Some Compatible and Weakly-Compatible Four Self-Mapping Results Approach to Nonlinear Integral Equations in Revised Fuzzy Cone Metric Spaces

Fixed-point theory plays a vital role in mathematical analysis, particularly in metric and fuzzy metric spaces. In recent years, researchers have explored various extensions of fuzzy metric spaces, including the concept of revised fuzzy cone metric (RFCM) spaces, which combine fuzzy metric structures with cone metric space properties. In this study, we investigate the existence of unique common fixed points for four self-mappings in RFCM spaces under specific contraction conditions. Objective The primary goal of this research is to establish unique common fixed-point theorems for four self-mappings that satisfy compatibility and weak compatibility conditions in RFCM spaces. Additionally, we aim to extend these results by proving rational contraction conditions with weaker assumptions on self-mapping continuity. Finally, we demonstrate the applicability of our theoretical results by solving nonlinear integral equations. Method We employ analytical techniques to establish fixed-point theorems in RFCM spaces. The study considers generalized rational contraction conditions and explores both strong and weak continuity assumptions on self-mappings. Our approach involves constructing proofs that leverage the properties of RFCM spaces, including their underlying topological and metric structures. The theoretical findings are then applied to demonstrate the existence of solutions for two nonlinear integral equations, showcasing the practical relevance of RFCM spaces. Result Our results confirm that under generalized rational contraction conditions, unique common fixed points exist for four self-mappings in RFCM spaces. We further demonstrate that even under weaker continuity assumptions, rational contraction results still hold. As an application, we successfully prove the existence of solutions to two nonlinear integral equations, reinforcing the significance of RFCM spaces in solving integral-type operator problems. Conclusion This study establishes fundamental fixed-point theorems in RFCM spaces, expanding upon existing results by incorporating compatibility conditions and rational contractions. By demonstrating the existence of solutions to nonlinear integral equations, we highlight the practical applications of RFCM spaces in mathematical analysis. These findings provide a foundation for further research in fixed-point theory and its applications in functional analysis and differential equations.