On the Analysis of Caputo–Hadamard Fractional Differential Equation with Delay and Pantograph

Abstract

This study investigates Caputo–Hadamard fractional differential equations with pantograph delay arguments.Such equations arise in the modeling of systems whose present state depends on both the current value and scaled paststates, reflecting memory and delay effects. The problem is formulated in an appropriate Banach space framework to analyzethe behavior of its solutions. By converting the given fractional differential equation into an equivalent integral equation,sufficient conditions for the existence of solutions are established using fixed-point techniques. Furthermore, the uniquenessof the solution is obtained under suitable Lipschitz-type conditions by applying the Banach contraction principle. Thestability of the system is investigated in the sense defined by Ulam–Hyers, showing that small perturbations in the systemlead to small deviations in the corresponding solutions. In addition, controllability results are derived using a fixed-pointapproach and a numerical illustration is provided to validate the theoretical conclusions.