@
Appl. Gen. Topol. 23, no. 2 (2022), 363-376

doi:10.4995/agt.2022.17418

© AGT, UPV, 2022

Fixed point theorems for F-contraction

mapping in complete rectangular M-metric
space

Mohammad Asim a , Samad Mujahid b and Izhar Uddin b

a Department of Mathematics, Faculty of Science, Shree Guru Gobind Singh Tricentenary Uni-

versity, Gurugram, Haryana, India. (mailtoasim27@gmail.com)
b Department of Mathematics, Jamia Millia Islamia, New Delhi-110025, India.

(mujahidsamad721@gmail.com, izharuddin1@jmi.ac.in)

Communicated by I. Altun

Abstract

In this paper, we prove a fixed point result for F -contraction princi-
ple in the framework of rectangular M-metric space. An example is
also adopted to exhibit the utility of our result. Finally, we apply our
fixed point result to show the existence of solution of Fredholm integral
equation.

2020 MSC: 47H10; 54E50.

Keywords: fixed point; F -contraction; rectangular M-metric space; integral
equation.

1. Introduction

Fixed point theory is an important and very active branch of functional anal-
ysis. It provides essential tools for solving problems arising in various branches
of mathematical analysis. It guarantees the uniqueness and existence of the
solution of integral and differential equations. In 1922, a polish mathematician
Stefan Banach gives a contraction principle [10], which is one of the most well-
known and important discovery in mathematics.

Received 31 March 2022 – Accepted 22 August 2022

http://dx.doi.org/10.4995/agt.2022.17418
https://orcid.org/0000-0002-2209-4488
https://orcid.org/0000-0002-5650-0296
https://orcid.org/0000-0002-2424-2798


M. Asim, S. Mujahid and I. Uddin

In the literature, there are two ways to generalized the Banach contraction prin-
ciple either change the contraction condition or alter the metric space. In fixed
point theory several contractions defined in metric space such that Boyd and
Wong’s nonlinear contraction principle [11], Meir-Keeler contraction [20, 1, 6],

Suzuki contraction [33], Kannan contraction [17], Ćirić generalized contraction

[14], Ćirić’s quasi contraction [15], weak-contraction [29], Chatterjea contrac-
tion [13], Zamfirescu contraction [35] and F-Suzuki contraction [27] and many
more [9, 25].
In 2012, Wardowski [34] introduced a new type of contraction for real-valued
mapping F defined on positive real numbers and satisfying some conditions
and obtained a fixed point theorem for it. After that several authors have
worked on F-contraction mapping in different metric space. In 2014, Kumam
and Piri [27] applied weaker condition on self map and extended the result
of Wardowski [34]. In 2014, Minak et al. [21] obtained result for generalized

F-contractions including Ćirić type generalized F-contraction and almost F-
contraction on complete metric space. In 2017, Kumam et al. [28] introduced
the F-contraction in the setting of complete asymmetric metric spaces and
extend several results. In 2018, Kadelburg and Radenović [16] obtained the
result on concerning F-contraction in b-metric space. In 2019, Luambano et
al. [18] introduced the fixed point theorem for F-contraction in partial metric
space and obtained certain results for it with suitable examples and many more
[31, 30, 23, 22].
In 2014, Asadi et al. [4] introduced M-metric space, which extends the p-metric
space given by Matthews [19] and proved the Banach contraction principle for
it. Several authors have worked in this metric space [23, 3, 24, 2]. In 2000,
Branciari [12] introduced rectangular metric space which is the another gen-

eralization of metric space. In 2018, Özg̈ur et al. [26] introduced rectangular
M-metric space. They were inspired by the work of Branciari [12] and Shukla
[32], who defined partial rectangular metric spaces which is the generalization
of rectangular metric space.
In 2019, Asim et al. [5, 7, 8] generalized the rectangular M-metric space as
rectangular Mb-metric space, extended rectangular Mrξ-metric space and Mν-
metric space.
In this article, we establish the fixed point theorem for F-contraction in rectan-
gular M- metric space. Throughout the article R is the set of all real numbers,
R+ is the set of all positive real numbers and N is the set of all natural numbers.

2. Preliminaries

In this section, we collect some basic notions, definitions, examples and
auxiliary results.
In 2000, Branciari [12] introduced rectangular metric space. The definition is
as follows:

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 364



Fixed point theorems for F-contraction mapping in complete rectangular M-metric space

Definition 2.1 ([12]). Let X be a non-empty set. A function r : X×X → R+
is said to be a rectangular metric on X, if it satisfies the following (for all
x,y ∈ X and for all distinct point u,v ∈ X \{x,y}):
(1) r(x,y) = 0, if and only if x = y,
(2) r(x,y) = r(y,x) and
(3) r(x,y) ≤ r(x,u) + r(u,v) + r(v,y).
Then, the pair (X,r) is called a rectangular metric space.

After that, Shukla [32] introduced partial rectangular metric space. The
definition is as follows:

Definition 2.2 ([32]). Let X be a non-empty set. A function ρ : X×X → R+
is said to be a partial rectangular metric on X, if it satisfies the following
conditions (for any x,y ∈ X and for all distinct point u,v ∈ X \{x,y}):
(1) x = y if and only if ρ(x,y) = ρ(x,x) = ρ(y,y),
(2) ρ(x,x) ≤ ρ(x,y),
(3) ρ(x,y) = ρ(y,x) and
(4) ρ(x,y) ≤ ρ(x,u) + ρ(u,v) + ρ(v,y) −ρ(u,u) −ρ(v,v).
Then, the pair (X,ρ) is called a partial rectangular metric space.

In 2014, Asadi et al. [4] generalized the partial metric space to M-metric
space and obtained certain theorems related to M-metric space.
Notation: The following notations are useful in the sequel:
(i) mxy := m(x,x) ∨m(y,y) = min{m(x,x),m(y,y)} and
(ii) Mxy := m(x,x) ∧m(y,y) = max{m(x,x),m(y,y)}.

Definition 2.3 ([4]). Let X be a non-empty set. A function m : X×X → R+
is called a m-metric, if it satisfying the following conditions:
(1) m(x,x) = m(y,y) = m(x,y) ⇐⇒ x = y,
(2) mxy ≤ m(x,y),
(3) m(x,y) = m(y,x) and
(4) (m(x,y) −mxy) ≤ (m(x,z) −mxz) + (m(z,y) −mzy).
Then, the pair (X,m) is called an M-metric space.

In 2018, Özg̈ur et al. [26] introduced rectangular M-metric space and defi-
nition are as follows:
Notation: The following notations are useful in the sequel:
(i) mrxy := mr(x,x) ∨mr(y,y) = min{mr(x,x),mr(y,y)} and
(ii) Mrxy := mr(x,x) ∧mr(y,y) = max{mr(x,x),mr(y,y)}.

Definition 2.4 ([26]). Let X be a non-empty set. A function mr : X×X → R+
is called mr-metric, if it satisfying the following conditions:
(RM1) mr(x,x) = mr(y,y) = mr(x,y) ⇐⇒ x = y,
(RM2) mrxy ≤ mr(x,y),
(RM3) mr(x,y) = mr(y,x) and
(RM4) (mr(x,y)−mrxy ) ≤ (mr(x,u)−mrxu) + (mr(u,v)−mruv ) + (mr(v,y)−
mrvy ) for all u,v ∈ X/{x,y}.
Then, the pair (X,mr) is called an rectangular M-metric space.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 365



M. Asim, S. Mujahid and I. Uddin

Example 2.5 ([26]). Let mr be an mr-metric. Put
(i) mwr (x,y) = mr(x,y) − 2mrxy + Mrxy
(ii) msr(x,y) = mr(x,y) −mrxy when x 6= y and msr(x,y) = 0 if x = y.
Then, mwr and m

s
r are ordinary metrics.

Definition 2.6 ([26]). Let (X,mr) be an rectangular M- metric space. Then,
(1) A sequence {xn} in X converges to a point x, if and only if

(2.1) lim
n→∞

(mr(xn,x) −mrxn,x) = 0.

(2) A sequence {xn} in X is said to be mr-Cauchy sequence, if and only if

(2.2) lim
n,m→∞

(mr(xn,xm) −mrxn,xm ) and limn,m→∞(Mr(xn,xm) −mrxn,xm )

exist and finite.
(3) An rectangular M-metric space is said to be mr-complete, if every mr-
Cauchy sequence {xn} converges to a point x such that

(2.3) lim
n→∞

(mr(xn,x) −mrxn,x) = 0 and limn→∞(Mr(xn,x) −mrxn,x) = 0.

Lemma 2.7 ([26]). Let (X,mr) be a rectangular M-metric space. Then,
(1) {xn} is an mr-Cauchy sequence in (X,mr) if and only if it is a Cauchy
sequence in the metric space (X,mwr ).
(2) (X,mr) is mr-complete if and only if the metric space (X,m

w
r ) is complete.

Furthermore,

lim
n→∞

(mwr (xn,x) = 0 ⇔ lim
n→∞

(mr(xn,x)−mrxn,x) = 0, limn→∞(Mr(xn,x)−mrxn,x) = 0.

Likewise the above definition holds also for msr.

Lemma 2.8 ([26]). Assume that xn → x as n → ∞ in an rectangular M-
metric space (X,mr). Then,

lim
n→∞

(mr(xn,y) −mrxn,y ) = mr(x,y) −mrx,y,∀ y ∈ X.

Lemma 2.9 ([26]). Assume that xn → x and yn → y as n → ∞ in an
rectangular M-metric space (X,mr). Then,

lim
n→∞

(mr(xn,yn) −mrxn,yn ) = mr(x,y) −mrx,y.

Lemma 2.10. [26] Assume that xn → x and yn → y as n → ∞ in an
rectangular M-metric space (X,mr). Then, mr(x,y) = mrx,y . Further if
mr(x,x) = mr(y,y), then x = y.

Lemma 2.11 ([26]). Let {xn} be a sequence in an rectangular M-metric space
(X,mr), such that there exists k ∈ [0, 1) such that

(2.4) mr(xn+1,xn) ≤ kmr(xn,xn−1) for all n ∈ N.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 366



Fixed point theorems for F-contraction mapping in complete rectangular M-metric space

Then,
(A) lim

n→∞
mr(xn,xn−1) = 0,

(B) lim
n→∞

mr(xn,xn) = 0,

(C) lim
n,m→∞

mrxn,xm = 0 and

(D) {xn} is an mr-Cauchy sequence.

Proof. [26] Using the definition of convergence and inequality (2.4), the proof
of the condition (A) follows easily. From the Condition (RM2) and the Condi-
tion (A), we get

lim
n→∞

min{mr(xn,xn),mr(xn−1,xn−1)} = lim
n→∞

mrxnxn−1

≤ lim
n→∞

mr(xn,xn−1) = 0.

Therefore, the Condition (B) holds. Since lim
n→∞

mr(xn,xn) = 0, the Condition

(C) holds. Using the previous conditions and the definition (2.6), we see that
the Condition (D) holds. �

In 2012, Wardowski [34] introduced F-contraction and the definition are as
follows:

Definition 2.12 ([34]). Let F : R+ → R be a mapping satisfying:
(F1) F is strictly increasing, i.e. for all α,β ∈ R+ such that α < β,F(α) <
F(β),
(F2) For each sequence {αn}n∈N of positive numbers lim

n→∞
αn = 0 if and only

if lim
n→∞

F(αn) = −∞,
(F3) There exists k ∈ (0, 1) such that lim

α→0+
αkF(α) = 0.

Denote 4F by the collection of all those functions which satisfy the conditions
(F1-F3).
A mapping T : X → X is said to be an F-contraction if there exists τ > 0 such
that

(2.5) d(Tx,Ty) > 0 ⇒ τ + F(d(Tx,Ty)) ≤ d(x,y) ∀ x,y ∈ X.

Some examples related to F-contraction [34] are:

Example 2.13. Let F : R+ → R be given by the formula F(α) = ln(α), it is
clear that F satisfies (F1)-(F3) ((F3) for any k ∈ (0, 1)).

Example 2.14. Let F : R+ → R be given by the formula F(α) = ln(α)+α,α >
0. Then, F satisfies (F1)-(F3).

Example 2.15. Let F : R+ → R be given by the formula F(α) = − 1√α, α > 0.
Then, F satisfies (F1)-(F3) ((F3) for any k ∈ (1/2, 1)).

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 367



M. Asim, S. Mujahid and I. Uddin

3. Main results

The following definition is new version of the F-contraction for a rectangular
M-metric space.

Definition 3.1. Let (X,mr) be an rectangular M-metric space. The mapping
T : X → X is said to be an F-contraction on X, if there exists τ > 0 and
F ∈4F such that ∀ x,y ∈ X
(3.1) mr(Tx,Ty) > 0 ⇒ τ + F(mr(Tx,Ty)) ≤ F(mr(x,y)).

Theorem 3.2. Let (X,mr) be a complete rectangular M-metric space and let
T : X → X be a continuous F -contraction. Then, T has a unique fixed point
x∗ ∈ X and for every x0 ∈ X a sequence {Tn(x0)}n∈N is convergent to x∗.

Proof. Let x0 ∈ X be arbitrary and fixed. We define a sequence {xn}n∈N ⊂ X,
xn+1 = Txn, n = 0, 1, · · · . Denote ξn = mr(xn,xn+1) − mrxn,xn+1 , n =
0, 1, · · · ,
if there exist n0 ∈ N for which xn0+1 = xn0. Then, Txn0 = xn0 and the proof
is finished.
Suppose that xn+1 6= xn for every n ∈ N. Then ξn > 0, for all n ∈ N.
Using(3.1), then following holds for every n ∈ N.

(3.2) F(ξn) ≤ F(ξn−1) − τ ≤ F(ξn−2) − 2τ ≤ ... ≤ F(ξ0) −nτ
From (3.2) we obtain lim

n→∞
F(ξn) = −∞ that together with condition (F2) gives

(3.3) lim
n→∞

mr(xn,xn+1) −mrxn+1,xn = 0.

We shall prove that
lim
n→∞

mr(xn,xn+2) = 0.

We assume that xn 6= xm for every n,m ∈ N,n 6= m.
Indeed, suppose that xn = xm for some n = m + k with k > 0.

F(mr(xm,xm+1) −mrxm,xm+1 ) = F(mr(xn,xn+1) −mrxn,xn+1 )
= F(mr(xm+k,xm+k+1) −mrxm+k,xm+k+1 )
≤ F(mr(xm,xm+1) −mrxm,xm+1 )) −kτ
< F(mr(xm,xm+1) −mrxm,xm+1 ))

a contradiction. Therefore, mr(xn,xm)−mrxn,xm > 0 for every n,m ∈ N with n 6=
m.

(3.4) τ + F(mr(xn,xn+2)) ≤ F(mr(xn−1,xn+1)), ∀ n ∈ N.
Hence

(3.5) F(mr(xn,xn+2)) ≤ F(mr(xn−1,xn+1)) − τ

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 368



Fixed point theorems for F-contraction mapping in complete rectangular M-metric space

F(mr(xn,xn+2)) ≤ F(mr(xn−2,xn)) − 2τ
≤ ···≤ F(mr(x0,x2)) −nτ.

Taking limit as n →∞ in above inequality, we get

lim
n→∞

F(mr(xn,xn+2)) = −∞.

Then, from the Condition (F2) of Definition (2.12), we conclude that

(3.6) lim
n→∞

(mr(xn,xn+2)) = 0.

Next, we shall show that {xn}n∈N is a mr-Cauchy sequence, that is
lim

n,m→∞
(mr(xn,xm)) −mrxn,xm = 0, n,m ∈ N.

Now, from Definition (2.12) there exist k ∈ (0, 1) such that

lim
n→∞

mr(xn,xn+1)
k
F(mr(xn,xn+1)) = 0.

F(mr(xn,xn+1)) ≤ F(mr(x0,x1)) −nτ.
We have

mr(xn,xn+1)
k
(F(mr(xn,xn+1))−F(mr(x0,x1))) ≤ mr(xn,xn+1)

k
(F(mr(x0,x1))−nτ)

(3.7)

mr(xn,xn+1)
k
(F(mr(xn,xn+1)) −F(mr(x0,x1))) ≤−mr(xn,xn+1)

k
nτ ≤ 0.

Taking limit as n →∞ in above inequality, we conclude that

lim
n→∞

mr(xn,xn+1)
k
nτ = 0.

Then, there exist n1 ∈ N such that nmr(xn,xn+1)
k ≤ 1, ∀ n ≥ n1

mr(xn,xn+1) ≤
1

n1/k
, ∀ n ≥ n1.

Now, from Definition (2.12) there exists k ∈ (0, 1) such that

(3.8) lim
n→∞

(mr(xn,xn+2))
kF(mr(xn,xn+2)) = 0.

Since

F(mr(xn,xn+2)) ≤ F(mr(x0,x2)) −nτ.
We have

(mr(xn,xn+2))
kF(mr(xn,xn+2)) ≤ (mr(xn,xn+2))k(F(mr(x0,x2)) −nτ)

(mr(xn,xn+2))
k(F(mr(xn,xn+2)) −F(mr(x0,x2))) ≤−nτ(mr(xn,xn+2))k

(mr(xn,xn+2)) ≤ 1n1/k . Next, we show that {xn} is mr-Cauchy sequence, that
is

lim
n→∞

(mr(xn,xn+p) −mrxn,xn+p ) = 0, ∀ p ∈ N.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 369



M. Asim, S. Mujahid and I. Uddin

The cases p = 1 and p = 2 are proved respectively by (3.3) and (3.4). Now, we
take p ≥ 3. It is sufficient to examine two cases.
Case 1. Firstly, let p is odd that is p = 2m + 1 for any m ≥ 1,n ∈ N. From
the Condition (RM4) of definition of the mr-metric, we get

mr(xn,xn+p) = mr(xn,xn+2m+1)

(mr(xn,xn+p) −mrxn,xn+p ) ≤ (mr(xn,xn+1) −mrxn,xn+1 )
+(mr(xn+1,xn+2) −mrxn+1,xn+2 )
+(mr(xn+2,xn+p) −mrxn+2,xn+p )

(mr(xn,xn+2m+1) −mrxn,xn+2m+1 ) ≤ (mr(xn,xn+1) −mrxn,xn+1 )
+(mr(xn+1,xn+2) −mrxn+1,xn+2 )
+(mr(xn+2,xn+2m+1) −mrxn+2,xn+2m+1 )

≤ (mr(xn,xn+1) −mrxn,xn+1 )
+(mr(xn+1,xn+2) −mrxn+1,xn+2 )
+(mr(xn+2,xn+3) −mrxn+2,xn+3 ) +
...

+(mr(xn+2m,xn+2m+1) −mrxn+2m,xn+2m+1 )
≤ [(mr(xn,xn+1) −mrxn,xn+1 )

+(mr(xn+2,xn+3) −mrxn+2,xn+3 ) +
...

+(mr(xn+2m−2,xn+2m−1) −mrxn+2m−2,xn+2m−1 )]
+[(mr(xn+1,xn+2) −mrxn+1,xn+2 ) +
...

+(mr(xn+2m−1,xn+2m) −mrxn+2m−1,xn+2m )
+(mr(xn+2m,xn+2m+1) −mrxn+2m,xn+2m+1 )]

(mr(xn,xn+p) −mrxn,xn+p ) ≤
n+p−1∑
i=n

(mr(xi,xi+1) −mrxi,xi+1 )

≤
∞∑
i=n

(mr(xi,xi+1) −mrxi,xi+1 )

≤
∞∑
i=n

1

i1/k
.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 370



Fixed point theorems for F-contraction mapping in complete rectangular M-metric space

From the above from the convergence of the series

n+p−1∑
i=n

1

i1/k
⇒ lim

n→∞
(mr(xn,xn+p) −mrxn,xn+p ) = 0.

Case 2. Secondly, assume p is even that is p = 2m for any m ≥ 1,n ∈ N.
From the Condition (RM4) of definition of the mr-metric, we get

(mr(xn,xn+p) −mrxn,xn+p ) ≤ (mr(xn,xn+2) −mrxn,xn+2 )
+(mr(xn+2,xn+3) −mrxn+2,xn+3 )
+(mr(xn+3,xn+2m) −mrxn+3,xn+2m )

(mr(xn,xn+2m) −mrxn,xn+2m ) ≤ (mr(xn,xn+2) −mrxn,xn+2 )
+(mr(xn+2,xn+3) −mrxn+2,xn+3 )
+(mr(xn+3,xn+4) −mrxn+3,xn+4 ) +
...

+(mr(xn+2m−3,xn+2m−2) −mrxn+2m−3,xn+2m−2 )
+(mr(xn+2m−2,xn+2m−1) −mrxn+2m−2,xn+2m−1 )
+(mr(xn+2m−1,xn+2m) −mrxn+2m−1,xn+2m )

≤ (mr(xn,xn+2) −mrxn,xn+2 )
+(mr(xn+2,xn+3) −mrxn+2,xn+3 )
+(mr(xn+3,xn+4) −mrxn+3,xn+4 )
+(mr(xn+4,xn+5) −mrxn+4,xn+5 ) +
...

+(mr(xn+2m−2,xn+2m−1) −mrxn+2m−2,xn+2m−1 )
+(mr(xn+2m−1,xn+2m) −mrxn+2m−1,xn+2m )

≤ (mr(xn,xn+2) −mrxn,xn+2 ) +
n+2m−1∑
i=n+2

(mr(xi,xi+1) −mrxi,xi+1 )

≤ (mr(xn,xn+2) −mrxn,xn+2 ) +
n+p−1∑
i=n+2

(mr(xi,xi+1) −mrxi,xi+1 )

≤ (mr(xn,xn+2) −mrxn,xn+2 ) +
∞∑

i=n+2

(mr(xi,xi+1) −mrxi,xi+1 )

≤ (mr(xn,xn+2) −mrxn,xn+2 ) +
∞∑

i=n+2

1

i1/k

≤
1

n1/k
+

∞∑
i=n+2

1

i1/k
.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 371



M. Asim, S. Mujahid and I. Uddin

From the above from the convergence of the series

n+2m−1∑
i=n+2

1

i1/k
⇒ lim

n→∞
(mr(xn,xn+p) −mrxn,xn+p ) = 0.

By Lemma (2.9), we obtain that for any n,m ∈ N,

msr(xn,xm) = mr(xn,xm) −mrxn,xm → 0 as n →∞.
This implies that {xn}n∈N is a mr-Cauchy sequence with respect to msr and
converges by Lemma (2.10). Thus,

lim
n,m→∞

msr(xn,xn+2m+1) = 0

and
lim

n,m→∞
msr(xn,xn+2m) = 0.

We received by Lemma (2.7) is that {xn} is an mr-Cauchy sequence.
From the completeness of X, there exist x∗ ∈ X such that

lim
n→∞

xn = x
∗.

Finally, the continuity of T yields

(mr(Tx
∗,x∗) −mrTx∗,x∗ ) = limn→∞(mr(Txn,xn) −mrTxn,xn )

= lim
n→∞

(mr(xn+1,xn) −mrxn+1,xn ) = 0.

Now, we show that the uniqueness of a fixed point of T . Assume that T has
two distinct fixed points x,y ∈ X, such that x = Tx,y = Ty.
From the Condition (3.1), we have

F(mr(x,y)) = F(mr(Tx,Ty)) < τ + F(mr(Tx,Ty)) ≤ F(mr(x,y)),
which is contradiction. Hence, T has unique fixed point. �

Example: Let X = [0, 1] and mr(x,y) =
|x|+|y|

2
, for all x,y ∈ X. Then,

(X,mr) is complete rectangular M-metric space. Define a mapping T : X → X
such that T(x) = x

2
, for all x ∈ X.

Define the function F : R+ → R by F(r) = ln(r), for all x,y ∈ X such that
mr(Tx,Ty) > 0 this implies that τ + F(mr(Tx,Ty)) = τ + ln(

|x|+|y|
4

).
Let τ ≤ ln 2. Then

τ + ln(
|x| + |y|

4
) ≤ ln 2 + ln(

|x| + |y|
4

) = F(mr(x,y)).

Thus, the contractive condition is satisfied for all x,y ∈ X. Hence, all hypothe-
ses of the Theorem (3.2) are satisfied and T has a unique fixed point x = 0.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 372



Fixed point theorems for F-contraction mapping in complete rectangular M-metric space

4. Applications

In this section, we apply Theorem (3.2) to investigate the existence and
uniqueness of solution of the Fredholm integral equation [7].
Let X = C([a,b],R) be the set of continuous real valued functions defined on
[a,b].
Now, we consider the following Fredholm type integral equation:

(4.1) x(p) =

∫ b
a

G(p,q,x(p))dq + h(p), for p,q ∈ [a,b]

where G,h ∈ C([a,b],R). Define mr : X ×X → R+ by

(4.2) mr(x(p),y(p)) = sup
p∈[a,b]

(|x(p)| + |y(p)|)
2

, ∀ x,y ∈ X.

Then, (X,mr) is an mr-complete in rectangular M-metric space.

Theorem 4.1. Suppose that there exist τ > 0 and for all x,y ∈ C([a,b],R)

|G(p,q,x(p)) + G(p,q,y(p)) + 2h(p)| ≤
e−τ

(b−a)
|x(p) + y(p)|, ∀ p,q ∈ [a,b].

Then, the integral equation (4.1) has a unique solution.

Proof. Define T : X → X by,

T(x(p)) =

∫ b
a

G(p,q,x(p))dq + h(p), ∀ p,q ∈ [a,b].

Observe that existence of a fixed point of the operator T is equivalent to the
existence of a solution of the integral equation (4.1). Now, for all x,y ∈ X. We

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 373



M. Asim, S. Mujahid and I. Uddin

have

mr(Tx,Ty) =

∣∣∣∣T(x(p)) + T(y(p))2
∣∣∣∣

=

∣∣∣∣∣
∫ b
a

(
G(p,q,x(p)) + G(p,q,y(p)) + 2h(p)

2
)dq

∣∣∣∣∣
≤
∫ b
a

∣∣∣∣(G(p,q,x(p)) + G(p,q,y(p)) + 2h(p)2 )dq
∣∣∣∣

≤
e−τ

(b−a)

∫ b
a

|x(p) + y(p)|
2

dq

≤
e−τ

(b−a)

∫ b
a

|x(p)| + |y(p)|
2

dq

≤
e−τ

(b−a)
sup
p∈[a,b]

(|x(p)| + |y(p)|)
2

(

∫ b
a

dq)

≤
e−τ

(b−a)
mr(x,y)(b−a)

≤ e−τmr(x,y).
Thus, the Condition (3.1) is satisfied with F(α) = ln(α). Therefore, all the
conditions of Theorem (3.2) are satisfied. Hence the operator T has a unique
fixed point, which means that the Fredholm integral equation (4.1) has a unique
solution. This completes the proof. �

5. Conclusion

As the rectangular M-metric is new generalization of m-metric and rect-
angular metric. In this article, we introduced F-contraction in rectangular
M-metric space and utilized a fixed point for it. We give a suitable example
which supported to the fixed point theorem. We give an application in Fred-
holm integral equation in rectangular M-metric space.

Acknowledgements. The authors are extremely grateful to the knowledge-
able referees for their insightful remarks and for pointing out numerous errors.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 2 374



Fixed point theorems for F-contraction mapping in complete rectangular M-metric space

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