Here we study the existence of solution and its continuous dependence of a constrained problem of a nonlinear functional integral equation subject to the constraint of the initial value problem of the pantograph differential equation. The Hyers-Ulam stability of the problem will be proved.
Here we study the existence of a unique solution y ϵ C[0,T] of (1) where the function u is the solution of the initial value problem of the pantograph equation
du/dt=f2 (t,u(t),u(ɤt)), a.e. t ϵ (0,T] and u(0)=u0 . (2)
Here, we prove the existence of a unique solution u ϵ C [0, T] of the problem (2) and study the continuous dependence of the solution u on ɤ and u_0.
Secondly, we prove the existence of a unique solution of the integral equation (1) and study the continuous dependence of y on u, β, λ.
Finally, we study Hyers-Ulam stability of our problem (1) , (2).