Global behavior of nonlinear difference equations of higher order with applications.

*(English)*Zbl 0787.39001
Mathematics and its Applications (Dordrecht). 256. Dordrecht: Kluwer Academic Publishers. xi, 228 p. (1993).

This interesting monograph bases mostly on the results obtained by the authors and their collaborators (other results are presented occasionally) and first of all refers to stability of selected types of autonomous difference equations of the form \(x_{n+1} =F(x_ n,x_{n- 1},\dots,x_{n-k})\). Quite a lot of the authors’ interest occupate rational recursive sequences (so-called by them), that is equations of the form \(x_{n+1}= f_ 1 (x_ n,\dots,x_{n-k})/f_ 2(x_ n,\dots,x_{n-k})\) where \(f_ i(x_ n,\dots,x_{n-k})=a_ i+\sum^ k_{j=0} a_{i,j}x_{n-j}\), \(i=1,2\); or their nonlinear generalizations with \(f_ i=a_ i+a_{i,1} x_ n+x_ n \exp (a_ 2+a_ 3x_{n-k})\) or \(f_ i=a_ i+a_{i,1} (x_{n+1-i})^ 2\).

Besides the stability problems the authors present (for some chosen types of equations) results on boundedness, permanence, periodicity, oscillatority. The reviewer, however, must point out that at least some of the considered problems possess essentially richer literature than seems to be understood while reading the book. Almost in every section the authors pose open questions, and every chapter includes notes and references about the presented material.

Contents: Chapter 1: General definitions, foundations, preliminary material. Chapter 2: Stability, permanence of the equations \(x_{n+1}=x_ nf(x_ n,\dots,x_{n-k})\) and \(x_{n+1}=\sum^ k_{j=0}a_{1,j} x_{n-j}+f(\sum^ k_{j=0} a_{2,j}x_{n-j})\). Chapter 3: Stability and other asymptotic properties of solutions of various particular cases of rational recursive sequences.

Chapter 4: Discrete models and discrete analogues of continuous ones taken from Mathematical Biology and Physics (Logistic, Simple Genotype Selection, Spread of an Epidemic, Dynamics of Nicholson’s Blowflies, Baleen Whale, Haematopoiesis Models, Emden-Fowler Equation). Chapter 5. Periodic cycles of some rational equations. Chapter 6: Open problems and conjectures (based on computer observations) also for Volterra summary equations, differential equations with piecewise constant argument, Fibonacci sequence. Appendix: Riccati difference equation, attractivity results for nonautonomous systems, brief view on qualitative results for systems of equations.

The book well exposes some branches of the subject, particularly stability and applications and could be an interesting complement of other recently published monographs like for example R. P. Agarwal [Difference Equations and Inequalities. Theory, Methods and Applications, Marcel Dekker Inc. New York (1992)].

Besides the stability problems the authors present (for some chosen types of equations) results on boundedness, permanence, periodicity, oscillatority. The reviewer, however, must point out that at least some of the considered problems possess essentially richer literature than seems to be understood while reading the book. Almost in every section the authors pose open questions, and every chapter includes notes and references about the presented material.

Contents: Chapter 1: General definitions, foundations, preliminary material. Chapter 2: Stability, permanence of the equations \(x_{n+1}=x_ nf(x_ n,\dots,x_{n-k})\) and \(x_{n+1}=\sum^ k_{j=0}a_{1,j} x_{n-j}+f(\sum^ k_{j=0} a_{2,j}x_{n-j})\). Chapter 3: Stability and other asymptotic properties of solutions of various particular cases of rational recursive sequences.

Chapter 4: Discrete models and discrete analogues of continuous ones taken from Mathematical Biology and Physics (Logistic, Simple Genotype Selection, Spread of an Epidemic, Dynamics of Nicholson’s Blowflies, Baleen Whale, Haematopoiesis Models, Emden-Fowler Equation). Chapter 5. Periodic cycles of some rational equations. Chapter 6: Open problems and conjectures (based on computer observations) also for Volterra summary equations, differential equations with piecewise constant argument, Fibonacci sequence. Appendix: Riccati difference equation, attractivity results for nonautonomous systems, brief view on qualitative results for systems of equations.

The book well exposes some branches of the subject, particularly stability and applications and could be an interesting complement of other recently published monographs like for example R. P. Agarwal [Difference Equations and Inequalities. Theory, Methods and Applications, Marcel Dekker Inc. New York (1992)].

Reviewer: J.Popenda (Poznań)

##### MSC:

39A10 | Additive difference equations |

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

39A11 | Stability of difference equations (MSC2000) |