International Journal of Analysis and Applications

Volume 19, Number 2 (2021), 280-287

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-19-2021-280

FIXED POINT RESULTS FOR ω-INTERPOLATIVE CHATTERJEA TYPE

CONTRACTION IN QUASI-PARTIAL B-METRIC SPACE

PRAGATI GAUTAM1,∗, SWAPNIL VERMA1, MANUEL DE LA SEN2, SEJAL

SUNDRIYAL1

1Department of Mathematics, Kamala Nehru College, University of Delhi, August Kranti Marg, New Delhi

110049, India

2Institute of Research and Development of Processes, University of Basque Country, Campus of

Leioa(Bizkaia)-Aptdo, 644-Bilbao, Bilbao, 48080, Spain

∗Corresponding author: pgautam@knc.du.ac.in

Abstract. The purpose of this paper is to revisit Chatterjea type contraction and determine some fixed

point results for interpolative Chatterjea type contraction mapping in the setting of quasi-partial b-metric

space using the concept of ω-admissibility introduced by Popescu. Also we present some useful examples to

elucidate relevance of the concept.

1. Introduction

In the diversified field of non-linear analysis, Banach [1] contraction principle holds a significant position.

The fixed point theorems are used to demonstrate the uniqueness of a solution of differential equations,

Fredholm integral equations and Picard theorem etc. Forging ahead Banach’s approach, many celebrated

authors [2–7] introduced distinctive concepts. In the year 1972, Chatterjea [8] inaugurated his contraction

defined as

Received February 7th, 2021; accepted March 1st, 2021; published March 17th, 2021.

2010 Mathematics Subject Classification. 46T99, 47H10, 54H25.

Key words and phrases. quasi-partial b-metric space; fixed point; ω admissible; interpolation; Chatterjea contraction.

©2021 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

280

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-19-2021-280


Int. J. Anal. Appl. 19 (2) (2021) 281

Let (X, d) be a complete metric space. A self mapping H : X → X is called Chatterjea type contraction if

d(Hσ, Hθ) ≤ δ[d(σ, Hθ) + d(θ, Hσ)]

for all σ, θ ∈ X, where δ ∈ (0, 1/2). Then he interestingly proved that it has a unique fixed point in

complete metric space. Additionally this result was proved by Mishra et al. [9] in complete quasi-partial

b-metric space for interpolative Chatterjea type contraction, which can be designated as,

Let (X, qpb) be a complete quasi-partial b-metric space. A self mapping H : X → X is called interpola-

tive Chatterjea type contraction if there exists δ ∈ [0, 1
s
),ρ ∈ (0, 1) such that

qpb(Hσ, Hθ) ≤ δ[qpb(σ, Hθ)]ρ
[

1

s2
qpb(θ, Hσ)

]1−ρ
for all σ, θ ∈ X.

Afterwards as a modification in the concept of α-admissible maps, Popescu [10] introduced ω-orbital admis-

sible maps.

In this research field, several authors [13–15] have made valuable contributions. In this paper, we commence

the concept of ω-interpolative Chatterjea contraction in quasi-partial b-metric space and deliver relevant

examples.

2. Preliminaries

Definition 2.1 [10]: Let ω : X × X → [0, ∞) be a mapping and X 6= φ. A self mapping H : X → X is

called ω-admissible if,

ω(σ, Hσ) ≥ 1 =⇒ ω(Hσ, H2σ) ≥ 1

for all σ ∈ X.

In order to ignore the continuity of contractive mappings, we often consider the following condition.

(M) If we take a sequence θn in X such that ω(θn, θn+1) ≥ 1, for all n. Also as n →∞, θn → θ ∈ X, then

from θn there exists θn(k) such that for all k, ω(θn(k), θ) ≥ 1.

Definition 2.2 [11] : A quasi-partial metric space on a set non-empty set X is a function qp: X×X → [0, ∞)

that satisfies the following properties :

[QP1] If qp(σ, σ) = qp(σ, θ) = qp(θ, θ) then σ = θ

[QP2] qp(σ, σ) ≤ qp(σ, θ)

[QP3] qp(σ, σ) ≤ qp(θ, σ)



Int. J. Anal. Appl. 19 (2) (2021) 282

[QP4] qp(σ, θ) + qp(δ, δ) ≤ qp(σ, δ) + qp(δ, θ) for all σ, θ, δ ∈ X.

Definition 2.3 [12] : A quasi-partial b-metric space on a set X 6= φ is a function qpb : X × X → [0, ∞)

such that for a real number s ≥ 1 satisfies the following properties :

[QPb1] If qpb(σ, σ) = qpb(θ, θ) = qpb(σ, θ) then σ = θ

[QPb2] qpb(σ, σ) ≤ qpb(σ, θ)

[QPb3] qpb(σ, σ) ≤ qpb(θ, σ)

[QPb4] qpb(σ, θ) ≤ s[qpb(σ, δ) + qpb(δ, θ)] −qpb(δ, δ) for all σ, θ, δ ∈ X.

Here, we present an example to show the usability of the concept.

Example 2.1 : Let X=
[
0, π

2k

]
. Define qpb(σ, θ) = sink|σ2 −θ2|, where k ≥ 1

Here, qpb(σ, σ) = sink|σ2 −σ2| = 0 = qpb(θ, θ), qpb(σ, θ) = sink|σ2 −θ2|

then 0 = sink|σ2 −θ2| =⇒ σ = θ i.e [QPb1] satisfied.

qpb(σ, σ) = 0 ≤ qpb(σ, θ) as 0 ≤ sink|σ2 −θ2| i.e [QPb2] satisfied.

qpb(σ, σ) = 0 ≤ qpb(θ, σ) as 0 ≤ sink|θ2 −σ2| i.e [QPb3] satisfied.

As σ, θ, δ ∈ X

|σ2 −θ2| ≤
π

2k
, |σ2 −θ2| + |δ2 −θ2| ≤

π

2k

k[|σ2 − δ2| + |δ2 −θ2|] ≤
π

2
,

Since sin k is increasing.

qpb(σ, θ) + qpb(δ, δ) = sink|σ2 −θ2|

≤ sink(|σ2 −δ2| + |δ2 −θ2|) ≤ k(|σ2 − δ2| + |δ2 −θ2|)

≤ k(sink(|σ2 − δ2| + |δ2 −θ2|)) ≤ k[qpb(σ, δ) + qpb(δ, θ)]

≤ s[qpb(σ, δ) + qpb(δ, θ)]

where, s ≥ k ≥ 1 i.e. [QPb4] satisfied.

Therefore, (X,qpb) is a quasi-partial-b metric space.



Int. J. Anal. Appl. 19 (2) (2021) 283

3. Main results

Here, we introduce the concept of ω-interpolative Chatterjea type contractions in quasi-partial b-metric space.

Definition 3.1 : Let (X, qpb) be a complete quasi-partial b-metric space. A self mapping H : X → X

is called ω-interpolative Chatterjea type contraction if there exists δ ∈ [0, 1
s
), ρ ∈ (0, 1) such that

ω(σ, θ)qpb(Hσ, Hθ) ≤ δ[qpb(σ, Hθ)]ρ
[

1

s2
qpb(θ, Hσ)

]1−ρ
for all σ, θ ∈ X.

Our main result is as follows :

Theorem 3.1. Let H : X → X be an ω-admissible self mapping which forms ω-interpolative Chatterjea

type contraction on a complete quasi-partial b-metric space (X, qpb). If there exists σ0 ∈ X such that

ω(σ0, Hσ0) ≥ 1, then H has a fixed point in X.

Proof : Let σ0 ∈ (X, qpb) such that ω(σ0, Hσ0) ≥ 1.

Let us consider a sequence σn defined as σn = H
n(σ0), n ≥ 0.

Considering for some n0,

If σn0 = σn0+1, this implies that σn0 is a fixed point of H.

If σn0 6= σn0+1, for all n ≥ 0 then,

qpb(σn, Hσn) = qpb(σn, Hσn+1) > 0

Also H is ω-admissible, ω(σ1, σ2) = ω(Hσ0, Hσ1) ≥ 1.

=⇒ ω(σn, σn+1) ≥ 1, for all n ≥ 0.

Taking σ = σn and θ = σn−1, we get

qpb(σn+1, σn) ≤ ω(σn, σn+1)qpb(Hσn, Hσn−1)

≤ δ[qpb(σn, Hσn−1)]ρ
[

1

s2
qpb(σn−1, Hσn)

]1−ρ
≤ δ[qpb(σn, σn)]ρ

[
1

s2
qpb(σn−1, σn+1)

]1−ρ
≤ δ[qpb(σn+1,σn)]ρ

[
1

s2
[s [qpb(σn−1, σn) + qpb(σn, σn+1)] −qpb(σn, σn)]

]1−ρ
≤ δ[qpb(σn+1, σn)]ρ

[ s
s2

[qpb(σn−1, σn) + qpb(σn, σn+1)]
]1−ρ

(3.1) ≤ δ[qpb(σn+1, σn)]ρ
[

1

s
[qpb(σn−1, σn) + qpb(σn, σn+1)]

]1−ρ



Int. J. Anal. Appl. 19 (2) (2021) 284

If qpb(σn−1, σn) ≤ qpb(σn, σn+1) for all n ≥ 1, then

1

s
[qpb(σn−1, σn) + qpb(σn, σn+1)]

1−ρ ≤ qpb(σn, σn+1)1−ρ

that is,

1

s
[qpb(σn−1, σn) + qpb(σn, σn+1)] ≤ qpb(σn, σn+1)

but qpb(σn+1, σn) ≤ qpb(σn−1, σn), which is a contradiction as per equation 3.1.

Thus qpb(σn−1, σn) is a decreasing sequence.

Now, let limn→∞qpb(σn−1,σn) = L.

As we have, from equation 3.1

qpb(σn+1, σn) ≤ ω(σn, σn+1)qpb(Hσn, Hσn−1)

≤ δ[qpb(σn+1, σn)]ρ
[

1

s2
qpb(σn−1, σn)

]1−ρ
ω(σ, θ)qpb(σn+1, σn)

1−ρ ≤ δ
[
[

1

s2
qpb(σn−1, σn)

]1−ρ
ω(σn, σn−1)qpb(σn+1, σn) ≤ δ

1
1−β

[
[

1

s2
qpb(σn−1, σn)

]

(3.2) ω(σn, σn−1)qpb(σn+1,σn) ≤ δqpb(σn−1, σn) ≤ ϕnqpb(σ0, σ1)

So putting n →∞ in equation 3.2, we get L = 0.

Now, to show σn is Cauchy sequence.

Let n, t ∈ N.

qpb(σn, σn+t) ≤ sqpb(σn, σn+1) + s2qpb(σn+1, σn+2) + . . . + stqpb(σn+t−1, σn+t)

≤ [sδn + s2δn+1 + . . . + stδn+t−1]qpb(σ0, σ1)

(3.3) ≤ st
n+t−1∑
i=n

δiqpb(σ0, σ1) ≤ st
∞∑
i=n

δiqpb(σ0, σ1) · · ·

From equation 3.3,

lim
n→∞

qpb(σn, σn+t) = lim
m→∞, n→∞

qpb(σn+m), σn+m+t) ≤ st lim
m→∞

∞∑
i=m

lim
n→∞

δiqpb(σn, σn+1) = 0

Now, if σn 6= Hσn.

qpb(σn+1, Hη) = qpb(Hσn, Hη) ≤ δ[qpb(σn, Hη)]ρ
[

1

s2
qpb(η, Hµn)

]1−ρ
≤ δ[qpb(σn, Hη)]β[qpb(η, σn+1)]1−ρ



Int. J. Anal. Appl. 19 (2) (2021) 285

for η ∈ X.

Here, for n →∞, qpb(η, Hη) = 0.

This is a contradiction and hence Hη = η.

Corollary 3.1 Let (X,qpb) be a complete quasi-partial b-metric space whose subsets ξ1 and ξ2 are closed.

Suppose that H : ξ1 ∪ ξ2 → ξ1 ∪ ξ2 satisfies :

ω(σ, θ)qpb(Hσ, Hθ) ≤ δ[qpb(σ, Hθ)]ρ
[

1

s2
qpb(θ, Hσ)

]1−ρ
for all σ ∈ ξ1 and θ ∈ ξ2 such that σ, θ ∈ X�Fix(H), where ρ ≥ 0, s ≥ 1. If H(ξ1) ⊆ ξ2 and H(ξ2) ⊆ ξ1,

then there exists a fixed point of H in ξ1 ∩ ξ2.

Proof : In Theorem 3.1, it is enough to take, ω(σ, θ) =




1 if(ξ1 × ξ2) ∪ (ξ2 × ξ1)

0 otherwise

Corollary 3.2 Suppose (X, qpb, �) be a complete partially-ordered quasi-partial b-metric space. Let

H : X → X be the mapping such that:

ω(σ, θ)qpb(Hσ, Hθ) ≤ δ[qpb(σ, Hθ)]ρ
[

1

s2
qpb(θ, Hσ)

]1−ρ
such that σ, θ ∈ X�Fix(H) where ρ ≥ 0, s ≥ 1.

Let us assume the following :

a) H is non decreasing with respect to partial order �;

b) There exists σ0 ∈ X such that σ0 � Hσ0;

c) H is continuous on (X, qpb).

Then H has a fixed point in X .

Proof : In Theorem 3.1, it is enough to take, ω(σ,θ) =




1 if(σ � θ)or(θ � σ)

0 otherwise

Example 3.1 : Let us consider the set X = [0, 3] with quasi-partial b-metric defined as qpb(σ, θ) =

sink|σ2 −θ2| and H be a self-mapping on X which is defined as

Hσ =




5
2

ifσ ∈ [2, 3]

2
7

ifσ ∈ [0, 2]
and taking, ω(σ, θ) =




1 if(σ, θ) ∈ [2, 3]

0 otherwise

Let σ, θ ∈ X be such that σ 6= Hσ, θ 6= Hθ and ω(σ, θ) ≥ 1 σ, θ ∈ [2, 3], and we have Hσ = Hθ = 5
2
.

Therefore, Definition 3.1 holds, for σ0 = 3.

ω(3, H3) = ω(3, 5
2
) = 1, ω(σ, θ) ≥ 1 for σ, θ ∈ X. So σ, θ ∈ [2, 3] and Hσ = Hθ ∈ [2, 3].

Thus H is ω-orbital admissible as ω(Hσ, Hθ) ≥ 1.

Now we show that condition (M) holds, let us take a sequence θn in X such that ω(θn, θn+1) ≥ 1 for all n,



Int. J. Anal. Appl. 19 (2) (2021) 286

Figure 1. The fixed points of H are 5
2

and 2
7
.

then θn ⊂ [2, 3]. Also as n →∞, θn → v ∈ X, we get |θn −v|→ 0. Therefore v ∈ [2, 3], ω(θn, v) = 1.

Hereby Theorem 3.1 holds true and the fixed points of H are 5
2

and 2
7

as shown in Figure 1.

4. Conclusion

The accession of this study is to commence the proposition of Interpolative Chatterjea type contraction on

ω-admissible mapping in quasi-partial b-metric space. ω-admissibility finds it’s real world applications in

varying fields be it in classical game theory for finding behaviour in multi-player games or even infinite games

that are played on graphs. Apart from this it is also used in deciding the extensions of DLs by concrete

domains. This concept has been conceded in many researches earlier. The current research can also be

exercised effectively in all these areas of study.

Acknowledgements: All authors are grateful to the Spanish Government for Grant RTI2018-094366-

B-I00 (MCIU/AEI/FEDER, UE) and to the Basque Government for Grant IT1207-19.

Funding: No external funding has been received in this research.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

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https://doi.org/10.1007/s41478-021-00309-6.

https://doi.org/10.1007/s41478-021-00309-6

	1. Introduction
	2. Preliminaries
	3. Main results
	4. Conclusion
	References