Fixed Point Theory in MR-Metric Spaces Fundamental Theorems and Applications to Integral Equations and Neutron Transport
DOI:
https://doi.org/10.29020/nybg.ejpam.v18i3.6440Keywords:
MR-metric spaces, fixed point theory, Banach contraction principle, integral equations, neutron transport theory, neural network optimizationAbstract
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 four
fundamental 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 contractions
The theoretical results are applied to:
• Nonlinear integral equations in neutron transport theory
• Optimization problems in neural networks
• Boundary value problems for nonlinear ODEs
Key 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.
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Copyright (c) 2025 Tariq Qawasmeh, Abed Al-Rahman Malkawi
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