A Darbo Fixed-Point Approach for Variable-Order Hybrid Caputo–Hadamard Sequential Differential Equations

In this study, we discuss the boundary value problem involving a hybrid Caputo–Hadamard fractional sequential differential equation with variable order. The use of a variable fractional order is motivated by the fact that, in many applications, memory effects and system behavior change over time rather than remaining constant. Therefore, fractional models with constant order may be inadequate for describing the dynamics of complex systems. Our primary tool for constructing the results is the Banach space of continuous functions. Existence and uniqueness results are obtained by applying Darbo’s fixed-point theorem in combination with Kuratowski’s measure of noncompactness. This approach allows us to deal with operators that are not compact while still ensuring the existence of solutions. The hybrid concept, in which Caputo and Hadamard fractional derivatives appear sequentially, offers a flexible formulation that can describe a wider class of problems than models involving a single fractional operator. A key idea in the approach is reformulating the considered boundary value problem as an equivalent operator equation. This representation enables the direct application of fixed-point theorems and facilitates a discussion of solution uniqueness under additional suitable assumptions that are easy to verify rationally. The stability of solutions is also examined. Using the concept of Ulam–Hyers stability, it is demonstrated that small perturbations in the problem data result in bounded variations in the corresponding solutions. To illustrate the theoretical results, an example supported by numerical computations is included. The results indicate that the proposed technique is suitable for problems in which system properties evolve with time.