Arbitrary binary relations, contraction mappings and $b$-metric spaces
UDC 517.9We prove some results on the existence and uniqueness of fixed points defined on a $b$-metric space endowed with an arbitrary binary relation.  As applications, we obtain some statements on coincidence points involving a pair of mappings.  Our results generalize, e...
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Institute of Mathematics, NAS of Ukraine
2020
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507015918387200 |
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| author | Chandok, S. Chandok, S. Chandok, S. |
| author_facet | Chandok, S. Chandok, S. Chandok, S. |
| author_sort | Chandok, S. |
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| description | UDC 517.9We prove some results on the existence and uniqueness of fixed points defined on a $b$-metric space endowed with an arbitrary binary relation.  As applications, we obtain some statements on coincidence points involving a pair of mappings.  Our results generalize, extend, modify and unify several well-known results especially those obtained by Alam and Imdad [J. Fixed Point Theory and Appl., 17, 693–702 (2015); Fixed Point Theory, 18, 415–432 (2017); Filomat, 31, 4421–4439 (2017)] and Berzig [J. Fixed Point Theory and Appl., 12, 221–238 (2012)].  Also, we provide an example to illustrate the suitability of results obtained. |
| doi_str_mv | 10.37863/umzh.v72i4.368 |
| first_indexed | 2026-03-24T02:02:36Z |
| format | Article |
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DOI: 10.37863/umzh.v72i4.368
UDC 517.54
S. Chandok (School Math., Thapar Univ., Patiala, India)
ARBITRARY BINARY RELATIONS, CONTRACTION MAPPINGS,
AND \bfitb -METRIC SPACES*
ДОВIЛЬНI БIНАРНI СПIВВIДНОШЕННЯ, СТИСКАЮЧI ВIДОБРАЖЕННЯ
ТА \bfitb -МЕТРИЧНI ПРОСТОРИ
We prove some results on the existence and uniqueness of fixed points defined on a b-metric space endowed with an
arbitrary binary relation. As applications, we obtain some statements on coincidence points involving a pair of mappings.
Our results generalize, extend, modify and unify several well-known results especially those obtained by Alam and Imdad
[J. Fixed Point Theory and Appl., 17, 693 – 702 (2015); Fixed Point Theory, 18, 415 – 432 (2017); Filomat, 31, 4421 – 4439
(2017)] and Berzig [J. Fixed Point Theory and Appl., 12, 221 – 238 (2012)]. Also, we provide an example to illustrate the
suitability of results obtained.
Доведено деякi результати про iснування та єдинiсть нерухомих точок на b-метричних просторах, що надiленi
довiльним бiнарним вiдношенням. В якостi застосувань отримано деякi твердження про точки збiгу для пар вi-
дображень. Цi результати узагальнюють, розширюють, модифiкують та унiфiкують деякi вiдомi результати Alam
i Imdad [J. Fixed Point Theory and Appl., 17, 693 – 702 (2015)]; Fixed Point Theory, 18, 415 – 432 (2017); Filomat, 31,
4421 – 4439 (2017)] та Berzig [J. Fixed Point Theory and Appl., 12, 221 – 238 (2012)]. Також наведено приклад для
iлюстрацiї застосовностi отриманих результатiв.
1. Relation theoretic notions and preliminary results. We begin with some preliminary definitions
and notations which will be required in the sequel.
Definition 1.1 [7, 11]. Let X be a (nonempty) set and k \geq 1 be a given real number. A function
d : X \times X \rightarrow [0,+\infty ) is a b-metric if and only if, for all x, y, z \in X, the following conditions are
satisfied:
(b1) d (x, y) = 0 if and only if x = y,
(b2) d (x, y) = d (y, x) ,
(b3) d (x, z) \leq k (d (x, y) + d (y, z)) .
The pair (X, d) is called a b-metric space.
It should be noted that, the class of b-metric spaces is effectively large than that of metric spaces,
since a b-metric is a metric when k = 1. Following example show that in general a b-metric need
not necessarily be a metric (see also [1, 14, 17]).
Example 1.1. Let (X, d) be a metric space, and \rho (x, y) = (d (x, y))p , p > 1 is a real number.
Then \rho is a b-metric with k = 2p - 1, but \rho is not metric on X.
Otherwise, for more concepts such as b-convergence, b-completeness, b-Cauchy sequence and
b-closed set in b-metric spaces, we refer the reader to [1, 13, 14, 17] and the references mentioned
therein. Also, for the concepts such as partial order, comparable, well ordered, nondecreasing,
increasing, dominated, dominating and other, we refer the reader to [1, 9, 12, 14, 17, 18, 21].
In the sequel, let \BbbN denote the set of all nonnegative integers, \BbbR the set of all real numbers.
Throughout this paper, \scrR stands for a nonempty binary relation but for the sake of simplicity, we
often write binary relation instead of nonempty binary relation.
* The author was supported by AISTDF, DST, India (project No. CRD/2018/000017).
c\bigcirc S. CHANDOK, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4 565
566 S. CHANDOK
Definition 1.2. Let \scrR be a binary relation defined on a nonempty set X and x, y \in X. We say
that x and y are \scrR -comparative if either (x, y) \in \scrR or (y, x) \in \scrR . We denote it by [x, y] \in \scrR .
Definition 1.3. A binary relation \scrR on a nonempty set X is called:
(1) reflexive if (x, x) \in \scrR for every x \in X,
(2) symmetric if whenever (x, y) \in \scrR , then (y, x) \in \scrR ,
(3) antisymmetric if whenever (x, y) \in \scrR and (y, x) \in \scrR , then x = y,
(4) transitive if whenever (x, y) \in \scrR and (y, z) \in \scrR , then (x, z) \in \scrR ,
(5) complete or connected or dichotomous if [x, y] \in \scrR for all x, y \in X,
(6) weakly complete or weakly connected or trichotomous if [x, y] \in \scrR or x = y for all
x, y \in X.
Definition 1.4. Let X be a nonempty set and \scrR a binary relation on X.
(1) The inverse or transpose or dual relation of \scrR , denoted by \scrR - 1, is defined by \scrR - 1 :=
:= \{ (x, y) \in X2 : (y, x) \in \scrR \} .
(2) The symmetric closure of \scrR , denoted by \scrR s, is defined to be the set \scrR \cup \scrR - 1 (i.e., \scrR s :=
:= \scrR \cup \scrR - 1). Indeed, \scrR s is the smallest symmetric relation on X containing \scrR .
Definition 1.5. Let X be a nonempty set and \scrR a binary relation on X. A sequence \{ xn\} \subseteq X
is called \scrR -preserving if (xn, xn+1) belongs to \scrR for all n \in \BbbN 0.
Definition 1.6 [2]. Assume that P,Q are self-mappings on a nonempty set X. A binary relation
\scrR on X is called (P,Q)-closed if for all x, y \in X, (Qx,Qy) \in \scrR , then (Px, Py) belongs to \scrR .
If we take Q =identity mapping, we have \scrR is P -closed.
If \scrR is P -closed, then Rs is also P -closed.
Definition 1.7. Let (X, d, k \geq 1) be a b-metric space and \scrR a binary relation on X. We say
that (X, d) is R-complete if every R-preserving b-Cauchy sequence in X converges.
Definition 1.8. Let (X, d, k \geq 1) be a b-metric space and \scrR a binary relation on X. A subset
E of X is called \scrR -closed if every \scrR -preserving b-convergent sequence in E converges to a point
of E.
In the following lines, we extend a weaker version of the notion of d-self-closeness of a partial
order \preceq (defined by Turinici [19]) to an arbitrary binary relation.
Definition 1.9. Let (X, d, k \geq 1) be a b-metric space and Q : X \rightarrow X. A binary relation
\scrR defined on X is called (Q, bd)-self-closed if whenever \{ xn\} is an \scrR -preserving sequence and
xn \rightarrow d x, then there exists a subsequence \{ xni\} of \{ xn\} with [Qxni , Qx] \in \scrR for all i \in \BbbN .
If Q is identity mapping, we have the following definitions.
Definition 1.10. Let (X, d, k \geq 1) be a b-metric space. A binary relation \scrR defined on X is
called bd-self-closed if whenever \{ xn\} is an \scrR -preserving sequence and xn \rightarrow d x, then there exists
a subsequence \{ xni\} of \{ xn\} with (xni , x) \in \scrR for all i \in \BbbN .
Definition 1.11 [20]. Let X be a nonempty set and \scrR a binary relation on X. A subset E of X
is called \scrR -directed if for each x, y \in E, there exists z \in X such that (x, z) \in \scrR and (y, z) \in \scrR .
Definition 1.12 [2]. Let X be a nonempty set and \scrR a binary relation on X. For x, y \in
\in X, a path of length k (where k is a natural number) in \scrR from x to y is a finite sequence
\{ z0, z1, z2, . . . , zk\} \subset X satisfying the following conditions:
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
ARBITRARY BINARY RELATIONS, CONTRACTION MAPPINGS, AND b-METRIC SPACES 567
(i) z0 = x and zk = y,
(ii) (zi, zi+1) \in R for each i (0 \leq i \leq k - 1).
Definition 1.13. Let (X, d, k \geq 1) be a b-metric space, \scrR a binary relation on X and P and
Q two self-mappings on X. We say that P and Q are \scrR -compatible if for any sequence \{ xn\} \subset X
such that \{ Pxn\} and \{ Qxn\} are \scrR -preserving and \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty Q(xn) = \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty P (xn), we have
\mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty d(QPxn, PQxn) = 0.
Following auxiliary results will be used in sequel.
Lemma 1.1 [15]. Let X be a nonempty set and T a self-mapping on X. Then there exists a
subset E \subseteq X such that T (E) = T (X) and T : E \rightarrow X is one-to-one.
Lemma 1.2 ([17], Lemma 3.1). Let \{ yn\} be a sequence in a b-metric space (X, d) with k \geq 1
such that
d (yn, yn+1) \leq \lambda d (yn - 1, yn) (1)
for some \lambda \in
\biggl[
0,
1
k
\biggr)
and each n = 1, 2, . . . . Then \{ yn\} is a b-Cauchy sequence in b-metric space
(X, d) .
Proposition 1.1 [2]. For a binary relation \scrR on a nonempty set X, (x, y) \in \scrR s if and only if
[x, y] \in \scrR .
Proposition 1.2. Let X be a nonempty set, \scrR a binary relation on X and P,Q are self-
mappings on X. If \scrR is (P,Q)-closed, then \scrR s is also (P,Q)-closed.
Proposition 1.3. If (X, d, k \geq 1) be a b-metric space and T, S : X \rightarrow X be self mappings,
then the following contractive conditions are equivalent:
d (Tx, Ty) \leq \lambda d (Sx, Sy) for all x, y \in X with (Sx, Sy) \in \scrR ,
d (Tx, Ty) \leq \lambda d (Sx, Sy) for all x, y \in X with [Sx, Sy] \in \scrR
for some \lambda \in
\biggl[
0,
1
k
\biggr)
.
For S =identity mapping, we have the following result.
Proposition 1.4. If (X, d, k \geq 1) be a b-metric space and T : X \rightarrow X be self mapping, then
the following contractive conditions are equivalent:
d (Tx, Ty) \leq \lambda d (x, y) for all x, y \in X with (x, y) \in \scrR ,
d (Tx, Ty) \leq \lambda d (x, y) for all x, y \in X with [x, y] \in \scrR .
for some \lambda \in
\biggl[
0,
1
k
\biggr)
.
In this paper, we use the following notations:
(i) F (T ) = the set of all fixed points of T,
(ii) X(T ;\scrR ) := \{ x \in X : (x, Tx) \in \scrR \} ,
(iii) \gamma (x, y,\scrR ) := the class of all paths in \scrR from x to y.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
568 S. CHANDOK
In this paper, we prove some results on the existence and uniqueness of fixed points defined
on a b-metric space endowed with an arbitrary binary relation. As applications, some results on
coincidence points involving a pair of mappings are also obtained. Our results generalize, extend,
modify and unify several well-known results especially those obtained in [2, 3, 5, 20].
2. Main results. First we introduce the concept of CT -relation theoretic contractive mappings
in the setting of b-metric space.
Definition 2.1. Let (X, d, k \geq 1) be a b-metric space and \scrR a binary relation on X. A mapping
T : X \rightarrow X is called a CT -contraction if there exists \lambda \in
\biggl[
0,
1
k
\biggr)
such that
d(Tx, Ty) \leq \lambda d(x, y) (2)
for every x, y \in X with (x, y) \in \scrR .
Now we are ready to state our first main result.
Theorem 2.1. Let (X, d, k \geq 1) be a b-complete b-metric space, \scrR a binary relation on X and
T : X \rightarrow X a CT -relation theoretic contraction satisfying the following conditions:
(i) X(T ;\scrR ) is nonempty,
(ii) \scrR is T -closed,
(iii) either T is b-continuous or \scrR is bd-self-closed.
Then T has a fixed point, that is, there exists x\ast \in X such that Tx\ast = x\ast .
Further, if
(iv) \gamma (x, y,\scrR s) is nonempty, for each x, y \in X, then T has a unique fixed point.
Proof. Let x0 \in X(T,\scrR ). Define the sequence \{ xn\} in X by
xn+1 = Txn for all n \geq 0.
Now, we shall show that the sequence \{ xn\} is \scrR -preserving. Then, by definition, we have
(x0, x1) \in \scrR . By T -closedness of \scrR , we obtain (Tx0, Tx1) = (x1, x2) \in \scrR . Repetition of this
argument gives
(xn, xn+1) \in \scrR for all n \in \BbbN .
Thus, the sequence \{ xn\} is \scrR -preserving. Since T is a CT -contractive mapping and sequence \{ xn\}
is \scrR -preserving, we obtain, for all n \in \BbbN ,
d(xn, xn+1) = d(Txn - 1, Txn) \leq \lambda d(xn - 1, xn).
Therefore, by using Lemma 1.2, it follows that \{ xn\} is a b-Cauchy sequence.
Since (X, d, k \geq 1) is b-complete, there exists x\ast \in X such that
xn \rightarrow x\ast as n \rightarrow \infty .
From the b-continuity of T, it follows that xn+1 = Txn \rightarrow Tx\ast as n \rightarrow \infty . Due to the
uniqueness of the limit, we derive that Tx\ast = x\ast , that is, x\ast is a fixed point of T.
Alternately, we assume that \scrR is bd-self-closed. Since \{ xn\} is \scrR -preserving and xn \rightarrow x, by
bd-self-closedness of \scrR there exists a subsequence \{ xnj\} of \{ xn\} such that [xnj , x] \in \scrR for all
j \in \BbbN .
Since T is a CT -contraction and using Proposition 1.4, we obtain
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
ARBITRARY BINARY RELATIONS, CONTRACTION MAPPINGS, AND b-METRIC SPACES 569
d(x\ast , Tx\ast ) \leq k[d(x\ast , xn+1) + d(xn+1, Tx
\ast )] =
= k[d(x\ast , xn+1) + d(Txn, Tx
\ast )] \leq
\leq k[d(x\ast , xn+1) + \lambda d(xn, x
\ast )] \rightarrow 0,
when n \rightarrow \infty . Hence Tx\ast = x\ast and x\ast is a fixed point of T.
Suppose that y\ast is another fixed point of T.
By assumption (iv), there exists a path (say \{ z0, z1, z2, . . . , zk\} ) of some finite length k \in \scrR s
from x to y so that z0 = x, zk = y, [zi, zi+1] \in \scrR for each i, 0 \leq i \leq k - 1.
As \scrR is T -closed, \scrR s is T -closed. Therefore, we have [Tnzi, T
nzi+1] \in \scrR for each i, 0 \leq i \leq
\leq k - 1, and for each n \in \BbbN .
Using the above arguments and inequality (2), we have
d(x\ast , y\ast ) = d(Tnx\ast , Tny\ast ) \leq
k - 1\sum
i=0
d(Tnzi, T
nzi+1) \leq
\leq \lambda
k - 1\sum
i=0
d(Tn - 1zi, T
n - 1zi+1) \leq . . . \leq \lambda n
k - 1\sum
i=0
d(zi, zi+1) \rightarrow 0 as n \rightarrow \infty .
Therefore, x\ast = y\ast . Hence, T has a unique fixed point.
Theorem 2.1 is proved.
Example 2.1. Let X = \BbbN \cup \{ \infty \} and d : X \times X \rightarrow \BbbR be defined by
d(m,n) =
\left\{
0, if m = n,\bigm| \bigm| \bigm| \bigm| 1m - 1
n
\bigm| \bigm| \bigm| \bigm| , if one of m,n is even and the other is even or \infty ,
5, if one of m,n is odd and the other is odd (and \not = n) or \infty ,
2, otherwise.
Then (X, d) is a b-metric space with k =
5
2
(see [16]).
Define a binary relation \scrR = \{ (x, y) \in R2 : x - y \geq 0\} on X. Consider a mapping T :
X \rightarrow X as
Tx =
\Biggl\{
5x, x \in \BbbN ,
\infty , x = \infty .
Clearly, \scrR is T -closed and T is continuous. Now, for x, y \in X with (x, y) \in \scrR , we have the
following cases.
Case I: x, y are even numbers. Then Tx = 5x, Ty = 5y, d(x, y) =
\bigm| \bigm| \bigm| \bigm| 1x - 1
y
\bigm| \bigm| \bigm| \bigm| , d(Tx, Ty) =
=
1
5
\bigm| \bigm| \bigm| \bigm| 1x - 1
y
\bigm| \bigm| \bigm| \bigm| , hence, d(Tx, Ty) =
1
5
\bigm| \bigm| \bigm| \bigm| 1x - 1
y
\bigm| \bigm| \bigm| \bigm| \leq 1
k
d(x, y).
Case II: x, y are odd numbers (and x \not = y). Then Tx = 5x, Ty = 5y, d(x, y) = 5, d(Tx, Ty) =
=
1
5
\bigm| \bigm| \bigm| \bigm| 1x - 1
y
\bigm| \bigm| \bigm| \bigm| , hence, d(Tx, Ty) =
1
5
\bigm| \bigm| \bigm| \bigm| 1x - 1
y
\bigm| \bigm| \bigm| \bigm| \leq 1
k
d(x, y).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
570 S. CHANDOK
Case III: x, y are natural numbers of different parity. Then Tx = 5x, Ty = 5y, d(x, y) = 2,
d(Tx, Ty) =
1
5
\bigm| \bigm| \bigm| \bigm| 1x - 1
y
\bigm| \bigm| \bigm| \bigm| , hence, d(Tx, Ty) =
1
5
\bigm| \bigm| \bigm| \bigm| 1x - 1
y
\bigm| \bigm| \bigm| \bigm| \leq 1
k
d(x, y).
Case IV: x is even, y = \infty . Then Tx = 5x, Ty = 5y, d(x, y) =
1
x
, d(Tx, Ty) =
1
5x
, hence,
d(Tx, Ty) =
1
5x
\leq 1
k
d(x, y).
Case V: x is odd and y = \infty . Then Tx = 5x, Ty = 5y, d(x, y) = 5, d(Tx, Ty) =
1
5x
, hence,
d(Tx, Ty) =
1
5x
\leq 1
k
d(x, y).
Hence, all the conditions of Theorem 2.1 are satisfied and T has a fixed point in X, that is \infty .
Theorem 2.2. Let (X, d, k \geq 1) be a b-complete b-metric space, \scrR a binary relation on X.
Suppose that P,Q : X \rightarrow X are self mappings on X. Further, following conditions hold:
(i) (X,P,\scrR ) , (X,Q,\scrR ) are nonempty and P (X) \subseteq Q(X),
(ii) \scrR is a (P,Q)-closed,
(iii) there exists \lambda \in
\biggl[
0,
1
k
\biggr)
, such that d (Px, Py) \leq \lambda d (Qx,Qy) for all x, y \in X with
(Qx,Qy) \in \scrR ,
(iv) either P is (Q,\scrR )-continuous or P and Q are continuous.
Then P,Q have a point of coincidence.
Proof. In view of assumption (i), let x0 be an arbitrary element of X(P,Q,\scrR ), then
(Qx0, Px0) \in \scrR . If Q(x0) = P (x0), then x0 is a coincidence point of P and Q and, hence,
we are through.
Otherwise, if Q(x0) \not = P (x0), then, from P (X) \subseteq Q(X), we can choose x1 \in X such that
Q(x1) = P (x0). Again from P (X) \subseteq Q(X), we can choose x2 \in X such that Q(x2) = P (x1).
Continuing this process, we construct a sequence \{ xn\} \subset X (of joint iterates) such that
Q(xn+1) = P (xn) for all n \in \BbbN . (3)
Now, we claim that \{ Qxn\} is \scrR -preserving sequence, i.e.,
(Qxn, Qxn+1) \in \scrR for all n \in \BbbN . (4)
We prove this fact by mathematical induction. By using equation (3) (with n = 0) and the fact
that x0 \in X(P,Q,\scrR ), we have (Qx0, Qx1) \in \scrR , which shows that (4) holds for n = 0.
Suppose that (4) holds for n = r > 0, i.e., (Qxr, Qxr+1) \in \scrR . As \scrR is (P,Q)-closed, we
have (Pxr, Pxr+1) \in \scrR , which, by using (1), yields that (Qxr+1, Qxr+2) \in \scrR , i.e., (4) holds for
n = r + 1. Hence, by induction, (4) holds for all n \in \BbbN .
In view of (3) and (4), the sequence \{ Pxn\} is also an \scrR -preserving, i.e.,
(Pxn, Pxn+1) \in \scrR for all n \in \BbbN
By using (3), (4) and assumption (iii), we obtain
d(Qxn, Qxn+1) = d(Pxn - 1, Pxn) \leq \lambda d(Qxn - 1, Qxn) for all n \in \BbbN .
Therefore, by using Lemma 1.2, it follows that \{ Qxn\} is a b-Cauchy sequence.
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ARBITRARY BINARY RELATIONS, CONTRACTION MAPPINGS, AND b-METRIC SPACES 571
Owing to (3), \{ Qxn\} \subseteq P (X) so that \{ Qxn\} is \scrR -preserving b-Cauchy sequence in X. As X
is b-complete, there exists u \in Q(X) such that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
Q(xn) = Q(u). (5)
By using (3) and (5), we obtain
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
P (xn) = Q(u). (6)
Now, we show that u is a coincidence point of P and Q.
Firstly, suppose that P is (Q,\scrR )-continuous, then, by using (4) and (5), we get
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
P (xn) = P (u). (7)
By using (6) and (7), we get Q(u) = P (u). Hence, we are done.
Secondly, suppose that P and Q are continuous. Owing to Lemma 1.1, there exists a subset
E \subseteq X such that Q(E) = Q(X) and Q : E \rightarrow X is one-to-one. Now, define T : Q(E) \rightarrow Q(X)
by T (Qa) = P (a) for all Q(a) \in Q(E) where a \in E.
As Q : E \rightarrow X is one-to-one and P (X) \subseteq Q(X), T is well defined. Again since P and Q
are continuous, it follows that T is continuous. By using the fact Q(X) = Q(E), P (X) \subseteq Q(X),
we have P (X) \subseteq Q(E), which follows that, without loss of generality, we are able to construct
\{ xn\} \subset E satisfying (3) and to choose u \in E. By using (5), (6) and continuity of T, we get
P (u) = T (Qu) = T (\mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty Qxn) = \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty T (Qxn) = \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty P (xn) = Q(u). Thus, u \in X
is a point of coincidence of P and Q and, hence, we have the result.
Theorem 2.2 is proved.
Theorem 2.3. In the hypotheses of Theorem 2.2, if instead of (iv), we take
(v) P and Q are \scrR -compatible, Q is \scrR -continuous, and either P is \scrR -continuous or \scrR is
(Q, bd)-self-closed.
Then P,Q have a point of coincidence.
Proof. On the lines of Theorem 2.2, we have \{ Qxn\} is a b-Cauchy sequence. Owing to (3),
\{ Qxn\} \subseteq P (X) so that \{ Qxn\} is \scrR -preserving b-Cauchy sequence in X. As X is b-complete,
there exists u \in Q(X) such that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
Q(xn) = Q(u). (8)
By using (3) and (8), we obtain
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
P (xn) = Q(u).
As Q is \scrR -continuous, we have
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
Q(Qxn) = Q( \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
Q(xn)) = Q(Q(u)).
Also, we get
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
Q(Pxn) = Q( \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
P (xn)) = Q(Q(u)).
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572 S. CHANDOK
As \{ Pxn\} and \{ Qxn\} are \scrR -preserving and \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty P (xn) = Q(u) = \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty Q(xn) and P and
Q are \scrR -compatible, we obtain \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty d(QP (xn), PQ(xn)) = 0.
Now, we show that Q(u) is a coincidence point of P and Q.
Firstly, we suppose that P is \scrR -continuous. By using (4), we have \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty P (Qxn) =
= P (\mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty Q(xn)) = P (Q(u)). Assume that Q(u) = z. By triangle inequality, we have
d(Qz, Pz) \leq k[d(Qz,Q(Pxn)) + d(Q(Pxn), P z)] \leq
\leq kd(Qz,Q(Pxn)) + k2[d(Q(Pxn), P (Qxn)) + d(P (Qxn), P z)].
Taking the limit as n \rightarrow \infty , we obtain d(Qz, Pz) \leq 0, which implies Qz = Pz, that is, z = Q(u)
is a coincidence point of P and Q.
Alternatively, suppose that \scrR is (Q, bd)-self closed. As \{ Qxn\} is \scrR -preserving and Qxn \rightarrow Qu,
therefore, by using (Q, bd)-self closedness of \scrR , there exists a subsequence \{ Qxni\} of \{ Qxn\} such
that [QQxni , QQu] belongs to \scrR for all i \in \BbbN \cup 0. Since Qxni \rightarrow Qu and using Proposition 1.3,
we obtain
d(PQxni , PQu) \leq \lambda d(QQxni , QQu) for all i \in \BbbN \cup \{ 0\} .
Take Qu = z. On using triangular inequality, we get
d(Qz, Pz) \leq k[d(Qz,Q(Pxni)) + d(Q(Pxni), P z)] \leq
\leq kd(Qz,Q(Pxni)) + k2[d(Q(Pxni), P (Qxni)) + d(P (Qxni), P z)] \leq
\leq kd(Qz,Q(Pxni)) + k2[d(Q(Pxni), P (Qxni)) + \lambda d(Q(Qxni), Qz)].
Taking the limit as i \rightarrow \infty , we obtain d(Qz, Pz) \leq 0, which implies Qz = Pz, that is, z = Q(u)
is a coincidence point of P and Q.
Theorem 2.3 is proved.
3. Consequences.
Definition 3.1 [8]. Let P and Q be two self-mappings on X. We say that P is Q-comparative
if for any x, y \in X, (Qx,Qy) \in \scrR s, then (Px, Py) \in \scrR s.
Remark 3.1. It is clear that P is Q-comparative if and only if \scrR s is (P,Q)-closed.
Definition 3.2 [20]. We say that (X, d,\scrR s) is regular if the following condition holds: if \{ xn\}
is nondecreasing sequence in X and the point x \in X are such that xn \rightarrow x, (xn, x) \in \scrR s for
all n.
Remark 3.2. Clearly, (X, d,\scrR s) is regular if and only if \scrR s is bd-self-closed.
We extend the result of Berzig [20] in the framework of b-metric space.
Corollary 3.1. Let (X, d,\leq ) be an b-complete b-metric space with k \geq 1 and \scrR a binary
relation on X. Assume that P and Q are two self-mappings on X. Suppose that the following
conditions hold:
(a) P (X) \subseteq Q(X) , Q(X) is closed,
(b) P is Q-comparative,
(c) there exists x0 \in X such that (Q(x0), P (x0)) \in \scrR s,
(d) there exists \lambda \in
\biggl[
0,
1
k
\biggr)
, such that d (Px, Py) \leq \lambda d (Qx,Qy) for all x, y \in X with
(Q(x), Q(y)) \in \scrR s,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
ARBITRARY BINARY RELATIONS, CONTRACTION MAPPINGS, AND b-METRIC SPACES 573
(e) (X, d,\scrR s) is regular.
Then P,Q have a point of coincidence.
Definition 3.3 [10]. Let (X,\leq ) be an ordered set and P and Q two self-mappings on X. We
say that P is Q-increasing if for any x, y \in X, Q(x) \leq Q(y), then P (x) \leq P (y).
Definition 3.4 [6]. Given a mapping Q : XX, we say that an ordered b-metric space (X, d, k \geq
\geq 1) has Q-ICU (increasing-convergence-upper bound) property if Q-image of every increasing
sequence \{ xn\} in X such that xn \rightarrow d x, is bounded above by Q-image of its limit (as an upper
bound), i.e., Q(xn) \leq Q(x), for all n \in \BbbN .
Notice that under the restriction Q = I, the identity mapping on X, Definition 3.4 transforms to
the notion of ICU property.
Remark 3.3. It is clear that if ordered b-metric space (X, d, k \geq 1) has ICU property (resp.,
Q-ICU property), then \leq is bd-self-closed (resp., (Q, bd)-self-closed).
Corollary 3.2. Let (X, d,\leq ) be an b-complete ordered b-metric space with k \geq 1. Assume that
P and Q are two self-mappings on X. Suppose that the following conditions hold:
(a) P (X) \subseteq Q(X),
(b) P is Q-increasing,
(c) there exists x0 \in X such that Q(x0) \leq P (x0),
(d) there exists \lambda \in
\biggl[
0,
1
k
\biggr)
, such that d (Px, Py) \leq \lambda d (Qx,Qy) for all x, y \in X with
Q(x) \leq Q(y),
(e) P and Q are compatible, Q is continuous, and either P is continuous or (X, d,\leq ) has
Q-ICU property.
Then P,Q have a point of coincidence.
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Received 08.12.16,
after revision — 31.01.20
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 4
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| id | umjimathkievua-article-368 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:36Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/8f/d42080b6442e3c6a53cb5d6bdc436b8f.pdf |
| spelling | umjimathkievua-article-3682022-03-26T11:01:32Z Arbitrary binary relations, contraction mappings and $b$-metric spaces Arbitrary binary relations, contraction mappings and $b$-metric spaces Arbitrary binary relations, contraction mappings and $b$-metric spaces Chandok, S. Chandok, S. Chandok, S. бінарні відношення стискаючи вiдображення Binary Relations Contraction Mappings UDC 517.9We prove some results on the existence and uniqueness of fixed points defined on a $b$-metric space endowed with an arbitrary binary relation.  As applications, we obtain some statements on coincidence points involving a pair of mappings.  Our results generalize, extend, modify and unify several well-known results especially those obtained by Alam and Imdad [J. Fixed Point Theory and Appl., 17, 693–702 (2015); Fixed Point Theory, 18, 415–432 (2017); Filomat, 31, 4421–4439 (2017)] and Berzig [J. Fixed Point Theory and Appl., 12, 221–238 (2012)].  Also, we provide an example to illustrate the suitability of results obtained. УДК 517.9 Доведено деякі результати про існування та єдиність нерухомих точок на $b$-метричних просторах, що наділені довільним бінарним відношенням.  В якості застосувань отримано деякі твердження про точки збігу для пар відображень.  Ці результати узагальнюють, розширюють, модифікують та уніфікують деякі відомі результати Alam i Imdad [J. Fixed Point Theory and Appl., 17, 693–702 (2015)]; Fixed Point Theory, 18, 415–432 (2017); Filomat, 31, 4421–,4439 (2017)] та Berzig [J. Fixed Point Theory and Appl., 12, 221–238 (2012)].  Також наведено приклад для ілюстрації застосовності отриманих результатів. Institute of Mathematics, NAS of Ukraine 2020-03-28 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/368 10.37863/umzh.v72i4.368 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 4 (2020); 565-574 Український математичний журнал; Том 72 № 4 (2020); 565-574 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/368/8708 |
| spellingShingle | Chandok, S. Chandok, S. Chandok, S. Arbitrary binary relations, contraction mappings and $b$-metric spaces |
| title | Arbitrary binary relations, contraction mappings and $b$-metric spaces |
| title_alt | Arbitrary binary relations, contraction mappings and $b$-metric spaces Arbitrary binary relations, contraction mappings and $b$-metric spaces |
| title_full | Arbitrary binary relations, contraction mappings and $b$-metric spaces |
| title_fullStr | Arbitrary binary relations, contraction mappings and $b$-metric spaces |
| title_full_unstemmed | Arbitrary binary relations, contraction mappings and $b$-metric spaces |
| title_short | Arbitrary binary relations, contraction mappings and $b$-metric spaces |
| title_sort | arbitrary binary relations, contraction mappings and $b$-metric spaces |
| topic_facet | бінарні відношення стискаючи вiдображення Binary Relations Contraction Mappings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/368 |
| work_keys_str_mv | AT chandoks arbitrarybinaryrelationscontractionmappingsandbmetricspaces AT chandoks arbitrarybinaryrelationscontractionmappingsandbmetricspaces AT chandoks arbitrarybinaryrelationscontractionmappingsandbmetricspaces |