##
**Qualitative analysis of delay partial difference equations.**
*(English)*
Zbl 1153.35078

Contemporary Mathematics and Its Applications 4. New York, NY: Hindawi Publishing Corporation (ISBN 978-977-454-000-4/hbk). viii, 374 p. (2007).

This is a monograph on qualitative analysis of delayed partial difference equations (PDEs). It is composed of five chapters, and the content includes oscillation theory (chapter 2 and 3), stability (chapter 4) and spatial chaos (chapter 5). Most of the material in this book are based on the research work carried out by the authors, their graduate students and some other research workers during the past ten years.

In the first chapter, several examples are given in order to illustrate a conclusion that a delay partial difference equation can be regarded as a discrete analog of a delay partial differential equation. Some basic concepts and theorem are also introduced in this chapter.

The classification of the delay partial difference equations in this monograph is similar to that in delay partial differential equations: linear and nonlinear, constant parameters and variable parameters, bounded delay and unbounded delay, initial value problem and initial boundary value problem, delay PDEs and neutral PDEs. In chapter 2 and chapter 3, various criteria on the sufficient conditions of oscillation and the existence of positive solutions are discussed for linear delay PDEs and nonlinear delay PDEs respectively. There are two kinds of definition for oscillation here. One is that a solution \(\{A_{ij}\}\) is said to be oscillatory if it is neither eventually positive nor eventually negative. Another is called the definition of frequent oscillation which is stronger than that of oscillation, that is, if a solution \(\{A_{ij}\}\) is frequently oscillatory, then it is oscillatory.

In chapter 4, the definitions and criteria for stability including “stable”, “linear stable”, “exponentially asymptotically stable” and “strongly exponentially asymptotically stable” are given. In chapter 5, the authors introduce some recent works on spatial chaos on delay PDEs.

The authors also consider a kind of delay PDEs with continuous arguments. Taking the following delay PDEs as an example: \[ p_1A_{m+1,n+1}+p_2A_{m+1,n}+p_3A_{m,n+1} -p_4A_{m,n}+p_{m,n}A_{m-k,n-l} =0,(m,n)\in \mathbb{N}_0^2,\tag{1} \]

a delay PDEs with continuous arguments is of the form \[ p_1A(x+a,y+b)+p_2A(x+a,y)+p_3A(x,y+b)-p_4A(x,y)+P(x,y)A(x-\tau,y-\sigma)=0,\tag{2} \]

where \(x\in \mathbb{R}^+, y\in \mathbb{R}^+, a,b,\tau,\sigma\in \mathbb{R}\). Most results on oscillation and stability are generalized to the delay PDEs with continuous arguments.

Like the (delay) partial differential equations, because of the multiple variables in the unknown function of the equations, the discussion of delay PDEs is complicate and delicate. This monograph gives abundance material and represents a pioneer work on delay PDEs. It seems that the theory of delay PDEs can be developed on some aspects if suitable techniques and methods will be applied to it.

In the first chapter, several examples are given in order to illustrate a conclusion that a delay partial difference equation can be regarded as a discrete analog of a delay partial differential equation. Some basic concepts and theorem are also introduced in this chapter.

The classification of the delay partial difference equations in this monograph is similar to that in delay partial differential equations: linear and nonlinear, constant parameters and variable parameters, bounded delay and unbounded delay, initial value problem and initial boundary value problem, delay PDEs and neutral PDEs. In chapter 2 and chapter 3, various criteria on the sufficient conditions of oscillation and the existence of positive solutions are discussed for linear delay PDEs and nonlinear delay PDEs respectively. There are two kinds of definition for oscillation here. One is that a solution \(\{A_{ij}\}\) is said to be oscillatory if it is neither eventually positive nor eventually negative. Another is called the definition of frequent oscillation which is stronger than that of oscillation, that is, if a solution \(\{A_{ij}\}\) is frequently oscillatory, then it is oscillatory.

In chapter 4, the definitions and criteria for stability including “stable”, “linear stable”, “exponentially asymptotically stable” and “strongly exponentially asymptotically stable” are given. In chapter 5, the authors introduce some recent works on spatial chaos on delay PDEs.

The authors also consider a kind of delay PDEs with continuous arguments. Taking the following delay PDEs as an example: \[ p_1A_{m+1,n+1}+p_2A_{m+1,n}+p_3A_{m,n+1} -p_4A_{m,n}+p_{m,n}A_{m-k,n-l} =0,(m,n)\in \mathbb{N}_0^2,\tag{1} \]

a delay PDEs with continuous arguments is of the form \[ p_1A(x+a,y+b)+p_2A(x+a,y)+p_3A(x,y+b)-p_4A(x,y)+P(x,y)A(x-\tau,y-\sigma)=0,\tag{2} \]

where \(x\in \mathbb{R}^+, y\in \mathbb{R}^+, a,b,\tau,\sigma\in \mathbb{R}\). Most results on oscillation and stability are generalized to the delay PDEs with continuous arguments.

Like the (delay) partial differential equations, because of the multiple variables in the unknown function of the equations, the discussion of delay PDEs is complicate and delicate. This monograph gives abundance material and represents a pioneer work on delay PDEs. It seems that the theory of delay PDEs can be developed on some aspects if suitable techniques and methods will be applied to it.

Reviewer: Peixuan Weng (Guangzhou)

### MSC:

35R10 | Partial functional-differential equations |

39A11 | Stability of difference equations (MSC2000) |

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |

35B35 | Stability in context of PDEs |

35B40 | Asymptotic behavior of solutions to PDEs |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |