GLOBAL ASYMPTOTIC STABILITY OF A HIGHER ORDER DIFFERENCE EQUATION

Global asymptotic stability of a higher order rational difference equation

In this note, we consider the following rational difference equation: xn+1 = f (xn−r1 , . . . , xn−rk )g(xn−m1 , . . . , xn−ml )+ 1 f (xn−r1 , . . . , xn−rk )+ g(xn−m1 , . . . , xn−ml ) , n= 0,1, . . . , where f ∈ C((0,+∞)k, (0,+∞)) and g ∈ C((0,+∞)l, (0,+∞)) with k, l ∈ {1,2, . . .}, 0 r1 < · · ·< rk and 0 m1 < · · ·<ml , and the initial values are positive real numbers. We give sufficient con...

Global asymptotic stability of the higher order equation

In this paper, we investigate the local and global stability and the period two solutions of all nonnegative solutions of the difference equation, xn+1 = axn + bxn−k A + Bxn−k where a, b, A, B are all positive real numbers, k ≥ 1 is a positive integer, and the initial conditions x−k, x−k+1, ..., x0 are nonnegative real numbers. It is shown that the zero equilibrium point is globally asymptotica...

Asymptotic Periodicity of a Higher-Order Difference Equation

We give a complete picture regarding the asymptotic periodicity of positive solutions of the following difference equation: xn = f (xn−p1 , . . . ,xn−pk ,xn−q1 , . . . ,xn−qm), n∈N0, where pi, i ∈ {1, . . . ,k}, and qj , j ∈ {1, . . . ,m}, are natural numbers such that p1 < p2 < ··· < pk, q1 < q2 < ··· < qm and gcd(p1, . . . , pk,q1, . . . ,qm) = 1, the function f ∈ C[(0,∞), (α,∞)], α > 0, is i...

Global Behavior of a Higher-order Rational Difference Equation

We investigate in this paper the global behavior of the following difference equation: xn+1 = (Pk(xn i0 ,xn i1 , . . . ,xn i2k ) + b)/(Qk(xn i0 ,xn i1 , . . . ,xn i2k ) + b), n = 0,1, . . ., under appropriate assumptions, where b [0, ), k 1, i0, i1, . . . , i2k 0,1, . . . with i0 < i1 < < i2k, the initial conditions xi 2k ,xi 2k+1, . . . ,x0 (0, ). We prove that unique equilibrium x = 1 of that...

Global behavior of a higher order difference equation