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2008 The Behavior of Positive Solutions of a Nonlinear Second-Order Difference Equation
Stevo Stević, Kenneth S. Berenhaut
Abstr. Appl. Anal. 2008: 1-8 (2008). DOI: 10.1155/2008/653243

Abstract

This paper studies the boundedness, global asymptotic stability, and periodicity of positive solutions of the equation x n = f ( x n 2 ) / g ( x n 1 ) , n 0 , where f , g C [ ( 0 , ) , ( 0 , ) ] . It is shown that if f and g are nondecreasing, then for every solution of the equation the subsequences { x 2 n } and { x 2 n 1 } are eventually monotone. For the case when f ( x ) = α + β x and g satisfies the conditions g ( 0 ) = 1 , g is nondecreasing, and x / g ( x ) is increasing, we prove that every prime periodic solution of the equation has period equal to one or two. We also investigate the global periodicity of the equation, showing that if all solutions of the equation are periodic with period three, then f ( x ) = c 1 / x and g ( x ) = c 2 x , for some positive c 1 and c 2 .

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Stevo Stević. Kenneth S. Berenhaut. "The Behavior of Positive Solutions of a Nonlinear Second-Order Difference Equation." Abstr. Appl. Anal. 2008 1 - 8, 2008. https://doi.org/10.1155/2008/653243

Information

Published: 2008
First available in Project Euclid: 9 September 2008

zbMATH: 1146.39018
MathSciNet: MR2393108
Digital Object Identifier: 10.1155/2008/653243

Rights: Copyright © 2008 Hindawi

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