International Journal of Analysis and Applications

Volume 19, Number 4 (2021), 619-632

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

DOI: 10.28924/2291-8639-19-2021-619

ON ω-INTERPOLATIVE BERINDE WEAK CONTRACTION IN QUASI-PARTIAL

B-METRIC SPACE

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

RAKESH MARWAHA1

1Department 0f 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 aim of this paper is to introduce interpolative weak contraction in the notion of Berinde

weak operator in quasi-partial b-metric space and to extend and generalize fixed point results by adopting

the condition of ω-admissibility. We also discussed convex contraction mapping and obtained a fixed point

result in the setting of Berinde weak operator in quasi-partial b-metric space. Consequently, we present

some examples to show the applicability of the concept.

1. Introduction and Preliminaries

In 1922, Banach [1] introduced the highly recognized Banach’s contraction principle in the field of non-

linear analysis. This result is used to prove the uniqueness of fixed point theorems as well as in Picard

theorems. The Banach’s contraction in metric space is stated as follow :

Theorem 1.1. [1] Let us consider (M,d) to be a complete metric space and T : M → M is the given self

mapping. Let ζ ∈ (0, 1) such that

d(Tτ,Tυ) ≤ ζd(τ,υ)

Received May 27th, 2021; accepted June 21st, 2021; published July 8th, 2021.

2010 Mathematics Subject Classification. 47H10, 49T99, 54H25.

Key words and phrases. quasi-partial b-metric space; ω-admissibility; interpolation; Berinde weak contractions; fixed point.

©2021 Authors retain the copyrights

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

619

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-19-2021-619


Int. J. Anal. Appl. 19 (4) (2021) 620

for all τ,υ ∈ M. Then T has a unique fixed point in M.

In 1992, Matthews [2] brought up the concept of partial metric space. Motivated by which in 1993,

Czerwik [3] introduced a more generalized form of Banach fixed point theorem in b-metric space. Many

authors including Miculescu et al. [4], Oltra et al [5], and Valero [6] introduced some fixed point results and

its topological properties. Later, Karapinar [7] and Shukla [8] introduced quasi-partial metric-space and

partial b-metric space respectively. In 1968, Kannan [9] removed the continuity condition from the Banach

contraction principle.

Theorem 1.2. [9] Suppose (M,d) is a complete metric space and T : M → M is called the Kannan

contraction mapping. Let ζ ∈
[
0, 1

2

)
such that

d(Tτ,Tυ) ≤ ζ[d(τ,Tτ) + d(υ,Tυ)]

for all τ,υ ∈ M. Then T has a unique fixed point in M.

In the year 2004, Berinde [10] brought up the concept of Berinde contraction also known as almost

contractions and stated that :

Theorem 1.3. [10] Let (M,d) is a complete metric space and T : M → M is almost contraction if there

exists ζ ∈ [0, 1) and some R ≥ 0 such that

d(Tτ,Tυ) ≤ ζd(τ,υ) + Rd(υ,Tτ)

for all τ,υ ∈ M. Hence, T has a unique fixed point in M and is called a weak contraction.

Additionally, Berinde obtained some fixed point theorems that serves as the most important results in

literature. Some results are the generalization of Banach, Kannan, Chatterjea, and C̀iric̀ etc. Others can

be found in [11–13]. Recently, Turkoglu [14] formulated a fixed point theorem consisting of four mappings

using the concept of Berinde contraction in partial metric spaces.

In 2018, the idea of interpolative Kannan-type contraction was introduced by Karapinar [15] which is

proved to be a very notable outcome for solving mathematical analysis.

Theorem 1.4. [15] Suppose a self mapping T : M → M in complete metric space (M,d) is an interpolative

Kannan-type contraction, i.e. there exist ζ ∈ [0, 1) and α ∈ (0, 1) such that

d(Tτ,Tυ) ≤ ζ[d(τ,Tυ)]α[d(υ,Tυ)]1−α

for all τ,υ ∈ M with τ 6= Tτ. Then T has a fixed point in M.

Taking this approach forward Ampadu [16] introduced the concept of interpolative Berinde weak contrac-

tion in metric space and expanded its approach in cone metric and partial metric space as well.



Int. J. Anal. Appl. 19 (4) (2021) 621

Definition 1.1. [16] Let us consider (M,d) is a metric space. Let T : M → M is an interpolative Berinde

weak operator if it satisfies,

(1.1) d(Tτ,Tυ) ≤ ζ[d(τ,υ)]α.[d(τ,Tτ)]1−α

for all τ,υ ∈ M and τ,υ /∈ Fix(T), where, ζ ∈ [0, 1) and α ∈ (0, 1) .

Remark 1.1. The Fix(T) denotes the set of all fixed points of T .

The idea of ω-orbital admissible maps was introduced by Popescu [17] as a refining of α-admissible maps

of Samet et al [18]. Further, Karapinar proposed that

Definition 1.2. [19] Let ω : M×M → [0,∞) and M is non-empty subset and T : M×M be a self mapping

is called ω-orbital admissible if for all v ∈ M, we have

(1.2) ω(v,Tv) ≥ 1 implies ω(Tv,Tv2) ≥ 1.

Recently in 2015, Gupta and Gautam [20, 21] brought up the concept of quasi-partial b-metric space and

its related topological properties. Several authors [22–29] have made valuable contributions in this research

field.

Definition 1.3. [20] A quasi-partial b-metric on a non-empty set M is a function qpb : M ×M → R+ such

that for some real number s ≥ 1 and for all τ,υ,χ ∈ M following condition hold:

(QPb1) qpb(τ,τ) = qpb(τ,υ) = qpb(υ,υ) implies τ = υ,

(QPb2) qpb(τ,τ) ≤ qpb(τ,υ),

(QPb3) qpb(τ,τ) ≤ qpb(υ,τ),

(QPb4) qpb(τ,υ) ≤ s[qpb(τ,χ) + qpb(χ,υ)] −qpb(χ,χ).

A quasi-partial b-metric space is a pair (M,qpb) such that M is a nonempty set and qpb is a quasi-partial

b-metric on M. The number s is called the coefficient of (M,qpb).

Example 1.1. Let M = [0,∞). Define qpb = ln(k) |τ −υ| + τ, where k ≥ 1

Here

qpb(τ,τ) = qpb(τ,υ) = qpb(υ,υ)

=⇒ ln(k) |τ − τ| + τ = ln(k) |τ −υ| + τ = ln(k) |υ −υ| + υ.

Therefore, (QPb1) holds for any (τ,υ ∈ M ×M).

Now as ln(k) |τ −υ| ≥ |τ −υ| ≥ 0 then qpb(τ,τ) = τ ≤ qpb(τ,υ)



Int. J. Anal. Appl. 19 (4) (2021) 622

and

qpb(τ,τ) = τ

= |τ −υ + υ|

≤ |τ −υ| + υ

≤ ln(k) |τ −υ| + υ

≤ ln(k) |υ − τ| + υ

≤ qpb(υ,τ).

Therefore, QPb2 and QPb3 both holds.

Now, qpb(τ,υ) + qpb(χ,χ) = ln(k) |τ −υ| + τ + χ.

Since

|τ −υ| ≥ 0

ln(k) |τ −υ| ≥ |τ −υ|

k(|τ −χ| + |χ−υ|) ≥ 0.

Since, ln(k) is an increasing function. Therefore,

qpb(τ,υ) + qpb(χ,χ) = ln(k) |τ −υ| + τ + χ

≤ ln(k)(|τ −χ| + |χ−υ|) + τ + χ

≤ k(|τ −χ| + |χ−υ|) + τ + χ

≤ (k)ln(k) |τ −χ| + (k)ln(k) |χ−υ| + τ + χ

≤ (k)(ln(k) |τ −χ| + |χ−υ|) + τ + χ

≤ s(qpb(τ,χ) + qpb(υ,χ)).

for all τ,υ,χ ∈ M and s ≥ k.

So (M,qpb) is a quasi-partial b-metric space with s ≥ 1.

Remark 1.2. Note that a metric space is included in the class of quasi-partial b-metric space. In fact, the

notions of convergent sequence, Cauchy sequence and complete space are defined as in metric spaces.

Miculescu and Mihail [4] (Lemma 2. 2.) obtain the following result for b-metric spaces.



Int. J. Anal. Appl. 19 (4) (2021) 623

Lemma 1.1. Every sequence {xn} of elements from a b-metric space (X,d,b), having the property that there

exists λ ∈ [0, 1) such that

(1.3) d(xn+2,xn+1) ≤ λd(xn+1,xn),

for any n ∈ N, is Cauchy.

Remark 1.3. Note that Lemma 1.1 holds in quasi-partial b-metric space (see proof of Lemma 2. 2 in [4]).

2. Main results

In this section, we introduce interpolative Berinde weak contractions in quasi-partial b-metric space and

adopted the condition of ω-admissibility to obtain a fixed point.

Definition 2.1. Let (M,qpb) be a complete quasi-partial b-metric space. We say that self-mapping T : M →

M is an interpolative Berinde weak operator if there exist ζ ∈
[
0, 1

s

)
and α ∈ (0, 1) such that

(2.1) qpb(Tτ,Tυ) ≤ ζ[qpb(τ,υ)]α
[

1

s2
qpb(τ,Tτ)

]1−α
,

for all τ,υ ∈ M\Fix(T).

Theorem 2.1. Let (M,qpb) be a complete quasi-partial b-metric space and T be an interpolative Berinde

weak operator. Then T has a fixed point in M.

Proof. Let τ0 ∈ (M,qpb). Consider a constructive sequence τn by τn+1 = Tn(τ0) for all n ∈ N ∪{0}. We

assume that τn = τn+1. Indeed if there exist n0 such that τn0 = τn0+1 = Tτn0 , then, τn0 forms a fixed point.

Thus, we have qpb(τn,Tτn) = qpb(τn,τn+1) > 0, for all n ∈ N∪{0}. Let τ = τn+1,υ = τn+2

qpb(τn+1,τn+2) = qpb(Tτn,Tτn+1)

≤ ζ[qpb(τn,τn+1)]α.
[

1

s2
qpb(τn,τn+1)

]1−α
≤ ζ[qpb(τn,τn+1)]α.

[
1

s2
qpb(τn−1,τn+1)

]1−α
≤ ζ[qpb(τn,τn+1)]α.

[
1

s2
[sqpb(τn−1,τn) + qpb(τn,τn+1) −qpb(τn,τn)

]1−α
≤ ζ[qpb(τn,τn+1)]α.

[
1

s
qpb(τn−1,τn) + qpb(τn,τn+1)

]1−α
(2.2)

By induction, for all n ∈ N∪{0} we get

qpb(τn,τn+1) ≤ ζnqpb(τ0,τ1).



Int. J. Anal. Appl. 19 (4) (2021) 624

On generalising the inequality,

qpb(τn,τn+1) =
sn−1τn

(1 −sτ)n
and qpb(τn+1,τn) =

sn−1τn

(1 −sτ)n
.

Now, we shall show that τn is Cauchy sequence. Let n,k ∈ N

qpb(τn,τn+k) ≤ sqpb(τn,τn+1) + s2qpb(τn+1,τn+2) + . . . + skqpb(τn+k−1,τn+k)

≤
s.sn−1.ζn.τn

(1 −sτ)n
+
s2.sn.ζn+1.τn+1

(1 −sτ)n+1
+ . . . +

sn−k.sn−2.ζn−1.τn−1

(1 −sτ)n−1

≤
sn.τn

(1 −sτ)n
+

sn+2.τn+1

(1 −sτ)n+1
+ . . . +

s2n−k−2.τn−1

(1 −sτ)n−1

≤
sn.τn

(1 −sτ)n

[
1 +

s2.τ

(1 −sτ)
+ . . . +

s2n−k−2.τn−k−1

(1 −sτ)n−k−1

]
The inequality, 0 ≤ τ ≤ 1

s2(s+1)
then s

2.τ
(1−sτ) ≤ 1.

qpb(τn,τn+k) ≤

(
sτ

1−sτ

)n {
1 −

(
s2.τ
1−sτ

)n−k}
(

1 − s2.τ
1−sτ

)
≤
(

sτ

a−sτ

)n
(1 −sτ)

1 −sτ −s2τ
.(2.3)

Therefore, we claim that τn is a Cauchy sequence in (M,qpb). Let m,n ∈ N. By triangle inequality in (2.2),

we deduce that

qpb(τn+m,τn+m+k) ≤
(

sτ

a−sτ

)n
(1 −sτ)

1 −sτ −s2τ

Since, sτ
1−sτ ≤ 1 and taking n →∞ in (2.3) and using limn→∞τ(t

n) = 0 for t ≥ 0, we get

qpb(τn+m,τn+m+k) ≤
(

sτ

a−sτ

)n
(1 −sτ)

1 −sτ −s2τ

Therefore,

lim
n→∞

qpb(τn,τn+k) = lim
m→∞,n→∞

qpb(τn+m,τn+m+k) = 0(2.4)

Since M is complete, so there exist χ ∈ M such that

lim
n→∞

τn = χ.

Suppose τn 6= Tτn for each n ≥ 0

qpb(τn+1,Tχ) = qpb(Tτn,Tχ)

≤ ζ[qpb(τn,χ)]α.
[

1

s2
qpb(τn,Tχ)

]1−α



Int. J. Anal. Appl. 19 (4) (2021) 625

Since 1
s
≤ 1, this implies 1

s2
≤ 1 therefore

≤ ζ[qpb(τn,χ)]α.
[

1

s2
qpb(τn,Tχ)

]1−α
≤ ζ[qpb(τn,χ)]α.[qpb(τn,Tχ)]1−α.(2.5)

Letting n →∞ in (2.5), we get

qpb(χ,Tχ) = 0

which is a contradiction. Thus Tχ = χ. �

Definition 2.2. [7] Let T : M → M be a map and α: M × M → R be a function. Then T is said to be

α-admissible if

(2.6) α(τ,υ) ≥ 1 implies α(Tτ,Tυ) ≥ 1.

We introduce ω-admissible interpolative Berinde weak contraction in quasi-partial b-metric space en-

grossed by Gupta and Gautam [20].

Definition 2.3. Let (M,qpb) be a complete quasi-partial b-metric space. The map T : M → M is said to be

an ω-interpolative Berinde weak contraction if there exists ζ, ω : M ×M → [0,∞) and α ∈ (0, 1) such that

(2.7) ω(τ,υ)qpb(Tτ,Tυ) ≤ ζ[qpb (τ,υ)]α
[

1

s2
qpb (τ,Tτ)

]1−α
,

for all τ,υ ∈ M\Fix(T).

Theorem 2.2. Let us consider (M,qpb) to be a complete quasi-partial b-metric space with self mapping

T : M → M is ω-orbital admissibile and forms an ω-interpolative Berinde weak contraction on a complete

quasi-partial b-metric space (M,qpb). If there exists τ0 ∈ M such that ω(τ0,Tτ1) ≥ 1, then T posses a fixed

point in M.

Proof. Let τ ∈ M be a point such that τn+1 = Tn(τ0) for all n ∈ N ∪{0}. If we have, τn0 = τn0+1 then τn

is a fixed point of T which ends the proof otherwise τn 6= τn+1 for all n ∈ N ∪{0}. We have ω(τ0,τ1) ≥ 1.

Since T is ω - orbital admissible,

ω(τ1,τ2) = ω(Tτ0,Tτ1) ≥ 1(2.8)



Int. J. Anal. Appl. 19 (4) (2021) 626

continuing ω(τn,τn+1) ≥ 1.

Let τ = τn+1, υ = τn+2, we have

qpb(τn,τn+1) ≤ ω(τn+1,τn+2)qpb(Tτn+1,Tτn+2)

qpb(τn+1,τn+2) = qpb(Tτn,Tτn+1)

≤ ζ[qpb(τn,τn+1)]α.
[

1

s2
qpb(τn,τn+1)

]1−α
≤ ζ[qpb(τn,τn+1)]α.

[
1

s2
qpb(τn−1,τn+1)

]1−α
≤ ζ[qpb(τn,τn+1)]α.

[
1

s2
[sqpb(τn−1,τn) + qpb(τn,τn+1) −qpb(τn,τn)

]1−α
≤ ζ[qpb(τn,τn+1)]α.

[
1

s
qpb(τn−1,τn) + qpb(τn,τn+1)

]1−α
(2.9)

By induction, for all n ∈ N∪{0} we get

ω(τn+1,τn+2)qpb(τn,τn+1) ≤ ζnqpb(τ0,τ1).

On generalising the inequality,

qpb(τn,τn+1) =
sn−1τn

(1 −sτ)n
and qpb(τn+1,τn) =

sn−1τn

(1 −sτ)n
.

Now we shall show that τn is Cauchy sequence. Let n,k ∈ N

qpb(τn,τn+k) ≤ sqpb(τn,τn+1) + s2qpb(τn+1,τn+2) + . . . + sn−kqpb(τn+k−1,τn+k)

≤
s.sn−1.ζn.τn

(1 −sτ)n
+
s2.sn.ζn+1.τn+1

(1 −sτ)n+1
+ . . . +

sn−k.sn−2.ζn−1.τn−1

(1 −sτ)n−1

≤
sn.τn

(1 −sτ)n
+

sn+2.τn+1

(1 −sτ)n+1
+ . . . +

s2n−k−2.τn−1

(1 −sτ)n−1

≤
sn.τn

(1 −sτ)n

[
1 +

s2.τ

(1 −sτ)
+ . . . +

s2n−k−2.τn−k−1

(1 −sτ)n−k−1

]
The inequality, 0 ≤ τ ≤ 1

s2(s+1)
then s

2.τ
(1−sτ) ≤ 1

qpb(τn,τn+k) ≤

(
sτ

1−sτ

)n {
1 −

(
s2.τ
1−sτ

)n−k}
(

1 − s2.τ
1−sτ

)
≤
(

sτ

a−sτ

)n
(1 −sτ)

1 −sτ −s2τ
.(2.10)

Therefore, we claim that τn is a Cauchy sequence in (M,qpb). Let m,n ∈ N. On account of the triangle

inequality in (2.9), we deduce that

qpb(τn+m,τn+m+k) ≤
(

sτ

a−sτ

)n
(1 −sτ)

1 −sτ −s2τ



Int. J. Anal. Appl. 19 (4) (2021) 627

Since, sτ
1−sτ ≤ 1, and taking n →∞ in (2.10) and using limn→∞τ(t

n) = 0 for t ≥ 0, we get

lim
n→∞

qpb(τn,τn+k) = lim
m→∞,n→∞

qpb(τn+m,τn+m+k) = 0

Since M is complete, so there exists χ ∈ M such that

lim
n→∞

τn = χ.

Suppose τn 6= Tτn for each n ≥ 0

qpb(τn+1,Tχ) = qpb(Tτn,Tχ)

≤ ζ[qpb(τn,χ)]α.[
1

s2
qpb(τn,Tχ)]

1−α.

Since 1
s
≤ 1, this implies 1

s2
≤ 1, therefore

≤ ζ[qpb(τn,χ)]α.[qpb(τn,Tχ)]1−α.(2.11)

Letting n →∞ in (2.11), we get

qpb(χ,Tχ) = 0

which is a contradiction. Thus Tχ = χ. �

Corollary 2.1. Let (M,qpb) be quasi-partial b-metric space. Let T : M → M be the mapping, such that

ω(τ,υ)qpb(Tτ,Tυ) ≤ ζ[qpb(τ,Tυ)]α.
[

1

s2
qpb(τ,Tτ)

]1−α
for all τ,υ ∈ M\Fix(T) with τ � υ,s ≥ 1 and α ∈ (0, 1).

Assume that :

(1) T is non-decreasing with respect to �,

(2) there exists τ0 ∈ X such that τ0 � Tτ0,

(3) T is continuous.

Then, T has a fixed point in M.

Proof. It suffices to take, in Theorem 2.1,

ω(τ,υ) =




1 if (τ � υ) or (υ � τ),

0 otherwise.

�

Corollary 2.2. Assume that the subsets B1 and B2 of a quasi-partial b-metric space (M,qpb) are closed.

Suppose that T : B1 ∩B2 → B1 ∩B2 satisfies,

ω(τ,υ)qpb(Tτ,Tυ) ≤ ζ[qpb(τ,Tυ)]α.
[

1

s2
qpb(τ,Tτ)

]1−α



Int. J. Anal. Appl. 19 (4) (2021) 628

for all τ ∈ B1 and υ ∈ B2, such that τ,υ /∈ Fix(T), where α ∈ (0, 1),s ≥ 1, If T(B1) ⊆ B2 and T(B2) ⊆ B1,

then there exists a fixed point of T in B1 ∩B2.

Proof. It suffices to take, in Theorem 2.1,

ω(τ,υ) =




1 if (τ � υ) or (υ � τ),

0 otherwise.

�

Example 2.1. Let a set M = [0, 4] with qpb(τ,υ) = Log(k)|τ −υ|+ τ. Let T be a self-mapping on M shown

as

Tτ =




10
3
, if τ ∈ [3, 4],

3
4
, if τ ∈ [0, 3].

We illustrate the self-mappings of T in the Fig1.

Figure 1. 10
3

and 3
4

are the fixed points of T.

Take,

ω(τ,υ) =




1, if τ,υ ∈ [3, 4],

0 otherwise.

Let τ,υ ∈ M such that τ 6= Tτ,υ 6= Tυ and ω(τ,υ) ≥ 1. Then τ,υ ∈ [3, 4] and τ,υ /∈ 10
3
. We have

Tτ = Tυ = 10
3
. For τ0 = 4, we have

ω(4,T4) = ω(4,
10

3
) = 1

Now, let τ,υ ∈ M be such that ω(τ,υ) ≥ 1. This shows that τ,υ ∈ [3, 4], so Tτ = Tυ in [3, 4]. Hence,

ω(Tτ,Tυ) ≥ 1, that is, T is ω-orbital admissible. Suppose τn to be a sequence in M such that ω(τn,τn+1) ≥ 1



Int. J. Anal. Appl. 19 (4) (2021) 629

for each n ∈ N. Then, τn ⊂ [3, 4]. If τn → w as n →∞, we have |τn −w|→ 0 as n →∞. Hence, w ∈ [3, 4]

and so, ω(τn,w) = 1. Therefore, Theorem 2.1 holds. So,
10
3

and 3
4

are the two fixed points of T .

In 2016, Berinde and Fukhar-ud-din [30] modified the concept of convex metric space and applied it to

obtain fixed point results of quasi-contractive operators. Motivated by this, Ampadu [16] introduced convex

interpolative Berinde weak operator in metric spaces. Forging this approach, we introduce the following

result in quasi-partial b-metric space.

Definition 2.4. Let (M,qpb) be a complete quasi-partial b-metric space with continuous mapping T : M → M

is convex interpolative Berinde weak operator if the following is true for all τ,υ ∈ X,τ,υ /∈ Fix(T),Fix(T2).

(2.12) qpb(T
2τ,T2υ) ≤ ζ1qpb(τ,υ)

1
2 + ζ2qpb(Tτ,Tυ)

1
2 qpb(Tτ,T

2τ)
1
2 ,

where ζ1,ζ2 ∈
[
0, 1

s

)
with ζ1 + ζ2 ≤ 1s for s ≥ 1.

Theorem 2.3. Let (M,qpb) be a complete quasi-partial b-metric space with self mapping T : M → M be a

convex interpolative Berinde weak operator. If (M,qpb) is complete, then the fixed point exists.

Proof. Let τn be a sequence in M such that τn+1 = Tτn = T
2τn−1, for all positive integers n. Now, we

observe that

qpb(τn+1,τn+2) = qpb(T
2τn−1,T

2τn)

≤ ζ1qpb(τn,τn−1)
1
2 qpb(τn−1,Tτn−1)

1
2 +

ζ2qpb(Tτn,Tτn−1)
1
2 qpb(Tτn−1,T

2τn−1)
1
2

= ζ1qpb(τn,τn−1)
1
2 qpb(τn−1,τn)

1
2 +

ζ2qpb(τn,τn+1)
1
2 qpb(τn,τn+1)

1
2

= ζ1qpb(τn,τn−1) + ζ2qpb(τn,τn+1)

≤ (ζ1 + ζ2)max{qpb(τn,τn−1),qpb(τn,τn−1)}

= (ζ1 + ζ2)qpb(τn,τn+1).

From the above, we deduce that

qpb(τn+1,τn+2) ≤ hqpb(τn,τn+1) =⇒ qpb(τn+1,τn+2) ≤
1

s
qpb(τn,τn+1).

where h := ζ1 + ζ2 ≤ 1s . By induction, the following is clear for all n ∈ N
⋃

0

qpb(τn,τn+1) ≤ hnqpb(τ0,τ1).



Int. J. Anal. Appl. 19 (4) (2021) 630

Now, we shall show that τn is a Cauchy sequence. For this, let n,m ∈ N with m ≥ n, and we have

qpb(τm,τn) ≤ qpb(τm,τm−1) + qpb(τm−1,τm−2) + . . . + qpb(τn+1,τn)

≤ sqpb(τm,τm−1) + s2qpb(τm−1,τm−2) + . . . + smqpb(τn+1,τn)

≤
[
s(ζ1 + ζ2)

n + s2(ζ1 + ζ2)
n+1 + . . . + sm(ζ1 + ζ2)

m+n−1]qpb(τ0,τ1)
≤

[
s(h)n + s2(h)n+1 + . . . + sm(h)m+n−1

]
qpb(τ0,τ1)

≤ sm
m+n−1∑
i=n

hiqpb(τ0,τ1)

≤ sm
∞∑
i=n

hiqpb(τ0,τ1).

Now, letting m,n → ∞ in the above inequality it follows that τn is a Cauchy sequence and since M is

complete χ ∈ M such that

lim
n→∞

τn = χ.

Suppose qpb(χ,Tχ) = 0, but qpb(χ,Tχ) ≥ 0. Therefore we observe that

0 ≥ qpb(χ,T2χ)

≥ qpb(χ,τn+1) + qpb(τn+1,T2χ)

= qpb(χ,τn+1) + qpb(T
2τn−1,T

2χ)

≥ qpb(χ,τn+1) + ζ1qpb(τn−1,χ)
1
2 qpb(τn−1,Tτn−1)

1
2

+ ζ2qpb(Tτn−1,Tχ)
1
2 qpb(Tτn−1,T

2τn−1)
1
2 .

Taking n → ∞ in the above inequality we find that qpb(χ,Tχ) = 0, which is a contradiction. Thus

Tχ = χ. �

3. Conclusion

The main contribution of this paper is to prove the existence of fixed points via interpolative Berinde

weak contraction. The interpolative weak contraction is extended to adapt various nonlinear self mappings

leading to achieve best topological and geometrical results. One of the major applications of Berinde weak

contraction is that it is used to solve multivalued mappings, that is, it can be used to get more than one

fixed point. Also, it is used to solve initial value problems in ordinary differential equations and integral

equations. Weak contraction merged large amount of contractive operators and formulated fixed points by

the means of Picard iteration.



Int. J. Anal. Appl. 19 (4) (2021) 631

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: This manuscript has received the funding from Spanish Government Grant RTI2018-094366-B-I00

(MCIU/AEI/FEDER, UE) in collaboration with Basque Government Grant IT1207-19.

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

of this paper.

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