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Periodicities in nonlinear difference equations. (English) Zbl 1078.39009
Advances in Discrete Mathematics and Applications 4. Boca Raton, FL: Chapman & Hall/CRC (ISBN 0-8493-3156-0/hbk). xiii, 379 p. (2005).
This interesting book presents old and new results concerning the periodicity of solutions of nonlinear difference equations, especially those which were discovered from the authors in the last ten years. It begins with basic definitions and general convergence theorems, followed by equations having only periodic solutions with the same period and equations with eventually periodic solutions. These difference equations are either rational ones, or they contain the maximum function and can be transformed into piecewise linear equations, where both the cases with constant and with variable coefficients are studied. Besides of the famous Sharkovsky’s theorem concerning the Sharkovsky ordering, in particular, the theorem of Li and Yorke that period 3 implies chaos, and the challenging Collatz problem concerning the \((3x+1)\)-conjecture, emphasis is laid on equations of higher order.
Further chapters concern equations with the property that every solution converges to a periodic solution, and investigations on boundedness and global asymptotic stability. A final chapter generalizes the Collatz problem to analogous problems of second order. The advantage of the book is not only the presentation of new results, but also the formulation of many open problems and conjectures which shall stimulate further investigations of researchers and graduate students.

39A11 Stability of difference equations (MSC2000)
39-02 Research exposition (monographs, survey articles) pertaining to difference and functional equations
39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type