EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5792
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Fixed Point Results for Enriched Interpolative Type
Multivalued Contractions via a Simulation Function

Amit Gangwar1, Shivam Rawat2, Hassen Aydi3,4, Sarah Aljohani5,∗,
Nabil Mlaiki5

1 H.N.B. Garhwal University, Srinagar Garhwal, Uttarakhand 246174, India.
2 Department of Mathematics, Graphic Era (Deemed to be) University, Dehradun,
Uttarakhand, 248002, India.
3 Institut Supérieur d’Informatique et des Techniques de Communication,
Université de Sousse, H. Sousse 4000, Tunisia
4 Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences
University, Ga-Rankuwa, South Africa
5 Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586,
Saudi Arabia

Abstract. In this paper, using a simulation function in the sense of Khojasteh, we define mul-
tivalued enriched interpolative Kannan-type and Hardy-Rogers-type contractions from a convex
metric space U to the collection of closed and bounded subsets of U . We establish two fixed points
for these types of multivalued contractions. Our results are illustrated by examples and followed
by some corollaries. As a consequence of each main result, a theorem on data dependence of fixed
point is proved.

2020 Mathematics Subject Classifications: 47H09, 47H10, 54H25

Key Words and Phrases: Convex metric space, simulation function, fixed point

1. Introduction

The Banach Contraction Principle [1] stands as a pivotal result in the field of fixed
point theory, furnishing a robust framework for comprehending the existence and unique-
ness of fixed points of mappings which are defined on a complete metric space. Due to
various applications, this result was generalized and extended in numerous ways (see, for
instance, [2–5] and references therein). Given the continuity of mappings satisfying the
Banach contraction principle, a natural question arose regarding the existence of fixed

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5792

Email addresses: amitgangwar069@gmail.com (A. Gangwar),
rawat.shivam09@gmail.com (S. Rawat), hassen.aydi@isima.rnu.tn (H. Aydi),
sjohani@psu.edu.sa (S. Aljohani), nmlaiki@psu.edu.sa; nmlaiki2012@gmail.com (N. Mlaiki)

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



A. Gangwar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5792 2 of 16

points of discontinuous mappings that satisfy similar contractive criteria. Kannan [6] pro-
vided an affirmative response to this inquiry by establishing a contractive condition for a
discontinuous map T , demonstrating the existence and uniqueness of fixed points within
the context of complete metric spaces.

In 2018, Karapinar [7] utilized the interpolation technique to revisit Kannan type
contractions. Prior to [7], interpolative techniques are used in interpolation theory, a field
of functional analysis. Karapinar [7] formulated the interpolative Kannan-type contraction
on a complete metric space (U , ρ) as follows:

ρ(Tx, Ty) ≤ λ([ρ(x, Tx)]α.[ρ(y, Ty)]1−α),

for each x, y ∈ U \Fix(T ), where Fix(T ) = {x ∈ U , Tx = x} and λ ∈ [0, 1). Recent
studies in the field of interpolative Ćirić-Reich-Rus type contractions [8–10] and Meir-
Keeler type contractions [11, 12] can be also referred to [13, 14].

Recently, Berinde [15, 16] has extended the literature related to Banach contraction
principle [1] in Banach spaces by introducing enriched contractions. Enriched contractions
[17] refer to self-mappings T on the structure U of a normed linear space (U , ∥ . ∥).
These mappings adhere to a symmetric contraction condition, expressed as ∥ b(x − y) +
Tx − Ty ∥≤ θ ∥ x − y ∥, where b ∈ [0,∞) and θ ∈ [0, b + 1), for each x, y ∈ U .
Undoubtedly, the category of enriched contractions is more extensive, encompassing not
only the conventional Banach contractions (where b = 0) but also incorporating Lipschitz-
type and non-expansive mappings. The broader scope of the enriched contraction, which
is an extension of Banach contractions, reinforces the assertion that within the Banach
space context, a fixed point x∗ is guaranteed to exist, and the Krasnoselskij iteration
offers an approach to approximate the fixed point. This assertion has been substantiated
by Berinde and Păcurar [17]. Additionally, it’s worth noting that contractive mappings of
Kannan type and of Chatterjea type, can similarly be enriched, as discussed in [18, 19]. In
2022, Rawat et al. [20] introduced the notion of an enriched ordered contraction to prove
some novel fixed point theorems in a convex noncommutative Banach space. Recently,
Gangwar et al. [21] defined λ-enriched multivalued nonexpansive mappings and (λ, θ)-
enriched multivalued contractions on a double controlled metric type space and deduced
some novel fixed point results along with an application to differential inclusions.

Nadler [22] presented a notable and widely acknowledged extension by introducing
the notion of Hausdorff metric, which is defined over a collection of bounded and closed
subsets on a complete metric space. He laid the groundwork for multivalued contrac-
tion mappings. To facilitate understanding, we revisit several standard notations and
terms. Consider a metric space (U , ρ). The set CB(U ) (resp. C(U)) denotes the
collection of those subsets of U which are nonempty bounded and closed (resp. com-
pact) subsets. For A,B ∈ CB(U ), H : CB(U ) × CB(U ) → [0,+∞) defined as
H(A,B) = max{D∗(A,B),D∗(B,A)}, where D∗(A,B) = sup

a∈A
inf
b∈B

ρ(a, b), is known as the

Hausdorff metric. Nadler formulated a theorem for fixed points applicable to set-valued
mappings satisfying a symmetric contraction condition. Takahashi [23] defined a convex
structure in a metric space and referred to it as a convex metric space. Takahashi also



A. Gangwar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5792 3 of 16

studied various characteristics of this metric space to conclude the that a fixed point exists
for nonexpansive mappings in the framework of a convex metric space.

Very recently, Rawat et al. [24] enriched three types of existing interpolative contrac-
tions (Kannan, Hardy-Rogers and Matkowski) in the context of a convex metric space.
In 2015, Khojasteh et al. [25] presented a novel approach to examining fixed points by
introducing a simulation function. They introduced a new type of contraction mappings
known as Z-contractions. Subsequently, other prominent researchers utilized the concept
of Z-contractions to explore common fixed points and coincidence points in various metric
space settings. De Hierro et al. [26] incorporated the notion of Z-contractions to establish
results on coincidence points in metric spaces. Additionally, Argoubi et al. [27] demon-
strated results within the framework of partially ordered metric spaces by employing some
non-linear contractions based on simulation functions.

Inspired by the results mentioned earlier, this study introduces enriched interpolative
Kannan type contractions (EIK-contractions) and enriched interpolative Hardy-Rogers
type contractions (EIHR-contractions) for multivalued mappings via a simulation function.
The research also establishes several fixed-point theorems utilizing multivalued mappings
by employing these contractions. To support our findings, illustrative examples are also
provided. As a consequence of each main result, a theorem on data dependence of fixed
point is proved.

2. Preliminaries

Khojasteh et. al. [25] introduced a new approach in fixed point theory by using the
concept of a simulation function and thus generalized many known results, starting with
Banach contraction principle. A simulation function is a function ζ : [0,∞)× [0,∞) → R
satisfying the following three conditions:

(ζ1) ζ(0, 0) = 0;
(ζ2) ζ(x, y) < y − x, ∀ x, y > 0;
(ζ3) If sequences {xn} and {yn} in the interval [0,∞) satisfy limn→∞ xn = limn→∞ yn > 0,
then

lim
n→∞

sup ζ(xn, yn) < 0.

The collection of all simulation functions will be denoted by Z.

Definition 1. Consider a self-mapping T on a metric space (U , ρ). If for each x, y ∈ U

ζ(ρ(Tx, Ty), ρ(x, y)) ≥ 0,

then T is known as a Z-contraction with respect to ζ .

The concept of a simulation function was broadened by Roldán et al. [28] by just
replacing the property (ζ3) with (ζ ′3):



A. Gangwar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5792 4 of 16

(ζ ′3) If {xn} and {yn} are sequences in [0,∞) such that limn→∞ xn = limn→∞ yn > 0 and
xn < yn for all n ∈ N, then

lim
n→∞

sup ζ(xn, yn) < 0.

A C-class function [29] G : [0,∞)× [0,∞) → R fulfills the following for each x, y ∈ [0,∞) :

(i) G(x, y) ≤ x;

(ii) G(x, y) = x implies either y = 0 or x = 0.

Definition 2. [30] Consider a mapping G : [0,∞) × [0,∞) → R. It is said to fulfill
property CG if for each x, y ∈ [0,∞), there is some constant CG ≥ 0 for which:

(i) G(x, y) > CG implies x > y;

(ii) G(y, y) ≤ CG, for each y.

Definition 3. [30] A ZG simulation function is any mapping ζ : [0,∞) × [0,∞) → R
which fulfills the following:

(i) ζ(0, 0) = 0;

(ii) ζ(x, y) < G(y, x), ∀ x, y > 0, where G is a C-class function;

(iii) If sequences {xn} and {yn} in the interval [0,∞) satisfy limn→∞ xn = limn→∞ yn >
0, then

lim
n→∞

sup ζ(xn, yn) < CG .

Lemma 1. [31] Consider a metric space (U , ρ) and A,B ⊆ U . Then, for every a ∈ A,
there is some b ∈ B so that for q > 1, we obtain

ρ(a, b) ≤ q H(A,B).

Rawat et al. [24] defined an enriched interpolative Kannan type contraction (EIK-contraction)
for single valued mappings as follows.

Definition 4. Let (U , d,W ) be a convex metric space. A self-mapping T : U → U is an
EIK-contraction if there exist λ ∈ [0, 1), c ∈ [0, 1) and α ∈ (0, 1), such that

d(W (x, Tx;λ),W (y, Ty;λ)) ≤ c[d(x,W (x, Tx;λ))]α.[d(y,W (y, Ty;λ))]1−α,

for all x, y ∈ X\Fix(T ).

They further demonstrated the next result.

Theorem 1. [24] Let (U , ρ) be a convex complete metric space and T : U → U be an
EIK-contraction mapping. Then T admits a fixed point.



A. Gangwar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5792 5 of 16

Karapinar et al. [32] defined a multivalued interpolative Hardy-Rogers type contraction
(IHR-contraction) as follows.

Definition 5. Let (U , d) be a metric space. We say that T : U → CB(U ) is a multivalued
interpolative HR-contraction via a simulation function ZG , if there exist k ∈ [0, 1) and
α, β, γ ≥≥ 0 with α+ β + γ < 1 such that

ζ(H(Tx, Ty), R(x, y)) ≥ CG,

where

R(x, y) = k[d(x, y)]α[D(x, Tx)]β[D(y, Ty)]γ [12(D(x, Ty) +D(x, Tx))]1−α−β−γ ,

for all x, y ∈ U \Fix(T ).
They further demonstrated the next result.

Theorem 2. [32] Let (U , ρ) be a complete metric space and T be a multivalued IHR-
contraction via a simulation function ZG. Then Fix(T ) ̸= ϕ.

Now, we define some basic preliminaries related to convex metric spaces.

Definition 6. [23] Let U be a metric space. A continuous function W : U ×U × [0, 1] →
U is known as a convex structure on U , if for every λ ∈ [0, 1] and x, y ∈ U , the next
inequality holds:

ρ(u,W (x, y;λ)) ≤ λρ(u, x) + (1− λ)ρ(u, y), for each x ∈ U . (1)

A metric space U with a convex structure W on U is called a Takahashi convex metric
structure, or simply with a convex metric structure and will be denoted as (U , ρ,W ).

The lemmas below outline some fundamental properties of a convex metric space.

Lemma 2. [23] Let (U , ρ,W ) be a convex metric space. For any x, y ∈ U and any
λ ∈ [0, 1], the following holds:

ρ(x, y) = ρ(x,W (x, y;λ)) + ρ(W (x, y;λ), y).

Lemma 3. [33] Let (U , ρ,W ) be a convex metric space. For any x, y ∈ U and any
λ, λ1, λ2 ∈ [0, 1], the following holds:

(i) W (x, x;λ) = x;W (x, y; 0) = y and W (x, y; 1) = x.

(ii) |λ1 − λ2|ρ(x, y) ≤ ρ(W (x, y;λ1),W (x, y;λ2)).

Lemma 4. [23] Let (U , ρ,W ) be a convex metric space. For any x, y ∈ U and any
λ ∈ [0, 1], the following holds:

ρ(x,W (x, y;λ)) = (1− λ)ρ(x, y) and ρ(W (x, y;λ), y) = λρ(x, y).

Lemma 5. [33] Let (U , ρ,W ) be a convex metric space and T : U → U be a mapping.
For each λ ∈ [0, 1),, define the mapping Tλ : U → U as follows:

Tλx = W (x, Tx;λ), x ∈ U . (2)

Then, Fix(T ) = Fix(Tλ).



A. Gangwar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5792 6 of 16

3. Main results

Definition 7. Let (U , ρ,W ) be a convex metric space. A multivalued mapping T : U →
CB(U ) is an EIK-contraction via a simulation function ZG, if there are α ∈ [0, 1) and
λ, p ∈ (0, 1) so that

ζ(H(W (x, Tx;λ),W (y, Ty;λ)), Q(x, y)) ≥ CG . (3)

Here,

Q(x, y) = p[D∗(x,W (x, Tx;λ)]α[D∗(y,W (y, Ty;λ)]1−α

for each x, y ∈ U /F ix(T ).

Theorem 3. Let (U , ρ,W ) be a convex complete metric space and T : U → CB(U ) be
a multivalued EIK-contraction via a simulation function ZG. Then Fix(T ) ̸= ϕ.

Proof. Using the multivalued EIK-contraction condition (3), the mapping Tλ : U →
CB(U ) given by (2) satisfies

ζ(H(Tλx, Tλy), Q(x, y)) ≥ CG , (4)

where

Q(x, y) = p[D∗(x, Tλx)]
α[D∗(y, Tλy)]

1−α

for each x, y ∈ U /F ix(T ), that is, Tλ is an interpolative Kannan type contraction. Let
y0 ∈ U and define a sequence yn ∈ Tλyn−1, for each n ≥ 1. If we have yn0 = yn0+1 for
some n0 ∈ N, then yn0 is fixed point of Tλ and thus a fixed point of T . So there is nothing
to prove.
Let yn ̸= yn+1 for each n ≥ 0. Since 0 < p < 1 and yn ∈ Tλyn−1 for each n ≥ 1, we can
choose q > 1 so that qp < 1, then from Lemma 1 there is yn+1 ∈ Tλyn, for each n ≥ 1 so
that

ρ(yn, yn+1) ≤ qH(Tλyn−1, Tλyn). (5)

Taking x = yn and y = yn−1, from equation (3), we obtain

ζ(H(Tλyn, Tλyn−1), Q(yn, yn−1) ≥ CG . (6)

By Definition 3, we obtain

CG ≤ ζ(H(Tλyn, Tλyn−1), Q(yn, yn−1)) < G(Q(yn, yn−1), H(Tλyn, Tλyn−1).

From Definition 2, we obtain

H(Tλyn, Tλyn−1) < Q(yn, yn−1)

= p[D∗(yn, Tλyn]
α.[D∗(yn−1, Tλyn−1]

1−α.



A. Gangwar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5792 7 of 16

Using equation (5) and substituting pq = θ < 1, we obtain

ρ(yn, yn+1) < pq [D∗(yn, Tλyn]
α.[D∗(yn−1, Tλyn−1]

1−α

= θ[D∗(yn, Tλyn]
α.[D∗(yn−1, Tλyn−1]

1−α.

Since we know yn ∈ Tλyn−1, one gets D∗(yn−1, Tyn−1) ≤ ρ(yn−1, yn), ∀ n ≥ 1. Therefore,
we obtain

ρ(yn, yn+1) < θ[ρ(yn, yn+1)]
α[ρ(yn−1, yn]

1−α. (7)

Suppose if possible ρ(yn−1, yn) < ρ(yn, yn+1) for some n ≥ 1, then

ρ(yn, yn+1) < θ[ρ(yn, yn+1)]
α[ρ(yn, yn+1)]

1−α

= θρ(yn, yn+1).

This leads to a contradiction as θ < 1. Therefore,

ρ(yn, yn+1) ≤ ρ(yn−1, yn).

From equation (7), we obtain

ρ(yn, yn+1) ≤ θρ(yn−1, yn), (8)

which further implies

ρ(yn, yn+1) < θρ(yn−1, yn)

< θ2ρ(yn−2, yn−1)

< . . .

< θnρ(y0, y1).

Taking n→ ∞, we get ρ(yn, yn+1) → 0. Let m,n ∈ N, m > n, then

ρ(yn, ym) ≤ ρ(yn, yn+1) + ρ(yn+1, yn+2) + · · ·+ d(ym−1, ym)

≤ θn

1− θ
ρ(y0, y1).

As n → ∞, we obtain ρ(yn, ym) → 0. This implies {yn} is a Cauchy sequence and using
the completeness of U , there is y∗ ∈ U so that lim

n→∞
yn = y∗. Suppose y /∈ Ty, then

y /∈ Tλy. Since Tλyn is closed for each n ≥ 0, therefore yn /∈ Tλyn. From equation (4), we
get

CG ≤ ζ(H(Tλyn, Tλyn−1), Q(yn, yn−1) < CG ,

which leads to a contradiction. Hence, y ∈ Ty, which implies Fix(T ) ̸= ϕ.

Now, we present an example in support of our first theorem.



A. Gangwar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5792 8 of 16

Example 1. Let U = R be equipped with the Euclidean metric ρ(x, y) = |x− y|, ∀ x, y ∈
U , and T : U → CB(U ) be given as

T (x) = {−x, 1− x}.

Also, let ζ(t, r) = 1
2r − t, G(r, t) = r − t for each r, t ∈ [0,∞) and CG = 0.

Taking λ = 1
2 , we obtain

T 1
2
(x) =

{
0,

1

2

}
.

Since Fix(T ) = {0, 12}, we get for each x, y ∈ U /{0, 12}, H(Tλx, Tλy) = 0 and for α ∈
(0, 1), Q(x, y) = c[D∗(x, {0, 12})]

α[D∗(y, {0, 12})]
1−α > 0, which implies

ζ(H(Tλx, Tλy), Q(x, y) =
1

2
Q(x, y)

≥ CG .

Also,

G(Q(x, y), H(Tλx, Tλy)) = Q(x, y),

and

CG ≤ ζ(H(Tλx, Tλy), Q(x, y) < G(Q(x, y), H(Tλx, Tλy)).

Thus, T is a multivalued EIK-contraction via a simulation function ZG and all require-
ments outlined in Theorem 3.1 are met. Here, Fix(T ) = {0, 12}.

Corollary 4. Let (U , ρ) be a complete metric space which. If T is a multivalued inter-
polative Kannan type contraction via a simulation function ZG, then Fix(T ) ̸= ϕ.

Proof. Taking λ = 0 and using the same method of proof as in Theorem 3.1, we achieve
the intended result.

Corollary 5. Let (U , ρ) be a convex complete metric space. If a self mapping T is an
EIK-contraction via a simulation function ZG, then T possesses a fixed point.

Definition 8. Let (U , ρ,W ) be a convex metric space. Then T : U → CB(U ) is a
multivalued EIHR-contraction via a simulation function ZG, if there is some p ∈ (0, 1) and
α, β, γ ∈ [0, 1) with α+ β + γ < 1 so that

ζ(H(W (x, Tx;λ),W (y, Ty;λ)), Q(x, y) ≥ CG . (9)

Here, CG ≥ 0 and

Q(x, y) = p.ρ(x, y)α.[D∗(x,W (x, Tx;λ))]β.[D∗(y,W (y, Ty;λ)))]γ .[
1

2
(D∗(x,W (y, Ty;λ)) +D∗(y,W (x, Tx;λ)))

]1−α−β−γ

for each x, y ∈ U /F ix(T ).



A. Gangwar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5792 9 of 16

Theorem 6. Let (U , ρ,W ) be a convex complete metric space. If T : U → CB(U ) is a
multivalued EIHR-contraction via a simulation function ZG, then Fix(T ) ̸= ϕ.

Proof. Using the multivalued EIHR-contraction condition (9), the mapping Tλ : U →
CB(U ) given by (2) satisfies

ζ(H(Tλx, Tλy), Q(x, y)) ≥ CG , (10)

where CG ≥ 0 and

Q(x, y) = p.ρ(x, y)α.[D∗(x, Tλx)]
β.[D∗(y,Tλy)]

γ .[
1

2
(D∗(x, Tλy) +D∗(y, Tλx)

]1−α−β−γ

for each x, y ∈ U /F ix(T ), that is, Tλ is an IHR-contraction. Let y0 ∈ U and define a
sequence yn ∈ Tλyn−1, for each n ≥ 1. If we have yn0 = yn0+1 for a particular n0 ∈ N,
then yn0 is a fixed point of Tλ and thus a fixed point of T . So there is nothing to prove.
Let yn ̸= yn+1 for each n ≥ 0. Since 0 < p < 1 and yn ∈ Tλyn−1 for each n ≥ 1, we can
choose q > 1 so that qp < 1, then from Lemma 1 there is some yn+1 ∈ Tλyn, for each
n ≥ 1 so that

ρ(yn, yn+1) ≤ qH(Tλyn−1, Tλyn). (11)

Taking x = yn and y = yn−1, from equation (3), we obtain

ζ(H(Tλyn, Tλyn−1), Q(yn, yn−1) ≥ CG . (12)

By Definition 3, we obtain

CG ≤ ζ(H(Tλyn, Tλyn−1), Q(yn, yn−1)) < G(Q(yn, yn−1), H(Tλyn, Tλyn−1).

From Definition 2, we obtain

H(Tλyn, Tλyn−1) < Q(yn, yn−1), (13)

where

Q(yn, yn−1) = p[ρ(yn−1, yn)
α[D∗(yn−1, Tλyn−1)]

β[D∗(yn, Tλyn))]
γ

× [
1

2
(D∗(yn−1, Tλyn) +D∗(yn, Tλyn−1))]

1−α−β−γ

≤ p[ρ(yn−1, yn)
α[ρ(yn−1, yn)]

β[ρ(yn, yn+1))]
γ

× [
1

2
(ρ(yn−1, yn+1) + ρ(yn, yn))]

1−α−β−γ

≤ p[ρ(yn−1, yn)
α[ρ(yn−1, yn)]

β[ρ(yn, yn+1))]
γ

× [
1

2
(ρ(yn−1, yn) + ρ(yn, yn+1) + d(yn, yn))]

1−α−β−γ .



A. Gangwar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5792 10 of 16

Suppose if possible ρ(yn−1, yn) < ρ(yn, yn+1) for some n ≥ 1, then

1

2
(ρ(yn−1, yn) + ρ(yn, yn+1) + ρ(yn, yn)) ≤ d(yn, yn+1),

which further implies

p[ρ(yn−1, yn)
α[ρ(yn−1, yn)]

β[ρ(yn, yn+1))]
γ .[

1

2
(ρ(yn−1, yn) + ρ(yn, yn+1) + ρ(yn, yn))]

1−α−β−γ

< p[ρ(yn, yn+1)]
α+β+γ .[ρ(yn, yn+1)]

1−α−β−γ

= pd(yn, yn+1)

i.e., R(yn, yn−1) < pd(yn, yn+1). From equation (9) and (11), we get

ρ(yn, yn+1) ≤ pqρ(yn, yn+1)

= θρ(yn, yn+1).

This leads to a contradiction as θ < 1. Therefore, we obtain

ρ(yn, yn+1) ≤ ρ(yn−1, yn).

From equations (11) and (13), one writes

ρ(yn, yn+1) ≤ qH(Tλyn−1, Tλyn) < pqρ(yn−1, yn). (14)

This implies that

ρ(yn, yn+1) < θρ(yn−1, yn)

< θ2ρ(yn−2, yn−1)

< . . .

< θnρ(y0, y1).

Taking n→ ∞, we obtain ρ(yn, yn+1) → 0. Let m,n ∈ N and m > n, then

ρ(yn, ym) ≤ ρ(yn, yn+1) + ρ(yn+1, yn+2) + · · ·+ d(ym−1, ym)

≤ θn

1− θ
ρ(y0, y1).

As n → ∞, we get ρ(yn, ym) → 0. This implies {yn} is a Cauchy sequence and using the
completeness of U there is some v∗ ∈ U so that lim

n→∞
yn = y∗. Suppose y /∈ Ty, then

y /∈ Tλy. Since Tλyn is closed for each n ≥ 0, yn /∈ Tλyn. From equation (10), we obtain

CG ≤ ζ(H(Tλyn, Tλyn−1), R(yn, yn−1) < CG ,

which leads to a contradiction. Hence, y ∈ Ty, which implies Fix(T ) ̸= ϕ.

The next example justifies the previous theorem.



A. Gangwar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5792 11 of 16

Example 2. Let U = [0, 1] be equipped with the Euclidean metric ρ(x, v) = |x − v|, for
each x, y ∈ U , and T : U → CB(U ) be given as

T (x) =

{
1− x

2
,
−x
2

}
.

Also, let ζ(t, r) = 1
2r − t, G(r, t) = r − t for each r, t ∈ [0,∞) and CG = 0.

Taking λ = 1
3 , we obtain

T 1
3
(x) =

{
0,

1

3

}
.

Since Fix(T ) = {0, 12}, we get for each x, y ∈ U /{0, 12}, H(Tλx, Tλy) = 0 and for α ∈
(0, 1), Q(x, y) = c[D∗(x, {0, 12})]

α[D∗(y, {0, 12})]
1−α > 0, which implies

ζ(H(Tλx, Tλy), Q(x, y) =
1

2
Q(x, y)

≥ CG .

Also,

G(R(x, y), H(Tλx, Tλy)) = Q(x, y).

One writes

CG ≤ ζ(H(Tλx, Tλy), Q(x, y) < G(Q(x, y), H(Tλx, Tλy)).

Thus, T is a multivalued EIHR-contraction via a simulation function ZG and all require-
ments outlined in Theorem 3.5 are met. Here, Fix(T ) = {0, 13}.

Corollary 7. Let (U , ρ) be a complete metric space. If T is a multivalued IHR-contraction
via simulation function ZG, then Fix(T ) ̸= ϕ.

Proof. Taking λ = 0, and using the same method of proof as in Theorem 3.5, we
achieve the intended result.

Corollary 8. Let (U , ρ) be a complete metric space. If a self mapping T : U → U is an
IHR-contraction via a simulation function ZG, then T possesses a fixed point.

4. Data Dependence Results

We propose data dependence results for multivalued EIK-contractions and multivalued
EIHR-contractions via a simulation function ZG . Here p1, p2 ∈ (0, 1) are the constants for
T1 and T2, respectively as given in Definition 7 and Definition 8.



A. Gangwar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5792 12 of 16

Theorem 9. Let (U , ρ) be a convex metric space and Ti : U → CB(U ) be two multivalued
EIK-contraction operators via a simulation function ZG. Suppose that there is some α > 0
so that H(T1y, T2y) ≤ α for each y ∈ U . Then
(i)Fix(Ti) is a closed subset of U for i ∈ {1, 2}.
(ii) H(Fix(T1, ), F ix(T2)) ≤ α

1−max{p1,p2} .

Proof. From Theorem 3.1, Fix(Ti) ̸= ϕ for i ∈ {1, 2}. Let {yn} be a sequence in
Fix(Ti) = Fix(Tiλ) for i ∈ {1, 2} so that yn → y∗ as n→ ∞, then for i ∈ {1, 2}

ζ(H(Tiλyn, Tiλyn−1), Q(yn, yn−1) ≥ CG .

By Definition 3, we obtain

CG ≤ ζ(H(Tiλyn, Tiλyn−1), Q(yn, yn−1) < G(Q(yn, yn−1), H(Tiλyn, Tiλyn−1)).

From Definition 2, we obtain

H(Tiλyn, Tiλyn−1) < Q(yn, yn−1),

which implies

D∗(yn, Tiλyn−1) < Q(yn, yn−1) = pi[D
∗(yn, Tiλyn]

α.[D∗(yn−1, Tiλyn−1]
1−α

= 0.

As n→ ∞, we get that D∗(x, Tiλx) = 0. Since Tiλx ∈ CB(U ), we have x ∈ Tiλx. Hence,
x ∈ Fix(Tiλ) = Fix(Ti) for i ∈ {1, 2}.
Let y0 ∈ Fix(T1) = Fix(T1λ) be arbitrary. Then for q > 1, there is some y1 ∈ T2λy0 so
that

ρ(y0, y1) ≤ qH(T1λy0, T2λy0).

Next, for y1 ∈ T2λy0 there is some y2 ∈ T2λy1 so that

ρ(y1, y2) ≤ qH(T2λy0, T2λy1).

Similarly, we derive the sequence of successive approximations for T2λ beginning from y0,
such that yn+1 ∈ T2λyn for each n ≥ 1 and

ρ(yn, yn+1) ≤ qH(T2λyn−1, T2λyn).

From equation (8) (taking p2 in place of θ), we obtain

ρ(yn, yn+1) ≤ qp2ρ(yn−1, yn) for each n ≥ 1.

Hence, for m ≥ 1 and n ∈ N, we obtain

ρ(yn+m, yn) ≤ ρ(yn, yn+1) + ρ(yn+1, yn+2) + · · ·+ ρ(vn+m−1, yn+m)



A. Gangwar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5792 13 of 16

≤ (qp2)
nρ(y0, y1) + (qp2)

n+1ρ(y0, y1) + · · ·+ (qp2)
n+m−1ρ(y0, y1)

≤ (qp2)
n

1− qp2
ρ(y0, y1).

Taking 1 < q < min{ 1
p1
, 1
p2
} and as n → ∞, we conclude that the {yn} is a Cauchy

sequence in (U , ρ). Therefore, there is some y∗ ∈ U so that yn → y∗ as n→ ∞.
Claim: y∗ is a fixed point of T2.
Suppose if possible y∗ /∈ T2y

∗, which implies y∗ /∈ T2λy
∗, then ynk

/∈ T2λynk
. From

Definition 3 and the contraction condition, we obtain

CG ≤ lim
n→∞

sup ζ(H(T2λy
∗, T2λynk

), Q(y∗, ynk
)) < CG .

This leads to a contradiction. Hence, y∗ is a fixed point of T2.
Taking m→ ∞, then for each n ∈ N we obtain

ρ(y∗, yn) ≤
(qp2)

n

1− qp2
ρ(y0, y1),

which implies

ρ(y0, y
∗) ≤ 1

1− qp2
d(y0, y1)

≤ qk′

1− qp2
.

Similarly, for every x0 ∈ Fix(T2), there is some x∗ ∈ Fix(T1), for which

ρ(x0, x
∗) ≤ 1

1− qp2
d(x0, x1)

≤ qk′

1− qp2
.

Hence,

H(Fix(T1), F ix(T2)) ≤
qk′

1−max{qp1, qp2}

Letting q → 1, we achieve the intended result.

Theorem 10. Let (U , ρ) be a metric space and Ti : U → CB(U ) be two multivalued
EIHR-contractions via a simulation function ZG. Suppose that there is some α > 0 so that
H(T1y, T2y) ≤ α for each y ∈ U . Then
(i)Fix(Ti) is a closed subset of U for i ∈ {1, 2}.
(ii) H(Fix(T1, ), F ix(T2)) ≤ α

1−max{p1,p2} .



A. Gangwar et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5792 14 of 16

Proof. From Theorem 3.5, Fix(Ti) ̸= ϕ for i ∈ {1, 2}. Let {yn} be a sequence in
Fix(Ti) = Fix(Tiλ) for i ∈ {1, 2} so that yn → y∗ as n→ ∞, then for i ∈ {1, 2}

ζ(H(Tiλyn, Tiλyn−1), Q(yn, yn−1)) ≥ CG

By Definition 3, we obtain

CG ≤ ζ(H(Tiλyn, Tiλyn−1), Q(yn, yn−1)) < G(Q(yn, yn−1), H(Tiλyn, Tiλyn−1)).

Using Definition 2, one writes

H(Tiλyn, Tiλyn−1) < Q(yn, yn−1)

which implies

D∗(yn, Tiλyn−1) < Q(yn, yn−1) = pi[D
∗(yn−1, yn)

α[D∗(yn−1, Tiλyn−1)]
β[D∗(yn, Tiλyn))]

γ

× [
1

2
(D∗(yn−1, Tiλyn) +D∗(yn, Tiλyn−1))]

1−α−β−γ .

Now, using the same method of proof as in Theorem 4.1, we achieve the intended result.

Acknowledgements

The authors S. Aljohani and N. Mlaiki would like to thank Prince Sultan University
for the support through the TAS research lab and for paying the APC. The authors
are thankful to the reviewers for their fruitful comments, which helped in improving the
manuscript.

Authors’ contributions

A.G., S.R., H.A., S.A. and N.M. wrote the main manuscript text. All authors reviewed
the manuscript.

Competing interests

The authors declare that they have no conflicts of interest.

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