Fixed Point Theorems in Complex-Valued b-Suprametric Spaces with Applications to Fractional Differential Equations

In this research article, we introduce and develop the notion of complex-valued b-suprametric spaces as a natural generalization of existing metric-type structures. Fundamental concepts, including convergence, Cauchy sequences, and completeness, are examined in this new setting. We establish new common fixed point theorems for generalized and cyclic rational contractive mappings. The obtained results extend and unify various known fixed point theorems available in the current literature. To demonstrate the applicability and effectiveness of our theoretical findings, illustrative nontrivial examples are provided. As an application, we investigate the existence and uniqueness of solutions for Caputo fractional differential equations, which naturally arise in systems with hereditary and memory effects, particularly in biomedical modeling of viscoelastic biological tissues such as arteries, cartilage, and brain tissue. This demonstrates both the mathematical strength and the practical relevance of the proposed framework.