Analysis of Existence, Uniqueness, Stability and Controllability in Pantograph with Caputo–Hadamard Volterra–Fredholm Fractional Integro–Differential Equations

This study addresses the analytical verification that a solution exists and that it is uniquely determined Boundary value problems involving the Caputo–Hadamard fractional operator that contain nonlinear Volterra–Fredholm type integrals and pantograph-type arguments under nonlocal boundary conditions, by making use of strategies that involve constructing upper and lower solutions. By converting the fractional differential equations into an equivalent integral form, a nonlinear operator is defined in a Banach space. Existence of a solution is shown through a fixed point theorem (FPT) argument, and uniqueness is obtained by applying the Banach fixed point theorem under suitable assumptions. The stability of the system is examined in the Ulam–Hyers sense, and controllability is verified using an appropriate fixed point framework. A illustrating example is provided to exhibit the practical relevance of the theoretical results.