On the Pantograph functional equation | ||
| Electronic Journal of Mathematical Analysis and Applications | ||
| Volume 12, Issue 1, January 2024, Pages 1-12 PDF (498.92 K) | ||
| Document Type: Regular research papers | ||
| DOI: 10.21608/ejmaa.2024.249812.1094 | ||
| Authors | ||
| Malak Ba-Ali* 1; Ahmed M. A. El-Sayed2; Eman Hamdallah3 | ||
| 1Faculty of Science, Princess Nourah Bint Abdul Rahman University,\\ Riyadh 11671, Saudi Arabia | ||
| 2Department of Mathematics, Faculty of Science, Alexandria University, Egypt. | ||
| 3Faculty ~of ~Science, Alexandria~University, ~Alexandria, ~Egypt | ||
| Abstract | ||
| 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. | ||
| Keywords | ||
| Pantograph equation; Schauder fixed point Theorem; existence of solutions; continuous dependence; Hyers–Ulam stability | ||
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