(untitled) (European Journal of Pure and Applied Mathematics, 2025)
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.