On a k-Order System of Lyness-Type Difference Equations

Abstract

We consider the following system of Lyness-type difference equations: x 1 (n + 1) = (a k x k (n) + b k)/x k−1 (n − 1), x 2 (n + 1) = (a 1 x 1 (n) + b 1)/x k (n − 1), x i (n + 1) = (a i−1 x i−1 (n) + b i−1)/x i−2 (n − 1), i = 3,4,...,k, where a i , b i , i = 1,2,...,k, are positive constants, k ≥ 3 is an integer, and the initial values are positive real numbers. We study the existence of invariants, the boundedness, the persistence, and the periodicity of the positive solutions of this system.

Boundary Value Problems for Some Important Classes of Recurrent Relations with Two Independent Variables

“…Since that time, related transformations have been frequently used on difference equations ( [14][15][16]), as well as on close to symmetric systems (see [15,17,18] and numerous references therein), an area essentially initiated by Papaschinopoulos and Schinas (see [19][20][21][22][23][24][25]). Somewhat more complex methods can be found in [26].…”

“…Somewhat more complex methods can be found in [26]. For some related topics, such as finding invariants, special types of solutions and applications of solvable difference equations (see, for example, [21][22][23]25,[27][28][29] and the references therein). Quite frequently, the solvability was essentially shown by using some special cases of the following difference equation:…”

General k-Dimensional Solvable Systems of Difference Equations

“…Since that time various modifications of the method have been often used (see [13,14] and the references therein for some related difference equations, as well as [15][16][17] and the references therein for some related systems of difference equations). It should be pointed out that the systems are usually symmetric or close-to-symmetric, whose study was popularized by Papaschinopoulos and Schinas ( [18][19][20][21][22][23][24]). In some of their papers, such as [19][20][21]23], they study the solvability and the long-term behaviour of solutions to the equations and systems by finding their invariants.…”

“…It should be pointed out that the systems are usually symmetric or close-to-symmetric, whose study was popularized by Papaschinopoulos and Schinas ( [18][19][20][21][22][23][24]). In some of their papers, such as [19][20][21]23], they study the solvability and the long-term behaviour of solutions to the equations and systems by finding their invariants. For some applications of solvability and related matters see [6,[25][26][27][28][29].…”

Global behaviour of solutions of cyclic systems ofqorder 2 or 3 generalized Lyness' difference equations and of other more general equations of higher order

“…In all these papers, the used method is based on complicated calculations, seems particular to the studied system, and gives only results about global periodicity (except in [13] where permanency is studied).…”