Singularity in nonlinear systems: differential inclusion model for the standard and transformed fractional pantograph equation.

This work develops the first unified differential inclusion model for singular fractional pantograph equations, a class of problems that simultaneously exhibit proportional delays, memory effects, singular behavior, and modeling uncertainties. While fractional pantograph models and singular differential inclusions have been studied independently, their combination has not been addressed in the literature. We address this gap by establishing a comprehensive existence theory for two classes of singular fractional inclusion problems: a standard formulation and a transformed singular formulation that absorbs strong singularities through appropriately weighted function spaces. Our methodology relies on fixed-point theory for multivalued maps and employs the Pompeiu-Hausdorff metric together with θ-δ contraction techniques to accommodate set-valued nonlinearities and non-smooth solution behaviors. Three numerical examples illustrate the applicability of the theoretical results. The examples demonstrate the validity of the derived conditions, the role of weighting transformations in handling strong singularities, and the emergence of entire solution families characteristic of differential inclusions. These findings highlight the flexibility of the proposed framework for modeling multiscale systems with proportional delays, scaling phenomena, and uncertain dynamics. The proposed approach provides a versatile foundation for analyzing multiscale systems and opens avenues for future work on stability, numerical schemes, and variable-order extensions.