FIXED POINT THEOREMS IN THE GENERALIZED RA-

TIONAL TYPE OF

-CLASS FUNCTIONS IN

-METRIC

SPACES WITH APPLICATION TO INTEGRAL EQUA-

TION

Mehdi Asadi

Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, (Iran)

E-mail:masadi.azu@gmail.com; masadi@iau.ac.ir

ORCID: https://orcid.org/0000-0003-2170-9919

Corresponding author

Mehdi Afshar

Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, (Iran)

E-mail:afshar1979@yahoo.com

ORCID: https://orcid.org/0000-0003-3764-2704

Reception: 14/08/2022 Acceptance: 29/08/2022 Publication: 29/12/2022

Suggested citation:

Medhi Asadi and Medhi Afshar (2022). Fixed point theorems in the generalized rational type of

-class functions in

b-metric spaces with Application to Integral Equation. 3C Empresa. Investigación y pensamiento crítico,11 (2), 64-74.

https://doi.org/10.17993/3cemp.2022.110250.64-74

ABSTRACT

In this paper, we study some results of existence and uniqueness of ﬁxed points for a

-class of mappings

satisfying an inequality of rational type in

-metric spaces. After deﬁnition of

-class functions covering

a large class of contractive conditions by Ansari [2]. Our results extend very recent results in the

literature; as well as Khan in [14] and later Fisher in [9] gave a revised improved version of Khan’s

result and Piri in [17] a new generalization of Khan’s Theorem. At the end, we present an example of

ﬁnding solutions for an integral equation.

KEYWORDS

Metric space; ﬁxed point, C-function.

https://doi.org/10.17993/3cemp.2022.110250.64-74

3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376

Ed. 50 Vol. 11 N.º 2 August - December 2022

FIXED POINT THEOREMS IN THE GENERALIZED RA-

TIONAL TYPE OF

-CLASS FUNCTIONS IN

-METRIC

SPACES WITH APPLICATION TO INTEGRAL EQUA-

TION

Mehdi Asadi

Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, (Iran)

E-mail:masadi.azu@gmail.com; masadi@iau.ac.ir

ORCID: https://orcid.org/0000-0003-2170-9919

Corresponding author

Mehdi Afshar

Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, (Iran)

E-mail:afshar1979@yahoo.com

ORCID: https://orcid.org/0000-0003-3764-2704

Reception: 14/08/2022 Acceptance: 29/08/2022 Publication: 29/12/2022

Suggested citation:

Medhi Asadi and Medhi Afshar (2022). Fixed point theorems in the generalized rational type of

-class functions in

b-metric spaces with Application to Integral Equation. 3C Empresa. Investigación y pensamiento crítico,11 (2), 64-74.

https://doi.org/10.17993/3cemp.2022.110250.64-74

ABSTRACT

In this paper, we study some results of existence and uniqueness of ﬁxed points for a

-class of mappings

satisfying an inequality of rational type in

-metric spaces. After deﬁnition of

-class functions covering

a large class of contractive conditions by Ansari [2]. Our results extend very recent results in the

literature; as well as Khan in [14] and later Fisher in [9] gave a revised improved version of Khan’s

result and Piri in [17] a new generalization of Khan’s Theorem. At the end, we present an example of

ﬁnding solutions for an integral equation.

KEYWORDS

Metric space; ﬁxed point, C-function.

https://doi.org/10.17993/3cemp.2022.110250.64-74

3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376

Ed. 50 Vol. 11 N.º 2 August - December 2022

1 INTRODUCTION

In 1989, Bakhtin [4] introduced

-metric spaces as a generalization of metric spaces. Since then, many

articles have been published in the ﬁeld of ﬁxed point theory and Banach generalization. The contraction

principle in such spaces, known as Banach’s contraction principle, states that every self-contracting

mapping in a complete metric space has a unique ﬁxed point. This principle has been generalized and

expanded in several ways.

In this paper, we study certain results of existence and uniqueness of ﬁxed points for a

-class of

mappings satisfying an inequality of rational type in

-metric spaces. The

-class functions cover a

large class of contractive conditions which applyed by Ansari [2]. Our results develop very recent results

in the literature; as well as Khan in [14] and later Fisher in [9] gave a revised improved version of

Khan’s result and Piri in [17] a new generalization of Khan’s Theorem. At the end of the paper, we

present examples of ﬁnding solutions for integral equations. For more detail and recent papers refer

to [3,7,8,11,16,18].

Deﬁnition 1 ( [10]).Let

be a (nonempty) set and

s≥

1be a given real number. A function

d:X×X→[0,∞)is called a b-metric on Xif the following conditions hold for all x, y, z ∈X:

(i)d(x, y)=0if and only if x=y,

(ii)d(x, y)=d(y, x),

(iii)d(x, y)≤s[d(x, z)+d(z, y)] (b-triangular inequality).

Then, the pair (X, d)is called a b-metric space with parameter s.

Example 1 ( [15]).Let (

X, d

)be a metric space and let

β>

,λ ≥

0and

µ>

0. For

x, y ∈X

, set

(

x, y

λd

(

x, y

µd

(

x, y

)

. Then (

X, ρ

)is a

-metric space with the parameter

β−1

and not a

metric space on X.

Deﬁnition 2 ( [5]).Let (X, d)be a b−metric space. Then a sequence {xn}in Xis called:

(i) b−

convergent if there exists

x∈X

such that

(

xn,x

)

→

0as

n→∞

. In this case, we write

lim

n→∞ xn=x.

(ii) A b−Cauchy sequence if d(xn,x

m)→0as n, m →∞.

Lemma 1 ( [1]).Let (

x, d

)be a

-metric space with

. If sequences

{xn}

and

{yn}

are

-convergent to

xand yin X. Then

s2d(x, y)≤lim inf

n→∞ d(xn,y

n)≤lim sup

n→∞ d(xn,y

n)≤s2d(x, y).

Specially if x=y, then lim

n→∞ d(xn,y

n)=0. Further if z∈Xwe have

sd(x, z)≤lim inf

n→∞ d(xn,z)≤lim sup

n→∞ d(xn,z)≤sd(x, z).

Deﬁnition 3 ( [6]).Let Ψdenote all functions ψ: [0,∞)→[0,∞)satisﬁed:

(i)ψis strictly increasing and continuous,

(ii)ψ(t)=0if and only if t=0.

We let Ψdenote the class of the altering distance functions.

Deﬁnition 4 ( [2]).An ultra altering distance function is a continuous, nondecreasing mapping

φ: [0,∞)→[0,∞)such that φ(t)>0for t>0.

https://doi.org/10.17993/3cemp.2022.110250.64-74

We let Φdenote the class of the ultra altering distance functions.

Zoran Kadelburg and et al. in [19], and Imdad and Ali [12,13] deﬁned and studied implicit functions

and utilized same to prove several ﬁxed point results for rational type condition. Also, in 2014 Ansari

in [2] introduced C-type functions as follows:

Deﬁnition 5 ( [2]).A mapping

: [0

,∞

)

2→R

is called

-class function if it is continuous and

satisﬁes following axioms:

1. F(s, t)≤s;

2. F(s, t)=simplies that either s=0or t=0.

It’s clear that F(0,0) = 0. We denote C-class functions by C.

Let

Ω := {φ∈C([0,∞),[0,∞)) : φ−1(0)=0}.

Example 2 ( [2]).The following functions F: [0,∞)2→Rare elements of C, for all s, t ∈[0,∞):

1. F(s, t)=s−t.

2. F(s, t)= s

(1+t)r;r∈(0,∞).

3. F(s, t)=s−φ(s)for φwhere φ∈Ω

2 RESULTS

We improve the recent results by

-class functions to generalization of contractions in the cases of

fractional contraction.

Theorem 1. Let (X, d)ab-metric space with sand T:X→Xa self mapping satisfying

ψ(d(T x, T y)) ≤g(x, y),A

xy =0;

0,otherwise (1)

for x, y ∈Xwhere x=yand

g(x, y) := F(ψ(m(x, y)),φ(m(x, y)))

m(x, y) := d(x, T x)d(x, T y)+d(y, Ty)d(y, T x)

d(x, T y)+d(T x, y),

Axy := max{d(x, T y),d(T x, y)}

and F∈C,φ∈Φ,ψ ∈Ψ. Then Thas a unique ﬁxed point.

Proof.

Let

x0∈X

. For

n∈N

put

n−1

. It’s clear when for some

n∈N

we have

xn+1

. So we

assume that

xn=xn+1 ∀n∈N.

Put

An,n−1= max{d(xn,Tx

n−1),d(Tx

n,x

n−1)}=0.

https://doi.org/10.17993/3cemp.2022.110250.64-74

3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376

Ed. 50 Vol. 11 N.º 2 August - December 2022

1 INTRODUCTION

In 1989, Bakhtin [4] introduced

-metric spaces as a generalization of metric spaces. Since then, many

articles have been published in the ﬁeld of ﬁxed point theory and Banach generalization. The contraction

principle in such spaces, known as Banach’s contraction principle, states that every self-contracting

mapping in a complete metric space has a unique ﬁxed point. This principle has been generalized and

expanded in several ways.

In this paper, we study certain results of existence and uniqueness of ﬁxed points for a

-class of

mappings satisfying an inequality of rational type in

-metric spaces. The

-class functions cover a

large class of contractive conditions which applyed by Ansari [2]. Our results develop very recent results

in the literature; as well as Khan in [14] and later Fisher in [9] gave a revised improved version of

Khan’s result and Piri in [17] a new generalization of Khan’s Theorem. At the end of the paper, we

present examples of ﬁnding solutions for integral equations. For more detail and recent papers refer

to [3,7,8,11,16,18].

Deﬁnition 1 ( [10]).Let

be a (nonempty) set and

s≥

1be a given real number. A function

d:X×X→[0,∞)is called a b-metric on Xif the following conditions hold for all x, y, z ∈X:

(i)d(x, y)=0if and only if x=y,

(ii)d(x, y)=d(y, x),

(iii)d(x, y)≤s[d(x, z)+d(z, y)] (b-triangular inequality).

Then, the pair (X, d)is called a b-metric space with parameter s.

Example 1 ( [15]).Let (

X, d

)be a metric space and let

β>

,λ ≥

0and

µ>

0. For

x, y ∈X

, set

(

x, y

λd

(

x, y

µd

(

x, y

)

. Then (

X, ρ

)is a

-metric space with the parameter

β−1

and not a

metric space on X.

Deﬁnition 2 ( [5]).Let (X, d)be a b−metric space. Then a sequence {xn}in Xis called:

(i) b−

convergent if there exists

x∈X

such that

(

xn,x

)

→

0as

n→∞

. In this case, we write

lim

n→∞ xn=x.

(ii) A b−Cauchy sequence if d(xn,x

m)→0as n, m →∞.

Lemma 1 ( [1]).Let (

x, d

)be a

-metric space with

. If sequences

{xn}

and

{yn}

are

-convergent to

xand yin X. Then

s2d(x, y)≤lim inf

n→∞ d(xn,y

n)≤lim sup

n→∞ d(xn,y

n)≤s2d(x, y).

Specially if x=y, then lim

n→∞ d(xn,y

n)=0. Further if z∈Xwe have

sd(x, z)≤lim inf

n→∞ d(xn,z)≤lim sup

n→∞ d(xn,z)≤sd(x, z).

Deﬁnition 3 ( [6]).Let Ψdenote all functions ψ: [0,∞)→[0,∞)satisﬁed:

(i)ψis strictly increasing and continuous,

(ii)ψ(t)=0if and only if t=0.

We let Ψdenote the class of the altering distance functions.

Deﬁnition 4 ( [2]).An ultra altering distance function is a continuous, nondecreasing mapping

φ: [0,∞)→[0,∞)such that φ(t)>0for t>0.

https://doi.org/10.17993/3cemp.2022.110250.64-74

We let Φdenote the class of the ultra altering distance functions.

Zoran Kadelburg and et al. in [19], and Imdad and Ali [12,13] deﬁned and studied implicit functions

and utilized same to prove several ﬁxed point results for rational type condition. Also, in 2014 Ansari

in [2] introduced C-type functions as follows:

Deﬁnition 5 ( [2]).A mapping

: [0

,∞

)

2→R

is called

-class function if it is continuous and

satisﬁes following axioms:

1. F(s, t)≤s;

2. F(s, t)=simplies that either s=0or t=0.

It’s clear that F(0,0) = 0. We denote C-class functions by C.

Let

Ω := {φ∈C([0,∞),[0,∞)) : φ−1(0)=0}.

Example 2 ( [2]).The following functions F: [0,∞)2→Rare elements of C, for all s, t ∈[0,∞):

1. F(s, t)=s−t.

2. F(s, t)= s

(1+t)r;r∈(0,∞).

3. F(s, t)=s−φ(s)for φwhere φ∈Ω

2 RESULTS

We improve the recent results by

-class functions to generalization of contractions in the cases of

fractional contraction.

Theorem 1. Let (X, d)ab-metric space with sand T:X→Xa self mapping satisfying

ψ(d(T x, T y)) ≤g(x, y),A

xy =0;

0,otherwise (1)

for x, y ∈Xwhere x=yand

g(x, y) := F(ψ(m(x, y)),φ(m(x, y)))

m(x, y) := d(x, T x)d(x, T y)+d(y, Ty)d(y, T x)

d(x, T y)+d(T x, y),

Axy := max{d(x, T y),d(T x, y)}

and F∈C,φ∈Φ,ψ ∈Ψ. Then Thas a unique ﬁxed point.

Proof.

Let

x0∈X

. For

n∈N

put

n−1

. It’s clear when for some

n∈N

we have

xn+1

. So we

assume that

xn=xn+1 ∀n∈N.

Put

An,n−1= max{d(xn,Tx

n−1),d(Tx

n,x

n−1)}=0.

https://doi.org/10.17993/3cemp.2022.110250.64-74

3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376

Ed. 50 Vol. 11 N.º 2 August - December 2022

by (1) we get

ψ(d(xn,x

n+1))

=ψ(d(Tx

n−1,Tx

n))

≤F(ψ(m(xn−1,x

n)),φ(m(xn−1,x

n)))

≤F(ψ(d(xn−1,x

n)),φ(d(xn−1,x

n)))

≤ψ(d(xn−1,x

n)).(2)

Now by increasing ψand relation (2)

d(xn,x

n+1)≤d(xn−1,x

n).

Since the {d(xn−1,x

n)}in decreasing therefore for some r

lim

n→∞ d(xn−1,x

n)=r. (3)

Again by (2)

ψ(r)≤F(ψ(r),φ(r)) ≤ψ(r).

So ψ(r)=0or φ(r)=0hence

lim

n→∞ d(xn−1,x

n)=r=0.(4)

It’s time to show that

{xn}

is a Cauchy sequence. If not, then for

ε>

0there exit

{xm(k)}

and

{xn(k)}

in {xn}and m(k)>n(k)>ksuch that

d(xm(k),x

n(k))≥ε, d(xm(k),x

n(k)−1)<ε

and

lim

n→∞ d(xm(k),x

n(k)+1) = lim

n→∞ d(xm(k)+1,x

n(k))=ε. (5)

∃N∈N∀k≥Nmax{d(xm(k),x

n(k)+1),d(xm(k)+1,x

n(k))≥ε

2>0.

By take a look at (1), for each k≥N

ψ(d(xm(k)+1,x

n(k)+1)) = ψ(d(Tx

m(k),Tx

n(k)))

≤F(ψ(m(xm(k),x

n(k))),φ(m(xm(k),x

n(k))))

≤ψ(d(xn−1,x

n)) (6)

even by the relations (3) and (5)

ψ(ε)≤F(ψ(ε),φ(ε)) ≤ψ(ε),

=0which is not. Thus

{xn}

is a Cauchy sequence in complete

-metric space. Therefore it is

convergent to some x∗∈X.

lim

n→∞ d(xn,x

∗)=0.(7)

If xn+1 =Tx

∗for some n, then

x∗= lim

n→∞ xn+1 = lim

n→∞ Tx

∗=Tx

∗.

∃N∈N∀n≥Nd(xn,Tx

∗)>0,

and also

max{d(xn,Tx

∗),d(x∗,Tx

n)>0.

https://doi.org/10.17993/3cemp.2022.110250.64-74

From (1)

ψ(d(xn+1,Tx

∗)) = ψ(d(Tx

n,Tx

∗))

≤F(ψ(m(xn,x

∗)).φ(m(xn,x

∗)))

≤ψ(d(xn−1,x

n)) (8)

where

m(xn,x

∗)=d(xn,Tx

n)d(xn,Tx

∗)+d(x∗,Tx

∗)d(x∗,Tx

d(xn,Tx

∗)+d(Tx

n,x

∗)

and or

m(xn,x

∗)=d(xn,x

n+1)d(xn,Tx

∗)+d(x∗,Tx

∗)d(x∗,x

n+1)

d(xn,Tx

∗)+d(xn+1,x

∗)

by taking lim inf and lim sup from (8) and relation (4) and (7) we have:

lim sup

n→∞ m(xn,x

∗)≤0×sd(x∗,Tx

∗)+sd(x∗,Tx

∗)×0=0

thus

ψ(1

sd(x∗,Tx

∗)) ≤F(ψ(0),φ(0)) ≤ψ(0)=0

and consequently x∗=Tx

∗.

For an other condition may be to happen: for some n∈N

An,n−1= max{d(xn,Tx

n−1),d(Tx

n,x

n−1)}=0.

So (1) thus d(Tx

n−1,Tx

n)=0Tx

n−1=Tx

nmeans that xn=Tx

For uniqueness, let x∗and y∗be two ﬁxed point of Tsuch that d(x∗,y

∗)>0. By (1)

ψ(d(x∗,y

∗)) = ψ(d(Tx

∗,Ty∗))

≤F(ψ(m(x∗,y

∗)),φ(m(x∗,y

∗)))

≤F(ψ(0),φ(0)) ≤ψ(0) = 0,

it should be d(x∗,y

∗)=0.

For the next result, it’s enough F(s, t)= s

2+t.

Corollary 1. Let (X, d)ab-metric space with sand T:X→Xa selfmapping satisfying

ψ(d(T x, T y)) ≤g(x, y),A

xy =0;

0,otherwise

for x, y ∈Xwhere x=yand

g(x, y) := ψ(m(x, y))

+φ(m(x, y))

m(x, y) := d(x, T x)d(x, T y)+d(y, Ty)d(y, T x)

d(x, T y)+d(T x, y),

Axy := max{d(x, T y),d(T x, y)}

and φ∈Φ,ψ ∈Ψ. Then Thas a unique ﬁxed point.

And if we put F(s, t)=s−t:

https://doi.org/10.17993/3cemp.2022.110250.64-74

3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376

Ed. 50 Vol. 11 N.º 2 August - December 2022

by (1) we get

ψ(d(xn,x

n+1))

=ψ(d(Tx

n−1,Tx

n))

≤F(ψ(m(xn−1,x

n)),φ(m(xn−1,x

n)))

≤F(ψ(d(xn−1,x

n)),φ(d(xn−1,x

n)))

≤ψ(d(xn−1,x

n)).(2)

Now by increasing ψand relation (2)

d(xn,x

n+1)≤d(xn−1,x

n).

Since the {d(xn−1,x

n)}in decreasing therefore for some r

lim

n→∞ d(xn−1,x

n)=r. (3)

Again by (2)

ψ(r)≤F(ψ(r),φ(r)) ≤ψ(r).

So ψ(r)=0or φ(r)=0hence

lim

n→∞ d(xn−1,x

n)=r=0.(4)

It’s time to show that

{xn}

is a Cauchy sequence. If not, then for

ε>

0there exit

{xm(k)}

and

{xn(k)}

in {xn}and m(k)>n(k)>ksuch that

d(xm(k),x

n(k))≥ε, d(xm(k),x

n(k)−1)<ε

and

lim

n→∞ d(xm(k),x

n(k)+1) = lim

n→∞ d(xm(k)+1,x

n(k))=ε. (5)

∃N∈N∀k≥Nmax{d(xm(k),x

n(k)+1),d(xm(k)+1,x

n(k))≥ε

2>0.

By take a look at (1), for each k≥N

ψ(d(xm(k)+1,x

n(k)+1)) = ψ(d(Tx

m(k),Tx

n(k)))

≤F(ψ(m(xm(k),x

n(k))),φ(m(xm(k),x

n(k))))

≤ψ(d(xn−1,x

n)) (6)

even by the relations (3) and (5)

ψ(ε)≤F(ψ(ε),φ(ε)) ≤ψ(ε),

=0which is not. Thus

{xn}

is a Cauchy sequence in complete

-metric space. Therefore it is

convergent to some x∗∈X.

lim

n→∞ d(xn,x

∗)=0.(7)

If xn+1 =Tx

∗for some n, then

x∗= lim

n→∞ xn+1 = lim

n→∞ Tx

∗=Tx

∗.

∃N∈N∀n≥Nd(xn,Tx

∗)>0,

and also

max{d(xn,Tx

∗),d(x∗,Tx

n)>0.

https://doi.org/10.17993/3cemp.2022.110250.64-74

From (1)

ψ(d(xn+1,Tx

∗)) = ψ(d(Tx

n,Tx

∗))

≤F(ψ(m(xn,x

∗)).φ(m(xn,x

∗)))

≤ψ(d(xn−1,x

n)) (8)

where

m(xn,x

∗)=d(xn,Tx

n)d(xn,Tx

∗)+d(x∗,Tx

∗)d(x∗,Tx

d(xn,Tx

∗)+d(Tx

n,x

∗)

and or

m(xn,x

∗)=d(xn,x

n+1)d(xn,Tx

∗)+d(x∗,Tx

∗)d(x∗,x

n+1)

d(xn,Tx

∗)+d(xn+1,x

∗)

by taking lim inf and lim sup from (8) and relation (4) and (7) we have:

lim sup

n→∞ m(xn,x

∗)≤0×sd(x∗,Tx

∗)+sd(x∗,Tx

∗)×0=0

thus

ψ(1

sd(x∗,Tx

∗)) ≤F(ψ(0),φ(0)) ≤ψ(0)=0

and consequently x∗=Tx

∗.

For an other condition may be to happen: for some n∈N

An,n−1= max{d(xn,Tx

n−1),d(Tx

n,x

n−1)}=0.

So (1) thus d(Tx

n−1,Tx

n)=0Tx

n−1=Tx

nmeans that xn=Tx

For uniqueness, let x∗and y∗be two ﬁxed point of Tsuch that d(x∗,y

∗)>0. By (1)

ψ(d(x∗,y

∗)) = ψ(d(Tx

∗,Ty∗))

≤F(ψ(m(x∗,y

∗)),φ(m(x∗,y

∗)))

≤F(ψ(0),φ(0)) ≤ψ(0) = 0,

it should be d(x∗,y

∗)=0.

For the next result, it’s enough F(s, t)= s

2+t.

Corollary 1. Let (X, d)ab-metric space with sand T:X→Xa selfmapping satisfying

ψ(d(T x, T y)) ≤g(x, y),A

xy =0;

0,otherwise

for x, y ∈Xwhere x=yand

g(x, y) := ψ(m(x, y))

+φ(m(x, y))

m(x, y) := d(x, T x)d(x, T y)+d(y, Ty)d(y, T x)

d(x, T y)+d(T x, y),

Axy := max{d(x, T y),d(T x, y)}

and φ∈Φ,ψ ∈Ψ. Then Thas a unique ﬁxed point.

And if we put F(s, t)=s−t:

https://doi.org/10.17993/3cemp.2022.110250.64-74

3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376

Ed. 50 Vol. 11 N.º 2 August - December 2022

Corollary 2. Let (X, d)ab-metric space with sand T:X→Xa selfmapping satisfying

ψ(d(T x, T y)) ≤g(x, y),A

xy =0;

0,otherwise

for x, y ∈Xwhere x=yand

g(x, y) := ψ(m(x, y)) −φ(m(x, y))

m(x, y) := d(x, T x)d(x, T y)+d(y, Ty)d(y, T x)

d(x, T y)+d(T x, y),

Axy := max{d(x, T y),d(T x, y)}

and φ∈Φ,ψ ∈Ψ. Then Thas a unique ﬁxed point.

And also we put F(s, t)=ks,ψ(t)=tand s=1, hence:

Corollary 3 ( [14]).Let (X, d)ab-metric space with sand T:X→Xa selfmapping satisfying

ψ(d(T x, T y)) ≤g(x, y),A

xy =0;

0,otherwise (9)

for x, y ∈Xwhere x=yand

g(x, y) := km(x, y)

m(x, y) := d(x, T x)d(x, T y)+d(y, Ty)d(y, T x)

d(x, T y)+d(T x, y),

Axy := max{d(x, T y),d(T x, y)}.

Then Thas a unique ﬁxed point.

Our results improve the following Corollary in [17] which it is extension of the Khan’s Theorem

with s=1.

Corollary 4 ( [17]).Let (X, d)ab-metric space with sand T:X→Xa self mapping satisfying

ψ(d(T x, T y)) ≤g(x, y),A

xy =0;

0,otherwise

for x, y ∈Xwhere x=yand

g(x, y) := m(x, y)−φ(m(x, y))

m(x, y) := d(x, T x)d(x, T y)+d(y, Ty)d(y, T x)

d(x, T y)+d(T x, y),

Axy := max{d(x, T y),d(T x, y)}.

where φ∈Φ. Then Thas a unique ﬁxed point.

3 Application to integral equation

In this section, we looking for the solutions of the following integral equation:

u(t)=g(t, u(t))+t

G(t, s, u(s))ds, t ∈[0,∞),(10)

where G: [0,∞)×[0,∞)×R→Rg: [0,∞)×R→Rare continuous.

Let

be complete space

([0

,∞

)) containing of bounded, continuous and real valued functions

on [0,∞)such that

∥u∥= sup{|u(t)|:t∈[0,∞)}.

https://doi.org/10.17993/3cemp.2022.110250.64-74

Example 3. We show that the following integral equation:

(Tu)(t)=g(t, u(t)) + t

G(t, s, u(s))ds, t ∈[0,∞),(11)

has unique solution, if we have the under following conditions.

Let d(u, v)=∥u−v∥2. So dis a b-metric with s=2.

1. ψ(t)=φ(t)=t

2. F(s, t)=s

3. ∥u−v∥≤min{∥u−Tu∥,∥v−Tv∥}

4. |g(t, u(t)) −g(t, v(t))|≤|u(t)−v(t)|

at∈[0,∞)

5. for t∈[0,∞)

|t

0G(t, s, u(s)) −G(t, s, v(s))ds|≤∥u−v∥

6. a, b > 0,c>2,c= min{a, b}and k=4

c2.

At ﬁrst we observe that

∥u−v∥2(∥u−Tv∥2+∥Tu−v∥2)=∥u−v∥2∥u−Tv∥2+∥u−v∥2∥Tu−v∥2

≤∥u−Tu∥2∥u−Tv∥2+∥Tv−v∥2∥Tu−v∥2,

d(u, v)=∥u−v∥2≤∥u−Tu∥2∥u−Tv∥2+∥Tv−v∥2∥Tu−v∥2

∥u−Tv∥2+∥Tu−v∥2=m(u, v).

And by (a+b)2≤2(a2+b2)

|(Tu)(t)−(Tv)(t)|≤|g(t, u(t)) −g(t, v(t))|+t

0|G(t, s, u(s)) −G(t, s, v(s))|ds

|(Tu)(t)−(Tv)(t)|2≤(|g(t, u(t)) −g(t, v(t))|+t

0|G(t, s, u(s)) −G(t, s, v(s))|ds)2

≤2(|g(t, u(t)) −g(t, v(t))|2+(

t

0|G(t, s, u(s)) −G(t, s, v(s))|ds)2)

≤2|u(t)−v(t)|

a)2+(∥u−v∥

b)2≤4∥u−v∥

c2

d(Tu,Tv)=∥Tu−Tv∥2

≤4∥u−v∥

c2

=kd(u, v)≤km(u, v)

≤F(ψ(m(u, v)),φ(m(u, v))).

Therefore integral equation (11) by the Theorem 1 has a unique answer.

Example 4. Let

Tu(t)=u(t)

2+t

e−(t−s)u(s)

3ds. (12)

https://doi.org/10.17993/3cemp.2022.110250.64-74

3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376

Ed. 50 Vol. 11 N.º 2 August - December 2022

Corollary 2. Let (X, d)ab-metric space with sand T:X→Xa selfmapping satisfying

ψ(d(T x, T y)) ≤g(x, y),A

xy =0;

0,otherwise

for x, y ∈Xwhere x=yand

g(x, y) := ψ(m(x, y)) −φ(m(x, y))

m(x, y) := d(x, T x)d(x, T y)+d(y, Ty)d(y, T x)

d(x, T y)+d(T x, y),

Axy := max{d(x, T y),d(T x, y)}

and φ∈Φ,ψ ∈Ψ. Then Thas a unique ﬁxed point.

And also we put F(s, t)=ks,ψ(t)=tand s=1, hence:

Corollary 3 ( [14]).Let (X, d)ab-metric space with sand T:X→Xa selfmapping satisfying

ψ(d(T x, T y)) ≤g(x, y),A

xy =0;

0,otherwise (9)

for x, y ∈Xwhere x=yand

g(x, y) := km(x, y)

m(x, y) := d(x, T x)d(x, T y)+d(y, Ty)d(y, T x)

d(x, T y)+d(T x, y),

Axy := max{d(x, T y),d(T x, y)}.

Then Thas a unique ﬁxed point.

Our results improve the following Corollary in [17] which it is extension of the Khan’s Theorem

with s=1.

Corollary 4 ( [17]).Let (X, d)ab-metric space with sand T:X→Xa self mapping satisfying

ψ(d(T x, T y)) ≤g(x, y),A

xy =0;

0,otherwise

for x, y ∈Xwhere x=yand

g(x, y) := m(x, y)−φ(m(x, y))

m(x, y) := d(x, T x)d(x, T y)+d(y, Ty)d(y, T x)

d(x, T y)+d(T x, y),

Axy := max{d(x, T y),d(T x, y)}.

where φ∈Φ. Then Thas a unique ﬁxed point.

3 Application to integral equation

In this section, we looking for the solutions of the following integral equation:

u(t)=g(t, u(t)) + t

G(t, s, u(s))ds, t ∈[0,∞),(10)

where G: [0,∞)×[0,∞)×R→Rg: [0,∞)×R→Rare continuous.

Let

be complete space

([0

,∞

)) containing of bounded, continuous and real valued functions

on [0,∞)such that

∥u∥= sup{|u(t)|:t∈[0,∞)}.

https://doi.org/10.17993/3cemp.2022.110250.64-74

Example 3. We show that the following integral equation:

(Tu)(t)=g(t, u(t)) + t

G(t, s, u(s))ds, t ∈[0,∞),(11)

has unique solution, if we have the under following conditions.

Let d(u, v)=∥u−v∥2. So dis a b-metric with s=2.

1. ψ(t)=φ(t)=t

2. F(s, t)=s

3. ∥u−v∥≤min{∥u−Tu∥,∥v−Tv∥}

4. |g(t, u(t)) −g(t, v(t))|≤|u(t)−v(t)|

at∈[0,∞)

5. for t∈[0,∞)

|t

0G(t, s, u(s)) −G(t, s, v(s))ds|≤∥u−v∥

6. a, b > 0,c>2,c= min{a, b}and k=4

c2.

At ﬁrst we observe that

∥u−v∥2(∥u−Tv∥2+∥Tu−v∥2)=∥u−v∥2∥u−Tv∥2+∥u−v∥2∥Tu−v∥2

≤∥u−Tu∥2∥u−Tv∥2+∥Tv −v∥2∥Tu−v∥2,

d(u, v)=∥u−v∥2≤∥u−Tu∥2∥u−Tv∥2+∥Tv −v∥2∥Tu−v∥2

∥u−Tv∥2+∥Tu−v∥2=m(u, v).

And by (a+b)2≤2(a2+b2)

|(Tu)(t)−(Tv)(t)|≤|g(t, u(t)) −g(t, v(t))|+t

0|G(t, s, u(s)) −G(t, s, v(s))|ds

|(Tu)(t)−(Tv)(t)|2≤(|g(t, u(t)) −g(t, v(t))|+t

0|G(t, s, u(s)) −G(t, s, v(s))|ds)2

≤2(|g(t, u(t)) −g(t, v(t))|2+(

t

0|G(t, s, u(s)) −G(t, s, v(s))|ds)2)

≤2|u(t)−v(t)|

a)2+(∥u−v∥

b)2≤4∥u−v∥

c2

d(Tu,Tv)=∥Tu−Tv∥2

≤4∥u−v∥

c2

=kd(u, v)≤km(u, v)

≤F(ψ(m(u, v)),φ(m(u, v))).

Therefore integral equation (11) by the Theorem 1 has a unique answer.

Example 4. Let

Tu(t)=u(t)

2+t

e−(t−s)u(s)

3ds. (12)

https://doi.org/10.17993/3cemp.2022.110250.64-74

3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376

Ed. 50 Vol. 11 N.º 2 August - December 2022

Tu(t)=u(t)

2+e−t∗u(t)

3=u∗δ(t)

2+e−t

3,g(t, u(t)) = u(t)

G(t, s, u(s)) = e−(t−s)u(s)

3,a=2,b=3,

where ∗is convolution of uand v; i.e.,

u(t)∗v(t)=t

u(t−s)v(s)ds.

We see that

u(t)−Tu(t)=u(t)∗δ(t)

2−e−t

3,v(t)−Tv(t)=v(t)∗δ(t)

2−e−t

3

and

t

0G(t, s, u(s)) −G(t, s, v(s))ds=t

0e−(t−s)u(s)

3−e−(t−s)v(s)

3ds

≤t

e−(t−s)∥u−v∥

3ds

≤∥u−v∥

3(1 −e−t)

≤∥u−v∥

where δ(t)is Dirichlet function with Laplace transformation L(δ)=1.

|Tu(t)−Tv(t)|=u(t)∗δ(t)

2+e−t

3−v(t)∗δ(t)

2+e−t

3

=(u(t)−v(t)) ∗δ(t)

2+e−t

3

≤5

6|u(t)−v(t)|,

without of loss of generality, let |u(t)|≤|v(t)|for all t∈[0,∞).

u(t)∗δ(t)

2−e−t

3≤v(t)∗δ(t)

2−e−t

3

therefore |Tu(t)−u(t)|≤|Tv(t)−v(t)|.

|u(t)−v(t)|≤|u(t)−Tu(t)|+|Tu(t)−Tv(t)|+|Tv(t)−v(t)|

≤|u(t)−Tu(t)|+5

6|u(t)−v(t)|+|Tv(t)−v(t)|

6|u(t)−v(t)|≤|u(t)−Tu(t)|+|Tv(t)−v(t)|

so for some positive number lwe have

∥u−v∥≤4lmin{∥u−Tu∥,∥v−Tv∥}.

All conditions of Example

(3)

with

(

r, s

)=4

hold, and integral equation

(12)

has a unique solution

u=0.

Availability of data and material

Not applicable.

https://doi.org/10.17993/3cemp.2022.110250.64-74

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors have read and approved the ﬁnal manuscript.

ACKNOWLEDGMENT

We would like to thank the referees for their comments and suggestions on the manuscript.

REFERENCES

[1]

Aghajani, A.,Abbas, M., and Roshan, J., ( 2014). “Common ﬁxed point of generalized weak

contractive mappings in partially ordered b-metric spaces.” Mathematica Slovaca, 64, 2 ), 941–960.

[2]

Ansari, A.H., (2014). “Note on

φ−ϕ

-contractive type mappings and related ﬁxed point. The 2nd

regional conference on mathematics and applications”, Payame Noor University, 377–380.

[3]

Asadi, M.,Moeini, B.,Mukheimer, A., and Hassen, A., (2019). Complex valued

–metric

spaces and related ﬁxed point results via complex

-class function. Journal of Inequalities and Special

Functions, 10, 1, 101–110.

[4]

Bakhtin, I. A., (1989). The contraction mapping principle in almost metric space. Funct. Anal.

Gos. Ped. Inst. Unianowsk, 30, 26–37.

[5]

Boriceanu, M.,Bota, M., and Petrusel, A., (2010). “Mutivalued fractals in

-metric spaces”.

Cent. Eur. J. Math., 367–377.

[6]

Choudhury, B. S., (2005). “A common unique ﬁxed point result in metric spaces involving

generalised altering distances.” Mathematical Communications, 10, 2, 105–110.

[7]

Eshraghisamani, M.,Vaezpour,S.M., and Asadi, M., (2-18). New ﬁxed point results with

αqsp

–

admissible contractions on

–Branciari metric spaces. Journal of Inequalities and Special Functions,

9, 3, 38–46.

[8]

Eshraghisamani, M.,Vaezpour, S.M., and Asadi, M., (2017). New ﬁxed point results on

Branciari metric spaces. Journal of Mathematical Analysis, 8, 6, 132–141.

[9] B. Fisher, (1978). “On a theorem of Khan”. Riv. Math. Univ. Parma. 4, 35–137.

[10]

Faraji, H.,Savic, D., and Radenovic, S., (2019). “Fixed Point Theorems for Geraghty Con-

traction Type Mappings in b-Metric Spaces and Applications”. Axioms, 8, 484–490.

[11]

Ghezellou, S.,Azhini,M., and Asadi, M., (2021). Best proximity point theorems by

k, c

and

mt types in b–metric spaces with an application. Int. J. Nonlinear Anal. Appl., 12, 2, 1317-1329.

[12]

Ali, J., and Imdad, M., (2008). An implicit function implies several contraction conditions.

Sarajevo J. Math. 4, 2, 269–285.

[13]

Ali, J. and Imdad, M., (2009). Common ﬁxed point theorems in symmetric spaces employing

a new implicit function and common property (E.A), Bull. Belg. Math. Soc. Simon Stevin 16(3),

421–433.

[14]

Khan, M.S. ,Swaleh, M. and Sessa, S., (1984). “Fixed point theorems by altering distances

between the points". Bull. Austral. Math. Soc. 30, 1, 1–9.

https://doi.org/10.17993/3cemp.2022.110250.64-74

3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376

Ed. 50 Vol. 11 N.º 2 August - December 2022

Tu(t)=u(t)

2+e−t∗u(t)

3=u∗δ(t)

2+e−t

3,g(t, u(t)) = u(t)

G(t, s, u(s)) = e−(t−s)u(s)

3,a=2,b=3,

where ∗is convolution of uand v; i.e.,

u(t)∗v(t)=t

u(t−s)v(s)ds.

We see that

u(t)−Tu(t)=u(t)∗δ(t)

2−e−t

3,v(t)−Tv(t)=v(t)∗δ(t)

2−e−t

3

and

t

0G(t, s, u(s)) −G(t, s, v(s))ds=t

0e−(t−s)u(s)

3−e−(t−s)v(s)

3ds

≤t

e−(t−s)∥u−v∥

3ds

≤∥u−v∥

3(1 −e−t)

≤∥u−v∥

where δ(t)is Dirichlet function with Laplace transformation L(δ)=1.

|Tu(t)−Tv(t)|=u(t)∗δ(t)

2+e−t

3−v(t)∗δ(t)

2+e−t

3

=(u(t)−v(t)) ∗δ(t)

2+e−t

3

≤5

6|u(t)−v(t)|,

without of loss of generality, let |u(t)|≤|v(t)|for all t∈[0,∞).

u(t)∗δ(t)

2−e−t

3≤v(t)∗δ(t)

2−e−t

3

therefore |Tu(t)−u(t)|≤|Tv(t)−v(t)|.

|u(t)−v(t)|≤|u(t)−Tu(t)|+|Tu(t)−Tv(t)|+|Tv(t)−v(t)|

≤|u(t)−Tu(t)|+5

6|u(t)−v(t)|+|Tv(t)−v(t)|

6|u(t)−v(t)|≤|u(t)−Tu(t)|+|Tv(t)−v(t)|

so for some positive number lwe have

∥u−v∥≤4lmin{∥u−Tu∥,∥v−Tv∥}.

All conditions of Example

(3)

with

(

r, s

)=4

hold, and integral equation

(12)

has a unique solution

u=0.

Availability of data and material

Not applicable.

https://doi.org/10.17993/3cemp.2022.110250.64-74

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors have read and approved the ﬁnal manuscript.

ACKNOWLEDGMENT

We would like to thank the referees for their comments and suggestions on the manuscript.

REFERENCES

[1]

Aghajani, A.,Abbas, M., and Roshan, J., ( 2014). “Common ﬁxed point of generalized weak

contractive mappings in partially ordered b-metric spaces.” Mathematica Slovaca, 64, 2 ), 941–960.

[2]

Ansari, A.H., (2014). “Note on

φ−ϕ

-contractive type mappings and related ﬁxed point. The 2nd

regional conference on mathematics and applications”, Payame Noor University, 377–380.

[3]

Asadi, M.,Moeini, B.,Mukheimer, A., and Hassen, A., (2019). Complex valued

–metric

spaces and related ﬁxed point results via complex

-class function. Journal of Inequalities and Special

Functions, 10, 1, 101–110.

[4]

Bakhtin, I. A., (1989). The contraction mapping principle in almost metric space. Funct. Anal.

Gos. Ped. Inst. Unianowsk, 30, 26–37.

[5]

Boriceanu, M.,Bota, M., and Petrusel, A., (2010). “Mutivalued fractals in

-metric spaces”.

Cent. Eur. J. Math., 367–377.

[6]

Choudhury, B. S., (2005). “A common unique ﬁxed point result in metric spaces involving

generalised altering distances.” Mathematical Communications, 10, 2, 105–110.

[7]

Eshraghisamani, M.,Vaezpour,S.M., and Asadi, M., (2-18). New ﬁxed point results with

αqsp

–

admissible contractions on

–Branciari metric spaces. Journal of Inequalities and Special Functions,

9, 3, 38–46.

[8]

Eshraghisamani, M.,Vaezpour, S.M., and Asadi, M., (2017). New ﬁxed point results on

Branciari metric spaces. Journal of Mathematical Analysis, 8, 6, 132–141.

[9] B. Fisher, (1978). “On a theorem of Khan”. Riv. Math. Univ. Parma. 4, 35–137.

[10]

Faraji, H.,Savic, D., and Radenovic, S., (2019). “Fixed Point Theorems for Geraghty Con-

traction Type Mappings in b-Metric Spaces and Applications”. Axioms, 8, 484–490.

[11]

Ghezellou, S.,Azhini,M., and Asadi, M., (2021). Best proximity point theorems by

k, c

and

mt types in b–metric spaces with an application. Int. J. Nonlinear Anal. Appl., 12, 2, 1317-1329.

[12]

Ali, J., and Imdad, M., (2008). An implicit function implies several contraction conditions.

Sarajevo J. Math. 4, 2, 269–285.

[13]

Ali, J. and Imdad, M., (2009). Common ﬁxed point theorems in symmetric spaces employing

a new implicit function and common property (E.A), Bull. Belg. Math. Soc. Simon Stevin 16(3),

421–433.

[14]

Khan, M.S. ,Swaleh, M. and Sessa, S., (1984). “Fixed point theorems by altering distances

between the points". Bull. Austral. Math. Soc. 30, 1, 1–9.

https://doi.org/10.17993/3cemp.2022.110250.64-74

3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376

Ed. 50 Vol. 11 N.º 2 August - December 2022

[15]

Kirk, W. and Shahzad, N., (2014). “Fixed point Theory in Distance Spaces”. Springer: Berlin,

Germany.

[16]

Monfared, H.,Azhini, M., and Asadi, M., (2017).

-class and

(

ψ, φ

)-contractions on

M–metric spaces. Int. J. Nonlinear Anal. Appl., 8, 1, 209-224.

[17]

H. Piri,S. Rahrovi and P. Kumam, (2017). “Generalization of Khan ﬁxed point theorem”. J.

Math. Computer Sci., 17, 76–83.

[18]

Younis, M.,Singh, D.,Asadi, M., and Joshi, V., (2019).. Results on contractions of Reich

type in graphical b–metric spaces with applications. FILOMAT, 33, 17, 5723–5735.

[19]

Kadelburg, Z.,Mohammad Imdad, and Sunny Chauhanm, (1999). Uniﬁed common ﬁxed

point theorems under weak reciprocal continuity or without continuity. Demonstratio Math., 32, 1,

157-163.

https://doi.org/10.17993/3cemp.2022.110250.64-74

3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376

Ed. 50 Vol. 11 N.º 2 August - December 2022