Results for Fuzzy Generalized Contractive Mappings Tailored with Directed Graph

Abstract

This paper introduces a novel class of graphic fuzzy generalized rational F-contractive mappings in controlled metric spaces endowed with a directed graph structure. Using these mappings, we establish the existence of fuzzy coincidence and fuzzy fixed points. Several significant consequences are derived for single-valued mappings in controlled metric spaces endowed with a directed graph structure. Similarly, important results are established for set-valued mappings in controlled metric spaces, tailored with a directed graph structure. To demonstrate the applicability of our results, we provide two examples in which all weights are explicitly computed for n = 4, confirming the existence of fuzzy coincidence and fixed points and the applicability of the proposed framework. Furthermore, we establish the existence of solutions for fractional differential equations and integral equations within the controlled metric space framework, highlighting the practical relevance of our approach. Our results generalize and extend several existing fixed point and fuzzy fixed point theorems in the literature, recovering many known cases as special instances.