On Global Periodicity of a Class of Difference Equations

Stevo Stevic

Discrete Dynamics in Nature and Society, 2007, vol. 2007, 1-10

Abstract:

We show that the difference equation x n = f 3 ( x n − 1 ) f 2 ( x n − 2 ) f 1 ( x n − 3 ) , n ∈ ℕ 0 , where f i ∈ C [ ( 0 , ∞ ) , ( 0 , ∞ ) ] , i ∈ { 1 , 2 , 3 } , is periodic with period 4 if and only if f i ( x ) = c i / x for some positive constants c i ,   i ∈ { 1 , 2 , 3 } or if f i ( x ) = c i / x when i = 2 and f i ( x ) = c i x if i ∈ { 1 , 3 } , with c 1 c 2 c 3 = 1 . Also, we prove that the difference equation x n = f 4 ( x n − 1 ) f 3 ( x n − 2 ) f 2 ( x n − 3 ) f 1 ( x n − 4 ) , n ∈ ℕ 0 , where f i ∈ C [ ( 0 , ∞ ) , ( 0 , ∞ ) ] ,   i ∈ { 1 , 2 , 3 , 4 } , is periodic with period 5 if and only if f i ( x ) = c i / x , for some positive constants c i ,   i ∈ { 1 , 2 , 3 , 4 } .

Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnddns:023503

DOI: 10.1155/2007/23503

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