A constrained problem of state dependent pantograph functional equation constrained by its conjugate

‎In this paper‎, ‎we define the state-dependent pantograph functional equation and study the existence and uniqueness of its solution and prove some data dependence theorems‎, ‎then we investigate the existence of the solution of a constrained problem of the state-dependent pantograph functional equation constrained by its conjugate equation‎. ‎Moreover‎, ‎we demonstrate the continuous dependence of the solution‎. ‎We also examine the Hyres-Ulam stability of our problem‎.