Commun. Korean Math. Soc. 2018; 33(4): 1141-1158
Online first article October 19, 2018 Printed October 31, 2018
https://doi.org/10.4134/CKMS.c170135
Copyright © The Korean Mathematical Society.
Marwa M. Abdelkhaliq, Alaa E. Hamza
Pyramids Higher Institute for Engineering and Technology, Cairo University
Hahn difference operator $D_{q,\omega}$ which is defined by \begin{equation*} D_{q,\omega}g(t)= \left\{ \begin{array}{ll} \frac {g(qt+\omega)-g(t)}{t(q-1)+\omega},&\text{if} \ \ t\neq \theta:=\frac{\omega}{1-q},\\[0.5em] g^{\prime}(\theta),&\text{if}\ \ t=\theta \end{array} \right. \end{equation*} received a lot of interest from many researchers due to its applications in constructing families of orthogonal polynomials and in some approximation problems. In this paper, we investigate sufficient conditions for stability of the abstract linear Hahn difference equations of the form \begin{equation*} D_{q,\omega}x(t)=A(t)x(t)+f(t),~ t\in I, \end{equation*} and \begin{equation*} D^{2}_{q,\omega}x(t)+A(t)D_{q,\omega}x(t)+R(t)x(t)=f(t), ~t\in I, \end{equation*} where $A,R:I\rightarrow \mathbb{X}$, and $f:I\rightarrow \mathbb{X}$. Here $\mathbb{X}$ is a Banach algebra with a unit element $\mathfrak{e}$ and $I$ is an interval of $\mathbb{R}$ containing $\theta$.
Keywords: Hahn difference operator, Jackson $q$-difference operator, stability theory
MSC numbers: Primary 39A13, 39A70
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd