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 continuous mappings \( S: \mathbb{X} \to \mathbb{X} \), where \( \mathbb{X} \) is a closed, bounded, and convex subset of a Banach space \( (E, \|\cdot\|) \). The study establishes that if \( S \) satisfies a contractive condition involving the \( MR \)-metric with a constant \( k \in [0, 1) \), and a measure of noncompactness condition governed by a function \( \phi \) where \( \phi(t) < t \) for \( t > 0 \), then \( S \) possesses a unique fixed point \( \upsilon^* \). The findings have significant 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.

In this paper, we present some important results for self-mappings in $MR-metric$ spaces, ensuring the existence and uniqueness of fixed points in contraction mappings. The study reveals the important role of contraction properties in achieving convergence. The paper continues to develop these concepts by providing concrete examples that demonstrate their importance in measurable spaces. Finally, these results lay the foundation for future investigations into the stability of fixed points and their applications via mathematical frameworks applicable to real life.

This paper explores the fundamental connections between Neutrosophic MR-Metric Spaces (NMR-MS) and classical Fuzzy Metric Spaces (FMS). We present three key theoretical contributions: (1) an embedding theorem showing how any FMS can be systematically incorporated into an NMR-MS framework, (2) a fixed point theorem for contraction mappings in complete NMR-MS that generalizes the fuzzy Banach contraction principle, and (3) a characterization of sequence convergence in NMR-MS that reveals its stricter requirements compared to FMS. Through concrete examples and applications in machine learning classification, robotic path planning, and medical image reconstruction, we demonstrate how the additional structure of NMR-MS - particularly its explicit handling of truth ($\mathcal{T}$), falsity ($\mathcal{F}$), and indeterminacy ($\mathcal{I}$) components offers enhanced modeling capabilities for uncertain systems. The compatibility conditions between the MR-metric ($M$) and neutrosophic components are shown to be crucial for maintaining theoretical consistency while enabling practical applications.

This paper establishes a comprehensive framework for fixed point theory in MR-metric spaces,a generalization of standard metric spaces that incorporates three-point relations. We present fourfundamental theorems:(1) A Banach contraction principle with optimal contraction constant k < 1/3R(2) A solvability theorem for Fredholm-type integral equations(3) A Krasnoselskii-type hybrid fixed point theorem(4) A Leray-Schauder alternative for generalized contractionsThe theoretical results are applied to:• Nonlinear integral equations in neutron transport theory• Optimization problems in neural networks• Boundary value problems for nonlinear ODEsKey innovations include the development of error estimates in the MR-metric framework and the derivation of precise existence conditions for operator equations. The work bridges theoretical mathematics with practical applications in physics and machine learning.

This paper introduces the concept of MR-metric spaces, a generalization of traditional metric spaces that operates on triples of points rather than pairs. We define MR-metrics and establish their fundamental properties, including non-negativity, identity, symmetry, and a generalized triangle inequality with a constant \( R > 1 \). Three main theorems are presented: (1) a construction of MR-metrics on weighted graphs via minimal spanning subtrees, with applications in network design and VLSI circuit optimization; (2) a set-valued fixed-point theorem for contractions in MR-metric spaces, applied to distributed consensus and fault-tolerant systems; and (3) an MR-metric based on coupling times in expander graphs, with implications for distributed storage and decentralized machine learning. The results are supported by rigorous proofs, illustrative examples, and performance analyses demonstrating practical advantages over traditional methods.

This paper investigates the role of MR-metric spaces in the fields of measure theory and convergence analysis. We analyze how measures defined through a triadic metric function reveal key characteristics such as $\sigma$-finiteness and absolute continuity. Furthermore, we explore the convergence behavior of sequences within the MR-metric framework, highlighting their relevance in areas like stochastic processes, optimization, and data science. The findings offer valuable perspectives on probability distributions that incorporate triadic dependencies, the conditions necessary for stability in machine learning, and the clustering behavior within complex networks. These discoveries pave the way for extending traditional metric space theory into more advanced settings, including those involving multi-agent systems and high-dimensional analyses.