Asymptotic Periodicity of a Higher-Order Difference Equation

Stevo Stevic

Discrete Dynamics in Nature and Society, 2007, vol. 2007, 1-9

Abstract:

We give a complete picture regarding the asymptotic periodicity of positive solutions of the following difference equation: x n = f ( x n − p 1 , … , x n − p k , x n − q 1 , … , x n − q m ) , n ∈ ℕ 0 , where p i ,  i ∈ { 1 , … , k } , and q j ,  j ∈ { 1 , … , m } , are natural numbers such that p 1 < p 2 < ⋯ < p k , q 1 < q 2 < ⋯ < q m and gcd ( p 1 , … , p k , q 1 , … , q m ) = 1 , the function f ∈ C [ ( 0 , ∞ ) k + m , ( α , ∞ ) ] ,  α > 0 , is increasing in the first k arguments and decreasing in other m arguments, there is a decreasing function g ∈ C [ ( α , ∞ ) , ( α , ∞ ) ] such that g ( g ( x ) ) = x ,  x ∈ ( α , ∞ ) , x = f ( x , … , x ︸ k , g ( x ) , … , g ( x ) ︸ m ) , x ∈ ( α , ∞ ) , lim x → α + g ( x ) = + ∞ , and lim x → + ∞ g ( x ) = α . It is proved that if all p i ,  i ∈ { 1 , … , k } , are even and all q j ,  j ∈ { 1 , … , m } are odd, every positive solution of the equation converges to (not necessarily prime) a periodic solution of period two, otherwise, every positive solution of the equation converges to a unique positive equilibrium.

Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnddns:013737

DOI: 10.1155/2007/13737

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