@
Appl. Gen. Topol. 23, no. 1 (2022), 157-167

doi:10.4995/agt.2022.15356

© AGT, UPV, 2022

Boyd-Wong contractions in F-metric spaces
and applications

Ashis Bera a, Lakshmi Kanta Dey a, Sumit Som b, Hiranmoy Garai a

and Wutiphol Sintunavarat c

a Department of Mathematics, National Institute of Technology Durgapur 713209, India

(beraashis.math@gmail.com,lakshmikdey@yahoo.co.in,hiran.garai24@gmail.com)
b Department of Mathematics, School of Basic and Applied Sciences, Adamas University, Barasat-

700126, West Bengal, India (somkakdwip@gmail.com)
c Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat

University Rangsit Center, Pathum Thani 12120, Thailand. (wutiphol@mathstat.sci.tu.ac.th)

Communicated by I. Altun

Abstract

The main aim of this paper is to study the Boyd-Wong type fixed point
result in the F-metric context and apply it to obtain some existence
and uniqueness criteria of solution(s) to a second order initial value
problem and a Caputo fractional differential equation. We substantiate
our obtained result by finding a suitable non-trivial example.

2020 MSC: 47H10; 54A20; 54E50.

Keywords: F-metric space; fractional differential equation; Boyd-Wong
fixed point theorem.

1. Introduction and Preliminaries

After the invention of metric spaces by Frèchet, many mathematicians have
generalized the metric structure by making some changes in the original def-
inition of a metric given by Frèchet. Most of the generalizations are made
by making some changes in the triangle inequality of the original definition.
Some well-known metrics of such generalizations are b-metric due to Czerwik
[10], rectangular metric due to Branciari [7], bv(s)- metric due to Mitrović and

Received 06 April 2021 – Accepted 15 December 2021

http://dx.doi.org/10.4995/agt.2022.15356
https://orcid.org/0000-0002-0932-1332


A. Bera, L. K. Dey, S. Som, H. Garai and W. Sintunavarat

Radenović [20], JS-metric due to Jleli and Samet [16] etc. After all such types
of generalizations, recently in 2018, Jleli and Samet [15] introduced another
such abstraction, which they denominate as F-metric. They defined this met-
ric structure by means of a certain class F, which contains the set of functions
f : (0,∞) → R satisfying the following conditions:

(F1) f is non-decreasing, i.e., 0 < s < t ⇒ f(s) ≤ f(t);
(F2) for every sequence (tn) ⊆ (0,∞),

lim
n→∞

tn = 0 ⇐⇒ lim
n→+∞

f(tn) = −∞.

The definition of an F-metric based on the idea of a class F is as follows:

Definition 1.1 ([15]). A function D : X ×X → [0,∞), X being a nonempty
set, is called an F-metric on X if there exists (f,α) ∈F× [0,∞) such that the
following conditions hold:

(D1) for (x,y) ∈ X ×X, D(x,y) = 0 ⇐⇒ x = y;
(D2) D(x,y) = D(y,x) for all (x,y) ∈ X ×X;
(D3) for every (x,y) ∈ X × X, for each N ∈ N,N ≥ 2, and for every

(ui)
N
i=1 ⊆ X with (u1,uN ) = (x,y), we have

D(x,y) > 0 =⇒ f(D(x,y)) ≤ f

(
N−1∑
i=1

D(ui,ui+1)

)
+ α.

Furthermore, Jleli and Samet [15] also introduced the concepts of F-openness,
F-convergence, F-Cauchyness and F-completeness as follows:

Definition 1.2 ([15]). A subset O of an F-metric space (X,D) is said to be
F-open if for every x ∈O, there is some r > 0 such that BD(x,r) ⊆O, where

BD(x,r) = {y ∈ X : D(x,y) < r}.

From the above definition, it is easy to see that the family of all F-open
subsets of an F-metric space (X,D) is a topology on X.

Definition 1.3 ([15]). Let (X,D) be an F-metric space and (xn) be a sequence
in X.

(1) We say that (xn) is F-convergent to x ∈ X if for every F-open subset
Ox of X containing x, there exists some N ∈ N such that xn ∈Ox for
all n ≥ N.

(2) We say that (xn) is an F-Cauchy sequence if lim
n,m→∞

D(xn,xm) = 0.

(3) We say that X is F-complete if every F-Cauchy sequence in X is F-
convergent to some point in X.

Remark 1.4. If (xn) is a sequence in an F-metric space (X,D), then (xn) is
F-convergent to x ∈ X if and only if lim

n→∞
D(xn,x) = 0. In addition, the limit

of an F-convergent sequence is unique.

Among all the mathematical theories, which make all such generalized struc-
tures interesting and important, fixed point theory is one of these. Throughout

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 158



Boyd-Wong contractions in F-metric spaces and applications

the last few decades, many renowned mathematicians have achieved a lot of
well-known metric fixed point theorems in these structures (see in [1, 3, 23, 25,
26] and references therein). However, if a distance space is metrizable, then
sometimes it may happen that some metric fixed point theorems directly fol-
low from the metrizability result of the spaces. Still, it is essential to note that
some well-known fixed point results can’t be obtained from the fact that the
space is metrizable.

In this paper, we deal with one such structure, F-metric space, which is
metrizable, and at the same time, some fixed point theorems like the Banach
contraction principle follows from its metric counterpart using metrizability re-
sult. However, the general non-linear contractions like Boyd-Wong contraction
[8] can’t be obtained from the metrizability result. Som et al. proved the above
facts in [24]. Thus the study of Boyd-Wong fixed point theorem in the context
of F-metric spaces seems to be interesting. We could not establish an analogous
result in this setting to that of usual metric spaces, i.e., it is not known whether
a mapping satisfying the Boyd-Wong contraction condition in an F-complete
F-metric space possesses a fixed point or not. We leave it as an open question
in Section 2. So it is challenging work to have Boyd-Wong type result in F-
metric spaces assuming some mild additional hypotheses. We succeed in this
direction and propose Boyd-Wong type result in F-metric spaces. Moreover,
we apply our result in the context of special types of ordinary and Caputo
fractional differential equations. We present the above-mentioned results in
Section 3.

2. Boyd-Wong type results in the F-metric structure

In the previous section, we have already mentioned that we need some addi-
tional hypotheses either on an F-metric space X or on a mapping T : X → X
to get a fixed point of T satisfying the Boyd-Wong contractive condition. In
this section, we present such additional hypotheses via the following theorem.

Theorem 2.1. Let (X,D) be an F-complete F-metric space with (f,α) ∈
F×[0,∞) such that f is continuous and ψ : [0,∞) → [0,∞) be a nondecreasing
upper semi-continuous mapping from right such that ψ(t) < t for all t > 0.
Suppose that T : X → X is a mapping such that

(2.1) D(Tx,Ty) ≤ ψ(D(x,y))

for all x,y ∈ X. Further, assume that

(2.2) f(t) > f(ψ(t)) + α

for all t ∈ (0,∞). Then T has a unique fixed point and (Tnx) converges to that
fixed point for all x ∈ X.

Proof. Let x0 be an arbitrary point in X. Define a sequence (xn) by setting
xn = T

nx0 for all n ∈ N. If xn∗−1 = xn∗ for some n∗ ∈ N, then the proof is
done. So, we now assume that xn−1 6= xn for all n ∈ N. Then from the given

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 159



A. Bera, L. K. Dey, S. Som, H. Garai and W. Sintunavarat

condition, we have

D(xn,xn+1) = D(Txn−1,Txn)

≤ ψ (D(xn−1,xn))
< D(xn−1,xn)

for all n ∈ N. Hence, the sequence (D(xn,xn+1)) is a strictly decreasing se-
quence and also this sequence is bounded below. Therefore, lim

n→∞
D(xn,xn+1)

exists. Let lim
n→∞

D(xn,xn+1) =: l ≥ 0. If l > 0, then by the property of ψ, we
get

l = lim
n→∞

D(xn,xn+1) ≤ lim sup
n→∞

ψ(D(xn−1,xn)) ≤ ψ(l) < l,

which is a contradiction. Therefore, we obtain

(2.3) l = lim
n→∞

D(xn,xn+1) = 0.

Next, we will show that (Tnx) is an F-Cauchy sequence. We will prove it
by contradiction. If possible let (Tnx) be not an F-Cauchy sequence. Then
there exist ε > 0 and subsequences (xmk ) and (xnk ) of (xn) with mk > nk ≥ k
such that

(2.4) D(xmk,xnk ) ≥ ε

for each k ∈ N. We Choose nk as the smallest number not exceeding mk for
which the equation (2.4) holds. Then we have

D(xmk,xnk−1) < ε.(2.5)

Now, from (D3) and (2.5), we have

f(ε) ≤ f(D(xmk,xnk ))
≤ f (D(xmk,xmk+1) + D(xmk+1,xnk )) + α
≤ f (D(xmk,xmk+1) + ψ(D(xmk,xnk−1)) + α
≤ f (D(xmk,xmk+1) + ψ(ε)) + α.

Now, letting k →∞ in both sides of the above equation, we get

f(ε) ≤ f(ψ(ε)) + α,

which is a contradiction. This shows that the sequence (xn) is an F-Cauchy
sequence. Since X is F-complete, there is a point x∗ ∈ X such that

lim
n→∞

D(xn,x
∗) = 0.(2.6)

Next, we prove that x∗ is a fixed point of T. We prove it by contradiction. Let
Tx∗ 6= x∗. Then by (D3), we have

(2.7) f(D(Tx∗,x∗)) ≤ f
(
D(Tx∗,Txn) + D(Txn,x

∗)
)

+ α

for all n ∈ N. If there are two natural numbers n1 and n2 such that D(xn1,x∗) =
0 and D(xn2,x

∗) = 0, we obtain xn1 = x
∗ = xn2 , which is a contradiction. So

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 160



Boyd-Wong contractions in F-metric spaces and applications

we may choose a subsequence {xnl} of {xn} such that D(xnl,x
∗) 6= 0 for all

l ∈ N. Using the given condition, (2.7) and (F1), we get

f(D(Tx∗,x∗)) ≤ f
(
D(Tx∗,Txnl ) + D(Txnl,x

∗)
)

+ α

≤ f
(
ψ(D(x∗,xnl )) + D(xnl+1,x

∗)
)

+ α

≤ f
(
D(x∗,xnl ) + D(xnl+1,x

∗)
)

+ α.(2.8)

Therefore, by the condition (F2) and the equation (2.6), we have

lim
l→∞

f
(
D(x∗,xnl ) + D(xnl+1,x

∗)
)

+ α = −∞,

which is a contradiction. Hence, Tx∗ = x∗. For the uniqueness, let T has two
fixed points, say x∗ and y∗ such that x∗ 6= y∗. Then

D(x∗,y∗) = D(Tx∗,Ty∗) ≤ ψ(D(x∗,y∗)) < D(x∗,y∗),
which is impossible. Hence, this theorem is proved. �

Next, we provide an example to validate our obtained result.

Example 2.2. Let X = N and consider the mapping D : X × X → [0,∞),
defined by

D(x,y) =

{
|x−y|, if x and y both are even or both are odd;
3|x−y| + 5, if any one of x and y is even and the other is odd.

Then (X,D) is an F-metric space with f(t) = ln t and α = ln 3. Also, X is
F-complete. Let us define a mapping T : X → X by

Tx =

{
2, if x is even;
4, if x is odd.

We define a mapping ψ : [0,∞) → [0,∞) by

ψ(t) =
1

4
t for all t ∈ [0,∞).

Then ψ is a nondecreasing upper semicontinuous from right on [0,∞) and
ψ(t) < t for all t > 0. In this case it is clear that f is continuous and satisfies
f(t) > f(ψ(t)) + α for all t ∈ (0,∞).

Let x,y ∈ X be arbitrary. If x,y both are even or both are odd, then
it is obvious that D(Tx,Ty) ≤ ψ(D(x,y)). If x is even and y is odd, then
D(Tx,Ty) = 2 and D(x,y) = 3|x−y|+ 5 ≥ 2, which implies that ψ(D(x,y)) ≥
2. Hence,

D(Tx,Ty) ≤ ψ(D(x,y))
for all x,y ∈ X. Thus, all conditions of Theorem 2.1 hold. So by this theorem,
T has a unique fixed point in X. Note that z = 2 is the unique fixed point of
T.

Now we give the open question, which we mentioned in the previous section,
as follows:

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 161



A. Bera, L. K. Dey, S. Som, H. Garai and W. Sintunavarat

Question 2.3. Does there exist a fixed point free self mapping T which satisfies
the Boyd-Wong contractive condition in an F-complete F-metric space?

3. Applications of Boyd-Wong contractions

The aims of this section is to give applications of the fixed result for Boyd-
Wong contractions in the previous section.

3.1. Application to a second-order IVP. In this part, we apply our result
to the following initial value problem of the second-order differential equation:

(3.1)




d2x

d2t
+ ω2x = g(t,x(t))

x(0) = a,x′(0) = b,

where x ∈ C([0,T],R) is an unknown function, ω( 6= 0),a,b ∈ R and g is a
continuous function from [0,T] ×R+ into R.

The above differential equation plays a crucial role in different engineering
problems of activation of a spring governed by an exterior force. It can be easily
shown that the given differential equation (3.1) is equivalent to the integral
equation

x(t) =

∫ t
0

G(t,u)g(u,x(u))du + a cos(wt) + b sin(wt), t ∈ [0,T],

where G(t,u) is the Green’s function defined by

G(x,t) = 1
ω

sin(ω(t−u))H(t−u),
where H is the Heaviside unit function.

We like to study the existence of solution(s) of the differential equation (3.1)
(studying the above equivalent integral equation) using our obtained result
(Theorem 2.1). For this, we need to consider an underlying F-metric space as
(X,D), where X is the set of all real-valued continuous functions defined on
[0,T], and D is defined by

(3.2) D(x,y) = ‖x−y‖∞ = max
t∈[0,T]

|x(t) −y(t)|

for all x,y ∈ X. Then clearly (X,D) is an F-metric space with f(t) = ln t and
α = 0.

Now we have the following theorem.

Theorem 3.1. Consider the following differential equation (3.1) under the
assumptions:

(1) g is a continuous function;
(2) there exists a nondecreasing function ψ : [0,∞) → [0,∞) such that ψ

is upper semi-continuous from right and ψ(t) < t for all t > 0, and

|g(t,r) −g(t,s)| ≤ ω2ψ(|r −s|)
for all t ∈ [0,T] and r,s ∈ R.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 162



Boyd-Wong contractions in F-metric spaces and applications

Then the differential equation (3.1) has a unique solution in X.

Proof. Consider the F-metric space (X,D) as in (3.2). Then by assumption
(2), it is clear that f(t) > f(ψ(t)) + α for all t ∈ (0,∞). Let us now define a
mapping T : X → X for each x ∈ X by

(Tx)(t) = a cos(wt) + b sin(wt) +

∫ t
0

G(t,u)g(u,x(u))du

for all t ∈ [0,T]. Then the existence of fixed point(s) of the mapping T is
equivalent to the existence of the solution(s) of the IVP (3.1).

Now, for each x,y ∈ X and each t ∈ [0,T], by applying the conditions (1)
and (2), we have

|(Tx)(t) − (Ty)(t)| = |a cos(wt) + b sin(wt) +
∫ t

0

G(t,u)g(u,x(u))du

−a cos(wt) − b sin(wt) −
∫ t

0

G(t,u)g(u,y(u))du|

=

∣∣∣∣
∫ t

0

G(t,u)g(u,x(u))du−
∫ t

0

G(t,u)g(u,y(u))du

∣∣∣∣
≤

∫ t
0

G(t,u)|g(u,x(u)) −g(u,y(u))|du

≤
∫ t

0

G(t,u)ω2ψ(|x(u) −y(u)|)du

≤ ω2ψ(‖x−y‖∞) sup
t∈[0,T]

∫ t
0

G(t,u)du

= ω2ψ(‖x(u) −y(u)‖∞) sup
t∈[0,T]

∫ t
0

1

ω
sin(ω(t−u))du

≤ ψ(‖x−y‖∞).

Therefore, we get

‖(Tx)(t) − (Ty)(t)‖∞ ≤ ψ(‖x−y‖∞).

This yields that

D(Tx,Ty) ≤ ψ(D(x,y)).

By Theorem 2.1, T has a unique fixed point, say x. Therefore, x is the unique
solution of the second-order differential equation (3.1) in X. �

3.2. Application to Caputo fractional differential equations. In this
part, we discuss on Caputo fractional differential equation and apply our result
to this equation. Before going further, we first recall some basic definitions of
fractional derivatives. We first start with the definition of fractional integral
operators as follows.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 163



A. Bera, L. K. Dey, S. Som, H. Garai and W. Sintunavarat

Definition 3.2.

• The fractional integral operator of order q ∈ (0,∞) (denoted by Iq0 ) is
defined as follows:

I
q
0f(t) =

1

Γ(q)

∫ t
0

f(u)

(t−u)1−q
du.(3.3)

• The Riemann-Liouville fractional derivative of order q > 0 is defined
as follows:

D
q
0f(t) =

{
1

Γ(m−q)
dm

dtm

∫ t
0

f(u)
(t−u)q−m+1 du, if m− 1 < q < m;

dmf(t)
dtm

, if q = m,

where m is a positive integer and m− 1 < q < m.
• The Caputo fractional derivative of order q > 0 with m−1 < q < m is

defined as follows:

D
q
0f(t) =

{
1

Γ(m−q)
dm

dtm

∫ t
0

f(m)(u)
(t−u)q−m+1 du, if m− 1 < q < m;

dmf(t)
dtm

, if q = m.

Now, we recall the following lemma due to [18], which will be needed for the
application.

Lemma 3.3 ([18]). For q > 0, the homogeneous fractional differential equation
D
q
0g(t) = 0 has a solution

g(t) = c1 + c2t + · · · + cntn−1,

where ci ∈ R for i = 1, 2, 3, . . . ,n and n = [q] + 1.

For more informations concerning the fractional calculus, one can see [12,
13, 14, 22] and the references therein.

Now, we consider the boundary value problem (BVP) as follows:

(3.4)




cD
q
0x(t) −g(t,x(t)) = 0, 0 ≤ t ≤ 1, 1 < q ≤ 2

x′(0) = 0,x(0) −βx(1) =
∫ r

0

h(u,x(u))du,r ∈ (0, 1),β 6= 1,

where x ∈ C([0, 1],R) is an unknown function and g,h : [0, 1] × R+ → R are
continuous functions.

Using Lemma 3.3 and the above BVP, we get

x(t) = I
q
0g(t,x(t)) + c1 + c2t.

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 164



Boyd-Wong contractions in F-metric spaces and applications

Using the boundary conditions, we have

x(t) = I
q
0g(t,x(t)) +

β

1 −β
I
q
0g(1,x(1)) +

1

1 −β

∫ r
0

h(u,x(u))du

=
1

Γ(q)

∫ t
0

g(u,x(u))

(t−u)1−q
du +

β

1 −β
1

Γ(q)

∫ 1
0

g(u,x(u))

(1 −u)1−q
du +

1

1 −β

∫ r
0

h(u,x(u))du

=
1

Γ(q)

[∫ t
0

g(u,x(u))

(t−u)1−q
du +

β

1 −β

∫ 1
0

g(u,x(u))

(1 −u)1−q
du

]
+

1

1 −β

∫ r
0

h(u,x(u))du

=
1

Γ(q)

∫ 1
0

G(t,u)g(u,x(u))du +
1

1 −β

∫ r
0

h(u,x(u))du,

where

G(t,u) =

{
1

(t−u)1−q +
β

1−β
1

(1−u)1−q , 0 ≤ u ≤ t ≤ 1;
β

1−β
1

(1−u)1−q , 0 ≤ t ≤ u ≤ 1.

Thus, we see that the BVP (3.4) is equivalent to the integral equation

(3.5) x(t) =
1

Γ(q)

∫ 1
0

G(t,u)g(u,x(u))du +
1

1 −β

∫ r
0

h(u,x(u))du.

Now we are in a position to present a result concerning the existence of a
solution of the above BVP.

Theorem 3.4. Consider the BVP (3.4) under the following assumptions:

(1) g,h are continuous functions;
(2) there exists a nondecreasing function ψ : [0,∞) → [0,∞) such that ψ

is upper semi-continuous mapping from right and ψ(t) < t for all t > 0

max{|g(t,a) −g(t,b)|, |h(t,a) −h(t,b)|}≤ Kψ(|a− b|)

for all t ∈ [0, 1] and a,b ∈ R, where K := Γ(q)−βΓ(q)
1+Γ(q)−β .

Then the equation (3.4) has a unique solution in X.

Proof. Consider the F-metric space (X,D) considered in (3.2). Then by as-
sumption (2), it is clear that f(t) > f(ψ(t)) + α for all t ∈ (0,∞). Note that α
is zero in this case. Let us now define a mapping T : X → X for each x ∈ X
by

(Tx)(t) =
1

Γ(q)

∫ 1
0

G(t,u)g(u,x(u))du +
1

1 −β

∫ r
0

h(u,x(u))du

for all t ∈ [0, 1]. Then a point x ∈ X is a solution of the BVP (3.4) if and only
if x is a fixed point of T .

© AGT, UPV, 2022 Appl. Gen. Topol. 23, no. 1 165



A. Bera, L. K. Dey, S. Som, H. Garai and W. Sintunavarat

Let x,y ∈ C[0, 1] and t ∈ [0, 1]. Then we have

|Tx(t) −Ty(t)| =
∣∣∣ 1
Γ(q)

∫ 1
0

G(t,u)g(u,y(u))du +
1

1 −β

∫ r
0

h(u,y(u))du

−
1

Γ(q)

∫ 1
0

G(t,u)g(u,y(u))du +
1

1 −β

∫ r
0

h(u,y(u))du
∣∣∣

=
1

Γ(q)

∫ 1
0

G(t,u)|g(u,x(u)) −g(u,y(u))|du

+
1

1 −β

∫ r
0

|h(u,x(u)) −h(u,y(u))|du

≤
1

Γ(q)

∫ 1
0

G(t,u)Kψ(|x−y|)du

+
1

1 −β

∫ r
0

Kψ(|x−y|)du

≤ Kψ(‖x−y‖∞) sup
t∈[0,1]

{
1

Γ(q)

∫ 1
0

G(t,u)du +
r

1 −β

}
≤ Kψ(‖x−y‖∞)

(
1

Γ(q)
+

1

1 −β

)
≤ ψ(‖x−y‖∞)

and so

‖Tx(t) −Ty(t)‖∞ ≤ ψ(‖x−y‖∞).
Then

D(Tx,Ty) ≤ ψ(D(x,y))
for all x,y ∈ X. Hence, from Theorem 2.1, T has a unique fixed point, and
hence the Caputo fractional differential equation (3.4) has a unique solution.

�

Acknowledgements. The Research is funded by the Ministry of Human Re-
source and Development, Government of India and by the Council of Scientific
and Industrial Research (CSIR), Government of India under the Grant Num-
ber: 25(0285)/18/EMR-II. This project is funded by National Research Council
of Thailand (NRCT) N41A640092.

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