CUBO, A Mathematical Journal

Vol.22, N◦02, (203–214). August 2020
http://dx.doi.org/10.4067/S0719-06462020000200203

Received: 14 February, 2020 | Accepted: 22 June, 2020

Contractive mapping theorems in Partially ordered metric
spaces

N.Seshagiri Rao1, K.Kalyani2 and Kejal Khatri3

1Department of Applied Mathematics, School of Applied Natural Sciences

Adama Science and Technology University

Post Box No.1888, Adama, Ethiopia.

2Department of Mathematics

Vignan’s Foundation for Science, Technology & Research

Vadlamudi-522213, Andhra Pradesh, India.

3Department of Mathematics

Government College Simalwara

Dungarpur 314403, Rajasthan, India.

seshu.namana@gmail.com, kalyani.namana@gmail.com, kejal0909@gmail.com

ABSTRACT

The purpose of this paper is to establish some coincidence, common fixed point the-

orems for monotone f-non decreasing self mappings satisfying certain rational type

contraction in the context of a metric spaces endowed with partial order. Also, the re-

sults involving an integral type of such classes of mappings are discussed in application

point of view. These results generalize and extend well known existing results in the

literature.

RESUMEN

El propósito de este art́ıculo es establecer teoremas de coincidencia y de punto fijo

común para auto mapeos monótonos f-no decrecientes satisfaciendo ciertas contrac-

ciones de tipo racional en el contexto de espacios métricos dotados de un orden parcial.

Adicionalmente, resultados que involucran clases de mapeos de tipo integral son dis-

cutidos desde un punto de vista de las aplicaciones. Estos resultados generalizan y

extienden resultados bien conocidos, existentes en la literatura.

Keywords and Phrases: Partially ordered metric spaces; Rational contractions; Compatible

mappings; Weakly compatible mappings.

2020 AMS Mathematics Subject Classification: 47H10; 54H25; 26A42; 46T99.

c©2020 by the author. This open access article is licensed under a Creative
Commons Attribution-NonCommercial 4.0 International License.

http://dx.doi.org/10.4067/S0719-06462020000200203


204 N.Seshagiri Rao, K.Kalyani & Kejal Khatri CUBO
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1 Introduction

Ever since in Fixed point theory and Approximation theory, the classical Banach contraction

principle plays a vital role to obtain an unique solution of the results. Of course, it is very

important and popular tool in different fields of mathematics to solve the existing problems in

nonlinear analysis. Since then a lot of variety of generalizations of this Banach contraction principle

[1] have been taken place in a metric fixed point theory by improving the underlying contraction

condition [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Thereafter vigorous research work has been obtained

by weakening its hypotheses on various spaces such as rectangular metric spaces, pseudo metric

spaces, fuzzy metric spaces, quasi metric spaces, quasi semi-metric spaces, probabilistic metric

spaces, D-metric spaces, G-metric spaces, F-metric spaces, cone metric spaces, and so on to prove

the existing results. Prominent work on the existence and uniqueness of a fixed point and common

fixed point theorems involving monotone mappings on cone metric spaces, partially ordered metric

spaces and others spaces [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]

generate natural interest to establish usable fixed point results.

The aim of this paper is to prove some coincidence, common fixed point results in the frame

work of partially ordered metric spaces for a pair of self-mappings satisfying a generalized contrac-

tive condition of rational type. These results generalize and extend the results of Harjani et al.[19]

and Chandok [28] in ordered metric space. Also the applications of these results are presented on

taking integral type contractions in the same space.

2 Preliminaries

The following definitions are frequently used in results given in upcoming sections.

Definition 1. The triple (X, d, �) is called a partially ordered metric space, if (X, �) is a partially

ordered set together with (X, d) is a metric space.

Definition 2. If (X, d) is a complete metric space, then the triple (X, d, �) is called a partially

ordered complete metric space.

Definition 3. Let (X, �) be a partially ordered set. A self-mapping f : X → X is said to be

strictly increasing, if f(x) ≺ f(y), for all x, y ∈ X with x ≺ y and is also said to be strictly

decreasing, if f(x) ≻ f(y), for all x, y ∈ X with x ≺ y.

Definition 4. A point x ∈ A, where A is a non-empty subset of a metric space (X, d) is called a

common fixed (coincidence) point of two self-mappings f and T if fx = T x = x(fx = T x).

Definition 5. The two self-mappings f and T defined over a subset A of a metric space (X, d)

are called commuting if fT x = T fx for all x ∈ A.



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Contractive mapping theorems in Partially ordered metric spaces 205

Definition 6. Two self-mappings f and T defined over A ⊂ X are compatible, if for any sequence

{xn} with lim
n→+∞

fxn = lim
n→+∞

T xn = µ, for some µ ∈ A then lim
n→+∞

d(T fxn, fT xn) = 0.

Definition 7. Two self-mappings f and T defined over A ⊂ X are said to be weakly compatible,

if they commute at their coincidence points. i.e., if fx = T x then fT x = T fx.

Definition 8. Let f and T be two self-mappings defined over a partially ordered set (X, �). A

mapping T is called a monotone f non-decreasing if

fx � fy implies T x � T y, for all x, y ∈ X.

Definition 9. Let A be a non-empty subset of a partially ordered set (X, �). If very two elements

of A are comparable then it is called well ordered set.

Definition 10. A partially ordered metric space (X, d, �) is called an ordered complete, if for each

convergent sequence {xn}
+∞
n=0 ⊂ X, one of the following condition holds

• if {xn} is a nondecreasing sequence in X such that xn → x implies xn � x, for all n ∈ N

that is, x = sup{xn} or

• if {xn} is a nonincreasing sequence in X such that xn → x implies x � xn, for all n ∈ N

that is, x = inf{xn}.

3 Main Results

In this section, we prove some coincidence, common fixed point theorems in the context of ordered

metric space.

Theorem 1. Let (X, d, �) be a complete partially ordered metric space. Suppose that the self-

mappings f and T on X are continuous, T is a monotone f-nondecreasing, T (X) ⊆ f(X) and

satisfying the following condition

d(T x, T y) ≤ α
d(fx, T x) d(fy, T y)

d(fx, fy)
+ β [d(fx, T x) + d(fy, T y)] + γd(fx, fy) (3.1)

for all x, y in X with f(x) 6= f(y) are comparable, where α, β, γ ∈ [0, 1) with 0 ≤ α + 2β + γ < 1.

If there exists a point x0 ∈ X such that f(x0) � T (x0) and the mappings T and f are compatible,

then T and f have a coincidence point in X.

Proof. Let x0 ∈ X such that f(x0) � T (x0). Since from hypotheses, we have T (X) ⊆ f(X) then,

we can choose a point x1 ∈ X such that fx1 = T x0. But T x1 ∈ f(X) then, again there exists

another point x2 ∈ X such that fx2 = T x1. By continuing the same way, we can construct a

sequence {xn} in X such that fxn+1 = T xn, for all n.



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Again, by hypotheses, we have f(x0) � T (x0) = f(x1) and T is a monotone f-nondecreasing

mapping then, we get T (x0) � T (x1). Similarly, we obtain T (x1) � T (x2), since f(x1) � f(x2)

and then by continuing the same procedure, we obtain that

T (x0) � T (x1) � T (x2) � .........T (xn) � T (xn+1) � ........

The equality T (xn+1) = T (xn) is impossible because f(xn+2) 6= f(xn+1) for all n ∈ N. Thus

d(T (xn), T (xn+1)) > 0 for all n ≥ 0 therefore, from contraction condition (3.1), we have

d(T xn+1, T xn) ≤ α
d(fxn+1, T xn+1) d(fxn, T xn)

d(fxn+1, fxn)
+ β [d(fxn+1, T xn+1) + d(fxn, T xn)]

+γd(fxn+1, fxn)

which intern implies that

d(T xn+1, T xn) ≤ αd(T xn, T xn+1) + β [d(T xn, T xn+1) + d(T xn−1, T xn)]

+ γd(T xn, T xn−1)

Finally, we arrive at

d(T xn+1, T xn) ≤

(

β + γ

1 − α − β

)

d(T xn, T xn−1)

Continuing the same process up to (n − 1) times, we get

d(T xn+1, T xn) ≤

(

β + γ

1 − α − β

)n

d(T x1, T x0)

Let k = β+γ
1−α−β

∈ [0, 1), then from triangular inequality for m ≥ n, we have

d(T xm, T xn) ≤ d(T xm, T xm−1) + d(T xm−1, T xm−2) + ........ + d(T xn+1, T xn)

≤
(

km−1 + km−2 + .......... + kn
)

d(T x1, T x0)

≤
kn

1 − k
d(T x1, T x0)

as m, n → +∞, d(T xm, T xn) → 0, which shows that the sequence {T xn} is a Cauchy sequence in

X. So, by the completeness of X, there exists a point µ ∈ X such that T xn → µ as n → +∞.

Again, by the continuity of T , we have

lim
n→+∞

T (T xn) = T

(

lim
n→+∞

T xn

)

= T µ.

But fxn+1 = T xn, then fxn+1 → µ as n → +∞ and from the compatibility for T and f, we have

lim
n→+∞

d(T (fxn), f(T xn)) = 0.

Further by triangular inequality, we have

d(T µ, fµ) = d(T µ, T (fxn)) + d(T (fxn), f(T xn)) + d(f(T xn), fµ)



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Contractive mapping theorems in Partially ordered metric spaces 207

On taking limit as n → +∞ in both sides of the above equation and using the fact that T and f

are continuous then, we get d(T µ, fµ) = 0. Thus, T µ = fµ. Hence, µ is a coincidence point of T

and f in X.

Corollary 1. Let (X, d, �) be a complete partially ordered metric space. Suppose that the self-

mappings f and T on X are continuous, T is a monotone f-nondecreasing, T (X) ⊆ f(X) and

satisfying the following condition

d(T x, T y) ≤ α
d(fx, T x) d(fy, T y)

d(fx, fy)
+ β [d(fx, T x) + d(fy, T y)]

for all x, y in X with f(x) 6= f(y) are comparable and for some α, β ∈ [0, 1) with 0 ≤ α + 2β < 1.

If there exists a point x0 ∈ X such that f(x0) � T (x0) and the mappings T and f are compatible,

then T and f have a coincidence point in X.

Proof. Set γ = 0 in Theorem 1.

Corollary 2. Let (X, d, �) be a complete partially ordered metric space. Suppose that the self-

mappings f and T on X are continuous, T is a monotone f-nondecreasing, T (X) ⊆ f(X) and

satisfying the following condition

d(T x, T y) ≤ β [d(fx, T x) + d(fy, T y)] + γd(fx, fy)

for all x, y in X with f(x) 6= f(y) are comparable and for some β, γ ∈ [0, 1) with 0 ≤ 2β + γ < 1.

If there exists a point x0 ∈ X such that f(x0) � T (x0) and the mappings T and f are compatible,

then T and f have a coincidence point in X.

Proof. The proof can be obtained by setting α = 0 in Theorem 1.

We may remove the continuity criteria of T in Theorem 1 is still valid by assuming the following

hypothesis in X:

If {xn} is a nondecreasing sequence in X such that xn → x, then xn � x for all n ∈ N.

Theorem 2. Let (X, d, �) be a complete partially ordered metric space. Suppose that f and T are

self-mappings on X, T is a monotone f-nondecreasing, T (X) ⊆ f(X) and satisfying

d(T x, T y) ≤ α
d(fx, T x) d(fy, T y)

d(fx, fy)
+ β [d(fx, T x) + d(fy, T y)] + γd(fx, fy) (3.2)

for all x, y in X with f(x) 6= f(y) are comparable and for some α, β, γ ∈ [0, 1) with 0 ≤ α+2β+γ <

1. If there exists a point x0 ∈ X such that f(x0) � T (x0) and {xn} is a nondecreasing sequence

in X such that xn → x, then xn � x for all n ∈ N.

If f(X) is a complete subset of X, then T and f have a coincidence point in X. Further, if

T and f are weakly compatible, then T and f have a common fixed point in X. Moreover, the set



208 N.Seshagiri Rao, K.Kalyani & Kejal Khatri CUBO
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of common fixed points of T and f is well ordered if and only if T and f have one and only one

common fixed point in X.

Proof. Suppose f(X) is a complete subset of X. As we know from the proof of Theorem 1, the

sequence {T xn} is a Cauchy sequence and hence {fxn} is also a Cauchy sequence in (f(X), d) as

fxn+1 = T xn and T (X) ⊆ f(X). Since f(X) is complete then there exists some fu ∈ f(X) such

that

lim
n→+∞

T (xn) = lim
n→+∞

f(xn) = f(u).

Also note that the sequences {T xn} and {fxn} are nondecreasing and from hypotheses, we have

T (xn) � f(u) and f(xn) � f(u) for all n ∈ N. But T is a monotone f-nondecreasing then, we get

T (xn) � T (µ) for all n. Letting n → +∞, we obtain that f(u) � T (u).

Suppose that f(u) ≺ T (u) then define a sequence {un} by u0 = u and fun+1 = T un for all

n ∈ N. An argument similar to that in the proof of Theorem 1 yields that {fun} is a nondecreasing

sequence and lim
n→+∞

f(un) = lim
n→+∞

T (un) = f(v) for some v ∈ X. So from hypotheses, it is clear

that sup f(un) � f(v) and sup T (un) � f(v), for all n ∈ N. Notice that

f(xn) � f(u) � f(u1) � ........ � f(un) � .... � f(v).

Case:1 Suppose if there exists some n0 ≥ 1 such that f(xn0) = f(un0) then, we have

f(xn0) = f(u) = f(un0) = f(u1) = T (u).

Hence, u is a coincidence point of T and f in X.

Case:2 Suppose that f(xn0) 6= f(un0) for all n then, from (3.2), we have

d(fxn+1, fun+1) =d(T xn, T un)

≤α
d(fxn, T xn) d(fun, T un)

d(fxn, fun)
+ β [d(fxn, T xn) + d(fun, T un)]

+γd(fxn, fun)

Taking limit as n → +∞ on both sides of the above inequality, we get

d(fu, fv) ≤γ d(fu, fv)

< d(fu, fv), since γ < 1.

Thus, we have

f(u) = f(v) = f(u1) = T (u).

Hence, we conclude that u is a coincidence point of T and f in X.



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Contractive mapping theorems in Partially ordered metric spaces 209

Now, suppose that T and f are weakly compatible. Let w be a coincidence point then,

T (w) = T (f(z)) = f(T (z)) = f(w), since w = T (z) = f(z), for some z ∈ X.

Now by contraction condition, we have

d(T (z), T (w)) ≤α
d(fz, T z) d(fw, T w)

d(fz, fw)
+ β [d(fz, T z) + d(fw, T w)] + γd(fz, fw)

≤γ d(T (z), T (w))

as γ < 1, then d(T (z), T (w)) = 0. Therefore, T (z) = T (w) = f(w) = w. Hence, w is a common

fixed point of T and f in X.

Now suppose that the set of common fixed points of T and f is well ordered, we have to show

that the common fixed point of T and f is unique. Let u and v be two common fixed points of T

and f such that u 6= v then from (3.2), we have

d(u, v) ≤α
d(fu, T u) d(fv, T v)

d(fu, fv)
+ β [d(fu, T u) + d(fv, T v)] + γd(fu, fv)

≤γ d(u, v)

< d(u, v), since γ < 1,

which is a contradiction. Thus, u = v. Conversely, suppose T and f have only one common fixed

point then the set of common fixed points of T and f being a singleton is well ordered. This

completes the proof.

Corollary 3. Let (X, d, �) be a complete partially ordered metric space. Suppose that f and T

are self-mappings on X, T is a monotone f-nondecreasing, T (X) ⊆ f(X) and satisfying

d(T x, T y) ≤ α
d(fx, T x) d(fy, T y)

d(fx, fy)
+ β [d(fx, T x) + d(fy, T y)]

for all x, y in X with f(x) 6= f(y) are comparable and for some α, β ∈ [0, 1) with 0 ≤ α + 2β < 1.

If there exists a point x0 ∈ X such that f(x0) � T (x0) and {xn} is a nondecreasing sequence in X

such that xn → x, then xn � x for all n ∈ N.

If f(X) is a complete subset of X, then T and f have a coincidence point in X. Further, if

T and f are weakly compatible, then T and f have a common fixed point in X. Moreover, the set

of common fixed points of T and f is well ordered if and only if T and f have one and only one

common fixed point in X.

Proof. Set γ = 0 in Theorem 2.

Corollary 4. Let (X, d, �) be a complete partially ordered metric space. Suppose that f and T

are self-mappings on X, T is a monotone f-nondecreasing, T (X) ⊆ f(X) and satisfying

d(T x, T y) ≤β [d(fx, T x) + d(fy, T y)] + γd(fx, fy)



210 N.Seshagiri Rao, K.Kalyani & Kejal Khatri CUBO
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for all x, y in X for which f(x) 6= f(y) are comparable and for some β, γ ∈ [0, 1) with 0 ≤ 2β +γ <

1. If there exists a point x0 ∈ X such that f(x0) � T (x0) and {xn} is a nondecreasing sequence

in X such that xn → x, then xn � x for all n ∈ N.

If f(X) is a complete subset of X, then T and f have a coincidence point in X. Further, if

T and f are weakly compatible, then T and f have a common fixed point in X. Moreover, the set

of common fixed points of T and f is well ordered if and only if T and f have one and only one

common fixed point in X.

Proof. Set α = 0 in Theorem 2.

Remark 1. (i). If β = 0, in Theorem 1 and Theorem 2, we obtain Theorem 2.1 and Theorem

2.3 of Chandok [28].

(ii). If f = I and β = 0, in Theorem 1 and Theorem 2, then we get Theorem 2.1 and Theorem

2.3 of Harjani et al. [19].

4 Applications

In this section, we state some applications of the main result for a self mapping involving the

integral type contractions.

Let us denote τ, a set of all functions ϕ defined on [0, +∞) satisfying the following conditions:

(1) each ϕ is Lebesgue integrable mapping on each compact subset of [0, +∞) and

(2) for any ǫ > 0, we have
∫ ǫ

0
ϕ(t)dt > 0.

Theorem 3. Let (X, d, �) be a complete partially ordered metric space. Suppose that the self-

mappings f and T on X are continuous, T is a monotone f-nondecreasing, T (X) ⊆ f(X) and

satisfying the following condition

∫ d(T x,T y)

0

ϕ(t)dt ≤ α

∫
d(fx,T x) d(fy,T y)

d(fx,fy)

0

ϕ(t)dt + β

∫ d(fx,T x)+d(fy,T y)

0

ϕ(t)dt

+ γ

∫ d(fx,fy)

0

ϕ(t)dt

(4.1)

for all x, y in X with f(x) 6= f(y) are comparable, ϕ(t) ∈ τ and for some α, β, γ ∈ [0, 1) such that

0 ≤ α + 2β + γ < 1. If there exists a point x0 ∈ X such that f(x0) � T (x0) and the mappings T

and f are compatible, then T and f have a coincidence point in X.

Similarly, we can obtain the following results in complete partially ordered metric space, by

putting γ = 0 and α = 0 in an integral contraction of Theorem 3.



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Contractive mapping theorems in Partially ordered metric spaces 211

Theorem 4. Let (X, d, �) be a complete partially ordered metric space. Suppose that the self-

mappings f and T on X are continuous, T is a monotone f-nondecreasing, T (X) ⊆ f(X) and

satisfying the following condition

∫ d(T x,T y)

0

ϕ(t)dt ≤ α

∫
d(fx,T x) d(fy,T y)

d(fx,fy)

0

ϕ(t)dt + β

∫ d(fx,T x)+d(fy,T y)

0

ϕ(t)dt (4.2)

for all x, y in X with f(x) 6= f(y) are comparable, ϕ(t) ∈ τ and where α, β ∈ [0, 1) such that

0 ≤ α + 2β < 1. If there exists a point x0 ∈ X such that f(x0) � T (x0) and the mappings T and

f are compatible, then T and f have a coincidence point in X.

Theorem 5. Let (X, d, �) be a complete partially ordered metric space. Suppose that the self-

mappings f and T on X are continuous, T is a monotone f-nondecreasing, T (X) ⊆ f(X) and

satisfying the following condition

∫ d(T x,T y)

0

ϕ(t)dt ≤ β

∫ d(fx,T x)+d(fy,T y)

0

ϕ(t)dt + γ

∫ d(fx,fy)

0

ϕ(t)dt (4.3)

for all x, y in X with f(x) 6= f(y) are comparable, ϕ(t) ∈ τ and for some β, γ ∈ [0, 1) such that

0 ≤ 2β + γ < 1. If there exists a point x0 ∈ X such that f(x0) � T (x0) and the mappings T and

f are compatible, then T and f have a coincidence point in X.

Corollary 5. By replacing β = 0 in Theorem 3, we obtain the Corollary 2.5 of Chandok [28].

We illustrate the usefulness of the obtained results for the existence of the coincidence point

in the space.

Example 1. Define a metric d : X × X → [0, +∞) by d(x, y) = |x − y|, where X = [0, 1] with

usual order ≤. Suppose that T and f be two self mappings on X such that T x = x
2

2
and fx = 2x

2

1+x
,

then T and f have a coincidence in point X.

Proof. By definition of a metric d, it is clear that (X, d) is a complete metric space. Obviously,

(X, d, ≤) is a partially ordered complete metric space with usual order. Let x0 = 0 ∈ X then

f(x0) ≤ T (x0) and also by definition; T , f are continuous, T is a monotone f- nondecreasing and

T (X) ⊆ f(X).

Now for any distinct x, y in X, we have

d(T x, T y) =
1

2
|x2 − y2| =

1

2
(x + y)|x − y|

<
α

4

x2y2

(x + y + xy)

|x − 3||y − 3|

|x − y|
+

β

2

x2(1 + y)|x − 3| + y2(1 + x)|y − 3|

(1 + x)(1 + y)

+ γ
2(x + y + xy)

(1 + x)(1 + y)
|x − y|



212 N.Seshagiri Rao, K.Kalyani & Kejal Khatri CUBO
22, 2 (2020)

< α

x
2|x−3|
2(1+x)

.
y
2|y−3|
2(1+y)

2|x − y| x+y+xy
(1+x)(1+y)

+ β

[

x2|x − 3|

2(1 + x)
+

y2|y − 3|

2(1 + y)

]

+ γ
2(x + y + xy)

(1 + x)(1 + y)
|x − y|

< α
d(fx, T x) d(fy, T y)

d(fx, fy)
+ β [d(fx, T x) + d(fy, T y)] + γd(fx, fy)

Then, the contraction condition in Theorem 1 holds by selecting proper values of α, β, γ in [0, 1)

such that 0 ≤ α + 2β + γ < 1. Therefore T , f have a coincidence point 0 ∈ X.

Similarly the following is one more example of main Theorem 1.

Example 2. A distance function d : X × X → [0, +∞) by d(x, y) = |x − y|, where X = [0, 1] with

usual order ≤. Define two self mappings T and f on X by T x = x2 and fx = x3, then T and f

have two coincidence points 0, 1 in X with x0 =
1
2
.

Acknowledgments

The authors would like to thank the editor and referees for their precise remarks which improve

the presentation of the paper.



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Contractive mapping theorems in Partially ordered metric spaces 213

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	Introduction
	Preliminaries
	Main Results
	Applications