Approximate Controllability Results for Stochastic Pantograph Integro‐Differential Inclusions With Hilfer Fractional Substantial Derivative

This paper studies the existence and approximate controllability for a class of stochastic pantograph‐type integro‐differential inclusions with fractional Hilfer substantial derivative in Hilbert spaces. The existence of mild solutions is established using the Bohnenblust‐Karlin fixed point theorem, in conjunction with techniques from fractional calculus, semi‐group theory, measure of noncompactness, and stochastic analysis. The study of approximate controllability is conducted by presuming that the related linear system exhibits this property, which in turn facilitates the derivation of sufficient criteria for governing the nonlinear system. A specific example is given to demonstrate the effectiveness and practicality of the theoretical findings.