A Note on the Periodicity of the Lyness Max Equation

Abstract

We investigate the periodic nature of solutions of a "max-type" difference equation sometimes referred to as the "Lyness max" equation. The equation we consider is x n 1 max{x n , A}/x n−1 . where A is a positive real parameter, x − 1 A r −1 , and x 0 A r 0 such that r −1 and r 0 are positive rational numbers. The results in this paper answer the Open Problem of Grove and Ladas 2005 .

Solution of the Maximum of Difference Equation xn+1=max{Axn−1,ynxn};yn+1=max{Ayn−1,xnyn}\matrix{ {x_{n + 1} = max \left\{ {{A \over {x_{n - 1} }},{{y_n } \over {x_n }}} \right\};} & {y_{n + 1} = max \left\{ {{A \over {y_{n - 1} }},{{x_n } \over {y_n }}} \right\}}}

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In the recent years, there has been a lot of interest in studying the global behavior of, the socalled, max-type difference equations; see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. The study of max type difference equations has also attracted some attention recently.

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