On the General Solution of x n + 1 = a x n + 1 − 2 k b + c x n + 1 − k x n + 1 − 2 k

A. Alameer, Bashir Al-Hdaibat (), Ahmad M. Adawi and Mohammad A. Safi
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A. Alameer: Department of Mathematics, University of Hafr Al-Batin, Hafr Al-Batin 31991, Saudi Arabia
Bashir Al-Hdaibat: Department of Mathematics, Faculty of Science, The Hashemite University, Zarqa, Jordan
Ahmad M. Adawi: Department of Mathematics, Faculty of Science, The Hashemite University, Zarqa, Jordan
Mohammad A. Safi: Department of Mathematics, Faculty of Science, The Hashemite University, Zarqa, Jordan

Mathematics, 2025, vol. 13, issue 19, 1-16

Abstract: This paper investigates the global dynamics of a broad class of nonlinear rational difference equations given by x n + 1 = a x n + 1 − 2 k b + c x n + 1 − k x n + 1 − 2 k , n = 0 , 1 , … , which generalizes several known models in the literature. We establish the existence of exactly three equilibrium points and show that the trivial equilibrium is globally asymptotically stable when the parameter ratio α = ( b / a ) lies in ( − 1 , 1 ) . The nontrivial equilibria are shown to be always unstable. An explicit general solution is derived, enabling a detailed analysis of solution behavior in terms of initial conditions and parameters. Furthermore, we identify and classify minimal period 2 k and 4 k solutions, providing necessary and sufficient conditions for the occurrence of constant and periodic behaviors. These analytical results are supported by numerical simulations, confirming the theoretical predictions. The findings generalize and refine existing results by offering a unified framework for analyzing a wide class of rational difference equations.

Keywords: difference equations; general solution; periodic solutions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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