Application of fixed point theory in artificial intelligence, mathematical modelling and optimization

Abstract

Fixed Point Theory (FPT) constitutes one of the most influential branches of nonlinear analysis, with deep roots in functional analysis, topology, and metric fixed point theory. In the present era of rapidly advancing Artificial Intelligence (AI), iterative computation, optimization-driven learning, and dynamical modelling have become indispensable. Remarkably, these modern computational processes mirror classical fixed point iterations in mathematics. This paper provides an extensive, structured, and unified treatment of the applications of classical and generalized fixed point theorems in machine learning, neural networks, reinforcement learning, fuzzy inference systems, game-theoretic multi-agent learning, mathematical modelling, and optimization theory. We present detailed theoretical discussions, new interpretations of convergence and stability conditions of AI algorithms, rigorous connections between fixed point principles and variational inequality problems, and expanded modelling frameworks in economics, epidemiology, and dynamical systems. This work aims to serve both as a comprehensive reference and a foundational research document bridging fixed point theory with modern intelligent systems.