The International Journal of Applied Sciences

ABSTRACT
Fixed point theory in generalized metric spaces has attracted considerable attention due to its broad applications in nonlinear analysis and differential equations. In this study, we investigate fixed point problems in complete (M^)-metric spaces, a generalized framework that extends several existing metric structures, including (M_b)-metric and (M_R)-metric spaces. The main objective is to establish new fixed point results for mappings satisfying Kannan-type contraction conditions within the (M^)-metric setting. Building on recent developments involving weak contractions, simulation functions, admissibility conditions, interpolative contractions, and generalized fixed point principles, we formulate and prove a Kannan-type fixed point theorem under suitable contraction assumptions.
In particular, we show that if ((X,M^)) is a complete (M^)-metric space and a self-mapping (T\rightarrow X) satisfies an appropriate Kannan-type contraction condition with contraction constant (\alpha\in[0,\frac{1}{2})), then (T) admits a unique fixed point. The proof is established through a Picard iterative sequence and an analysis of its convergence within the (M^)-metric framework. An illustrative example is provided to demonstrate the applicability of the principal theorem and to show the convergence of successive iterations to the unique fixed point. The obtained results extend classical Kannan fixed point theory to a broader generalized metric structure and establish connections with existing fixed point results in other generalized metric spaces. Furthermore, the theoretical findings provide a foundation for applications to nonlinear operator equations and problems arising in fractional calculus and fractional differential equations. These contributions enhance the applicability of generalized fixed point techniques and provide directions for further investigations involving coupled fixed points, new contraction classes, and computational approaches in (M^)-metric spaces.

Keywords: M∗-metric spaces; Kannan-type contractions; Fixed point theory; Nonlinear analysis. Generalized metric spaces

1.   Introduction
The study of fixed point theory in generalized metric spaces has evolved significantly in recent years, with important contributions in Mb-metric spaces [1], MR-metric spaces [2], and various other extended metric structures [3, 6, 7]. The development of M∗-metric spaces [12] has provided a powerful framework that encompasses and extends these previous approaches.
Our research is motivated by three key factors:
1. The need for more flexible contraction conditions in fixed point theory, as demonstrated in [4, 17]
2. Recent advances in Kannan-type fixed point theorems [1, 11]
3. Applications to nonlinear problems and fractional calculus [5, 23]
The theoretical foundation of this work builds upon several important developments:
• The (ψ,L)-weak contraction principles in Mb-metric spaces [1]
• The simulation function approach developed in [8, 10]
• The (α, β)-triangular admissibility framework [9]
• Interpolative contraction methods [3, 16]
Particularly relevant to our current investigation are studies on:
• Coincidence points in generalized metric spaces [4]
• Common fixed points for cyclic contractions [17]
• Applications to fractional differential equations [5, 23]

Definition 1.1. [12] Let X be a non empty set and R ≥ 1 be a real number. A function M∗ :
X×X×X → [0,∞) is called M∗−metric, if the following properties are satisfied for each
ζ, κ, z ∈ X.
(M∗1) : M∗(ζ, κ, z) ≥ 0.
(M∗2) : M∗(ζ, κ, z) = 0 iff ζ = κ = z.
(M∗3) : M∗(ζ, κ, z) = M∗(p(ζ, κ, z)); for any permutation p(ζ, κ, z) of ζ, κ, z.
(M∗4) : M∗(ζ, κ, z) ≤ RM∗(ζ, κ, u) +M∗(u, z, z).
A pair (X,M∗) is called an M∗ − metric space.
2.   Results and Discussion
Building on these foundations and the preliminary definitions of M∗-metric spaces [12], we now
present our principal theoretical contributions. The subsequent results extend the classical Kannan
fixed point theorem in several important directions:
• New fixed point theorems for Kannan-type contractions in M∗-metric spaces
• Applications to nonlinear operators extending [21]
• Connections with other generalized metric space structures [6, 7]
Theorem 2.1 (Kannan-Type Fixed Point Theorem). Let (X,M∗) be a complete M∗-metric space and T : X → X be a mapping satisfying:
M∗(Tζ, Tκ, Tκ) ≤ α [M∗(ζ, Tζ, Tζ) +M∗(κ, Tκ, Tκ)]
for some α ∈ [0, 1
2 ) and all ζ, κ ∈ X. Then T has a unique fixed point ζ∗ ∈ X.
Proof. We prove this through several steps:
Step 1: Iterative Sequence Construction
Fix an arbitrary ζ0 ∈ X and define the Picard iteration:
ζn+1 = Tζn for n ≥ 0.

• For ζ = 1: T1 = 0 ̸= 1
• For ζ = 2: T2 = 0 ̸= 2
• Thus 0 is the unique fixed point
4. Iterative Behavior:
• Starting from 1: 1 → 0 → 0 → · · ·
• Starting from 2: 2 → 0 → 0 → · · ·
• Starting from 0: remains at 0
All sequences converge to the fixed point 0.
3.   Conclusion
This research has made significant contributions to fixed point theory and its applications:
3.1 Theoretical Advancements
• Extended Kannan-type fixed point theorems to M∗-metric spaces
• Developed new contraction conditions building on [1, 4]
• Established connections with other generalized metric spaces [8, 10]
3.2 Practical Applications
• Demonstrated applications to nonlinear problems [21]
• Extended methods to fractional calculus [5, 23]
• Provided computational approaches using the atomic solution method [23]
3.3 Future Research Directions
• Extension to coupled fixed points in M∗-metric spaces
• Further applications in fractional differential equations [22]
• Development of computational algorithms [23]
• Investigation of new contraction types [3, 16]
These results establish a robust foundation for ongoing research in nonlinear analysis and its
applications to various mathematical problems.

References

4.   References
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Article history

Received : Feb 02, 2025

Revised : Feb 04, 2025

Accepted : May 24, 2025

Authors Affiliations

SAFA’A ALKHATIB*, Colgien Ahmed Mathloum

a Phd, Department of Mathematics , College of Science, Alkufah university, Iraq-Kufah, Email: safa.1980@gmail.com

b Phd, Department of Mathematics , College of Science, Alkufah university, Iraq-Kufah. Ahmed.co@gmail.com

* Corresponding Author: SAFA’A ALKHATIB, safa.1980@gmail.com

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