Analytic Study and Solutions on Some Classes for Difference Equations

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DOI: https://doi.org/10.28924/2291-8639-24-2026-173
Published: May 29, 2026
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Fahad D. Alrehaili, Elsayed M. Elsayed, Analytic Study and Solutions on Some Classes for Difference Equations, Int. J. Anal. Appl., 24 (2026), 173.

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Fahad D. Alrehaili, Elsayed M. Elsayed

Abstract

This paper is devoted to finding the explicit form of the solutions of the rational difference equation \[x_{n+1} = \frac{x_{n-4}x_{n-6}}{x_{n-1}\bigl(\pm 1 \pm x_{n-4}x_{n-6}\bigr)},\qquad n=0,1,\ldots,\] where the initial conditions \(x_{-6}\), \(x_{-5}\), \(x_{-4}\), \(x_{-3}\), \(x_{-2}\), \(x_{-1}\) and \(x_0\) are arbitrary positive constants. Specific closed-form expressions for the solutions of four distinct special cases of the equation are derived, corresponding to the choices of the signs in the denominator. For the cases with a positive constant term (\(+1\)), the solutions are expressed as infinite products whose factors depend on the initial data in a structured periodic pattern. For the cases with a negative constant term (\(-1\)), the solutions are shown to be periodic with period 20. In all four cases, the unique equilibrium point \(\bar{x}=0\) is identified and its local asymptotic stability is analyzed; it is proven that the zero equilibrium is not locally asymptotically stable. Numerical simulations are provided to illustrate the theoretical results and to confirm the non-convergence behavior of the solutions.

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