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**Difference equations: an introduction with applications.**
*(English)*
Zbl 0733.39001

Boston, MA etc.: Academic Press, Inc. xi, 455 p. (1991).

This fine and interesting book is devoted to different problems of the theory of finite difference equations. Contents: 1. Introduction; 2. Difference calculus (properties of the difference, shift, and summation operators, generating function, Euler summation formula); 3. Linear difference equations (general results for n-th order linear equations, Casoratian, the method of variation of parameters, annihilator and other various methods for equations with variable coefficients, Riccati equation, z-transforms with properties and applications to difference, Volterra, and Fredholm summation equations).

4. Stability theory (initial value problem for linear systems, stability of linear and general autonomous systems, staircase method, chaotic behavior); 5. Asymptotic methods (Poincaré’s and Perron’s theorems, asymptotic methods for scalar autonomous equations); 6. Second order self-adjoint equation (Cauchy function, Sturm separation theorem, boundary value problem, Green’s function, disconjugacy, disconjugacy and Riccati equation, Sturm comparison theorem, oscillation).

7. The Sturm-Liouville problem (eigenpairs, finite Fourier analysis, nonhomogeneous boundary value problem); 8. Discrete calculus of variations (the simplest variational problem, discrete Euler-Lagrange equation, necessary conditions for various variational problems to have a local extremum, sufficient conditions and disconjugacy); 9. Boundary value problem: \(\Delta^ 2y(t-1)=f(t,y(t)),\) \(y(a)=A,\) \(y(b+2)=B\) (Lipschitz case, existence results, uniqueness, relationship between boundary value problems for differential and difference equations); 10. Partial difference equations (discretization of partial differential equations, some methods for solving linear partial difference equations).

Including many examples (e.g. Fibonacci equation, Airy equation, predator-prey model, tiling problem, crystal lattice, logistic model, and many others which arise not only in mathematics but also from biology, economy, engineering), illustrative figures, and almost 400 exercises this book can be used as a textbook at a variety of different levels. At the end of the book there is long list (252) of books and papers on difference equations. In the main part of the book the authors concentrate on second order equations, especially around the concept of disconjugacy (speciality of the second author).

The reviewer hopes that in the next enlarged issue, more paragraphs will be devoted to various aspects of qualitative theory like: periodicity, monotonicity, summability, boundedness, and also oscillation. Notice, that the solution of the second order linear equation with variable coefficients can be found in some kind of closed form (compare p. 91).

4. Stability theory (initial value problem for linear systems, stability of linear and general autonomous systems, staircase method, chaotic behavior); 5. Asymptotic methods (Poincaré’s and Perron’s theorems, asymptotic methods for scalar autonomous equations); 6. Second order self-adjoint equation (Cauchy function, Sturm separation theorem, boundary value problem, Green’s function, disconjugacy, disconjugacy and Riccati equation, Sturm comparison theorem, oscillation).

7. The Sturm-Liouville problem (eigenpairs, finite Fourier analysis, nonhomogeneous boundary value problem); 8. Discrete calculus of variations (the simplest variational problem, discrete Euler-Lagrange equation, necessary conditions for various variational problems to have a local extremum, sufficient conditions and disconjugacy); 9. Boundary value problem: \(\Delta^ 2y(t-1)=f(t,y(t)),\) \(y(a)=A,\) \(y(b+2)=B\) (Lipschitz case, existence results, uniqueness, relationship between boundary value problems for differential and difference equations); 10. Partial difference equations (discretization of partial differential equations, some methods for solving linear partial difference equations).

Including many examples (e.g. Fibonacci equation, Airy equation, predator-prey model, tiling problem, crystal lattice, logistic model, and many others which arise not only in mathematics but also from biology, economy, engineering), illustrative figures, and almost 400 exercises this book can be used as a textbook at a variety of different levels. At the end of the book there is long list (252) of books and papers on difference equations. In the main part of the book the authors concentrate on second order equations, especially around the concept of disconjugacy (speciality of the second author).

The reviewer hopes that in the next enlarged issue, more paragraphs will be devoted to various aspects of qualitative theory like: periodicity, monotonicity, summability, boundedness, and also oscillation. Notice, that the solution of the second order linear equation with variable coefficients can be found in some kind of closed form (compare p. 91).

Reviewer: J.Popenda (Poznań)

### MSC:

39Axx | Difference equations |

39-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to difference and functional equations |

39-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to difference and functional equations |

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