Fixed‐Point Theorems for α‐ψ‐Contractions in Fuzzy b‐Metric Spaces and Applications

ABSTRACT This article introduces novel fixed‐point theorems for a generalized class of ‐ contractive mappings within complete fuzzy ‐metric spaces. Our results significantly extend and unify several known theorems from the literature, including standard fuzzy contractive principles and their ‐metric variants. The key innovations lie in the introduction of the ‐ framework, which provides a more flexible contraction condition, and the integration of auxiliary structures such as partial orders and graphs into these spaces. We further establish comprehensive fixed‐point theorems for cyclic contractive mappings. To demonstrate the practical relevance of our theoretical contributions, we apply these theorems to solve a nonlinear integral equation and a fractional differential equation, showcasing their effectiveness in addressing real‐world problems. Our work provides a substantial extension of fixed‐point theory in fuzzy ‐metric spaces, offering valuable tools for researchers working with generalized contractive mappings and their diverse applications.