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**Advanced topics in difference equations.**
*(English)*
Zbl 0878.39001

Mathematics and its Applications (Dordrecht). 404. Dordrecht: Kluwer Academic Publishers. ix, 507 p. (1997).

The theory of finite difference equations is in a process of continuous development and it has become significant for its various applications. Many physical phenomena in nature are most naturally described by systems of difference equations. In the past few decades the study of difference equations and their applications has received a wide attention of a number of researchers interested both in the theory and applications.

In a book published in 1992 entitled “Difference equations and inequalities”, the first author has made a detailed survey of the important results in the field of finite difference equations and their applications. In the present book the authors collect some of the results which have been obtained by themselves in the last few years, some of which are yet unpublished ones, as well as the related recent results obtained by various investigators which reflect the major advances in the field and the diversity of the subject.

The book is divided into forty sections. Each section contains major results, and, in some cases these sections, taken together, form a survey of an important area of research in difference equations. Specifically, Sections 1 to 5 deal with the results related to the periodic solutions of various types of finite difference equations, Sections 6 and 7 deal with the nonlinear variation of parameters methods for the equations involving one or more independent variables and their applications, Sections 8 to 10 deal with various results concerning the convergence of equilibria, asymptotic behavior and stability of difference equations, Sections 11 to 26 deal with the oscillation of solutions of various types of finite difference equations in one and two independent variables, Sections 27 to 37 deal with various results related to different boundary value problems for difference equations, Sections 38, 39 and 40 deal with the linear square optimal control problem for stochastic difference equations, symmetries of difference systems on manifolds, and discrete polar coordinates, respectively.

The book contains a collection of recent results and it will serve as a reference book for researchers in discrete dynamical systems and their applications and the reader will also find material, which is not available in other books on difference equations. It will also be of interest to graduate students interested in the theory of finite difference equations and their applications. The presentation is clear and it is a welcome addition to the literature.

In a book published in 1992 entitled “Difference equations and inequalities”, the first author has made a detailed survey of the important results in the field of finite difference equations and their applications. In the present book the authors collect some of the results which have been obtained by themselves in the last few years, some of which are yet unpublished ones, as well as the related recent results obtained by various investigators which reflect the major advances in the field and the diversity of the subject.

The book is divided into forty sections. Each section contains major results, and, in some cases these sections, taken together, form a survey of an important area of research in difference equations. Specifically, Sections 1 to 5 deal with the results related to the periodic solutions of various types of finite difference equations, Sections 6 and 7 deal with the nonlinear variation of parameters methods for the equations involving one or more independent variables and their applications, Sections 8 to 10 deal with various results concerning the convergence of equilibria, asymptotic behavior and stability of difference equations, Sections 11 to 26 deal with the oscillation of solutions of various types of finite difference equations in one and two independent variables, Sections 27 to 37 deal with various results related to different boundary value problems for difference equations, Sections 38, 39 and 40 deal with the linear square optimal control problem for stochastic difference equations, symmetries of difference systems on manifolds, and discrete polar coordinates, respectively.

The book contains a collection of recent results and it will serve as a reference book for researchers in discrete dynamical systems and their applications and the reader will also find material, which is not available in other books on difference equations. It will also be of interest to graduate students interested in the theory of finite difference equations and their applications. The presentation is clear and it is a welcome addition to the literature.

Reviewer: B.G.Pachpatte (Aurangabad)

### MSC:

39Axx | Difference equations |

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

37-XX | Dynamical systems and ergodic theory |