity
point for contractive mappings on a metric space endowed with a graph.
As an application of this theorem, we obtain a result on the existence of
unique best proximity point for uniformly locally contractive mappings.
Moreover, our theorem subsumes and generalizes many recent fixed
point and best proximity point results.

2010 MSC: 54H25; 47H10.

Keywords: Fixed point; best proximity point; contraction; graph; metric
space; P -property.

1. Introduction

Fixed point theory plays an important role for solving equations of the form
Tx = x where T is defined on a subset of a metric space, partially ordered
metric space, topological vector space or some suitable space. Given two non-
empty subsets A and B of a metric space (X,d), consider a non-self mapping
T : A → B. If T(A) ∩A = ∅, there does not exist a solution of the equation
Tx = x. Then it is interesting to find a point x ∈ A that is closest to Tx in
some sense. Best approximation and best proximity point results have been
established in this direction. The well-known best approximation theorem due
to Ky Fan [3] states that for a given non-empty compact convex subset C of

∗Corresponding author.

Received 25 November 2014 – Accepted 29 September 2016

http://dx.doi.org/10.4995/agt.2017.3424


A. Sultana and V. Vetrivel

a normed linear space E and a continuous mapping F : C → E, there exists
x∗ ∈ C such that ‖x∗−Fx∗‖ = d(Fx∗,C) = inf{‖Fx∗−x‖ : x ∈ C}. Though
this result gives the existence of an approximate solution of Fx = x, such
solution need not be optimal in the sense that ‖x−Fx‖ is minimum.

Naturally, for the map T , one can think of finding an element x∗ ∈ A such
that d(x∗,Tx∗) = min{d(x,Tx) : x ∈ A}. Since for all x ∈ A, d(x,Tx) ≥
d(A,B) = inf{d(a,b) : a ∈ A, b ∈ B}. An optimal solution of min{d(x,Tx) :
x ∈ A} is one for which the value d(A,B) is attained. An element x∗ ∈ A is
called a best proximity point for the mapping T if d(x∗,Tx∗) = d(A,B). Hence
a best proximity point of the map T is not only an approximate solution of
Tx = x, but also optimal in the sense that d(x,Tx) is minimum. Clearly, a best
proximity point theorem is a natural generalization of a fixed point theorem.
Some interesting best proximity point results can be found in [7, 11, 14] and
for applications, one can refer to [5, 6].

Recently, Jachymski [4] established the existence of fixed points for contrac-
tive mappings on a metric space endowed with a graph. This result unified
various fixed point theorems for contractive mappings on metric spaces and
partially ordered metric spaces. For some more fixed point results on a metric
space with a graph, one can refer to [1, 13].

1.1. Our contribution. Following Jachymski [4], in this article we prove an
existence and uniqueness theorem on best proximity point for non-self contrac-
tive mappings on a metric space endowed with a graph. As an application of
this result, we obtain a generalization of the fixed point theorem for uniformly
locally contractive mappings due to Edelstein [2, Theorem 5.2]. Also, our re-
sult enables us to obtain a best proximity point result for non-self mappings
on partially ordered metric spaces. Further, our result subsumes a very recent
result on existence of a unique best proximity point on a metric space due to
V. Sankar Raj [11, Theorem 3.1].

2. Preliminaries

In this section, let us recall some definitions and notations which are needed
for our results.

Let (X,d) be a metric space. For given non-empty subsets A and B of (X,d),
we denote by A0 and B0 the following sets:

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}.

For sufficient conditions which ensure the non-emptiness of A0 and B0, one can
refer to [7].

Let (A,B) be a pair of non-empty subsets of (X,d) such that A0 6= ∅. Then
the pair (A,B) is said to have the P-property [11] if and only if

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 14



Best proximity points of contractive mappings

d(x1,y1) = d(A,B)

d(x2,y2) = d(A,B)

}
⇒ d(x1,x2) = d(y1,y2),

where x1,x2 ∈ A0 and y1,y2 ∈ B0.
It is easy to verify that for a non-empty subset A of (X,d), the pair (A,A)

has the P-property. Every pair of non-empty closed convex subsets of a real
Hilbert space H has the P-property (see [11]).

Consider a directed graph G where the set V (G) of its vertices coincides
with X, the set E(G) of its edges is such that E(G) ⊇ ∆ (where ∆ = {(x,x) :
x ∈ X}) and E(G) has no parallel edges. We denote by G̃ the undirected graph
obtained from G by ignoring the direction of edges. For given two vertices x
and y, we say that there is a path in G of length N (where N ∈ N ∪ {0})
between them if there exists a sequence (xi)Ni=0 such that x

0 = x, xN = y and
(xi−1,xi) ∈ E(G) ∀ i = 1, 2, . . . ,N. The graph G is called connected if there is
a path between any two vertices and weakly connected if G̃ is connected. For
x ∈ V (G) = X, we denote

[x]NG = {y ∈ X : there is a path in G of length N from x to y} .

3. Main results

Throughout this section we assume that (X,d) is a metric space endowed
with a directed graph G where V (G) = X, E(G) ⊇ ∆ and G has no parallel
edges. We now introduce a notion of Banach contraction (for non-self map)
with respect to the graph G for which we prove our main results.

Definition 3.1. Let A and B be two non-empty subsets of (X,d). A mapping
T : A → B is said to be a Banach G -contraction or simply G-contraction if
for all x,y ∈ A, x 6= y with (x,y) ∈ E(G):
(a) d(Tx,Ty) ≤ αd(x,y) for some α ∈ [0, 1);

(b)
d(x1,Tx) = d(A,B)

d(y1,Ty) = d(A,B)

}
⇒ (x1,y1) ∈ E(G), for all x1,y1 ∈ A.

Theorem 3.2. Let (X,d) be complete metric space, A and B be two non-empty
closed subsets of (X,d) such that (A,B) has the P -property. Let T : A → B be
a G-contraction such that T(A0) ⊆ B0. Assume that for some N ∈ N,

(i) there exist x0 and x1 in A0 such that there is a N-length path (y
i
0)

N
i=0 ⊆ A0

in G between them and d(x1,Tx0) = d(A,B);
(ii) for any sequence {sn}n∈N in A with sn → s and sn+1 ∈ [sn]NG , there is a

subsequence (snk )k∈N such that (snk,s) ∈ E(G) ∀k ∈ N.
Then there exists a sequence {xn}n∈N with d(xn+1,Txn) = d(A,B) for n ∈ N,
converging to a best proximity point of T . Furthermore, T has a unique best
proximity point if for any two elements x and y in A0, there exists a path
(yi)li=0 ⊆ A0 in G̃ between them.

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 15



A. Sultana and V. Vetrivel

Proof. By (i), there exist two points x0,x1 ∈ A0 such that d(x1,Tx0) = d(A,B)
and a sequence (yi0)

N
i=0 containing points of A0 such that y

0
0 = x0, y

N
0 = x1

and (yi−10 ,y
i
0) ∈ E(G) ∀1 ≤ i ≤ N. As y10 ∈ A0 and T(A0) ⊆ B0, there exists

y11 ∈ A0 such that d(y11,Ty10) = d(A,B). Similarly, for i = 2, · · · ,N, there
exists yi1 ∈ A0 such that d(yi1,Tyi0) = d(A,B).

As (y00 = x0,y
1
0) ∈ E(G) and T is a G-contraction, it follows from the above

that (x1,y
1
1) ∈ E(G). In a similar way, it follows that (y

i−1
1 ,y

i
1) ∈ E(G) for

i = 2, · · · ,N. Let x2 = yN1 . Thus (yi1)Ni=0 is a path from x1(= y
0
1) to x2(= y

N
1 ).

Again, for each i = 1, 2, · · · ,N, since yi1 ∈ A0 and Tyi1 ∈ T(A0) ⊆ B0, there
exists yi2 ∈ A0 such that d(yi2,Tyi1) = d(A,B). Also, we have d(x2,Tx1) =
d(A,B). As shown in the previous paragraph, it follows that (x2,y

1
2) ∈ E(G)

and (yi−12 ,y
i
2) ∈ E(G) ∀i = 2, · · · ,N. Set x3 = yN2 . Thus (yi2)Ni=0 is a path

from x2(= y
0
2) to x3(= y

N
2 ).

Continuing in this manner for all n ∈ N, we obtain a sequence {xn}n∈N
where xn+1 ∈ [xn]NG and d(xn+1,Txn) = d(A,B) by producing a path (y

i
n)

N
i=0

from xn(= y
0
n) to xn+1(= y

N
n ) in such way that

(3.1) d(yin+1,Ty
i
n) = d(A,B) ∀ i = 0, · · · ,N.

Using the P-property of (A,B), it follows from equation (3.1) that for each
n ∈ N,

(3.2) d(yi−1n ,y
i
n) = d(Ty

i−1
n−1,Ty

i
n−1) ∀ 1 ≤ i ≤ N.

Now for any positive integer n,

d(xn,xn+1) = d(y
0
n,y

N
n )

≤ d(y0n,y
1
n) + d(y

1
n,y

2
n) + · · · + d(y

N−1
n ,y

N
n )

=

N∑
i=1

d(yi−1n ,y
i
n) =

N∑
i=1

d(Tyi−1n−1,Ty
i
n−1).

Since for all n ∈ N and 1 ≤ i ≤ N, (yi−1n−1,y
i
n−1) ∈ E(G) and T is a G-

contraction, it follows from the above inequalities that for n ∈ N,

d(xn,xn+1) ≤ α
N∑
i=1

d(yi−1n−1,y
i
n−1) for some α ∈ [0, 1).

Repeating the process, it follows that for all n ∈ N,

d(xn,xn+1) ≤ αn
N∑
i=1

d(yi−10 ,y
i
0) = Mα

n where M =

N∑
i=1

d(yi−10 ,y
i
0).

Now for m ≥ n, n ∈ N,

d(xn,xm) ≤ d(xn,xn+1) + · · · + d(xm−1,xm)
≤ Mαn + · · · + Mαm−1

= Mαn[1 + · · · + αm−n−1] ≤ M
αn

1 −α
.

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 16



Best proximity points of contractive mappings

Hence {xn}n∈N is a Cauchy sequence. Therefore {xn}n∈N converges to some
point x∗ ∈ A as n → ∞. By (ii), there is a subsequence (xnk )k∈N such that
(xnk,x

∗) ∈ E(G) ∀k ∈ N. Hence,

d(Txnk,Tx
∗) ≤ αd(xnk,x

∗) for k ∈ N.

Thus taking k →∞, Txnk → Tx
∗. Using the continuity of the metric function,

we get d(xnk+1,Txnk ) → d(x
∗,Tx∗) as k → ∞. Now {d(xnk+1,Txnk )} is

nothing but a constant sequence with value d(A,B). Therefore d(x∗,Tx∗) =
d(A,B).

Suppose that p and q are two best proximity points of T . Consider two
sequences {pn}n∈N and {qn}n∈N where pn = p and qn = q for all n ≥ 1.
Clearly, d(pn+1,Tpn) = d(A,B) and d(qn+1,Tqn) = d(A,B) for all n ≥ 1. As
p,q ∈ A0, it follows from the hypothesis that there is a path (yi1)Mi=0 ⊆ A0 in
G̃ between p1 = p and q1 = q. For each i = 1, 2, · · · ,M − 1, since yi1 ∈ A0
and T(yi1) ∈ T(A0) ⊆ B0, we can obtain {yin}n∈N such that d(yin+1,Tyin) =
d(A,B) ∀n ∈ N. It is easy to verify that T is also a G̃-contraction. Also, we
have (yi−11 ,y

i
1) ∈ E(G̃) for 1 ≤ i ≤ M. Thus it follows that (yi2)Mi=0 is a path in

G̃ between p2(= y
0
2) and q2(= y

M
2 ). Similarly, it follows that ∀n ∈ N, (yin)Mi=0

is a path in G̃ from pn(= y
0
n) to qn(= y

M
n ). Now for n ∈ N,

d(p,q) = d(pn+1,qn+1) ≤
M∑
i=1

d(yi−1n+1,y
i
n+1) =

M∑
i=1

d(Tyi−1n ,Ty
i
n)

≤ α
M∑
i=1

d(yi−1n ,y
i
n) ≤ ···≤ α

n
M∑
i=1

d(yi−11 ,y
i
1). [where α ∈ [0, 1)]

This implies that p = q and this completes the proof. �

Remark 3.3. Theorem 3.2 still holds true if we replace the condition (ii) by the
continuity of the function T on the set A.

The above Theorem 3.2 yields the following result due to Jachymski [4].

Theorem 3.4 (see [4]). Let (X,d) be complete and f : X → X be a map such
that for all x,y ∈ X with (x,y) ∈ E(G), (fx,fy) ∈ E(G) and d(fx,fy) ≤
kd(x,y) where k ∈ [0, 1). Assume that for any {yn}n∈N in X with yn → y∗
and (yn+1,yn) ∈ E(G) ∀n ≥ 1, there exists a subsequence {ynp}p∈N such that
(ynp,y

∗) ∈ E(G) for all p ∈ N. Then the following statements hold:
(i) {fn(x)}n∈N converges to a fixed point of f if (x,fx) ∈ E(G);

(ii) if G is weakly connected and there exists x0 ∈ X such that (x0,fx0) ∈
E(G), then ∀x ∈ X, {fn(x)}n∈N converges to a unique fixed point of f.

Further, we get the following result due to V. Sankar Raj [11] as a corollary
to the Theorem 3.2 by taking E(G) = X ×X.

Corollary 3.5 ([11, Theorem 3.1]). Let (X,d) be a complete metric space, A
and B be two non-empty closed subsets of (X,d) such that A0 6= ∅ and (A,B)

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 17



A. Sultana and V. Vetrivel

satisfies P -property. Suppose that T : A → B is such that T(A0) ⊆ B0 and

(3.3) d(Tx,Ty) ≤ kd(x,y) ∀x,y ∈ A and for some k ∈ [0, 1).

Then there exists a unique x∗ in A such that d(x∗,Tx∗) = d(A,B). Further,
for any fixed x0 ∈ A0, there exists a sequence {xn}n∈N with d(xn,Txn−1) =
d(A,B) for n ∈ N, converging to x∗.

The following example shows that our Theorem 3.2 is an extension of the
above result due to V. Sankar Raj [11].

Example 3.6. Consider X = R2 with usual metric and suppose that

A =
{(

0, 1n

)
: n ∈ N

}
∪{(0, 0)},

B =
{(

1, 1n

)
: n ∈ N

}
∪{(1, 0)}.

It is easy to check that the pair (A,B) has the P-property. Suppose that a
map T : A → B is defined as follows:

T((0,x)) =
(

1, x
2

)
, for all (0,x) ∈ A with x 6= 1,

T((0, 1)) = (1, 1).

Consider a graph G with V (G) = X and E(G) = {(x,y) ∈ X×X : d(x,y) < 1
2
}.

Let x = (0,x′) and y = (0,y′) be two elements in A with (x,y) ∈ E(G). Then,

d(T(x),T(y)) = d

((
1,
x′

2

)
,

(
1,
y′

2

))
≤

1

2
d(x,y).

If x1 = (0,x
′
1) and y1 = (0,y

′
1) are two elements in A such that

d(x1,T(x)) = d(y1,T(y)) = dist(A,B).

Then by using the P-property of (A,B), it follows from the above equation
that d(x1,y1) = d(T(x),T(y)) ≤ 12d(x,y) <

1
2
. Hence the pair (x1,y1) ∈ E(G).

This proves that T is a non-self G-contraction with α = 1
2
. Clearly, (X,d) is

complete and A and B are closed subsets of X. Also, note that in this case
A0 = A,B0 = B and T(A0) = T(A) ⊆ B = B0. Let x0 = (0, 12 ), x1 = (0,

1
4
)

and N = 1. Then d(x1,T(x0)) = dist(A,B) = 1 and the pair (x1,x0) ∈ E(G).
Hence, the condition (i) of Theorem 3.2 holds. Also, let {sn}n∈N be a sequence
in A such that sn → s as n →∞. Then there exists a positive integer M such
that d(sn,s) <

1
2
∀n ≥ M. Let nk = M + k for k ≥ 1. Consequently, {snk}k∈N

is a subsequence of the sequence {sn}n∈N such that (snk,s) ∈ E(G) ∀k ∈ N.
This implies that the condition (ii) of Theorem 3.2 is also satisfied. Therefore
Theorem 3.2 guarantees the existence of a best proximity point of T . Note that
(0, 0) and (0, 1) are two best proximity points. However,

d(T(0, 0),T(0, 1)) = d((1, 0), (1, 1)) = 1 > kd((0, 0), (0, 1)),

for any k ∈ [0, 1). This proves that T does not satisfy the contractive condition
(3.3).

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 18



Best proximity points of contractive mappings

4. Applications

Let A and B be two non-empty subsets of a metric space (X,d). A mapping
f : A → B is called (�,k)-uniformly locally contractive [2] (where k ∈ [0, 1)
and � > 0) if d(fx,fy) ≤ kd(x,y) for all x,y ∈ A with d(x,y) < �. An (�,k)-
uniformly locally contractive mapping need not be a contraction, for example
one can refer to [2, 8]. As an application of Theorem 3.2, we now establish the
following result for uniformly locally contractive mappings.

Theorem 4.1. Let (X,d) be complete metric space, A and B be closed subsets
of (X,d) such that A0 6= ∅ and (A,B) satisfies P -property. Suppose that
T : A → B is an (�,k)-uniformly locally contractive mapping satisfying
T(A0) ⊆ B0. Then T has a unique best proximity point if the space (A0,d) is
�-chainable, that is, given a,b ∈ A0, there exist N ∈ N and a sequence (yi)Ni=0
in A0 such that y

0 = a, yN = b and d(yi−1,yi) < � for each i = 1, 2, · · · ,N.

Proof. Consider the graph G where V (G) = X and E(G) as follows:

E(G) = {(x,y) ∈ X ×X : d(x,y) < �}.
It is clear that E(G) ⊇ ∆ and G has no parallel edges. Also, in this case
G = G̃. Let x,y ∈ A be such that (x,y) ∈ E(G) and for all x1,y1 ∈ A,

d(x1,Tx) = d(A,B) and d(y1,Ty) = d(A,B).

Since (x,y) ∈ E(G), d(Tx,Ty) ≤ kd(x,y) where k ∈ [0, 1). Hence and by the
P-property of (A,B), we have d(x1,y1) < �. Therefore T is a G-contraction.
Since A0 6= ∅ and T(A0) ⊆ B0, there exist x0 and x1 in A0 such that
d(x1,Tx0) = d(A,B). The �-chainability of (A0,d) implies that there exist
a natural number N and a sequence (yi)Ni=0 containing points of A0 such that
y0 = x0, y

N = x1 and d(y
i−1,yi) < � for i = 1, · · · ,N. Thus (yi)Ni=0 ⊆ A0

is a path in G between x0 and x1. If {sn}n∈N is a sequence in A such that
sn → s, then there exists M ∈ N such that d(sn,s) < � ∀n ≥ M. Hence we can
obtain a subsequence {snp}p∈N such that (snp,s) ∈ E(G) ∀p ∈ N. Also, it is
clear from the �-chainability of (A0,d) that for every x,y ∈ A0, there is a path
(qi)li=0 ⊆ A0 in G̃ (i.e., G) between them. Thus T has a unique best proximity
point by Theorem 3.2. �

As a corollary to the above theorem, we get the following theorem due to
Edelstein [2] by considering A = B = X.

Theorem 4.2 ([2, Theorem 5.2]). Let (X,d) be a complete metric space. An
(�,k)- uniformly locally contractive mapping f : X → X has a unique fixed
point if (X,d) is �-chainable.

In the last part of this section we establish the following result for non-self
contractive mapping on a partially ordered metric space.

Let (X,d) be a metric space endowed with a partial order � and A and B
be two non-empty subsets of (X,d). By X�, we denote the following set:

X� = {(x,y) ∈ X ×X : x � y or x � y}.

c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 19



A. Sultana and V. Vetrivel

Following [10], we say that a mapping T : A → B is a proximally monotone
mapping if for all x1,x2 ∈ A with x1 � x2:

d(y1,Tx1) = d(A,B)

d(y2,Tx2) = d(A,B)

}
⇒ (y1,y2) ∈ X�, for all y1,y2 ∈ A.

Theorem 4.3. Let (X,d) be complete metric space, A and B be two closed
subsets of (X,d) such that (A,B) has the P -property. Let T : A → B be a
proximally monotone map such that T(A0) ⊆ B0 and

d(Tx,Ty) ≤ kd(x,y) for all x � y and for some k ∈ [0, 1).
Assume that either T is continuous on A or for any {yn}n∈N in A with yn → y∗
and (yn,yn+1) ∈ X� for n ∈ N, there exists (ynp )p∈N such that (ynp,y∗) ∈
X� for p ∈ N. Then T has a best proximity point if there exist x0 and x1
in A0 such that d(x1,Tx0) = d(A,B) and (x0,x1) ∈ X�. Moreover, the best
proximity point of T is unique if for x,y ∈ A0, there exists z ∈ A0 such that
(x,z), (y,z) ∈ X�.

Proof. By considering the graph G where V (G) = X and

E(G) := {(x,y) ∈ X ×X : x � y ∨y � x},
the proof follows by Theorem 3.2 and Remark 3.3. �

The above result includes the fixed point results for mappings on a partially
ordered metric space due to Ran and Reurings [12] and J. J. Nieto and R. R.
López [9].

Acknowledgements. The authors are grateful to the referees for their valu-
able comments and suggestions to improve this manuscript. The first author
is thankful to University Grants Commission

(
F.2 − 12/2002(SA − I)

)
, New

Delhi, India for the financial support.

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