EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 6077
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Existence and Uniqueness of Fixed Points in
MR−Metric Spaces and Their Applications

Abed Al-Rahman M. Malkawi

Department of Mathematics, Faculty of Arts and Science, Amman Arab University, Amman
11953, Jordan

Abstract. This paper investigates fixed-point theorems within MR-metric spaces, an extension
of standard metric spaces, emphasizing the existence and uniqueness of fixed points for continu-
ous mappings S : X → X, where X is a closed, bounded, and convex subset of a Banach space
(E, ∥ · ∥). The study establishes that if S satisfies a contractive condition involving the MR-metric
with a constant k ∈ [0, 1), and a measure of noncompactness condition governed by a function
ϕ where ϕ(t) < t for t > 0, then S possesses a unique fixed point υ∗. The findings have signifi-
cant applications in solving nonlinear integral equations, ensuring stability of iterative processes,
optimization, game theory, economic equilibria, and boundary value problems, showcasing the
versatility of MR-metric spaces in addressing noncompact settings and fixed-point problems.

2020 Mathematics Subject Classifications: 47H10, 54H25, 46T99, 47H09

Key Words and Phrases: MR −metric Fixed-point theorems, MR-metric spaces, measure of
noncompactness, iterative processes, optimization, game theory, nonlinear integral equations,boundary
value problems, Banach spaces

1. Introduction

Fixed-point theory represents a fundamental aspect of functional analysis, playing a
pivotal role in mathematics and numerous scientific fields. The idea of a fixed point,
where a function maps a specific point to itself, is essential for solving equations, studying
dynamic systems, and addressing optimization problems. In this framework, the introduc-
tion of generalized metric spaces, including MR-metric spaces, offers a versatile structure
for tackling fixed-point challenges, particularly in noncompact contexts.

This study explores MR-metric spaces, which generalize the classical metric space
framework to encompass a wider array of applications. By defining a generalized metric M
and utilizing the measure of noncompactness, we derive fixed-point theorems for mappings
S : X → X, where X represents a closed, bounded, and convex subset of a Banach space.
These results ensure the existence and uniqueness of fixed points under certain contractive
conditions and noncompactness criteria.

DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6077

Email address: a.malkawi@aau.edu.jo and math.malkawi@gmail.com (A. Malkawi)

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



A. Malkawi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6077 2 of 16

The importance of these findings stems from their wide-ranging applications in both
mathematical theory and practical problems, such as nonlinear integral equations, iterative
methods, optimization, game theory, and boundary value challenges. This work seeks to
establish a solid theoretical framework for employing MR-metric spaces within fixed-
point theory, emphasizing their capability to handle intricate and noncompact situations
effectively.

For additional information, we direct readers to [1–27].

Definition 1. [28] Let X ̸= ∅ denote a non-empty set, and let R > 1 be a given real
number. A function M : X × X × X → [0,∞) is called an MR-metric if it fulfills the
following conditions for all υ, ξ,ℑ ∈ X:

• (M1) : M(υ, ξ,ℑ) ≥ 0.

• (M2) : M(υ, ξ,ℑ) = 0 if and only if υ = ξ = ℑ.

• (M3) : M(υ, ξ,ℑ) = M(p(υ, ξ,ℑ)), for any permutation p(υ, ξ,ℑ) of υ, ξ,ℑ.

• (M4) : M(υ, ξ,ℑ) ≤ R [M(υ, ξ, ℓ1) +M(υ, ℓ1,ℑ) +M(ℓ1, ξ,ℑ)] .

A pair (X,M) that satisfies these properties is called an MR-metric space.

Definition 2. Let {υin} be a sequence in an MR-metric space (Y,M). The sequence is
termed MR-convergent if there exists an element υi1 ∈ Y such that for every ϵ > 0, there
exists a positive integer N satisfying M(υin , υim , υi1) < ϵ for all m,n ≥ N . In this case,
the sequence {υin} is said to MR-converge to υi1, where υi1 is considered the limit of the
sequence.

Definition 3. A sequence {υin} in an MR-metric space (Y,M) is called MR-Cauchy if
for every ϵ > 0, there exists a positive integer N such that M(υin, υim, υip) < ϵ for all
m,n, p ≥ N .

Definition 4. An MR-metric space (X,M) is considered bounded if there exists a real
number L > 0 such that M(υ, ξ,ℑ) ≤ L for all υ, ξ,ℑ ∈ X. In this case, M is referred to
as an MR-bound for the metric.

Definition 5. Let E be a set subset of an MR-metric space (X,M) is said to be M -
bounded if there exists L > 0 such that M(υ, ξ,ℑ) ≤ L for all υ, ξ,ℑ ∈ E.

Definition 6. [29] A Banach space is a vector space X over the field R or C equipped
with a norm ∥ · ∥ : X → R that satisfies the following conditions:

(i) Positivity: ∥υ∥ ≥ 0 for all υ ∈ X, and ∥υ∥ = 0 if and only if υ = 0.

(ii) Scalar Multiplication: ∥αυ∥ = |α|∥υ∥ for all α ∈ R or C, and υ ∈ X.

(iii) Triangle Inequality: ∥υ + ξ∥ ≤ ∥υ∥+ ∥ξ∥ for all υ, ξ ∈ X.



A. Malkawi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6077 3 of 16

Moreover, X is complete with respect to the norm, meaning that every Cauchy sequence
in X converges to a limit in X.

Definition 7. [30] In measure theory, a measurable set is a subset of a set X that
belongs to a σ-algebra A over X. This means:

• The set X is associated with a measure µ, which is a function defined on a collection
A.

• A is a family of subsets of X satisfying the following conditions:

(i) X ∈ A.

(ii) If A ∈ A, then X \A ∈ A (closure under complements).

(iii) If {An}∞n=1 ⊂ A, then
⋃∞

n=1An ∈ A (closure under countable unions).

Thus, a measurable set is any element of the σ-algebra A.

2. Fixed Point Theorem in MR-Metric Space within a Banach Space

Fixed-point theory within MR-metric spaces broadens traditional fixed-point results
by addressing mappings in more generalized and intricate settings, including noncompact
spaces. Here, we focus on MR-metric spaces where X is a closed, bounded, and convex
subset of a Banach space (E, ∥ · ∥). Conditions are derived to ensure that a continuous
mapping S : X → X possesses a unique fixed point. This result forms a crucial theoretical
foundation for applications in both pure and applied mathematical contexts.

Theorem 1. Consider an MR-metric space (X,M), where X is a closed, bounded, and
convex subset of a Banach space (E, ∥ · ∥). Let S : X → X satisfy the following conditions:

(i) S is continuous.

(ii) For all υ, ξ,ℑ ∈ X,

M(S(υ), S(ξ), S(ℑ)) ≤ k ·M(υ, ξ,ℑ),

where k ∈ [0, 1) is a constant.

Under these conditions, S has a unique fixed point υ∗ ∈ X, such that S(υ∗) = υ∗.

Proof. Let υ0 ∈ X be an arbitrary element, and define a sequence {υn} by υn+1 = S(υn)
for all n ≥ 0.

Using the contraction condition satisfied by S, we have for all n ≥ 0:

M(υn+1, υn+2, υn+3) = M(S(υn), S(υn+1), S(υn+2)) ≤ k ·M(υn, υn+1, υn+2).

Iterating this inequality yields:

M(υn+1, υn+2, υn+3) ≤ kn ·M(υ0, υ1, υ2),



A. Malkawi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6077 4 of 16

where M(υ0, υ1, υ2) is a finite constant since M is non-negative.
As n → ∞, the term kn → 0 because k ∈ [0, 1). Thus:

lim
n→∞

M(υn+1, υn+2, υn+3) = 0.

This implies that the sequence {υn} is a Cauchy sequence with respect to the MR-metric
M .

Since (X,M) is an MR-metric space and X is closed in the Banach space E, the
completeness of E ensures that the sequence {υn} converges to some υ∗ ∈ X. That is,

lim
n→∞

υn = υ∗.

To show that υ∗ is a fixed point of S, we use the continuity of S. By definition of S
and the convergence of {υn}, we have:

lim
n→∞

S(υn) = S
(
lim
n→∞

υn

)
.

Substituting υn+1 = S(υn), we obtain:

υ∗ = S(υ∗).

To prove the uniqueness of the fixed point, suppose there exists another fixed point
ξ∗ ̸= υ∗ such that S(ξ∗) = ξ∗. Using the contraction condition for S, we have:

M(υ∗, υ∗, υ∗) = M(S(ξ∗), S(ξ∗), S(ξ∗)) ≤ k ·M(ξ∗, ξ∗, ξ∗).

Since M(υ∗, υ∗, υ∗) = 0 by the properties of the MR-metric, it follows that M(ξ∗, ξ∗, ξ∗) =
0. The second axiom of the MR-metric implies υ∗ = ξ∗, which contradicts the assumption
that υ∗ ̸= ξ∗. Hence, υ∗ is the unique fixed point of S.

Theorem 2. Let (X,M) be an MR-metric space, where X is a closed, bounded, and convex
subset of a Banach space (E, ∥ · ∥). Assume S : X → X is a continuous operator satisfying
the following conditions:

(i) For all υ, ξ,ℑ ∈ X,

M(S(υ), S(ξ), S(ℑ)) ≤ k ·M(υ, ξ,ℑ),

where k ∈ [0, 1) is a constant.

(ii) The measure of noncompactness µ satisfies:

µ(S(A)) ≤ ϕ(µ(A)),

for every bounded subset A ⊂ X, where ϕ : [0,∞) → [0,∞) is a non-decreasing
function such that ϕ(t) < t for all t > 0 and ϕ(0) = 0.



A. Malkawi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6077 5 of 16

Then S has a unique fixed point υ∗ ∈ X, such that S(υ∗) = υ∗.

Proof. Let υ0 ∈ X be arbitrary, and define a sequence {υn} by υn+1 = S(υn) for all
n ≥ 0.

Using the contraction condition of S with respect to the MR-metric M , we have:

M(υn+1, υn+2, υn+3) = M(S(υn), S(υn+1), S(υn+2)) ≤ k ·M(υn, υn+1, υn+2).

By induction, it follows that:

M(υn+1, υn+2, υn+3) ≤ kn ·M(υ0, υ1, υ2),

where M(υ0, υ1, υ2) is a finite constant since M is non-negative. As n → ∞, kn → 0, and
hence:

lim
n→∞

M(υn+1, υn+2, υn+3) = 0.

Thus, the sequence {υn} is a Cauchy sequence under the MR-metric.
Since X is closed and (X,M) is complete, the sequence {υn} converges to some υ∗ ∈ X.
By the continuity of S, we have:

lim
n→∞

S(υn) = S
(
lim
n→∞

υn

)
,

which implies υ∗ = S(υ∗). Thus, υ∗ is a fixed point of S.
Now, for uniqueness, assume there exists another fixed point ξ∗ ̸= υ∗. Using the

contraction condition for S with respect to the MR-metric, we have:

M(υ∗, υ∗, υ∗) = M(S(ξ∗), S(ξ∗), S(ξ∗)) ≤ k ·M(ξ∗, ξ∗, ξ∗).

Since M(υ∗, υ∗, υ∗) = 0 and k ∈ [0, 1), it follows that M(ξ∗, ξ∗, ξ∗) = 0. By the properties
of M , this implies υ∗ = ξ∗. Hence, the fixed point is unique.

Finally, consider the measure of noncompactness µ. Since X is closed and bounded,
we have:

µ(S(A)) ≤ ϕ(µ(A)),

for any bounded subset A ⊂ X. Since ϕ(t) < t for all t > 0, repeated application of
S reduces the measure of noncompactness, ensuring the existence of a fixed point. The
uniqueness follows from the argument above.

Example 1. Fixed Point in MR-Metric Space
Let X = R and define the MR-metric M : X× X× X → [0,∞) by:

M(υ, ξ,ℑ) = |υ − ξ|+ |ξ −ℑ|+ |ℑ − υ|.

Let S : R → R be the mapping defined by:

S(υ) =
υ

2
.



A. Malkawi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6077 6 of 16

We verify that S satisfies the contraction condition:

M(S(υ), S(ξ), S(ℑ)) = 1

2
M(υ, ξ,ℑ),

where k = 1
2 < 1. Hence, S satisfies the conditions of the MR-metric space fixed point

theorem.
Let υ0 ∈ R. Construct the sequence υn+1 = S(υn). By iteration:

υn =
υ0
2n

.

As n → ∞, υn → 0. Thus, the fixed point of S is υ∗ = 0.

Example 2. Fixed Point in Banach Space with Measure of Noncompactness
Let X = C[0, 1], the Banach space of continuous functions on [0, 1] with the norm:

∥f∥ = max
υ∈[0,1]

|f(υ)|.

Define the measure of noncompactness µ for a bounded set A ⊂ X by:

µ(A) = inf{δ > 0 : A can be covered by a finite number of sets of diameter δ}.

Let S : C[0, 1] → C[0, 1] be defined by:

(Sf)(υ) =
1

2
f(υ).

We verify that S satisfies:
µ(S(A)) ≤ ϕ(µ(A)),

where ϕ(t) = t
2 is non-decreasing with ϕ(t) < t for t > 0.

For f ∈ X, consider the sequence fn+1(υ) = S(fn)(υ). Iterating S gives:

fn(υ) =
1

2n
f0(υ).

As n → ∞, fn(υ) → 0, showing that the fixed point of S is f∗(υ) = 0.

Example 3. Fixed Point in MR-Metric Space with Measure of Noncompactness
Let X = C[0, 1], the Banach space of continuous functions on [0, 1] with the norm:

∥f∥ = max
υ∈[0,1]

|f(υ)|.

Define the MR-metric M : X× X× X → [0,∞) by:

M(f, g, h) = ∥f − g∥+ ∥g − h∥+ ∥h− f∥.

Consider the operator S : C[0, 1] → C[0, 1] defined by:

(Sf)(υ) =
1

2
f(υ).



A. Malkawi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6077 7 of 16

We verify the conditions of the theorem:
1. Contraction Condition in MR-Metric: For all f, g, h ∈ C[0, 1],

M(S(f), S(g), S(h)) = ∥S(f)− S(g)∥+ ∥S(g)− S(h)∥+ ∥S(h)− S(f)∥.

Since S(f)(υ) = 1
2f(υ), we have:

∥S(f)− S(g)∥ =
1

2
∥f − g∥.

Similarly for other terms, giving:

M(S(f), S(g), S(h)) =
1

2
M(f, g, h).

Thus, the contraction constant is k = 1
2 < 1.

2. Measure of Noncompactness Condition: Define the measure of noncompactness µ
on a bounded set A ⊂ C[0, 1] as:

µ(A) = inf{δ > 0 : A can be covered by a finite number of sets of diameter δ}.

For S(A) ⊂ C[0, 1], we compute:

µ(S(A)) = µ

({
1

2
f : f ∈ A

})
.

By the linearity of S and the definition of µ, we get:

µ(S(A)) =
1

2
µ(A).

Define ϕ(t) = 1
2 t, which satisfies ϕ(t) < t for all t > 0 and ϕ(0) = 0. Thus:

µ(S(A)) ≤ ϕ(µ(A)).

Since both conditions are satisfied, S has a unique fixed point f∗ ∈ C[0, 1]. Iteratively
applying S, starting from any f0 ∈ C[0, 1], we find that:

fn(υ) =
1

2n
f0(υ).

As n → ∞, fn(υ) → 0. Therefore, the fixed point of S is f∗(υ) = 0.

3. Applications of the Theorem

The concept of anMR-metric space plays a critical role in the following applications, as
it provides a generalization of the standard metric and enables the handling of noncompact
settings, essential for various fixed-point problems.



A. Malkawi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6077 8 of 16

1. Solving Nonlinear Integral Equations

Example 4. Consider the nonlinear integral equation:

υ(t) =

∫ b

a
K(t, s, υ(s)) ds,

where K : [a, b] × [a, b] × R → R is a continuous function that satisfies a Lipschitz-type
condition with respect to the third variable, i.e., there exists a constant L > 0 such that
for all t, s ∈ [a, b] and υ1, υ2 ∈ R,

|K(t, s, υ1)−K(t, s, υ2)| ≤ L|υ1 − υ2|.

Step 1: Define the Space Let X = C([a, b]), the space of continuous functions on the
interval [a, b], which is a Banach space when equipped with the supremum norm:

∥υ∥∞ = sup
t∈[a,b]

|υ(t)|.

Now, equip X with the MR-metric:

M(υ, ξ,ℑ) = sup
t∈[a,b]

|υ(t)− ξ(t)|+ sup
t∈[a,b]

|ξ(t)−ℑ(t)|.

The MR-metric generalizes the usual metric by considering the distance between three
functions simultaneously, which provides a richer structure to analyze the operator.

Step 2: Define the Operator Define the operator S : X → X by:

(Sυ)(t) =

∫ b

a
K(t, s, υ(s)) ds.

The operator S maps a function υ ∈ X to another function Sυ, obtained by evaluating the
integral.

Step 3: Verify Continuity of S Since K(t, s, υ) is continuous and satisfies the Lipschitz
condition in X, the operator S is well-defined and continuous on X. Specifically, for any
υ, ξ ∈ X,

|(Tυ)(t)− (Tξ)(t)| =
∣∣∣∣∫ b

a

(
K(t, s, υ(s))−K(t, s, ξ(s))

)
ds

∣∣∣∣ .
Using the Lipschitz condition:

|(Sυ)(t)− (Sξ)(t)| ≤
∫ b

a
L|υ(s)− ξ(s)| ds ≤ L∥υ − ξ∥∞(b− a).

Thus, S is a contraction in the supremum norm.
Step 4: Verify Contraction in MR-Metric For υ, ξ,ℑ ∈ X,

M(Sυ, Sξ, Sℑ) = sup
t∈[a,b]

|(Sυ)(t)− (Sξ)(t)|+ sup
t∈[a,b]

|(Sξ)(t)− (Sℑ)(t)|.



A. Malkawi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6077 9 of 16

Using the contraction property derived above:

sup
t∈[a,b]

|(Sυ)(t)− (Sξ)(t)| ≤ k sup
t∈[a,b]

|υ(t)− ξ(t)|,

and similarly:
sup
t∈[a,b]

|(Sξ)(t)− (Sℑ)(t)| ≤ k sup
t∈[a,b]

|ξ(t)−ℑ(t)|.

Therefore:
M(Sυ, Sξ, Sℑ) ≤ k ·M(υ, ξ,ℑ),

where k = L(b− a) < 1 ensures the contraction condition in the MR-metric.
Step 5: Existence and Uniqueness of Fixed Point By the theorem, since S satisfies the

contraction condition in the MR-metric and X is closed, bounded, and convex, there exists
a unique fixed point υ∗ ∈ X such that:

υ∗(t) = (Sυ∗)(t) =

∫ b

a
K(t, s, υ∗(s)) ds.

This fixed point υ∗ is the unique solution to the nonlinear integral equation.

2. Stability of Iterative Processes

Example 5. Consider an iterative scheme represented by the update rule:

υn+1 = S(υn),

where S : X → X is a mapping on a Banach space X equipped with a norm ∥ · ∥. The space
X is further structured with an MR-metric defined as:

M(υ, ξ,ℑ) = ∥υ − ξ∥+ ∥ξ −ℑ∥.

Assumptions: 1. S satisfies a contraction condition in the MR-metric:

M(S(υ), S(ξ), S(ℑ)) ≤ k ·M(υ, ξ,ℑ),

for all υ, ξ,ℑ ∈ X, where k ∈ [0, 1) is a constant.
2. The norm ∥ · ∥ ensures that X is a complete metric space.
Step-by-Step Analysis: Step 1: Understanding the MR-Metric The MR-metric gen-

eralizes the standard metric by simultaneously measuring the ”distances” between three
elements. In this case:

M(υ, ξ,ℑ) = ∥υ − ξ∥+ ∥ξ −ℑ∥.

This metric allows the analysis to track how the distances between successive iterations
evolve in a dynamic system.

Step 2: Contraction Property of S For any υ, ξ,ℑ ∈ X, the contraction condition
ensures:

M(S(υ), S(ξ), S(ℑ)) = ∥S(υ)− S(ξ)∥+ ∥S(ξ)− S(ℑ)∥ ≤ k(∥υ − ξ∥+ ∥ξ −ℑ∥),



A. Malkawi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6077 10 of 16

where k ∈ [0, 1). This guarantees that the operator S ”brings points closer together” under
the MR-metric.

Step 3: Iterative Sequence Start with an initial guess υ0 ∈ X and define the sequence:

υn+1 = S(υn), n ≥ 0.

For any n, consider three successive iterates υn−1, υn, υn+1:

M(υn+1, υn, υn−1) = ∥υn+1 − υn∥+ ∥υn − υn−1∥.

Using the contraction property of S:

M(υn+1, υn, υn−1) ≤ k ·M(υn, υn−1, υn−2).

By induction:
M(υn+1, υn, υn−1) ≤ kn ·M(υ1, υ0, υ−1),

where υ−1 can be taken as the initial condition.
Step 4: Convergence to the Fixed Point Since k ∈ [0, 1), kn → 0 as n → ∞. This

implies:
M(υn+1, υn, υn−1) → 0.

Decomposing M(υn+1, υn, υn−1), we observe that:

∥υn+1 − υn∥ → 0 and ∥υn − υn−1∥ → 0.

Thus, the sequence {υn} is Cauchy in X. Since X is a Banach space, {υn} converges to a
unique point υ∗ ∈ X.

Step 5: Verification of the Fixed Point From the continuity of S, the limit υ∗ satisfies:

υ∗ = lim
n→∞

υn = lim
n→∞

S(υn) = S(υ∗).

Hence, υ∗ is the unique fixed point of S.

3. Optimization Problems

Example 6. Consider an iterative scheme represented by the update rule:

υn+1 = S(υn),

where S : X → X is a mapping on a Banach space X equipped with a norm ∥ · ∥. The space
X is further structured with an MR-metric defined as:

M(υ, ξ,ℑ) = ∥υ − ξ∥+ ∥ξ −ℑ∥.

Assumptions: 1. S satisfies a contraction condition in the MR-metric:

M(S(υ), S(ξ), S(ℑ)) ≤ k ·M(υ, ξ,ℑ),



A. Malkawi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6077 11 of 16

for all υ, ξ,ℑ ∈ X, where k ∈ [0, 1) is a constant.
2. The norm ∥ · ∥ ensures that X is a complete metric space.
Step-by-Step Analysis: Step 1: Understanding the MR-Metric The MR-metric gen-

eralizes the standard metric by simultaneously measuring the ”distances” between three
elements. In this case:

M(υ, ξ,ℑ) = ∥υ − ξ∥+ ∥ξ −ℑ∥.

This metric allows the analysis to track how the distances between successive iterations
evolve in a dynamic system.

Step 2: Contraction Property of S For any υ, ξ,ℑ ∈ X, the contraction condition
ensures:

M(S(υ), S(ξ), S(ℑ)) = ∥S(υ)− S(ξ)∥+ ∥S(ξ)− S(ℑ)∥ ≤ k(∥υ − ξ∥+ ∥ξ −ℑ∥),

where k ∈ [0, 1). This guarantees that the operator S ”brings points closer together” under
the MR-metric.

Step 3: Iterative Sequence Start with an initial guess υ0 ∈ X and define the sequence:

υn+1 = S(υn), n ≥ 0.

For any n, consider three successive iterates υn−1, υn, υn+1:

M(υn+1, υn, υn−1) = ∥υn+1 − υn∥+ ∥υn − υn−1∥.

Using the contraction property of S:

M(υn+1, υn, υn−1) ≤ k ·M(υn, υn−1, υn−2).

By induction:
M(υn+1, υn, υn−1) ≤ kn ·M(υ1, υ0, υ−1),

where υ−1 can be taken as the initial condition.
Step 4: Convergence to the Fixed Point Since k ∈ [0, 1), kn → 0 as n → ∞. This

implies:
M(υn+1, υn, υn−1) → 0.

Decomposing M(υn+1, υn, υn−1), we observe that:

∥υn+1 − υn∥ → 0 and ∥υn − υn−1∥ → 0.

Thus, the sequence {υn} is Cauchy in X. Since X is a Banach space, {υn} converges to a
unique point υ∗ ∈ X.

Step 5: Verification of the Fixed Point From the continuity of S, the limit υ∗ satisfies:

υ∗ = lim
n→∞

υn = lim
n→∞

S(υn) = S(υ∗).

Hence, υ∗ is the unique fixed point of S.



A. Malkawi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6077 12 of 16

4. Game Theory and Economic Equilibria

Example 7. In game theory, consider a strategic game where N players aim to optimize
their individual payoffs. Let X = X1 × X2 × · · · × XN represent the strategy space, where
Xi is the strategy set of player i, assumed to be a closed, bounded, and convex subset of
a Banach space. The goal is to find a Nash equilibrium υ∗ = (υ∗1, υ

∗
2, . . . , υ

∗
N ), where no

player has an incentive to unilaterally deviate.
Best-Response Mapping Define the best-response operator T as:

S(υ) = BestResponse(υ) = (S1(υ), S2(υ), . . . , SN (υ)),

where Si(υ) is the best-response of player i given the strategies of all other players υ−i.
Mathematically:

Si(υ) = argmax
ξ∈Xi

ui(ξ, υ−i),

where ui is the utility function of player i, and υ−i denotes the strategy profile of all players
except i.

MR-Metric Equip the strategy space X with the MR-metric:

M(υ, ξ,ℑ) = ∥υ − ξ∥+ ∥ξ −ℑ∥,

where ∥ · ∥ is a norm defined on X. This metric is particularly useful for capturing the
distances between successive strategy profiles υ, ξ,ℑ during iterative updates.

Assumptions and Contraction Property 1. Convexity of Strategy Sets: The strategy
sets Xi are convex, ensuring the existence of well-defined best responses for each player.

2. Lipschitz Continuity of S: Assume that S satisfies a Lipschitz-type condition in the
MR-metric:

M(S(υ), S(ξ), S(ℑ)) ≤ k ·M(υ, ξ,ℑ),

for some constant k ∈ [0, 1).
3. Existence and Uniqueness: By the conditions of the MR-metric theorem, S has a

unique fixed point υ∗ ∈ X, ensuring a unique Nash equilibrium.
Iterative Process Starting from an initial strategy profile υ0 ∈ X, the sequence {υn} is

generated iteratively using:
υn+1 = S(υn).

The contraction property of S ensures that {υn} converges to the fixed point υ∗, which is
the Nash equilibrium:

S(υ∗) = υ∗.

Interpretation of the Fixed Point At the Nash equilibrium υ∗ = (υ∗1, υ
∗
2, . . . , υ

∗
N ), each

player’s strategy υ∗i satisfies:

ui(υ
∗
i , υ

∗
−i) ≥ ui(ξ, υ

∗
−i), ∀ξ ∈ Xi.

This implies that no player can improve their utility by unilaterally changing their strategy.



A. Malkawi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6077 13 of 16

5. Boundary Value Problems

Example 8. Consider the boundary value problem (BVP):

−u′′(υ) = f(υ, u(υ)), υ ∈ (0, 1),

u(0) = u(1) = 0.

This type of problem often arises in physics and engineering, such as in heat conduction,
elastic deformation, and electrostatics.

Function Space Define the function space X = C([0, 1]), the space of continuous func-
tions on the interval [0, 1], equipped with the MR-metric:

M(u, v, w) = ∥u− v∥∞ + ∥v − w∥∞,

where ∥u− v∥∞ = supυ∈[0,1] |u(υ)− v(υ)| is the supremum norm.
Integral Operator Representation The solution of the boundary value problem can be

represented using an integral operator S, defined as:

(Su)(υ) =

∫ 1

0
G(υ, s)f(s, u(s)) ds,

where G(υ, s) is the Green’s function for the BVP:

G(υ, s) =

{
s(1− υ), if s ≤ υ,

υ(1− s), if s > υ.

The Green’s function satisfies the boundary conditions u(0) = u(1) = 0 and accounts for
the second-order differential operator.

Assumptions for f(υ, u) 1. Continuity: The function f(υ, u) is continuous in both υ
and u, ensuring the well-posedness of the integral operator S.

2. Lipschitz Condition: There exists a constant L > 0 such that:

|f(υ, u1)− f(υ, u2)| ≤ L|u1 − u2|, ∀υ ∈ [0, 1], u1, u2 ∈ R.

This condition ensures that S is a contraction mapping in the MR-metric.
Contraction Property in the MR-Metric Let u, v, w ∈ X. For the operator S, we have:

∥S(u)− S(v)∥∞ = sup
υ∈[0,1]

∣∣∣∣∫ 1

0
G(υ, s)

(
f(s, u(s))− f(s, v(s))

)
ds

∣∣∣∣ .
Using the Lipschitz condition for f :

∥S(u)− S(v)∥∞ ≤ L sup
υ∈[0,1]

∫ 1

0
|G(υ, s)| ds · ∥u− v∥∞.

The boundedness of G(υ, s) ensures that the contraction property is satisfied. Similarly,
for the MR-metric:

M(S(u), S(v), S(w)) = ∥S(u)− S(v)∥∞ + ∥S(v)− S(w)∥∞.



A. Malkawi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6077 14 of 16

Existence and Uniqueness By the MR-metric fixed-point theorem, the contraction prop-
erty of S guarantees that S has a unique fixed point u∗ ∈ X. This fixed point satisfies:

u∗(υ) =

∫ 1

0
G(υ, s)f(s, u∗(s)) ds,

which is the unique solution to the original boundary value problem.
Iterative Approximation Starting with an initial guess u0(υ) ∈ X, the solution can be

approximated iteratively using:

un+1(υ) = (Tun)(υ) =

∫ 1

0
G(υ, s)f(s, un(s)) ds.

The sequence {un} converges to the fixed point u∗(υ) under the MR-metric.
Applications 1. Heat Equation: Modeling steady-state heat distribution in a one-

dimensional rod. 2. Elasticity: Determining the deflection of a beam under a distributed
load. 3. Electrostatics: Solving for potential distributions in one-dimensional domains.

The use of the MR-metric provides a robust framework for analyzing the convergence
of solutions and stability of the problem.

4. Conclusion

This paper introduces new fixed-point theorems in the context of MR-metric spaces,
which extend classical metric space theory. The analysis focuses on continuous self-
mappings S : X → X, where X is a closed, bounded, and convex subset of a Banach
space. The main results establish the existence and uniqueness of fixed points under two
key conditions:

(i) A contraction condition with a constant k ∈ [0, 1).

(ii) A noncompactness condition controlled by a function φ satisfying φ(t) < t for
all t > 0.

These theorems provide a significant generalization of classical fixed-point theory, with
applications spanning multiple fields, including:

• Solutions to nonlinear integral equations

• Stability analysis of iterative methods

• Optimization problems

• Game theory and economic equilibria

• Boundary value problems

The framework of MR-metric spaces offers a versatile and powerful structure for ana-
lyzing noncompact and complex systems, enhancing both theoretical insights and practical
applications.



A. Malkawi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6077 15 of 16

References

[1] M. S. Alsauodi, G. M. Gharib, A. Malkawi, A. M. Rabaiah, and W. A. Shatanawi.
Fixed point theorems for monotone mappings on partial M∗-metric spaces. Italian
Journal of Pure and Applied Mathematics, 44:154–172, 2023.

[2] I. A. Bakhtin. The contraction mapping principle in almost metric spaces. Functional
Analysis, 30:26–37, 1989.

[3] S. Czerwik. Contraction mappings in b-metric spaces. Acta Mathematica et Infor-
matica Universitatis Ostraviensis, 1:5–11, 1993.

[4] Y. J. Cho, P. P. Murthy, and G. Jungck. A common fixed point theorem of Meir
and Keeler type. International Journal of Mathematics and Mathematical Sciences,
16(4):669–674, 1993.

[5] R. O. Davies and S. Sessa. A common fixed point theorem of Gregus type for compat-
ible mappings. Facta Universitatis, Series: Mathematics and Informatics, 7:51–58,
1992.

[6] B. C. Dhage. Generalized metric spaces and mappings with fixed points. Bulletin of
the Calcutta Mathematical Society, 84:329–336, 1992.

[7] G. Gharib, A. Malkawi, A. Rabaiah, W. Shatanawi, and M. Alsauodi. A common
fixed point theorem in M∗-metric space and an application. Nonlinear Functional
Analysis and Applications, 27(2):289–308, 2022.

[8] A. Malkawi, A. Tallafha, and W. Shatanawi. Coincidence and fixed point results for
(Ψ, L)-M -weak contraction mapping on Mb-metric spaces. Italian Journal of Pure
and Applied Mathematics, 47:751–768, 2022.

[9] A. Malkawi, A. Tallafha, and W. Shatanawi. Coincidence and fixed point results
for generalized weak contraction mapping on b-metric spaces. Nonlinear Functional
Analysis and Applications, 26(1):177–195, 2021.

[10] T. Qawasmeh, A. Bataihah, K. Bataihah, A. Qazza, and R. Hatamleh. Nth composite
iterative scheme via weak contractions with application. International Journal of
Mathematics and Mathematical Sciences, 2023:6953499, 2023.

[11] A. Bataihah and T. Qawasmeh. A new type of distance spaces and fixed point results.
Journal of Mathematical Analysis, 15(4):81–90, 2024.

[12] A. Bataihah and T. Qawasmeh. A new type of distance spaces and fixed point results.
Journal of Mathematical Analysis, 15(4):81–90, 2024.

[13] W. Shatanawi, T. Qawasmeh, A. Bataihah, and A. Tallafha. New contractions and
some fixed point results with application based on extended quasi b-metric spaces.
U.P.B. Scientific Bulletin, Series A, 83(2):53–64, 2021.

[14] T. Qawasmeh, W. Shatanawi, A. Bataihah, and A. Tallafha. Fixed point results and
(α, β)-triangular admissibility in the frame of complete extended b-metric spaces and
application. U.P.B. Scientific Bulletin, Series A, 83(1):113–124, 2021.

[15] A. Bataihah, W. Shatanawi, and A. Tallafha. Fixed point results with simulation
functions. Nonlinear Functional Analysis and Applications, 25(1):13–23, 2020.

[16] K. Abodayeh, W. Shatanawi, A. Bataihah, and A. H. Ansari. Some fixed point and
common fixed point results through Ω-distance under nonlinear contractions. Gazi



A. Malkawi / Eur. J. Pure Appl. Math, 18 (2) (2025), 6077 16 of 16

University Journal of Science, 30(1):293–302, 2017.
[17] A. Bataihah, A. Tallafha, and W. Shatanawi. Fixed point results with Ω-distance

by utilizing simulation functions. Italian Journal of Pure and Applied Mathematics,
43:185–196, 2020.

[18] K. Abodayeh, A. Bataihah, and W. Shatanawi. Generalized Ω-distance mappings
and some fixed point theorems. U.P.B. Scientific Bulletin, Series A, 79(4):223–232,
2017.

[19] T. Qawasmeh, W. Shatanawi, and A. Bataihah. Common fixed point results for ratio-
nal (α, β)ϕ-mω contractions in complete quasi metric spaces. Mathematics, 7(5):392,
2019.

[20] A. Rabaiah, A. Tallafha, and W. Shatanawi. Common fixed point results for map-
pings under nonlinear contraction of cyclic form in b-metric spaces. Advances in
Mathematics: Scientific Journal, 10(2):289–301, 2021.

[21] A. Rabaiah, A. Malkawi, A. Al-Rawabdeh, D. Mahmoud, and M. Qousini. Fixed point
theorems in MR-metric space through semi-compatibility. Advances in Mathematics:
Scientific Journal, 10(6):2831–2845, 2021.

[22] B. E. Rhoades. A fixed point theorem for generalized metric spaces. International
Journal of Mathematics and Mathematical Sciences, 19(1):145–153, 1996.

[23] S. Sedghi, D. Turkoglu, and N. Shobe. Common fixed point theorems for six weakly
compatible mappings in D∗-metric spaces. Thai Journal of Mathematics, 7(2):381–
391, 2009.

[24] A. Branciari. A fixed point theorem for mappings satisfying a general contractive
condition of integral type. International Journal of Mathematics and Mathematical
Sciences, 29(9):531–536, 2002.

[25] T. Qawasmeh. (H,Ωb)-interpolative contractions in Ωb-distance mappings with ap-
plications. European Journal of Pure and Applied Mathematics, 16(3):1717–1730,
2023.

[26] K. Abodayeh, T. Qawasmeh, W. Shatanawi, and A. Tallafha. ϵφ-contraction and some
fixed point results via modified ω-distance mappings in the frame of complete quasi
metric spaces and applications. International Journal of Electrical and Computer
Engineering, 10(4):3839–3853, 2020.

[27] T. Qawasmeh, A. Tallafha, and W. Shatanawi. Fixed and common fixed point the-
orems through modified ω-distance mappings. Nonlinear Functional Analysis and
Applications, 24(2):221–239, 2019.

[28] A. Malkawi, A. Rabaiah, W. Shatanawi, and A. Tallafha. MR-metric spaces and an
application. Preprint, 2021.

[29] W. Rudin. Functional Analysis. McGraw-Hill, New York, 2 edition, 1991.
[30] H. L. Royden and P. M. Fitzpatrick. Real Analysis. Pearson, Upper Saddle River,

NJ, 4 edition, 2010.