Stability of a fractional difference equation of high order

Volume 12, Issue 2, pp 65--74 http://dx.doi.org/10.22436/jnsa.012.02.01

Publication Date: October 12, 2018 Submission Date: July 16, 2018 Revision Date: September 24, 2018 Accteptance Date: September 26, 2018

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Authors

M. A. El-Moneam - Mathematics Department, Faculty of Science, Jazan University, Saudi Arabia. Tarek F. Ibrahim - Mathematics Department, College of Sciences and Arts for Girls in sarat Abida, King Khalid University, Saudi Arabia. - Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt. S. Elamody - Mathematics Department, Faculty of Science, Jazan University, Saudi Arabia.

Abstract

In this paper we investigate the local stability, global stability, and boundedness of solutions of the recursive sequence% \[ x_{n+1}=x_{n-p}\ \left( \frac{2\ x_{n-q}\ +a\ x_{n-r}}{x_{n-q}\ +a\ x_{n-r}}% \right), \] where \(x_{-q+k}\ \neq -a\ x_{-r+k} \) for \( k=0,1,\dots,\min (q,r) , a\in \mathbb{R},\ p ,q, r \geq 0\) with the initial condition \(x_{-p},x_{-p+1} ,\dots, x_{-q},\) \(x_{-q+1} ,\dots, x_{-r},x_{-r+1} ,\dots, x_{-1}\) and \(x_{0}\in (0,\infty )\). Some numerical examples will be given to illustrate our results.

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Keywords

• Difference equations
• prime period two solution
• boundedness character
• locally asymptotically stable
• global attractor
• global stability
• high orders

MSC

•  39A10
•  39A11
•  39A99
•  34C99

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