Fixed-Point Results in Classified Fuzzy Metric Spaces with Applications to Nonlinear Fuzzy Integral Equations

Abstract

This paper develops new fixed-point results by introducing an extended generalized (α,β)–contractive mapping in multiple classified fuzzy metric spaces including GV-FMS, IFMS, PFMS, G-FMS, and b-FMS. New existence and uniqueness theorems are established under weaker contractive conditions, improving upon classical Banach and Ciric-type results. The work also derives conditions for common fixed points of hybrid pair mappings, which have not been previously explored in these classifications simultaneously. Comparative results show that PFMS provides stronger convergence behavior due to probabilistic distance structures, whereas IFMS better handles dual membership uncertainty. The developed fixed-point theorems are then applied to prove the existence of solutions to a class of nonlinear fuzzy Volterra integral equations, demonstrating the practical utility of the theoretical framework. Numerical examples and analytical comparisons show that the newly proposed approach yields faster convergence rates and broader generalizability compared to existing results. This result-based paper contributes both theoretical advancement and applied significance.