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Title: Dynamics of higher order rational difference equation xn+1= (α+βxn)/(A+Bxn+Cxn−k)
Authors: Abu Alhalawa, Muna
Saleh, Mohammad
Keywords: Differentiable dynamical systems;Stability;Invariants;Difference equations;Oscillations;Nonlinear difference equations - Numerical solutions
Issue Date: 2017
Abstract: The main goal of this paper is to investigate the periodic character, invariant intervals, oscillation and global stability and other new results of all positive solutions of the equation xn+1 = α + βxn A + Bxn + Cxn−k , n = 0, 1, 2, . . . , where the parameters α, β, A, B and C are positive, and the initial conditions x−k, x−k+1, . . . , x−1, x0 are positive real numbers and k ∈ {1, 2, 3, . . .}. We give a detailed description of the semi-cycles of solutions and determine conditions under which the equilibrium points are globally asymptotically stable. In particular, our paper is a generalization of the rational difference equation that was investigated by Kulenovic et al. [The Dynamics of xn+1 = α+βxn A+Bxn+Cxn−1 , Facts and Conjectures, Comput. Math. Appl. 45 (2003) 1087–1099]
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