EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6113
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Interpolative Contractions for b-metric Spaces and
Their Applications

Dasari Ratna Babu1, Naga Koteswara Rao Koduru2,∗

1 Department of Mathematics, PSCMR College of Engineering and Technology, Vijayawada,
Andhra Pradesh, India
2 Department of Mathematics, Andhra Loyola College, Vijayawada, Andhra Pradesh, India

Abstract. In this research, we introduce the interpolative contractions for a pair of maps in b-
metric spaces and we utilize the idea of interpolation in complete b-metric spaces to prove the
associated common fixed point theorems. Our findings generalize and expand the findings of
Edraoui et al. [7] and Karapınar [10] from the metric space setting to b-metric spaces. We provide
instances to support our conclusions. We offer implementations to solve Fredholm-type nonlinear
integral equations and functional equations that emerge in dynamic programming in order to
illustrate the importance of our theoretical findings.

2020 Mathematics Subject Classifications: 47H10, 54H25

Key Words and Phrases: Common fixed points, b-metric space, integral equation, functional
equation

1. Introduction

The successive approximation methods that were first developed by a number of prior
mathematicians, including well-known figures like Cauchy, Liouville, Picard, Lipschitz, and
others, are successfully encapsulated and reinterpreted by the Banach contraction princi-
ple. Czerwik [5] developed the idea of b-metric space, often known as metric type space, as
a generalization of metric space.. Regarding the Hardy-Rogers fixed point theorem’s gen-
eralization to the interpolative Hardy-Rogers type contractive mapping. Interestingly, this
new kind of mapping was first developed by Karapınar, who integrated the interpolation
notion into the Hardy-Rogers framework. This method probably broadens the original
theorem’s usefulness by generating intermediate points between known data points.

It is true that interpolation is frequently used in mathematical study to generalize
different types of contractions. Researchers can broaden the application of current theo-
rems and offer a more adaptable framework for examining fixed points in metric spaces

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6113

Email addresses: ratnababud@gmail.com (D. R. Babu), nagakoteswararao.k@gmail.com (K. N. K. Rao)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



D. R. Babu, K. N. K. Rao / Eur. J. Pure Appl. Math, 18 (3) (2025), 6113 2 of 14

by including interpolation techniques into contraction mappings. It appears that the in-
terpolative approach has been used to generalize various contraction types and in other
studies. This illustrates the interpolation approach’s adaptability and efficiency in ex-
tending the notion of fixed points and offering fresh perspectives on the existence and
uniqueness of solutions.

I suggest consulting the relevant paper [6, 8–16, 18] and looking into similar research
in the subject to learn more about the particulars and ramifications of Karapınar’s work
as well as the generalization of other contraction forms utilizing the interpolative method.
These resources ought to offer a more thorough comprehension of the interpolative con-
tractive mapping of the Hardy-Rogers type and its uses in fixed point theory.

In 2018, E. Karapınar [10] introduced the notion of interpolative Kannan type con-
traction and established corresponding fixed point theorem in complete metric spaces.

Definition 1. [10] Let (E, d) be a metric space. A mappings T : E → E is said to be
interpolative Kannan type contraction if there exist a constant λ ∈ [0, 1) and α ∈ (0, 1)
such that

d(Ta, Tb) ≤ λ[d(Ta, a)]α[d(Tb, b)]1−α

for all a, b ∈ X such that Ta ̸= a.

Theorem 1. [10] Suppose that (E, d) be a complete metric space, and T is a interpolative
Kannan type contraction. Then, T has a unique common fixed point.

Definition 2. [16] Let (E, d) be a metric space. A mapping T : E → E is said to be
interpolative Hardy-Rogers contraction if there exist a constant k ∈ [0, 1) and α, β, γ ∈
(0, 1) with α+ β + γ < 1, such that

d(Ta, Tb) ≤ k[d(a, b)β][d(Ta, a)]γ [d(Tb, b)]α
[
d(Ta, b) + d(a, T b)

2

]1−α−β−γ

for all a, b ∈ X such that Ta ̸= a.

Theorem 2. [16] Suppose that (E, d) be a complete metric space, and T is a interpolative
Hardy-Rogers pair. Then, T has a unique common fixed point.

Recently, Mohamed Edraoui [7] proved the following theorem using interpolative
Hardy-Rogers pair contraction in complete metric spaces.

Definition 3. [7] Let (E, d) be a metric space. A pair of mappings T, S : E → E is said to
be interpolative Hardy-Rogers pair contraction if there exist k ∈ [0, 1) and α, β, γ ∈ (0, 1)
with α+ β + γ < 1, such that

d(Ta, Sb) ≤ k[d(a, b)β][d(Ta, a)]γ [d(Sb, b)]α
[
d(Ta, b) + d(a, Sb)

2

]1−α−β−γ

for all a, b ∈ X such that Ta ̸= a whenever Sb ̸= b.

Theorem 3. [7] Suppose that (E, d) be a complete metric space, and (T, S) is a interpola-
tive Hardy-Rogers pair contraction. Then, S and T have a unique common fixed point.



D. R. Babu, K. N. K. Rao / Eur. J. Pure Appl. Math, 18 (3) (2025), 6113 3 of 14

2. Main Results

In the following, we introduce interpolative contraction maps in b-metric spaces.
The definition of Hardy-Rogers-type contraction has been generalized by adding the

notion of interpolation. The goal is to find new qualities and broaden the definition of a
Hardy-Rogers-type contraction by interpolation.

Definition 4. Let (E, d, s) be a b-metric space. A pair of mappings T, S : X → X is said
to be interpolative Hardy-Rogers-type contraction if there exist λ ∈ [0, 1) and α, β, γ ∈ (0, 1)
with α+ β + γ < 1, such that

d(Ta, Sb) ≤ λ[d(a, b)β][d(Ta, a)]γ [d(Sb, b)]α
[
d(Ta, b) + d(a, Sb)

2s

]1−α−β−γ

(2.1)

for all a, b ∈ X such that Ta ̸= a whenever Sb ̸= b.

Proposition 1. Allow (E, d, s) to be a b-metric space with two self-maps T, S : E → E
and a coefficient s ≥ 1. The pair (T, S) is assumed to be an interpolative Hardy-Rogers-
type contraction. In the event that a

′
is a fixed point of S, then a

′
is a fixed point of T

because of this. Additionally, in this instance, a
′
is unique.

Theorem 4. Suppose that (E, d, s) be a complete b-metric space, and (T, S) is an inter-
polative Hardy-Rogers-type contraction. Both S and T have a unique common fixed point
if either T or S is b-continuous.

Proof. Let a0 ∈ E be an arbitrary point. Consider {an}, given as a2n+1 = Ta2n and
a2n+2 = Sa2n+1 for each positive integer n.
Take a = a2n and b = a2n+1 in (2.1), we get

d(a2n+1, a2n+2) = d(Ta2n, Sa2n+1)
≤ λ[d(a2n, a2n+1)

β][d(Ta2n, a2n)]
γ [d(Sa2n+1, a2n+1)]

α

[ 12sd(Ta2n, a2n+1) + d(a2n, Sa2n+1)]
1−α−β−γ

= λ[d(a2n, a2n+1)
β][d(a2n+1, a2n)]

γ [d(a2n+2, a2n+1)]
α

[ 12sd(a2n+1, a2n+1) + d(a2n, a2n+2)]
1−α−β−γ

Then

[d(a2n+1, a2n+2)]
1−α ≤ λ[d(a2n, a2n+1)]

β+γ [ 12sd(a2n, a2n+2)]
1−α−β−γ

≤ λ[d(a2n, a2n+1)]
β+γ [12 [d(a2n, a2n+1) + d(a2n+1, a2n+2)]]

1−α−β−γ

(2.2)
Suppose that d(a2n, a2n+1) < d(a2n+1, a2n+2).
Therefore, the inequality (2.2) produces that

[d(a2n+1, a2n+2)]
1−α ≤ λ[d(a2n, a2n+1)]

β+γ [d(a2n+1, a2n+2)]
1−α−β−γ

implies that
[d(a2n+1, a2n+2)]

β+γ ≤ λ[d(a2n, a2n+1)]
β+γ



D. R. Babu, K. N. K. Rao / Eur. J. Pure Appl. Math, 18 (3) (2025), 6113 4 of 14

which implies that

d(a2n+1, a2n+2) ≤ λ
1

β+γ d(a2n, a2n+1) < d(a2n, a2n+1),

which is a contradiction. Thus, we have d(a2n+1, a2n+2) ≤ d(a2n, a2n+1).
From (2.2), we have

[d(a2n+1, a2n+2)]
1−α ≤ λ[d(a2n, a2n+1)]

β+γ [d(a2n, a2n+1)]
1−α−β−γ

= λ[d(a2n, a2n+1)]
1−α

which implies that

d(a2n+1, a2n+2) ≤ λ
1

1−αd(a2n, a2n+1)
= κd(a2n, a2n+1)
...
= κ2n+1d(a0, a1).

(2.3)

Therefore, d(a2n+1, a2n+2) ≤ κ2n+1d(a0, a1). Now, take a = a2n and b = a2n−1 in (2.1),
we get

d(a2n+1, a2n) = d(Ta2n, Sa2n−1)
≤ λ[d(a2n, a2n−1)

β][d(Ta2n, a2n)]
γ [d(Sa2n−1, a2n−1)]

α

[ 12sd(Ta2n, a2n−1) + d(a2n, Sa2n−1)]
1−α−β−γ

= λ[d(a2n, a2n−1)
β][d(a2n+1, a2n)]

γ [d(a2n, a2n−1)]
α

[ 12sd(a2n+1, a2n−1) + d(a2n, a2n)]
1−α−β−γ

Then

[d(a2n+1, a2n)]
1−γ ≤ λ[d(a2n, a2n−1)]

β+α[ 12sd(a2n+1, a2n−1)]
1−α−β−γ

≤ λ[d(a2n, a2n−1)]
β+α[12(d(a2n−1, a2n) + d(a2n, a2n+1))]

1−α−β−γ

(2.4)
Suppose that d(a2n−1, a2n) < d(a2n, a2n+1).
Thus, the inequality (2.4) produces that

[d(a2n+1, a2n)]
1−γ ≤ λ[d(a2n, a2n−1)]

β+α[d(a2n+1, a2n)]
1−α−β−γ

implies that
[d(a2n, a2n+1)]

β+α ≤ λ[d(a2n−1, a2n)]
β+α

which implies that

d(a2n, a2n+1) ≤ λ
1

β+αd(a2n−1, a2n) < d(a2n−1, a2n),

which is a contradiction. Thus, we have d(a2n, a2n+1) ≤ d(a2n−1, a2n).
From (2.4), we have

[d(a2n, a2n+1)]
1−γ ≤ λ[d(a2n−1, a2n)]

β+α[d(a2n−1, a2n)]
1−α−β−γ

= λ[d(a2n−1, a2n)]
1−γ



D. R. Babu, K. N. K. Rao / Eur. J. Pure Appl. Math, 18 (3) (2025), 6113 5 of 14

which implies that

d(a2n, a2n+1) ≤ λ
1

1−γ d(a2n−1, a2n)
= ιd(a2n−1, a2n)
...
= ι2nd(a0, a1).

(2.5)

Therefore, d(a2n, a2n+1) ≤ ι2nd(a0, a1).
It follows from (2.3) and (2.5), we deduce that

d(an, an+1) ≤ knd(a0, a1) for all n ∈ N
where k = min{κ, ι} < 1.
For m > 0. By employing the b-triangular inequality, we arrive at

d(an, an+m) ≤ sd(an, an+1) + s2d(an+1, an+2) + . . .+ smd(an+m−1, an+m)
≤ sknd(a0, a1) + s2kn+1d(a0, a1) + . . .+ smkn+m−1d(a0, a1)
= skn[1 + sk + s2k2 + . . .+ sm−1km−1]d(a0, a1)
≤ skn[1 + sk + (sk)2 + . . .+ (sk)m−1 + . . .]d(a0, a1)

= skn

1−skd(a0, a1) → 0 as n → ∞.

Therefore, {an} is a b-Cauchy sequence in (E, d, s) and by completeness, there exists a
′

such that lim
n→∞

an = a
′
.

Hence, a
′
= lim

n→∞
a2n+1 = lim

n→∞
Ta2n, and a

′
= lim

n→∞
a2n+2 = lim

n→∞
Sa2n+1

so that a
′
= lim

n→∞
Ta2n = lim

n→∞
Sa2n+1.

We assume that T is b-continuous.
Since a2n → a

′
as n → ∞, we have Ta2n → Ta

′
as n → ∞.

Now, 0 ≤ d(a
′
, Ta

′
) ≤ s[d(a

′
, Ta2n) + d(Ta2n, Ta

′
)] → 0 as n → ∞.

a
′
is a fixed point of T as a result.

According to Proposition 1, a
′
is a unique common fixed point of T and S.

An example supporting Theorem 4 is shown below.

Example 1. Let E = [0, 1]. We define d : E × E → R+ by

d(a, b) =


0, if a = b,
11
15 , if a, b ∈ [0, 23 ],

23
25 + a+b

26 , if a, b ∈ (23 , 1],
121
250 , otherwise.

When (E, d, s) has the coefficient s = 51
49 , it is evident that it is a complete b-metric space.

Here we observe that when a = 9
10 , b = 1 and c ∈ (0, 23 ], we have

d(a, b) = 23
25 + a+b

26 = 1291
1300 ̸= 121

125 = 121
250 + 121

250 = d(a, c) + d(c, b) so that d is not a metric.
We specify T, S : E → E by

T (a) =

{
a, if a ∈ [0, 23),

4
3 − a, if a ∈ [23 , 1]

and S(a) =

{
a2+3
4 , if a ∈ [0, 23),

1− a
2 , if a ∈ [23 , 1].



D. R. Babu, K. N. K. Rao / Eur. J. Pure Appl. Math, 18 (3) (2025), 6113 6 of 14

Clearly, T is b-continuous.
We take λ = 99

100 , α = β = γ = 1
20 . Then clearly α+ β + γ < 1.

Keeping generality intact, we suppose that a ≥ b.

Case(i): a, b ∈ [0, 23).

d(Ta, Sb) = 121
250 ≤ 99

100

[
11
15

] 1
20
[
11
15

] 1
20
[
121
250

] 1
20
[
223685
382500

] 17
20

= λ [d(a, b)]β [d(a, Ta)]γ [d(b, Sb)]α
[
d(b,Ta)+d(a,Sb)

2s

]1−α−β−γ

Case(ii): a, b ∈ (23 , 1].

d(Ta, Sb) = 11
15 ≤ 99

100

[
23
25 + a+b

26

] 1
20
[
121
250

] 1
20
[
121
250

] 1
20
[
223685
382500

] 17
20

= λ [d(a, b)]β [d(a, Ta)]γ [d(b, Sb)]α
[
d(b,Ta)+d(a,Sb)

2s

]1−α−β−γ

Case(iii): a ∈ (23 , 1], b ∈ [0, 23).

d(Ta, Sb) = 121
250 ≤ 99

100

[
121
250

] 1
20
[
121
250

] 1
20
[
121
250

] 1
20
[
3038
3825

] 17
20

= λ [d(a, b)]β [d(a, Ta)]γ [d(b, Sb)]α
[
d(b,Ta)+d(a,Sb)

2s

]1−α−β−γ

From all above cases we conclude that (T, S) is a pair of interpolative Hardy-Rogers con-
traction maps.
As a result, T and S satisfy every hypothesis of Theorem 4, and 2

3 is the only joint fixed
point.

Remark 1. Theorem 4 and Example 1 extend and generalize Theorem 3 to b-metric
spaces.

Theorem 5. Suppose that (E, d, s) be a complete b-metric space, and the maps T, S satisfy
the following condition: there exist a constant λ ∈ [0, 1) and α ∈ (0, 1) such that

d(Ta, Sb) ≤ λ[d(Ta, a)]α[d(Sb, b)]1−α

∀ a, b ∈ X such that Ta ̸= a whenever Sa ̸= a. Both S and T have a unique common fixed
point if either T or S is b-continuous.

Proof. Since the proof resembles Theorem 4, we left it out.

Example 2. Let E = R+. We define d : E × E → R+ by

d(a, b) =


0, if a = b,
4, if a, b ∈ [0, 1],

5 + 1
a+b , if a, b ∈ (1,∞),

27
10 , otherwise.



D. R. Babu, K. N. K. Rao / Eur. J. Pure Appl. Math, 18 (3) (2025), 6113 7 of 14

When (E, d, s) has the coefficient s = 489
480 , it is evident that it is a complete b-metric space.

Here we observe that when a = 11
10 , b =

12
10 and c ∈ (0, 1], we have

d(a, b) = 5 + 1
a+b = 125

23 ̸= 27
5 = 27

10 + 27
10 = d(a, c) + d(c, b) so that d is not a metric. We

define T, S : E → E by

T (a) =

{
log(1 + a), if a ∈ [0, 1),

2
a2+1

, if a ∈ [1,∞)
and S(a) =

{
exp a, if a ∈ [0, 1),
1+a
2 , if a ∈ [1,∞).

Clearly, T is b-continuous.
We take λ = 99

100 , α = 4
5 . Assuming a ≥ b, we maintain generality.

Case(i): a, b ∈ [0, 1).

d(Ta, Sb) = 27
10 ≤ 99

100 [4]
4
5
[
27
10

] 1
5

= λ [d(a, Ta)]α [d(b, Sb)]1−α

Case(ii): a, b ∈ (1,∞).

d(Ta, Sb) = 27
10 ≤ 99

100 [4]
4
5
[
27
10

] 1
5

= λ [d(a, Ta)]α [d(b, Sb)]1−α

Case(iii): a ∈ (1,∞), b ∈ [0, 1).

d(Ta, Sb) = 27
10 ≤ 99

100 [4]
4
5
[
27
10

] 1
5

= λ [d(a, Ta)]α [d(b, Sb)]1−α

From all above cases we conclude that (T, S) is an interpolative Kannan-type contraction
maps.
Thus, 1 is the only common fixed point, and T and S satisfy every hypothesis of Theorem
5.

Corollary 1. Suppose that (E, d, s) be a complete b-metric space, and the map T satisfies
the following condition: there exist a constant λ ∈ [0, 1) and α, β, γ ∈ (0, 1) with α+β+γ <
1, such that

d(Ta, Tb) ≤ λ[d(a, b)β][d(Ta, a)]γ [d(Tb, b)]α
[
d(Ta, b) + d(a, Tb)

2s

]1−α−β−γ

for all a, b ∈ X such that Ta ̸= a. Then T has a unique fixed point in E.

Remark 2. Corollary 1 extend and generalize Theorem 2 to b-metric spaces.

Corollary 2. Suppose that (E, d, s) be a complete b-metric space, and the map T satisfies
the following condition: there exist a constant λ ∈ [0, 1) and α ∈ (0, 1) such that

d(Ta, Tb) ≤ λ[d(Ta, a)]γ [d(Tb, b)]1−α

for all a, b ∈ X such that Ta ̸= a. Then T has a unique fixed point in E.

Remark 3. Corollary 2 extend and generalize Theorem 1 to b-metric spaces.



D. R. Babu, K. N. K. Rao / Eur. J. Pure Appl. Math, 18 (3) (2025), 6113 8 of 14

3. Nonlinear integral equations: An Approach

The primary objective of this section is to determine the solution to an integral prob-
lem. If [a, b] is a closed and bounded integral in R, then Ω = C[a, b] is a set of real valued
continuous functions on [a, b].d : Ω × Ω → R+ is what we define. For every ξ, η ∈ Ω,
d(ξ, η) = max

t∈[a,b]
|ξ(t)− η(t)|p, where p > 1 is a real number. Therefore (Ω, d) is a complete

b-metric space with s = 2p−1. Several authors have studied the unique solution of a sys-
tem of nonlinear integral equations [1–3, 17]. We demonstrate the existence of a single
common solution for a system of two nonlinear integral equations of Fredholm type, which
is defined by 

ξ(t) = f(t) + µ
b∫
a
D1(t, r, ξ(r))dr,

ζ(t) = f(t) + µ
b∫
a
D2(t, r, ζ(r))dr

(3.1)

where ξ ∈ C[a, b], µ ∈ R, t, r ∈ [a, b],D1,D2 : [a, b] × [a, b] × R → R and f : [a, b] → R are
continuous functions.
Consider two mappings F1,F2 : Ω → Ω that are specified by

F1(ξ(t)) = f(t) + µ
b∫
a
D1(t, r, ξ(r))dr,

F2(ξ(t)) = f(t) + µ
b∫
a
D2(t, r, ξ(r))dr

(3.2)

Make the following assumptions:

(i) there exists a continuous function γ : [a, b]× [a, b] → R+, such that

max
r∈[a,b]

b∫
a
γ(t, r)dr ≤ 1;

(ii) there exists a constant K ∈ (0, 1) such that for all t, r ∈ [a, b], ξ, ζ ∈ R, and α, β, γ ∈
(0, 1) with α+ β + γ < 1, the following condition is satisfied:
|D1(t, r, ξ1(r))−D2(t, r, ξ2(r)|p ≤ K

(b−a)p−126p−6γ(t, r)∆(ξ1, ξ2),

where

∆(ξ1, ξ2) = [|ξ1(r)− ξ2(r)|p]β [|ξ1(r)−F1ξ1(r)|p]γ [|ξ1(r)−F1ξ1(r)|p]α[
|ξ1(r)−F2ξ2(r)|p+|ξ2(r)−F1ξ1(r)|p

2p

]1−α−β−γ

(iii) |µ| ≤ 1.

Theorem 6. The requirements (i) − (iii) hold if (3.2) is used to define F1,F2 : Ω → Ω.
Next, there is a unique common solution in Ω for the system of nonlinear integral equations
(3.1).



D. R. Babu, K. N. K. Rao / Eur. J. Pure Appl. Math, 18 (3) (2025), 6113 9 of 14

Proof. Let ξ, η ∈ Ω and let q ∈ R such that 1
p + 1

q = 1 using Hölder’s inequality and
from the conditions (i)− (iii), for all t, we have

d(F1ξ1,F2ξ2) = max
t∈[a,b]

|F1ξ1(t)−F2ξ2(t)|p

= |µ|p max
t∈[a,b]

∣∣∣∣∣ b∫a D1(t, r, ξ1(r)−
b∫
a
D2(t, r, ξ2(r)dr

∣∣∣∣∣
p

= |µ|p max
t∈[a,b]

∣∣∣∣∣ b∫a (D1(t, r, ξ1(r)−D2(t, r, ξ2(r))dr

∣∣∣∣∣
p

≤

|µ|p max
t∈[a,b]

(
b∫
a
1pdr

) 1
q
(

b∫
a
|(D1(t, r, ξ1(r)−D2(t, r, ξ2(r))|p dr

) 1
p

p

≤ (b− a)
p
q max
t∈[a,b]

(
b∫
a
|(D1(t, r, ξ1(r)−D2(t, r, ξ2(r))|p dr

)

= (b− a)p−1 max
t∈[a,b]

(
b∫
a
|(D1(t, r, ξ1(r)−D2(t, r, ξ2(r))|p dr

)
≤ (b− a)p−1 max

t∈[a,b]

b∫
a

K
(b−a)p−126p−6γ(t, r)∆(ξ1, ξ2)

which implies that

d(F1ξ1,F2ξ2) ≤ K
s6

[|ξ1(r)− ξ2(r)|p]β [|ξ1(r)−F1ξ1(r)|p]γ [|ξ1(r)−F1ξ1(r)|p]α[
|ξ1(r)−F2ξ2(r)|p+|ξ2(r)−F1ξ1(r)|p

2p

]1−α−β−γ

= λ∆(ξ1, ξ2)

where λ = K
s2

∈ (0, 1).
As a result, F1,F2 have a unique common solution of the system of nonlinear integral

equations specified in (3.1), since all the requirements of Theorem 4 are satisfied.

4. Application to dynamic programming

The decision space is D ⊆ X1, and the state space is S ⊆ X2. X1 and X2 are assumed to
be two Banach spaces in this section. All bounded real valued functions on S have a Banach
space called Ω(S), whose b-metric is defined as follows: d(ξ, ζ) = sup

t∈S
| ξ(t) − ζ(t) |p, ∀

ξ, ζ ∈ Ω(S) with coefficient s = 2p−1 and the norm is defined as ∥F∥ = sup{| F(t) |: t ∈ S},
where F ∈ Ω(S).

Ω(S, d) is obviously a complete b-metric space. According to Bellman et al. [4], the
functional equation in dynamic programming has the following basic form:
f(ξ) = H

ζ∈D̃
(ξ, ζ, f(T (ξ, ζ))), ξ ∈ S, where T indicates the process transformation, f(ξ)

indicates the optimal return function with the initial state ξ, and opt stands for sup or



D. R. Babu, K. N. K. Rao / Eur. J. Pure Appl. Math, 18 (3) (2025), 6113 10 of 14

inf. The state and decision vectors are denoted by ξ and ζ, respectively.
We look at the functional equation system

f1(νs) = opt
νd∈D̃

(η1(νs, νd) + ξ1(νs, νd, f1(ρ1(νs, νd))))∀νs ∈ S,

f2(νs) = opt
νd∈D̃

(η2(νs, νd) + ξ2(νs, νd, f2(ρ2(νs, νd))))∀νs ∈ S (4.1)

where the state vector is νs, the decision vector is νd, the process transformations are
represented by ρ1, ρ2, and the optimal return functions with initial state νs are indicated
by f1(νs), f2(νs).
Let F1,F2 : Ω(S) → Ω(S) be two mappings defined by;

F1f1(νs) = opt
νd∈D̃

(η1(νs, νd) + ξ1(νs, νd, f1(ρ1(νs, νd))))∀νs ∈ S,

F2f2(νs) = opt
νd∈D̃

(η2(νs, νd) + ξ2(νs, νd, f2(ρ2(νs, νd))))∀νs ∈ S (4.2)

Assume the following:

(Da) for all (νs, νd, f1, f2) ∈ S×D×Ω(S)×Ω(S)×Ω(S)×Ω(S) and there exist 0 < h < 1
and 0 < α < 1, such that;

| ξ1(νs, νd, f1(ρ1(νs, νd)))− ξ2(νs, νd, f2(ρ2(νs, νd))) | + | η1(νs, νd)− η2(νs, νd) |
≤ [ h

24p−4M(f1, f2)]
1
p

where
M(f1, f2) = [|f1 −F1f1|p]α [|f2 −F2f2|p]1−α

(Db) ρi, ξi are bounded i = 1, 2.

Theorem 7. Assume F1,F2 : Ω(S) → Ω(S) be defined by (4.2) for which the conditions
Da and Db are satisfied. Then, there is a unique bounded common solution in Ω(S) for
the system of functional equations provided by (4.1).

Proof. Let νs ∈ S, f1, f2 ∈ Ω(S) and ϵ > 0.
As ρi, ξi are bounded for i = 1, 2 ∃ L ≥ 0 ∋

sup{||ρ1(νs, νd)||, ||ρ2(νs, νd)||, ||ξ2(νs, νd, t)|| : (νs, νd, t) ∈ S × D × R} ≤ L. (4.3)

From the inequalities (4.2) and (4.3), we conclude that F1,F2 are self mappings of Ω(S)
First assume that opt

νs∈D̃
= inf

νd∈D
.

The inequality (4.2) allows us to determine νd ∈ D and (νs, f, g) ∈ S ×Ω(S)×Ω(S) such
that

F1f1(νs) > ξ1(νs, νd, f1(ρ1(νs, νd))) + η1(νs, νd)− ϵ (4.4)

F1f2(νs) > ξ2(νs, νd, f2(ρ2(νs, νd))) + η2(νs, νd)− ϵ (4.5)



D. R. Babu, K. N. K. Rao / Eur. J. Pure Appl. Math, 18 (3) (2025), 6113 11 of 14

F1f1(νs) ≤ ξ1(νs, νd, f1(ρ1(νs, νd))) + η1(νs, νd) (4.6)

F1f2(νs) ≤ ξ2(νs, νd, f2(ρ2(νs, νd))) + η2(νs, νd) (4.7)

By using the inequalities (4.4) and (4.7), we get that
F1f1(νs)−F1f2(νs) > ξ1(νs, νd, f1(ρ1(νs, νd)))− ξ2(νs, νd, f2(ρ2(νs, νd)))

+η1(νs, νd)− η2(νs, νd)− ϵ
≥ −{| ξ1(νs, νd, f1(ρ1(νs, νd)))− ξ2(νs, νd, f2(ρ2(νs, νd))) |

+ | η1(νs, νd)− η2(νs, νd) | +ϵ}

(4.8)

Also, from (4.5) and (4.6), we have
F1f1(νs)−F1f2(νs) ≤ ξ1(νs, νd, f1(ρ1(νs, νd)))− ξ2(νs, νd, f2(ρ2(νs, νd)))

+η1(νs, νd)− η2(νs, νd) + ϵ
≤| ξ1(νs, νd, f1(ρ1(νs, νd)))− ξ2(νs, νd, f2(ρ2(νs, νd))) |

+ | η1(νs, νd)− η2(νs, νd) | +ϵ

(4.9)

By using (4.8) and (4.9), we get that

| F1f1(νs)−F1f2(νs) |< ξ1(νs, νd, f1(ρ1(νs, νd)))− ξ2(νs, νd, f2(ρ2(νs, νd)))
+η1(νs, νd)− η2(νs, νd) + ϵ

≤ ξ1(νs, νd, f1(ρ1(νs, νd)))− ξ2(νs, νd, f2(ρ2(νs, νd)))
+η1(νs, νd)− η2(νs, νd) + ϵ

Now, we support that opt
νd∈D̃

= sup
νd∈D

.

From (4.2), we can determine νd ∈ D and (νs, f, g) ∈ S × Ω(S)× Ω(S) such that

F1f1(νs) < ξ1(νs, νd, f1(ρ1(νs, νd))) + η1(νs, νd) + ϵ (4.10)

F1f2(νs) < ξ2(νs, νd, f2(ρ2(νs, νd))) + η2(νs, νd) + ϵ (4.11)

F1f1(νs) < ξ1(νs, νd, f1(ρ1(νs, νd))) + η1(νs, νd) (4.12)

F1f2(νs) < ξ2(νs, νd, f2(ρ2(νs, νd))) + η2(νs, νd) (4.13)

Using the inequalities (4.10) and (4.13), we have
F1f1(νs)−F1f2(νs) < ξ1(νs, νd, f1(ρ1(νs, νd)))− ξ2(νs, νd, f2(ρ2(νs, νd)))

+η1(νs, νd)− η2(νs, νd) + ϵ
≤| ξ1(νs, νd, f1(ρ1(νs, νd)))− ξ2(νs, νd, f2(ρ2(νs, νd))) | + | η1(νs, νd)− η2(νs, νd) | +ϵ

(4.14)
Also, from the inequalities (4.11) and (4.12), we get that

F1f1(νs)−F1f2(νs) ≥ ξ1(νs, νd, f1(ρ1(νs, νd)))− ξ2(νs, νd, f2(ρ2(νs, νd)))
+η1(νs, νd)− η2(νs, νd)− ϵ

≥ −{| ξ1(νs, νd, f1(ρ1(νs, νd)))− ξ2(νs, νd, f2(ρ2(νs, νd))) | + | η1(νs, νd)− η2(νs, νd) + ϵ}
(4.15)



D. R. Babu, K. N. K. Rao / Eur. J. Pure Appl. Math, 18 (3) (2025), 6113 12 of 14

From (4.14) and (4.15), we have
| F1f1(νs)−F1f2(νs) |< ξ1(νs, νd, f1(ρ1(νs, νd)))− ξ2(νs, νd, f2(ρ2(νs, νd)))

+η1(νs, νd)− η2(νs, νd)− ϵ
≤| ξ1(νs, νd, f1(ρ1(νs, νd)))− ξ2(νs, νd, f2(ρ2(νs, νd))) | + | η1(νs, νd)− η2(νs, νd) + ϵ

(4.16)
On taking ϵ → 0 in (4.16), we obtain that

| F1f1(νs)−F1f2(νs) |≤| ξ1(νs, νd, f1(ρ1(νs, νd)))− ξ2(νs, νd, f2(ρ2(νs, νd))) |
+ | η1(νs, νd)− η2(νs, νd) |

From the condition (Db), we have

| F1f1(νs)−F1f2(νs) |≤| ξ1(νs, νd, f1(ρ1(νs, νd)))− ξ2(νs, νd, f2(ρ2(νs, νd))) |
+ | η1(νs, νd)− η2(νs, νd) |

≤ [ h
24p−4M(f1, f2)]

1
p

≤
[
sup
νs∈S

( h
24p−4 [|f1 −F1f1|p]α [|f2 −F2f2|p]1−α

] 1
p

which implies that

sup
νs∈S

| F1f1(νs)−F1f2(νs) |p ≤ h
24p−4 sup

νs∈S
[|f1 −F1f1|p]α [|f2 −F2f2|p]1−α .

Now, for all f1, f2 ∈ Ω(S), we have

d(F1f1,F1f2) ≤ h
24p−4 [d(f1,F1f1)]

α [d(f2,F2f2)]
1−α .

Consequently, F1,F2 have a unique bounded common solution to the system of functional
equations (4.1) since all the requirements of Theorem 5 are met.

5. Conclusion and future work

In this paper, we studied fixed point results for interpolative contraction mappings
in b-metric spaces. Using similar approaches, it can be studied new fixed point results on
metric and some generalized metric spaces. The investigation of certain circumstances to
exclude the identity map of E from Theorem 4 and Theorem 5 and related results is a
worthwhile problem for future effort.

Acknowledgements

The authors are sincerely thankful to the anonymous referee for the valuable sugges-
tions which helped us to improve the quality of the paper.



D. R. Babu, K. N. K. Rao / Eur. J. Pure Appl. Math, 18 (3) (2025), 6113 13 of 14

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