##
**An introduction to difference equations.**
*(English)*
Zbl 0840.39002

Undergraduate Texts in Mathematics. New York, NY: Springer-Verlag. xiii, 389 p. (1996).

This textbook on difference equations is addressed to undergraduates and, in particular, to physicists and engineers. It deals with difference equations of one discrete variable. Emphasis is laid on linear equations of first and higher order, and on systems of such equations, but also some nonlinear difference equations appear.

The main topics from the theory of dynamical systems are treated in discrete form from iteration and a short indication of bifurcation and chaos up to stability theory (method of Lyapunov), perturbations (stability by linear approximation), oscillations (Sturm’s separation theorem), asymptotics (PoincarĂ© and Levison theorem), control theory (observability, feedback), transformation methods (power series, \(Z\)-transform), and also the discrete versions of the Riccati, the Volterra and the heat equation.

The book includes numerous applications in economics, social sciences, biology, physics, engineering, neural networks, probability and information theory etc., as well as numerous exercises, some of them with solutions.

Reviewer’s remark: Some of the Exercises 7.1 are erroneous, so 10. (i) for \(g_n (t)\) with different signs, 12. for \(g(t) = \sqrt t\), 14. for \(g(t) = f(t) = 2^{- n}\) for \(2^{n - 1} < t \leq 2^n\), and Exercise 7.2.8 (ii) for \(u(n) = \lambda^n\). The expression on p. 305 that “\(f = O(h) \) is a better approximation of \(f\) than \(f = O(g)\) in case of \(g = O(h)\)” does not make sense. On p. 306, the formula \(O(g) : = \{f : f = O(g)\}\), the analogous one with \(o\) instead of \(O\) and the pair of formulas “\(o(f) = O(f)\) but \(O(f) \neq o(f)\)” cannot be used without further comment. Some expression are unexplained as \(x^{(n)}\) on p. 7, the connection between \(f\) and \(Q\) on p. 29 and p. 30, and \(G\) on p. 307. Finally, there are misprints in the last three integrals on p. 308, and in the Exercises 7.2: \(u\) must be \(x\) in 8., \(4^n\) must be \(4n\) in 10., = must be cancelled in \(12(d)\), and (7.2.20) is completely wrong in \(12(e)\).

The main topics from the theory of dynamical systems are treated in discrete form from iteration and a short indication of bifurcation and chaos up to stability theory (method of Lyapunov), perturbations (stability by linear approximation), oscillations (Sturm’s separation theorem), asymptotics (PoincarĂ© and Levison theorem), control theory (observability, feedback), transformation methods (power series, \(Z\)-transform), and also the discrete versions of the Riccati, the Volterra and the heat equation.

The book includes numerous applications in economics, social sciences, biology, physics, engineering, neural networks, probability and information theory etc., as well as numerous exercises, some of them with solutions.

Reviewer’s remark: Some of the Exercises 7.1 are erroneous, so 10. (i) for \(g_n (t)\) with different signs, 12. for \(g(t) = \sqrt t\), 14. for \(g(t) = f(t) = 2^{- n}\) for \(2^{n - 1} < t \leq 2^n\), and Exercise 7.2.8 (ii) for \(u(n) = \lambda^n\). The expression on p. 305 that “\(f = O(h) \) is a better approximation of \(f\) than \(f = O(g)\) in case of \(g = O(h)\)” does not make sense. On p. 306, the formula \(O(g) : = \{f : f = O(g)\}\), the analogous one with \(o\) instead of \(O\) and the pair of formulas “\(o(f) = O(f)\) but \(O(f) \neq o(f)\)” cannot be used without further comment. Some expression are unexplained as \(x^{(n)}\) on p. 7, the connection between \(f\) and \(Q\) on p. 29 and p. 30, and \(G\) on p. 307. Finally, there are misprints in the last three integrals on p. 308, and in the Exercises 7.2: \(u\) must be \(x\) in 8., \(4^n\) must be \(4n\) in 10., = must be cancelled in \(12(d)\), and (7.2.20) is completely wrong in \(12(e)\).

Reviewer: L.Berg (Rostock)

### MSC:

39Axx | Difference equations |

93-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory |

00A06 | Mathematics for nonmathematicians (engineering, social sciences, etc.) |

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