Difference equations. An introduction with applications. 2nd ed.

*(English)*Zbl 0970.39001
San Diego, CA: Harcourt/Academic Press. x, 403 p. (2001).

[See also the review of the first ed. (1991; Zbl 0733.39001).]

This book is devoted to the following problems of the theory of finite difference equations: 1. Introduction. 2. The difference calculus (the difference operator, summation, generating functions and approximate summation). 3. Linear difference equations (first order equations, general results for linear equations, solving linear equations, applications, equations with variable coefficients, nonlinear equations that can be linearized, the \(Z\)-transform). 4. Stability theory (initial value problems for linear systems, fundamental matrices and Floquet theory, chaotic behaviour).

5. Asymptotic methods (introduction, asymptotic analysis of sums, linear equations, nonlinear equations). 6. The selfadjoint second order linear equation (introduction, Sturmian theory, Green’s functions, disconjugacy, the Riccati equation, oscillation). 7. The Sturm-Liouville problem (introduction, finite Fourier analysis, nonhomogeneous problem). 8. Discrete calculus of variations (introduction, necessary conditions, sufficient conditions and disconjugacy). 9. Boundary value problems for nonlinear equations (introduction, the Lipschitz case, existence of solutions, boundary value problems for differential equations). 10. Partial difference equations (discretization of partial differential equations, solutions of partial difference equations).

This is a fine and interesting book. No prior familiarity with difference equations is assumed. The goal is to present an overview of the various facts of difference equations that can be studied by elementary mathematical tools.

This book is devoted to the following problems of the theory of finite difference equations: 1. Introduction. 2. The difference calculus (the difference operator, summation, generating functions and approximate summation). 3. Linear difference equations (first order equations, general results for linear equations, solving linear equations, applications, equations with variable coefficients, nonlinear equations that can be linearized, the \(Z\)-transform). 4. Stability theory (initial value problems for linear systems, fundamental matrices and Floquet theory, chaotic behaviour).

5. Asymptotic methods (introduction, asymptotic analysis of sums, linear equations, nonlinear equations). 6. The selfadjoint second order linear equation (introduction, Sturmian theory, Green’s functions, disconjugacy, the Riccati equation, oscillation). 7. The Sturm-Liouville problem (introduction, finite Fourier analysis, nonhomogeneous problem). 8. Discrete calculus of variations (introduction, necessary conditions, sufficient conditions and disconjugacy). 9. Boundary value problems for nonlinear equations (introduction, the Lipschitz case, existence of solutions, boundary value problems for differential equations). 10. Partial difference equations (discretization of partial differential equations, solutions of partial difference equations).

This is a fine and interesting book. No prior familiarity with difference equations is assumed. The goal is to present an overview of the various facts of difference equations that can be studied by elementary mathematical tools.

Reviewer: S.Balint (Timişoara)

##### MSC:

39Axx | Difference 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 |