Fixed Point Theory in MR-Metric Spaces Fundamental Theorems and Applications to Integral Equations and Neutron Transport

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.