On the solvability of a delay tempered-fractal differential equation

Document Type : Regular research papers

Authors

1 Department of Mathematics, Faculty of Science, Alexandria University, Egypt.

2 Department of Mathematics, Faculty of Science, Alexandria University, Alexandria, Egypt.

Abstract

In this paper we define the tempered-fractal derivative
$$e^{-\lambda t}\frac{d }{dt^{\beta}} ( f(t) ~e^{\lambda t})$$
and study the initial-value problem of the delay tempered-fractal differential equation
$$e^{-\lambda t}\frac{d }{dt^{\beta}} (x(t)~e^{\lambda t})~=~f(t,x(\phi(t))),~~~~a.e.,~~~t\in (0,T],~~~~~~~x(0)~=~x_o.$$

We discuss the existence of at least one solution $~x \in C[0,T]$. The Uniqueness of the solution will be proved. The continuous dependence on the initial data $x_0$,the delay function $\phi$ and on the function $f$ is proved. The Hyers - Ulam stablity of the problem itself will be established.

This research paper focuses on investigate the existence of solutions for the delay tempered fractal differential problem (5) and properties associated with these solutions. Firstly, we examined the equivalence between the problem (5) and the integral equation (6), then we studied the existence of at least one solution $x \in C(I)$ of (6) by applying Schauder’s fixed point Theorem [3]. Furthermore, we established sufficient conditions to ensure the uniqueness of the solution and its dependence on the initial data $x_0$, the delay function $\phi$ and on the function $f$. We also studied investigated the Hyers-Ulam stability of the problem (5). Finally, we discussed the special cases.

Keywords

Files

Share

How to cite

Statistics