Graph-Based Fixed Points in 𝐹-Metric Spaces: A New Approach 🔢🔗 | #Sciencefather #researchers #Metric

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$Fixed point theory is a cornerstone of modern mathematics, influencing fields like functional analysis, topology, differential equations, and numerical analysis. In this study, we explore new fixed point results in F-metric spaces equipped with graph structures, presenting novel contraction conditions that extend classical results to more generalized spaces.$

 Mathematical Foundation: Fixed Points in F-Metric Spaces

In standard metric spaces (X,d), a function f:X→X has a fixed point x∗ if:

f(x∗)=x∗

A fundamental result in this area is Banach’s Contraction Principle, which guarantees the existence of a unique fixed point for contractive mappings. However, this principle does not directly apply to F-metric spaces, where the metric function takes a more generalized form:

F:X×X×X→R+

subject to specific axioms, generalizing distance measurement beyond traditional metrics.

This study extends fixed point theorems to F-metric spaces by introducing graphic contractions, where mappings satisfy contractive conditions along the edges of an associated graph G(X,E). This combination of graph theory and fixed point analysis enables us to study convergence in spaces with complex connectivity patterns.

 Graph-Based Contraction Mappings

Consider a directed graph G(X,E), where:

• Nodes represent elements of the F-metric space X.

• Edges define the contraction conditions between pairs of points.

A mapping f:X→X is G-contractive if there exists α∈(0,1) such that:

$                                                                                                                            F(f(x),f(y),z)≤α⋅F(x,y,z)∀(x,y)∈E$

where F satisfies specific contraction properties. The graph constraint allows us to refine classical results, ensuring convergence even when the entire space is not contractive.

 Real-World Applications: Fixed Points & Graph Theory

Fixed point theory is essential in solving nonlinear equations, optimization problems, and dynamic systems. Some key applications include:

 Mathematical Modeling  – Used in complex systems with non-traditional distances.
 Computational Methods  – Convergence of iterative algorithms in machine learning.
 Physics & Engineering  – Stability analysis in chaotic and dynamical systems.
 Network Theory  – Used in graph-based data structures and communication networks.

 Solving Fractional Differential Equations (FDEs) with Fixed Points

Fractional calculus extends classical derivatives to non-integer orders, leading to fractional differential equations (FDEs) of the form:

Dαy(t)=f(t,y(t))

where Dα represents the fractional derivative. These equations model memory-dependent and non-local processes in physics, biology, and finance.

By applying F-metric space fixed point results, we establish existence and uniqueness criteria for solutions to boundary value problems involving FDEs. This generalization provides new insights into long-term stability and convergence in systems with fractional dynamics.

 Future Directions & Mathematical Impact

This research bridges fixed point theory, graph structures, and fractional calculus, paving the way for new advances in:

 Nonlinear Analysis & Functional Equations 
 Mathematical Optimization & Numerical Methods 
 Graph-Based Computational Models 

By extending fixed point results to F-metric spaces with graphs, this study contributes to the broader mathematical landscape, opening new doors for mathematical modeling, applied analysis, and theoretical advancements.

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