ANALYSIS OF EXISTENCE, UNIQUENESS, AND ULAM-HYERS STABILITY FOR CAPUTO CONFORMABLE FRACTIONAL DIFFERENTIAL PANTOGRAPH EQUATIONS WITH BOUNDARY CONDITIONS

This study investigates a novel class of boundary value problems governed by Caputo conformable fractional differential pantograph equations, characterized by a proportional delay of the form ϕ(γt), with 0 < γ < 1. Such models effectively describe phenomena in physics, engineering, and biologi-cal systems that exhibit both memory effects and self-similar dynamics. We formulate a general nonlinear problem and establish the existence and uniqueness of solutions through fixed-point techniques, including Banachs contraction principle and Schauders fixed-point theorem, under appropriate assumptions on the nonlinear term Φ(x, ϕ(t), ϕ(γt)). Furthermore, the Ulam–Hyers stability of the proposed problem is analyzed, offering insights into the robustness of solutions with respect to perturbations in the initial data.