Dynamics of third-order rational difference equations with open problems and conjectures.

*(English)*Zbl 1129.39002
Advances in Discrete Mathematics and Applications 5. Boca Raton, FL: Chapman & Hall/CRC (ISBN 978-1-58488-765-2/hbk). 554 p. (2008).

This book is about the global character of solutions of the third-order rational difference equation
\[
x_{n+1}=\frac{\alpha+\beta x_n+\gamma x_{n-1}+\delta x_{n-2}} {A+Bx_n+Cx_{n-1}+Dx_{n-2}},\quad n=0, 1, \dots
\]
with nonnegative parameters \(\alpha, \beta, \gamma, \delta, A, B, C, D\) and with arbitrary nonnegative initial conditions \(x_{-2}, x_{-1}, x_0\) such that the denominator is always positive. The authors are primarily concerned with the “boundedness nature of solutions, the stability of the equilibrium points, the periodic character of the equation, and with convergence to periodic solutions including periodic trichotomies. The book also provides numerous thought-provoking open problems and conjectures on the boundedness character, global stability, and periodic behavior of solutions of rational difference equations.

After introducing several basic definitions and general results, the authors examine 135 special cases of rational difference equations that have only bounded solutions and the equations that have unbounded solutions in some range of their parameters. They then explore the seven known nonlinear periodic trichotomies of third order rational difference equations. The main part of the book presents the known results of each of the 225 special cases of third order rational difference equations. In addition, the appendices supply tables that feature important information on these cases as well as on the boundedness character of all fourth order rational difference equations.

The theory and techniques developed in this book to understand the dynamics of rational difference equations will be useful in analyzing the equations in any mathematical model that involves difference equations. Moreover, the stimulating conjectures will promote future investigations in this fascinating, yet surprisingly little known area of research” (cited from the publisher’s description).

After introducing several basic definitions and general results, the authors examine 135 special cases of rational difference equations that have only bounded solutions and the equations that have unbounded solutions in some range of their parameters. They then explore the seven known nonlinear periodic trichotomies of third order rational difference equations. The main part of the book presents the known results of each of the 225 special cases of third order rational difference equations. In addition, the appendices supply tables that feature important information on these cases as well as on the boundedness character of all fourth order rational difference equations.

The theory and techniques developed in this book to understand the dynamics of rational difference equations will be useful in analyzing the equations in any mathematical model that involves difference equations. Moreover, the stimulating conjectures will promote future investigations in this fascinating, yet surprisingly little known area of research” (cited from the publisher’s description).

Reviewer: Alexander O. Ignatyev (Donetsk)

##### MSC:

39A11 | Stability of difference equations (MSC2000) |

39A20 | Multiplicative and other generalized difference equations, e.g., of Lyness type |

39-02 | Research exposition (monographs, survey articles) pertaining to difference and functional equations |