Analysis of delay differential equations with dual caputo-type fractional derivatives using laplace transform methods.

This paper investigates a class of initial value problems arising in the modeling of systems with memory and delayed feedback, described by nonlinear fractional differential equations with finite delay, governed by the generalized Caputo-Katugampola fractional derivative. The presence of the parameter [Formula: see text] in this operator allows interpolation between different fractional behaviors and provides additional flexibility in modeling the intensity and scaling of memory effects in delay systems. By employing ρ-Laplace transform techniques, we first derive an equivalent integral formulation of the considered problem. We then establish the existence and uniqueness of solutions to the proposed Cauchy problem by employing the Banach contraction principle and Schauder's fixed point theorem. The use of both fixed-point approaches enables us to address existence and uniqueness under complementary sets of assumptions, thereby enlarging the class of admissible nonlinearities. Moreover, we examine the Ulam-Hyers stability of the solutions under suitable conditions, demonstrating that small variations in the initial data result in proportionally small deviations in the solution. This stability property reflects the robustness of the model with respect to perturbations and is closely related to the contraction condition imposed on the associated operator. To illustrate the theoretical results and confirm the applicability of the method, numerical examples are provided and discussed. The numerical simulations are carried out using the L1 scheme, which is known for its stability and effectiveness in approximating Caputo-type fractional derivatives.