Existence and multiplicity of solutions for fuzzy fractional order initial value problems on time scales

This paper establishes a new theoretical framework for analyzing fuzzy fractional dynamic equations on time scales. We introduce a novel Caputo–Hukuhara-type fractional delta derivative for fuzzy-valued functions, which simultaneously incorporates memory effects, uncertainty quantification, and hybrid time domains. Our study focuses on the initial value problem for a resulting nonlinear equation under general growth conditions. The problem is reformulated using a fixed-point approach with a decomposition into two operators, P and Q. We employ two distinct fixed-point theorems to obtain our main results: first, using Krasnoselskii's fixed point theorem, we prove the existence of at least one nonnegative solution. Second, by applying a more sophisticated fixed-point theorem for expansive and completely continuous operators, we establish the existence of at least two distinct nonnegative solutions. The paper provides complete verification of all required conditions for both theorems. These results represent a significant contribution to the emerging field of fuzzy fractional calculus on time scales, offering new existence and multiplicity theory for this class of equations.