Some properties of the solutions of the difference equation \(x_{n+1}=a x_{n}+\dfrac{b x_{n} x_{n-4}}{c x_{n-3}+dx_{n-4}}\)

Abstract

In this article, we study some properties of the solutions of the following difference equation:, n = 0, 1, ... where the initial conditions x −4 , x −3 , x −2 , x −1 , x 0 are arbitrary positive real numbers and a, b, c, d are positive constants. Also, we give specific form of the solutions of four special cases of this equation.

General solution to a difference equation and the long-term behavior of some of its solutions

“…Remark 2.2. From Theorem 2.1, some calculation and a representation in [51] are obtained the closed-form formulas in [38] for solutions to the special cases (i)-(iv) of Eq.(1.4). We omit the standard problem.…”

“…We also analyze the claims on the long-term behavior of solutions to Eq. (1.4) formulated in [38] and show that they are false.…”

“…where a, b, c, d ∈ R, x −j ∈ R, j = 0, 4, was studied recently in [38], where some formulas for solutions in the cases:…”

“…were presented. Besides, [38] gives some claims on the long-term behavior of solutions to Eq. (1.4).…”

Dynamical Behavior and Solutions of Nonlinear Difference Equations of Twentieth Order Lama Sh. Aljoufi and M. B. Almatrafi

“…We mention some of them. Sanbo and Elsayed [2] studied the local and global behavior, periodicity, boundedness and some solutions of a fifth order recursive equation. Alayachi et al [3] discussed the stability, periodicity and the solutions of a sixth order recursive equation.…”