Almost periodic oscillations and waves.

*(English)*Zbl 1163.34002
New York, NY: Springer (ISBN 978-0-387-09818-0/hbk; 978-0-387-09819-7/ebook). viii, 308 p. (2009).

The book by C. Corduneanu is devoted to various applications of the theory of almost periodic function (a.p.f.) to ordinary and partial differential equations (ODE and PDE). Its main goal is to make the concept of almost periodicity more familiar to its users: applied mathematicians and engineers.

Chapter 2 contains some rudiments of Functional Analysis needed for the study of a.p.f. The main part of the book opens with Chapter 3 which concerns the definitions and basic properties of the various types of a.p.f.: Bohr’s (uniform) a.p.f., Stepanov’s and Besicovich’s a.p.f. and a.p.f. with the values in Banach spaces. The Fourier series of a.p.f. and the problems of their convergence and summability are discussed in the next chapter.

Chapters 5-7 contain applications of the general theory of a.p.f. to oscillations and waves. Classical problems such as oscillations of a pendulum or the vibrations of an elastic string are sufficient to emphasize that almost periodicity has a much higher occurrence than simple periodicity. The author begins with the existence of almost periodic solutions (a.p.s.) of linear differential equations \(x'(t)=A(t)x(t)+f(t)\) and then goes over to various non-linear differential equations, in particular, Lienard’s type and gradient type equations, equations with monotone operators and others. The last section of the book deals with a.p.s. of some PDE.

The reader is taken from elementary and well-known facts through the latest results in almost periodic oscillations and waves. This is the first text to present these latest results. The presentation level and inclusion of clear and lucid proofs make the book ideal for graduate students in science and engineering.

Chapter 2 contains some rudiments of Functional Analysis needed for the study of a.p.f. The main part of the book opens with Chapter 3 which concerns the definitions and basic properties of the various types of a.p.f.: Bohr’s (uniform) a.p.f., Stepanov’s and Besicovich’s a.p.f. and a.p.f. with the values in Banach spaces. The Fourier series of a.p.f. and the problems of their convergence and summability are discussed in the next chapter.

Chapters 5-7 contain applications of the general theory of a.p.f. to oscillations and waves. Classical problems such as oscillations of a pendulum or the vibrations of an elastic string are sufficient to emphasize that almost periodicity has a much higher occurrence than simple periodicity. The author begins with the existence of almost periodic solutions (a.p.s.) of linear differential equations \(x'(t)=A(t)x(t)+f(t)\) and then goes over to various non-linear differential equations, in particular, Lienard’s type and gradient type equations, equations with monotone operators and others. The last section of the book deals with a.p.s. of some PDE.

The reader is taken from elementary and well-known facts through the latest results in almost periodic oscillations and waves. This is the first text to present these latest results. The presentation level and inclusion of clear and lucid proofs make the book ideal for graduate students in science and engineering.

Reviewer: Leonid Golinskii (Kharkov)

##### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

11K70 | Harmonic analysis and almost periodicity in probabilistic number theory |

34C27 | Almost and pseudo-almost periodic solutions to ordinary differential equations |

35B15 | Almost and pseudo-almost periodic solutions to PDEs |

34K14 | Almost and pseudo-almost periodic solutions to functional-differential equations |

42A75 | Classical almost periodic functions, mean periodic functions |