Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

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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.

Stability of Hahn difference equations in Banach algebras

Marwa M. Abdelkhaliq, Alaa E. Hamza

Pyramids Higher Institute for Engineering and Technology, Cairo University

Abstract

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

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