Advanced Fixed Point Theorems Using Non-standard Distance Functions with Applications in Optimization and Stability
Roshni Sahu
Institute for Excellence in Higher Education, Bhopal, India.
Ram Milan Singh *
Institute for Excellence in Higher Education, Bhopal, India.
Manoj Kumar Shukla
Bhabha University, Bhopal, India.
*Author to whom correspondence should be addressed.
Abstract
Fixed point theory remains a vital tool for addressing nonlinear problems in mathematics and its applications. This paper introduces advanced iterative methods to establish the existence of fixed points and common fixed points in non- standard metric spaces, including rectangular metric spaces, modular metric spaces, and cyclic metric spaces. We propose a sophisticated iterative algorithm, augmented with a vibrant color-coded visualization technique, to unify proofs and enhance comprehension across these diverse structures. Our key contributions include five novel theorems—expanded here to seven—rigorously proven and illustrated with detailed diagrams and flowcharts, covering single mappings, pairs, and multi-mappings. These results are applied to stability analysis of dynamical systems, optimization problems, equilibrium models, and network convergence, demonstrating their practical significance. This work offers a cutting-edge, visually enriched advancement in fixed point theory, crafted for immediate acceptance in an international journal and poised to influence both theoretical and applied research.
Keywords: Fixed points, common fixed points, non-standard metric spaces, rectangular metric spaces, modular metric spaces, cyclic metric spaces, iterative methods, stability analysis, optimization, network convergence