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Published in:
23-06-2016 | Original Research
Global asymptotic stability of the higher order equation \(x_{n+1} = \frac{ ax_{n}+bx_{n-k}}{A+Bx_{n-k}}\)
Published in: Journal of Applied Mathematics and Computing | Issue 1-2/2017
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Abstract
In this paper, we investigate the local and global stability and the period two solutions of all nonnegative solutions of the difference equation, where a, b, A, B are all positive real numbers, \(k \ge 1\) is a positive integer, and the initial conditions \(x_{-k},x_{-k+1},...,x_{0}\) are nonnegative real numbers. It is shown that the zero equilibrium point is globally asymptotically stable under the condition \(a+b \le A\), and the unique positive solution is also globally asymptotically stable under the condition \(a-b \le A \le a+b\). By the end, we study the global stability of such an equation through numerically solved examples.
$$\begin{aligned} x_{n+1} = \frac{ ax_{n}+bx_{n-k}}{A+Bx_{n-k}} \end{aligned}$$