Abstract
In this paper we study the qualitative properties and the periodic nature of the solutions of the difference equation x n + 1 = α x n - 2 + β x n - 2 2 γ x n - 2 + δ x n - 5 , n = 0 , 1 , . . . , where the initial conditions x - 5 , x - 4 , x - 3 , x - 2 , x - 1 , x 0 are arbitrary positive real numbers and α , β , γ , δ are positive constants. In addition, we derive the form of the solutions of some special cases of this equation.
Highlights
IntroductionElabbasy et al [8] studied the boundedness, global stability, periodicity character and gave the solution of some special cases of the difference equation
This paper deals with behavior of the solutions of the difference equation xn+1 = αxn−2 +, γxn−2 + δxn−5 n = 0, 1, ..., (1.1)where the initial conditions x−5, x−4, x−3, x−2, x−1, x0 are arbitrary positive real numbers and α, β, γ, δ are constants
In this paper we study the qualitative properties and the periodic nature of the solutions of the difference equation xn+1 = αxn−2 +
Summary
Elabbasy et al [8] studied the boundedness, global stability, periodicity character and gave the solution of some special cases of the difference equation. Elabbasy and Elsayed [9] investigated the local and global stability, boundedness, and gave the solution of some special cases of the difference equation x n +1 =. The equilibrium point x of Equation (1.2) is locally asymptotically stable if x is locally stable solution of Equation (1.2) and there exists γ > 0, such that for all x−k , x−k+1 , ..., x−1 , x0 ∈ I with. The equilibrium point x0 ∈ I, we have x of Equation (1.2) is global attractor if for all x−k , x−k+1 , ..., x−1 , lim xn = x. From Theorem B, it follows that x is a global attractor of Equation (1.1) and the proof is complete.
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