On nonlinear $(p,q)$-fractional hybrid pantograph equations: Existence, and uniqueness

We consider a nonlinear class of hybrid pantograph equations governed by Caputo-type $(p,q)$-fractional derivatives and proportional delay arguments. By rewriting the problem in an equivalent integral form, we first established the existence of solutions through Dhage's hybrid fixed point theorem in a Banach algebra. Under additional Lipschitz conditions, the Banach contraction principle was employed to guarantee the existence and uniqueness of solutions. The results generalize and extend studies on fractional and hybrid pantograph equations within the $(p,q)$-fractional setting.