On the Periodicity of a Difference Equation with Maximum

Abstract

We investigate the periodic nature of solutions of the max difference equationxn+1=max⁡{xn,A}/(xnxn−1),n=0,1,…, whereAis a positive real parameter, and the initial conditionsx−1=Ar−1andx0=Ar0such thatr−1andr0are positive rational numbers. The results in this paper answer the Open Problem 6.2 posed by Grove and Ladas (2005).

Well-Defined Solutions of a Three-Dimensional System of Difference Equations

“…Assume that {( , , )} ≥−1 are well-defined solutions of system (4). Then for = 0,5 ̅̅̅̅ , the following statements hold.…”

“…that is the sequence ( ) ≥−1 is periodic with period 12 and takes the form 4 , − −1 + (5) , − 0 + (0) , − 1 + (1) , − 2 + (2) , − 3 + (3) , − 4 + (4) , −1 , 0 , … ).…”

“…Difference equations and their systems are related to many real life models in different branches of modern science such as biology, physics, economics, etc, in [1,2]. This is due to the fact that these models are expressed by discrete equations and this explain why difference equations and their systems have attracted the attention of many researchers in recent years in [3][4][5][6]. One of the most popular subject associated with difference equations and their system, especially the non-linear ones, is to examine their solvability and the behavior of their solutions in .…”

“…If one of the terms 0 , 0 and 0 for 0 ≥ 1, is equal to zero, then from system (4) either 0 so, it follows that, 0 +2 is not defined or 0 and so, it follows that, 0 +2 is not defined. Thus, well-defined solutions of system(4) satisfy . .…”

On a max type recursive sequence of order three

“…The study of max-type difference equations attracted a considerable attention recently, see, for example, [1,5,7,[13][14][15][16] and the references therein. This type of difference equations stem from, for example, certain models in automatic control theory (see [17] and [18]).…”