Asymptotic Periodicity of a Higher-Order Difference Equation
نویسنده
چکیده
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 increasing in the first k arguments and decreasing in otherm arguments, there is a decreasing function g ∈ C[(α,∞),(α,∞)] such that g(g(x)) = x, x ∈ (α,∞), x = f (x, . . . ,x
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