This research paper focuses on the definition of the pantograph functional equation and the existence of its solutions in two cases: firstly, the existence of solution $x \in C[0,T]$, we employ we use the technique of the Banach fixed point theorem and, secondly, the existence of solution $x \in L_1[0,T]$, in this case we use Schauder fixed point Theorem. In both cases we study the continuous dependence of the unique solution on the Pantograph functional equation. Furthermore, we delve into the study of the Hyers–Ulam stability. Additionally, we give an example to illustrate our outcomes.
It is well known that the pantograph differential equations create an important branch of nonlinear analysis and have numerous applications in describing of miscellaneous real world problems. For papers studying such kind of problems (see \cite{122,123,124}) and therein.\\
Pantograph differential equations have been studied in many papers and monographs \cite{125,126}.\\

Here, we define the pantograph functional equation with parameter as
\begin{eqnarray}\label{eq1}
x(t) = f\bigg(t,~x(t), ~\lambda ~x(\gamma t)~\bigg), ~~t \in [0, T].
\end{eqnarray}
where $\lambda> 0$ and $\gamma \in (0,~1]$. Our aim here is to establish the solvability of the solution $x \in C[0, T]$ and $x \in L_1[0, T]$ of (\ref{eq1}). Furthermore, the continuous dependence of the unique solution on the function $f$, $\gamma$ and on the parameter $\lambda> 0$ will be proved. The Hyers – Ulam stability of (\ref{eq1}) will be studied.