Generalized Fixed Point Theorems in Nonconvex Spaces with Applications to Complex Functional Systems

Abstract

Fixed point theory has been a cornerstone of nonlinear analysis and functional systems. However, classical results often rely on convexity and compactness assumptions, limiting their applicability to real-world nonconvex problems. This study developed generalized fixed point theorems for nonconvex metric and topological spaces, employing admissibility conditions, control functions, and relaxed contraction principles to extend classical frameworks. The research adopted a deductive and analytical methodology, formulating new theorems, proving existence and uniqueness of fixed points, and analyzing convergence behavior of iterative sequences. The generalized results were applied to various complex functional systems, including nonlinear integral equations, functional differential equations, coupled operator systems, and fractional equations. Comparative analyses demonstrated that the generalized framework preserved solution existence, ensured uniqueness, enhanced stability, and maintained convergence without relying on convexity or compactness. Tables and figures highlighted improvements over classical theorems in terms of convergence reliability, perturbation tolerance, iterative efficiency, and applicability to diverse functional systems. The findings confirmed that generalized fixed point theory provided a robust and versatile analytical tool capable of addressing practical problems in applied mathematics and computational modeling. The study contributes to theoretical literature by broadening the scope of fixed point theorems and offers a foundation for future research in stochastic, hybrid, and high-dimensional systems.