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**Oscillation theory for second order linear, half-linear, superlinear and sublinear dynamic equations.**
*(English)*
Zbl 1073.34002

Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-0802-3/hbk). xiv, 672 p. (2002).

There has been a striking progress in the development of the theory of oscillations after the excellent book on the subject by [C. A. Swanson, Comparison and oscillation theory of linear differential equations. New York-London: Academic Press, (1968; Zbl 0191.09904)] was published. The monograph under review is a very successful attempt to collect and systematically present numerous oscillation or nonoscillation criteria for various classes of linear and nonlinear differential equations and related results on asymptotic behavior of solutions published during the last few decades, many of which have been obtained by the authors. The material in the book is divided into ten chapters. In Chapter 1, which has an auxiliary character, sufficient conditions for solutions of a general class of nonlinear differential equations to be continuable, bounded, and converge to zero are presented along with a number of useful fixed-point theorems. Fundamental results on oscillation and nonoscillation of second-order linear differential equations, including Sturm-type comparison theorems, are collected in Chapter 2. Oscillation and nonoscillation theorems for second-order half-linear differential equations are presented in Chapter 3, where the reader can find Sturm- and Levin-type comparison theorems, an oscillation criterion for almost-periodic Sturm-Liouville equations, interval oscillation criteria, as well as a number of interesting results dealing with differential equations with a damping term, external forces and deviating argument. Sufficient conditions for oscillation and sufficient and necessary conditions for nonoscillation of superlinear differential equations, including damped and forced differential equations, can be found in Chapter 4. In addition to standard techniques involving integrals and weighted integrals, more general weighted and iterated averages are employed. Chapter 5 is concerned with oscillation and nonoscillation results for sublinear differential equations, including criteria derived for the celebrated Emden-Fowler equations arising in many applied problems. Linearized oscillation theorems and comparison results are also discussed here. Special techniques required for the study of oscillatory behavior of solutions of different classes of differential equations are developed in Chapter 6, where the reader can learn the method due to C. Olech, Z. Opial and T. Waẓewski [Bull. Acad. Pol. Sci., Cl. III 5, 621–626 (1957; Zbl 0078.07701)], explore the use of variational inequalities and the direct Lyapunov method for the study of oscillation. Oscillatory behavior of two-dimensional superlinear, linear, and sublinear systems and second-order matrix differential equations is discussed in Chapter 7 by using Riccati and variational techniques. Oscillations of systems with a functionally commutative matrix, operator-valued equations, and systems with a forcing term are also considered. Asymptotic behavior of solutions to second-order differential equations is addressed in Chapter 8 with a special attention paid to the study of existence, uniqueness and asymptotic behavior of positive solutions of Emden-Fowler equations and systems of equations. Chapter 9 is a collection of miscellaneous interesting results including the extension of the Sturm-Picone theorem, criteria for nonoscillation of forced differential equations, limit circle criteria and related results. Finally, a number of nonoscillation results for differential inclusions are proved in Chapter 10 by using specific fixed-point theorems and a compactness criterion.

The book is nicely written and the selection of the material is very interesting, although several important recent results on oscillatory and asymptotic behavior of second-order linear and nonlinear differential equations and matrix differential equations were not noticed by the authors. Since the research in the theory of oscillation has been very intensive during the last decades, it is reasonable that the choice of the material reflects primarily the preferences of the authors and their contributions to the subject. The book is a welcome addition to the existing literature on oscillations and shall be useful for researchers and graduate students in differential equations who are interested not only in oscillations of various classes of equations but also in the asymptotic behavior of solutions in general.

The book is nicely written and the selection of the material is very interesting, although several important recent results on oscillatory and asymptotic behavior of second-order linear and nonlinear differential equations and matrix differential equations were not noticed by the authors. Since the research in the theory of oscillation has been very intensive during the last decades, it is reasonable that the choice of the material reflects primarily the preferences of the authors and their contributions to the subject. The book is a welcome addition to the existing literature on oscillations and shall be useful for researchers and graduate students in differential equations who are interested not only in oscillations of various classes of equations but also in the asymptotic behavior of solutions in general.

Reviewer: Yuri V. Rogovchenko (Famagusta)

### MSC:

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

34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |

34C11 | Growth and boundedness of solutions to ordinary differential equations |

34K11 | Oscillation theory of functional-differential equations |