Asymptotic Stability of Linear Delay Difference Equations Including Generalized Difference Operator

Year 2019, Volume: 11 , 141 - 148 , 30.12.2019

Murat Gevgeşoğlu , Yaşar Bolat

https://izlik.org/JA56GP22XJ

Abstract

In this study, some necessary and sufficient conditions are given for the stability of linear delay difference equations involving generalized difference operator. For the root analysis Schur-Cohn criteria is used and some examples are given to verify the results.

Keywords

generalized difference operator , Schur Cohn Criteria , stability

References

• Agarwal, R.P., Difference Equations and Inequalities, Marcel Dekker, New York, 2000.
• \v{C}erm\'{a}k, J., J\'{a}nsk\i , J.\ \& Kundr\'{a}t, P., \textit{On necessary and sufficient conditions for the asymptotic stability of higher order linear difference equations}, Journal of Difference Equations and Applications, \textbf{18(11)}(2011), 1781--1800.
• Camouzis, E., Ladas, G., Dynamics of Third Order Rational Difference Equations with Open Problems and Conjectures, Chapman \&Hall, 2008.
• Clark,C. W., \textit{A delay-recruitment model of populations dynamics with application to baleen whale populations}, J. Math. Biol., \textbf{3}(1976), 381--391.
• Dannan, F.M., Elaydi, S., \textit{Asymptotic stability of linear difference equation of advanced type}, J. Comput. Anal. Appl., \textit{6}(2004), 173--187.
• Elaydi, S., An Introduction to Difference Equations, 3nd ed., Springer, 2000.
• Kelley, W.G., Peterson, A.C., Difference Equations. An Introduction with Applications, Academic Press inc, 1991.
• Kuruklis, S.A., \textit{The asymptotic stability of x(n+1) - ax(n) +bx(n-k) = 0}, J. Math. Anal. Appl., \textbf{188}(1994), 719--731.
• Levin, S., May, R., \textit{A note on difference-delay equations}, Theoretical Population Biol., \textbf{9}(1976), 178--187.
• Liz, E., \textit{On explicit conditions for the asymptotic stability of linear higher order difference equations}, J. Math. Anal. Appl., \textbf{303}(2005), 492--498.
• Matsunaga, H., Hara, T., \textit{The asymptotic stability of a two-dimensional linear delay difference equation}, Dynam. Contin. Discrete Impuls. Systems, \textbf{6}(1999), 465--473.
• Matsunaga, H., Ogita, R., Murakami, K., \textit{Asymptotic behavior of a system of higher order linear difference equations}, Nonlinear Analysis, \textbf{47}(2001), 4667-4677.
• Mickens, R.E., Difference Equations, Van Nostrand Reinhold Company, New York, 1990.
• Popenda, J., Szmanda, B., \textit{On the oscillation of solutions of certain difference equations}, Demonstratio Mathematica, \textbf{XVII}(1984), 153--164.
• Popenda, J., \textit{Oscillation and nonoscillation theorems for second-order difference equations}, J. Math. Anal. Appl., \textbf{123(1)}(1987), 34--38.