EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 3, 2023, 1717-1730
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

(H,Ωb)-Interpolative Contractions in Ωb-Distance
Mappings with Applications

Tariq Qawasmeh

Department of Mathematics, Faculty of Science and Information Technology,
Jadara University, Jordan

Abstract. Interpolative Kannan contractions are a refinement of Kannan contraction, which is
considered as one of the significant notions in fixed point theory. Gb-metric spaces is considered
as a generalized concept of both concepts b-metric and G-metric spaces therefore, the significant
fixed and common fixed point results of the contraction based on this concept is generalized results
for both concepts. The purpose of this manuscript, is to take advantage to interpolative Kannan
contraction together with the notion of Ωb which equipped with Gb-metric spaces and H simulation
functions to formulate two new interpolative contractions namely, (H,Ωb)-interpolative contraction
for self mapping f and generalized (H,Ωb)-interpolative contraction for pair of self mappings
(f1, f2). We discuss new fixed and common fixed point theorems. Moreover, to demonstrate the
applicability and novelty of our theorems, we formulate numerical examples and applications to
illustrate the importance of fixed point theory in applied mathematics and other sciences.

2020 Mathematics Subject Classifications: 54H25, 47H10, 34B15

Key Words and Phrases: Ωb distance mappings, Interpolative Kannan contractions,H-simulation
functions, Gb-metric spaces

1. Introduction and Mathematical Preliminaries

The study of fixed point theory has gained increasing importance and interest in pure
and applied mathematics [8]–[17] ever since Banach came up with his result (Banach
contraction principle) [4] which is considered to be one of the most important results in
mathematics as well as other sciences. Since then, many mathematicians refined the result
of Banach in two directions; some by replacing the frame of distance space such as b, G-
metric spaces, modified ω, Ω-distance mappings (see [5]–[16]), and the others refined the
contraction condition (for example see [18]–[15]).
Kannan contraction principle [12] is the first outstanding result after Banach contraction
principle, and it is important to mention that this contraction characterizes the metric
completeness. Many mathematicians improved this contraction; an interesting example of
this improving is interpolative Kannan contractions [10, 13]. Since then, many significant

DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4819

Email addresses: ta.qawasmeh@jadara.edu.jo, jorqaw@yahoo.com (T. Qawasmeh)

https://www.ejpam.com 1717 © 2023 EJPAM All rights reserved.



T. Qawasmeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1717-1730 1718

contractions formulated based on interpolative contractions which utilized in the literature
to investigate significant fixed and common fixed point results such as Debnath et.al. [7, 9].
In this study, our purpose is to formulate two significant interpolative contractions in the
framework of Ωb distance mappings which equipped withGb-metric spaces where nontrivial
generalisations are possible and as such, application of the results in relevant fields becomes
feasible and easier.

Definition 1. [10, 13] Suppose (C, d) is a metric space and f, g are two self mappings on
C and λ ∈ [0, 1), α, β ∈ (0, 1) where β + α < 1. Then

1. We call f a (λ, α, β)-interpolative Kannan contraction if

d(fc1, fc2) ≤ λd(c1, fc1)
αd(c2, fc2)

β, (1)

∀ c1, c2 ∈ C with fc1 ̸= c1 and fc2 ̸= c2.

2. We call the pair (f, g) a (λ, α, β)-interpolative Kannan contraction pair if

d(fc1, gc2) ≤ λd(c1, fc2)
αd(c2, gc2)

β, (2)

∀ c1, c2 ∈ C with fc1 ̸= c1 and gc2 ̸= c2.

The concept of Gb space has been formulated by a pioneer mathematician, Aghajani et
al. [2], providing a generalization of the standard concepts of G-metric space which are
formulated by Mustafa and Sims [14] and b-metric space , which is formulated by Bakhtin
[3] as follows:

Definition 2. [2] Let C be a non-empty set and b ∈ [1,+∞). Assume that the function
Gb : C × C × C → [0,+∞) fulfills the following conditions:

1. Gb(c, c
′
, c

′′
) = 0 if and only if c = c

′
= c

′′
;

2. Gb(c, c, c
′
) ≥ 0 for all c, c

′ ∈ C with c ̸= c
′
;

3. Gb(c, c
′
, c

′
) ≤ Gb(c, c

′
, c

′′
) for all c, c

′
, c

′′ ∈ C with c
′ ̸= c

′′
;

4. Gb(c, c
′
, c

′′
) = Gb(p{c, c

′
, c

′′}) where p is a permutation of c, c
′
, c

′′
;

5. Gb(c, c
′
, c

′′
) ≤ b[Gb(c, a, a) +Gb(a, c

′
, c

′′
)] for all c, c

′
, c

′′
, a ∈ C.

Then Gb is called Gb-metric on C and the pair (C, Gb) is called Gb-metric spaces.

Example 1. [2] If (C, G) is G-metric space and p ∈ (1,+∞). Define Gb : C × C × C →
[0,+∞) via Gb(c1, c2, c3) = (G(c1, c2, c3))

p. Then Gb is Gb-metric space with the base
b = 2p−1.

Henceforth, (C, Gb) refers to Gb-metric spaces on the set C.
In the sub-sequence, C refers to non empty set and Λf refers to the set of all fixed points
of f in C.
The concepts of Gb-completeness and Gb-convergence are as below:



T. Qawasmeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1717-1730 1719

Definition 3. [2] Assume (cn) be a sequence in (C, Gb). Then the sequence (cn) is a:

1. Gb-Cauchy sequence if ∀ϵ > 0 there is N ∈ N such that ∀n,m, l ≥ N,
G(cn, cm, cl) < ϵ;

2. Gb-convergent sequence to c if ∀ϵ > 0 there is N ∈ N such that ∀n,m ≥ N,
G(c, cn, cm) < ϵ;

3. Gb-complete if ∀ Gb-Cauchy sequence, then Gb is convergent.

Remark 1. A sequence (cn) in (C, Gb) is Gb-convergent sequence if one of the following
conditions is true:

(1) Gb(cnc, c) → 0 as n → +∞;

(2) Gb(cn, cn, c) → 0 as n → +∞.

The concept Ωb distance mappings (Generalized Ω distance mappings) was introduced
by Abodayeh et.al. [1] and they utilized this concept to unify some fixed point results in
the literature.

Definition 4. [1] An Ωb-distance mappings on (C, Gb) is a function Ωb : C × C × C →
[0,+∞) fulfill:

1. Ωb(c, c
′
, c

′′
) ≤ b[Ωb(c, a, a) + Ωb(a, c

′
, c

′′
)] for all c, c

′
, c

′′
, a ∈ C, b ∈ [0,+∞);

2. ∀c, c′ ∈ C, Ωb(c, c
′
, .),Ωb(c, ., c

′
) : C → C are lower semi-continuous;

3. ∀ϵ > 0 there is an α > 0, if Ωb(c, a, a) ≤ α and Ωb(a, c
′
, c

′′
) ≤ α,

then Gb(c, c
′
, c

′′
) ≤ ϵ, ∀ c, c

′
, c

′′ ∈ C.
Definition 5. If Ωb distance mappings is equipped with (C, Gb), then we call C bounded
w.r.t. Ωb if there exists L ≥ 1 with Ωb(c1, c2, c3) ≤ L for all c1, c2, c3 ∈ C.
The concept of H-simulation functions which formulated by Bataihah et.al in 2020 is as
belows:

Definition 6. [6] A set of functions {h : [1,+∞)× [1,+∞) → R} is called H-simulation
functions if

h(c, c
′
) ≤ c

′

c
∀c, c′ ∈ [1,+∞). (3)

Remark 2. [6] If h ∈ H and (cn), (c
′
n) are sequences in [1,+∞) with

1 ≤ lim
n→+∞

c
′
n < lim

n→+∞
cn, then

lim sup
n→+∞

h(cn, c
′
n) < 1. (4)

Definition 7. [6, 11] The class of functions: {θ : [0,+∞) → [1,+∞)}, θ is continuous
and none decreasing functions fulfill the condition:
∀(cn) a sequence in [0,+∞), lim

n→+∞
θ(cn) = 1 if and only if lim

n→+∞
cn = 0.

Is said to be Θ class

Remark 3. [11] If θ ∈ Θ, then θ−1 ({1})=0.



T. Qawasmeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1717-1730 1720

2. Main Results

We start our main results with the following concepts and definitions

Definition 8. Suppose (C, Gb) is equipped with Ωb-distance mappings. A mapping f : C →
C is said to be (H,Ωb)-interpolative contraction if there are b ∈ [1,+∞), λi ∈ (0, 1) with
i ∈ {1, 2, 3} and λ2 + λ3 < 1 , θ ∈ Θ and h ∈ H such that ∀ c1, c2, c3 ∈ C we have:

1 ≤ h

(
θbΩb(fc1, f

2c1, fc2), θλ1Γ(c1, c2, c3)

)
. (5)

Where

Γ(c1, c2, c3) = max

{
Ωb(c1, fc1, c2), [Ωb(c1, fc1, fc1)]

λ2 [Ωb(c2, fc2, fc2)]
λ3

}
.

Lemma 1. Suppose the self function f : C → C fulfills the conditions of (H,Ωb)-interpolative
contraction.Then

1. Γ(c1, c2, c3) > 0 =⇒ Ωb(fc1, f
2c1, fc2) ≤

λ1

b
Γ(c1, c2, c3);

2. Γ(c1, c2, c3) = 0 =⇒ Ωb(fc1, f
2c1, fc2) = 0.

Proof. (1) If Γ(c1, c2, c3) > 0, then

1 ≤ H(θbΩb(fc1, f
2c1, fc2), θλ1Γ(c1, c2, c3))

≤ θλ1Γ(c1, c2, c3)

θbΩb(fc1, f2c1, fc2)
.

This implies that, θbΩb(fc1, f
2c1, fc2) ≤ θλ1Γ(c1, c2, c3). Due to the fact that the set Θ

is a non-decreasing function, we conclude:

Ωb(fc1, f
2c1, fc2) ≤

λ1

b
Γ(c1, c2, c3). Hence the result.

(2) If Γ(c1, c2, c3) = 0, then by utilizing condition (1), we have:

1 ≤ θbΩb(fc1, f
2c1, fc2) ≤ θλΓ(c1, c2, c3) = 1.

Thus, Ωb(fc1, f
2c1, fc2) = 0.

Lemma 2. Suppose the self function f : C → C fulfills the conditions of (H,Ωb)-interpolative
contraction. Then Λf has at most one element.

Proof. To prove that Λf has at most one element, first we claim that, Ωb(α, α, α) = 0
∀α ∈ Λf . Assume Ωb(α, α, α) > 0 for some α ∈ Λf , then by employing Lemma 1 we get:

Ωb(fα, f
2α, fα) ≤ λ1

b
Γ(α, α, α)

=
λ1

b
max{Ωb(α, fα, α), [Ωb(α, fα, fα)]

λ2 [Ωb(α, fα, fα)]
λ3}

< Ωb(α, α, α).



T. Qawasmeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1717-1730 1721

A contradiction. Hence the result.
Now assume that there is c∗, α ∈ Λf with c∗ ̸= α, assume that Ωb(c

∗, c∗, α) > 0, so by
Lemma 1 we have:

Ωb(c
∗, c∗, α) = Ωb(fc

∗, f2c∗, fα) ≤ λ1

b
Γ(c∗, c∗, α)

=
λ1

b
max{Ωb(c

∗, fc∗, α), [Ωb(c
∗, c∗, c∗)]λ2 [Ωb(α, α, α)]

λ3}
< Ωb(c

∗, c∗, α).

A contradiction. Therefore, Ωb(c
∗, c∗, α) = 0 and by utilizing the definition of of Ωb

(condition (3)) and since Ωb(c
∗, c∗, c∗) = 0, we conclude that Gb(c

∗, c∗, α) = 0 therefore,
c∗ = α.

For an arbitrary point c0 ∈ C the Picard sequence is defined by iterating f : C → C
where cn+1 = f(cn) = fn(c0). Henceforth, we mean by the sequence cn the Picard sequence
unless otherwise stated.

Lemma 3. Suppose the self function f : C → C fulfills the conditions of (H,Ωb)-interpolative
contraction and suppose that for some k ∈ N we have Ωb(ck−1, ck, ck) = 0. Then,
Λf = {ck}

Proof. Note that

Γ(ck−1, ck, ck) =
λ1

b
max

{
Ωb(ck−1, ck, ck), [Ωb(ck−1, ck, ck)]

λ2 [Ωb(ck, ck+1, ck+1)]
λ3

}
= 0.

So, by Lemma 1, we get that Ωb(ck, ck+1, ck+1) = Ωb(ck−1, ck, ck) = 0. In a similar manner,
we can verify that Ωb(ck+1, ck+2, ck+2) = 0. By utilizing the definition of Ωb, we conclude
that Gb(ck−1, ck+1, ck+1) = 0 and so ck−1 = ck+1. In a typical way, we can prove that
ck = ck+2.
Now, by employing the triangle inequality of Ωb, we get

Ωb(ck, ck, ck) ≤ b[Ωb(ck, ck+1, ck+1) + Ωb(ck+1, ck, ck)]
= b[Ωb(ck, ck+1, ck+1) + Ωb(ck+1, ck+2, ck+2)]
= 0.

(6)

From inequality (6) and Ωb(ck, ck+1, ck+1) = 0, we conclude that ck ∈ Λf and Lemma 2
ensures that ck is the unique element in Λf .

Theorem 1. Suppose (C, Gb) is Gb-complete equipped with Ωb distance mappings with the
base b ∈ [1,+∞) and C is bounded w.r.t. Ωb. Suppose there are λi ∈ (0, 1), i ∈ {1, 2, 3}
with λ2+λ3 < 1, θ ∈ Θ, h ∈ H such that the mapping f : C → C is a (H,Ωb)-interpolative
contraction if one of the following conditions is fulfilled:
1. The self mapping f is a continuous;
2. For all c∗ ∈ C if fc∗ ̸= c∗, then 0 < inf{Ωb(c, fc, c

∗) : c ∈ C},
then Λf has only one element.



T. Qawasmeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1717-1730 1722

Proof. Let c0 ∈ C and start by the Picard sequence (cn). Without lose of generality,
we may assume that ∀n ∈ N, we have Ωb(cn, cn+1, cn+1) > 0. So, by Lemma 1, we have

Ωb(cn, cn+1, cn+1) ≤
λ1

b
max

{
Ωb(cn−1, cn, cn), [Ωb(cn−1, cn, cn)]

λ2 [Ωb(cn, cn+1, cn+1)]
λ3

}
.

(7)

If max

{
Ωb(cn−1, cn, cn), [Ωb(cn−1, cn, cn)]

λ2 [Ωb(cn, cn+1, cn+1)]
λ3

}
= Ωb(cn−1, cn, cn).

Therefore, we get

Ωb(cn, cn+1, cn+1) ≤
λ1

b
Ωb(cn−1, cn, cn); (8)

else, we have

[Ωb(cn, cn+1, cn+1)]
1−λ3 ≤ λ1

b
[Ωb(cn−1, cn, cn)]

λ2 <
λ1

b
[Ωb(cn−1, cn, cn)]

1−λ3 . (9)

From the inequalities (8) and (9), we conclude

Ωb(cn, cn+1, cn+1) ≤ λ1

b
Ωb(cn−1, cn, cn)

...

≤ (
λ1

b
)nΩb(c0, c1, c1).

(10)

Then there is L ≥ 1 such that

Ωb(cn, cn+1, cn+1) ≤ (
λ1

b
)nL. (11)

To show that the iterative sequence (cn) is Gb-Cauchy, first we prove that ∀ m, l ∈ N with
m ≤ l we have:

Ωb(cm−1, cm, cl) ≤ (
λ1

b
)m−1L. (12)

Now,

Ωb(cm−1, cm, cl) ≤
λ1

b
max

{
Ωb(cm−2, cm−1, cl−1), [Ωb(cm−2, cm−1, cm−1)]

λ2 [Ωb(cl−1, cl, cl)]
λ3

}
.

(13)
Assume that l = m+ t for some t ∈ N. Then

Ωb(cl−1, cl, cl) ≤ λ1

b
max

{
Ωb(cl−2, cl−1, cl−1), [Ωb(cl−2, cl−1, cl−1)]

λ2 [Ωb(cl−1, cl, cl)]
λ3

}
=

λ1

b
Ωb(cl−2, cl−1, cl−1)

≤ (
λ1

b
)tΩb(cm−1, cm, cm).

(14)



T. Qawasmeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1717-1730 1723

Now,

Ωb(cm−1, cm, cl) ≤ λ1

b
max

{
Ωb(cm−2, cm−1, cl−1), [Ωb(cm−2, cm−1, cm−1)]

λ2+λ3

}
≤ λ1

b
max

{
λ1

b
max{Ωb(cm−3, cm−2, cl−2), [Ωb(cm−3, cm−2, cm−2)]

λ2+λ3},

[Ωb(cm−2, cm−1, cm−1)]
λ2+λ3

}
≤ (

λ1

b
)2
{
Ωb(cm−3, cm−2, cl−2), [Ωb(cm−3, cm−2, cm−2)]

λ2+λ3

}
...

≤ (
λ1

b
)m−1

{
Ωb(c0, c1, ct), [Ωb(c0, c1, c1)]

λ2+λ3

}
≤ (

λ1

b
)m−1L.

(15)
Now, by employing inequalities (11), (12) and condition (1) of the the definition of Ωb

∀n < m ≤ l, we get:

Ωb(cn, cm, cl) ≤ bΩb(cn, cn+1, cn+1) + bΩb(cn+1, cm, cl)
≤ bΩb(cn, cn+1, cn+1) + b2Ωb(cn+1, cn+2, cn+2) + b2Ωb(cn+2, cm, cl)
...
≤ bΩb(cn, cn+1, cn+1) + b2Ωb(cn+1, cn+2, cn+2) + · · ·
+bm−n−1Ωb(cm−2, cm−1, cm−1) + bm−n−1Ωb(cm−1, cm, cl)

≤ b(
λ1

b
)nL+ b2(

λ1

b
)n+1L+ · · ·+ bm−n−1(

λ1

b
)m−1L

= bL(
λ1

b
)n
[
1 + λ1 + λ2

1 + · · ·+ λm−n−1
1

]

= bL(
1− λm−n

1

1− λ1
)(
λ1

b
)n.

(16)
By taking the limit as n → +∞ in above inequality, we find out that (cn) is a Gb-Cauchy
sequence, and since (C, Gb) is Gb- complete, then there is c∗ ∈ C s.t. the sequence (cn) is
Gb-convergent to c∗. If f is any continuous mapping, then fc∗ = c∗. Else, by utilizing the
lower semi continuity of Ωb, we obtain:

Ωb(cn, cm, c∗) ≤ lim
t→+∞

Ωb(cn, cm, ct) < ϵ for all n,m ≥ N ∀ ϵ > 0. (17)

Suppose that m = n+ 1. Then Ωb(cn, cn+1, c
∗) ≤ lim

t→+∞
Ωb(cn, cn+1, ct) < ϵ ∀n ≥ N.

If fc∗ ̸= c∗, we obtain:

0 < inf{Ωb(c, fc, c
∗) : c ∈ C} ≤ inf{Ωb(cn, cn+1, c

∗) : n ∈ N} < ϵ ∀ ϵ > 0, (18)



T. Qawasmeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1717-1730 1724

a contradiction. Hence, c∗ ∈ Λf , the uniqueness follows from Lemma 2. This is complete
the proof.

In the next two examples we consider the following:

Define h : [1,+∞) × [1,+∞) → [0,+∞), θ : [0,+∞) → [1,+∞) via h(c1, c2) =
c2
c1
,

θ(ω) = eω, ∀ω ∈ C respectively, then h ∈ H and θ ∈ Θ.
Also, define: Gb : C ×C ×C → [0,+∞) by Gb(c1, c2, c3) = (|c1− c2|+ |c2− c3|+ |c1− c3|)2,
then , Gb is a complete with the base b = 2.
Moreover, define Ωb : C × C × C → [0,+∞) by Ωb(c1, c2, c3) = (|c1 − c2|+ |c1 − c3|)2, Ωb is
a generalized Ω-distance mapping equipped with Gb.

Example 2. Suppose C = {0, 1, ..., 10}, define mapping f : C → C via :

fc =


0, c ∈ {0, 1, 2};
1, c ∈ {3, 4, 5};
2, c ∈ {6, 7, ..., 10}.

Then Λf has only one element.
To prove this, we need to show that ∀ c1, c2 ∈ C, we have

1 ≤ h(θbΩb(fc1, f
2c1, fc2), θλ1Γ(c1, c2, c3)).

First it is not hard to prove

Ωb(fc1, f
2c1, fc2) ≤ 0.45max

{
Ωb(c1, fc1, c2), [Ωb(c1, fc1, fc1)]

0.45[Ωb(c2, fc2, fc2)]
0.45

}
.

Now,

Ωb(fc1, f
2c1, fc2) ≤

λ1

b
max

{
Ωb(c1, fc1, c2), [Ωb(c1, fc1, fc1)]

λ2 [Ωb(c2, fc2, fc2)]
λ3

}
⇐⇒ θbΩb(fc1, f

2c1, fc2) ≤ θλ1Γ(c1, c2, c3))

⇐⇒ 1 ≤ H(θbΩb(fc1, f
2c1, fc2), θλ1Γ(c1, c2, c3)).

Consequently, f satisfy all conditions of (H,Ωb)-interpolative contraction. Theorem 1
confirms that Λf has only one element.

Example 3. Consider the following mapping

f(c) =
1− cm

B + cm
where m ∈ N− {1} and B ≥

√
2m.



T. Qawasmeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1717-1730 1725

Then Λf has only one element on [0, 1].
To prove this, let C = [0, 1] for all c1, c2, c3 ∈ C, assume fc = s. Then

Ωb(fc1, f
2c1, fc2) =

[∣∣∣∣ 1− cm1
B + cm1

− 1− sm

B + sm

∣∣∣∣+ ∣∣∣∣ 1− cm1
B + cm1

− 1− cm2
B + cm2

∣∣∣∣]2

=

[
1

(B + cm1 )(B + sm)

∣∣∣∣(1− cm1 )(B + sm)− (1− sm)(B + cm1 )

∣∣∣∣
+

1

(B + cm1 )(B + cm2 )

∣∣∣∣(1− cm1 )(B + cm2 )− (1− cm2 )(B + cm1 )

∣∣∣∣]2

≤ (B − 1)2

B4

[
|cm1 − sm|+ |cm1 − cm2 |

]2

=
(B − 1)2m2

B4

[
|c1 − s|+ |c1 − c2|

]2

≤ (B − 1)

2B2

[
|c1 − fc1|+ |c1 − c2|

]2

=
λ1

b
Ωb(c1, fc1, c2).

Notice that λ1 = (
B − 1

B
)2 and the base b = 2.

Now,
bΩb(fc1, f

2c1, fc2) ≤ λ1Ωb(c1, fc1, c2) ≤ λ1Γ(c1, c2, c3)

⇐⇒ ebΩb(fc1,f
2c1,fc2) ≤ eλ1Γ(c1,c2,c3)

⇐⇒ 1 ≤ eλ1Γ(c1,c2,c3)

ebΩb(fc1,f2c1,fc2)

⇐⇒ 1 ≤ H(θbΩb(fc1, f
2c1, fc2), θλ1Γ(c1, c2, c3).

Consequently, f satisfy all conditions of (H,Ωb)-interpolative contraction. Theorem 1
confirms that Λf has only one element.

Definition 9. Suppose that (C, Gb) is equipped with Ωb-distance mappings and f1, f2 are
two self mapping on C. We called the pair (f1, f2) is a generalized (H,Ωb)-interpolative
contraction if there exist b ∈ [1,+∞), λi ∈ (0, 1) with i ∈ {1, 2, 3} and λ2 + λ3 < 1,
θ ∈ Θ and h ∈ H s.t. ∀ c1, c2, c3 ∈ C we have:

1 ≤ h

(
θbΩb(f1c1, f2(f1c1), f2c2), θλ1Γ1(c1, c2, c3)

)
; (19)



T. Qawasmeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1717-1730 1726

and

1 ≤ h

(
θbΩb(f2c1, f1(f2c1), f1c2), θλ1Γ2(c1, c2, c3)

)
. (20)

Where

Γ1(c1, c2, c3) = max

{
Ωb(c1, f2c1, c2), [Ωb(c1, f1c1, f1c1)]

λ2 [Ωb(c2, f2c2, f2c2)]
λ3

}
;

and

Γ2(c1, c2, c3) = max

{
Ωb(c1, f1c1, c2), [Ωb(c1, f2c1, f2c1)]

λ2 [Ωb(c2, f1c2, f1c2)]
λ3

}
.

Theorem 2. Suppose (C, Gb) is Gb-complete equipped with Ωb distance mappings with the
base b ∈ [1,+∞) and C is bounded w.r.t. Ωb. Suppose there are λi ∈ (0, 1), i ∈ {1, 2, 3}
with λ2+λ3 < 1, θ ∈ Θ, h ∈ H s.t. the pair of self mappings f1, f2 : C → C is a generalized
(H,Ωb)-interpolative contraction if one of the following fulfilled:
1. If the mappings f1, f2 are continuous;
2. If one of the self mappings is continuous and for all c∗ ∈ C if f∗c∗ ̸= c∗,
then 0 < inf{Ωb(c, f

∗c, c∗) : c ∈ C}, where f∗ refers to non-continuous function f1 or f2.
then Λf has only one element.

Proof. We start our proof our by setting a constructive sequence (cn) ∈ C by iterating
c2n+1 = f1c2n and c2n+2 = f2c2n+1 for n ∈ N for some arbitrary element c0 ∈ C. So we
have
Ωb(c2n+1, c2n+2, c2n+2) = Ωb(f1c2n, f2(f1c2n), f2c2n+1), and so

1 ≤ H

(
θbΩb(c2n+1, c2n+2, c2n+2), θλ1Γ(c2n, c2n, c2n+1)

)

≤
θλ1max

{
Ωb(c2n, c2n+1, c2n+1), [Ωb(c2n, c2n+1, c2n+1)]

λ2 [Ωb(c2n+1, c2n+2, c2n+2)]
λ3

}
θbΩb(c2n+1, c2n+2, c2n+2)

.

(21)
Therefore,

Ωb(c2n+1, c2n+2, c2n+2) ≤ λ1

b
max

{
Ωb(c2n, c2n+1, c2n+1),

[Ωb(c2n, c2n+1, c2n+1)]
λ2 [Ωb(c2n+1, c2n+2, c2n+2)]

λ3

}
.

By employing the inequalities (8) and (9), we conclude that

Ωb(c2n+1, c2n+2, c2n+2) ≤
λ1

b
Ωb(c2n, c2n+1, c2n+1). (22)



T. Qawasmeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1717-1730 1727

By utilizing typical way, we can easily show that

Ωb(c2n+2, c2n+3, c2n+3) ≤
λ1

b
Ωb(c2n+1, c2n+2, c2n+2). (23)

Hence, we get

Ωb(cn+1, cn+2, cn+2) ≤
λ1

b
Ωb(cn, cn+1, cn+1). (24)

The completion of the proof of this Theorem is identical to the Theorem 1, and this is
complete the proof.

3. Application

Throughout this application, we will emphasize the significant idea that the solution
of a fixed point equation (uniqueness and existence) under certain conditions is often
comparable to that of other equations.
Consider the following equation:

cm+1 + cm +Bc− 1, where B ≥
√
2 m, m ∈ N− {1}, (25)

has a unique solution in the unit interval [0, 1].
To prove this, it is typical to prove that the following self mapping f has a unique fixed
point in [0, 1].

f(c) =
1− cm

B + cm
, B ≥

√
2 m, m ∈ N− {1}.

Example 3 confirms that the self mapping f has a unique fixed point and hence, the
Equation (25) has a unique solution.
Next, we discuss an application on Theorem 1. We employ Theorem 1 to prove the
uniqueness and existence of a solution for Volterra type integral equation:

η(t) = η0 +

∫ t

t0

H(r, η(r))dr. (26)

Suppose that ∥.∥∞ is the superior norm on C[0, 1] which is defined by ∥v∥∞ = sup
t∈[0,1]

v(t).

In this application, we consider that C = C[0, 1] and Gb,Ωb as follows:

Gb(u, v, w) = (∥u−v∥∞+∥v−w∥∞+∥u−w∥∞)2, Ωb(u, v, w) = (∥u−v∥∞+∥u−w∥∞)2.
(27)

Next, we have the following theorem:

Theorem 3. Suppose that H : [0, 1] × R → R is a continuous function on [0, 1] × R and
t0 is the interior point in [0, 1] and suppose that α0 > 0 such that the function H fulfills
the following:

|H(t, u)−H(t, v)| ≤ α0|u− v| for all u, v ∈ R and for all t ∈ [0, 1]. (28)

Then the integral equation fη(t) = η0 +
∫ t
t0
H(r, η(r))dr has a unique solution.



T. Qawasmeh / Eur. J. Pure Appl. Math, 16 (3) (2023), 1717-1730 1728

Proof. Let ϵ > 0 with ϵ <

√
λ1

bα2
0

. Define the self mapping f : C[0, 1] → C[0, 1] via

fη(t) = η0 +

∫ t

t0

H(r, η(r))dr. (29)

Then we show that f satisfies the condition (8) on the interval C0 = [t0, t0 + ϵ].
It suffices to show that:

Ωb(fu, f
2u, fv) ≤ λ1

b
Ωb(u, fu, v). (30)

Now, for all u, v ∈ C[0, 1], we obtain:

∥fu− fv∥∞ = sup
t∈C0

|fu(t)− fv(t)|

= sup
t∈C0

|
∫ t
t0
(H(r, u(r))−H(r, v(r)))dr|

≤ sup
t∈C0

∫ t
t0
|(H(r, u(r))−H(r, v(r)))dr|

≤ sup
t∈C0

α0|u(t)− v(t)|
∫ t
t0
dr

= α0∥u− v∥∞(t− t0)
= ϵα0∥u− v∥∞.

Therefore,

(∥fu− f2u∥∞ + ∥fu− fv∥∞)2 = (sup
t∈C0

|fu(t)− f2u(t)|+ sup
t∈C0

|fu(t)− fv(t)|)2

=

(
sup
t∈C0

|
∫ t
t0
(H(r, u(r))−H(r, fu(r))dr|

+sup
t∈C0

|
∫ t
t0
(H(r, u(r))−H(r, v(r))dr|

)2

≤ (ϵα0)
2(∥u− fu∥∞ + ∥u− v∥∞)2.

Now, set
λ1

b
= (ϵα0)

2, we get the desire result.

4. Conclusion

In this manuscript, we formulated two significant interpolative contractions namely,
(H,Ωb)-interpolative contraction for self mapping f and generalized (H,Ωb)-interpolative
contraction for pair of self mappings (f1, f2). By employing these contractions we unify new
fixed and common fixed results. We formulated some numerical examples and applications
to show the novelty of our results; one of these applications based on the significant idea



REFERENCES 1729

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equation. we utilized this idea to prove that this equation not only has solution as the
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improved by utilizing the concept of extended Gb-metric spaces.

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