##
**Oscillation theory for difference and functional differential equations.**
*(English)*
Zbl 0954.34002

Dordrecht: Kluwer Academic Publishers. 337 p. Dfl. 274.00; $ 152.00; £91.00 (2000).

This is a good monograph on oscillation of difference and functional-differential equations. Although its appearance has been preceded by the books on oscillation of functional-differential equations by I. Gÿori and G. Ladas [Oscillation theory of delay differential equations: with applications. Oxford: Clarendon Press. (1991; Zbl 0780.34048)], D. D. Bainov and D. P. Mishev [Oscillation theory for neutral differential equations with delay. Bristol etc.: Adam Hilger. (1991; Zbl 0747.34037)], G. S. Ladde, V. Lakshmikantham and B. G. Zhang [Oscillation theory of differential equations with deviating arguments. New York, NY: Marcel Dekker, Inc. (1987; Zbl 0832.34071)], K. Gopalsamy [Stability and oscillations in delay differential equations of population dynamics. Dordrecht etc.: Kluwer Academic Publishers. (1992; Zbl 0752.34039)], and L. H. Erbe, Q. Kong and B. G. Zhang [Oscillation theory for functional differential equations. New York: Marcel Dekker, Inc. (1994; Zbl 0821.34067)] and expositions by R. P. Agarwal and P. J. Y. Wong [Advanced topics in difference equations. Dordrecht: Kluwer Academic Publishers (1997; Zbl 0878.39001)] and by W. G. Kelley and A. C. Peterson [Difference equations: an introduction with applications. Boston, MA etc.: Academic Press Inc. (1991; Zbl 0733.39001)] dealing (among other topics) with oscillation of difference equations, the new text is a welcome addition to the literature on the subject. It provides a nice and systematic exposition of the recent oscillation results both for difference equations and functional-differential equations with deviating arguments and functional-differential equations of neutral type giving an overview on an important area of research such as oscillation theory.

The monograph is divided into two chapters, each containing 20 sections, dealing with difference and functional-differential equations. The first section in each chapter introduces the reader to the subject and explains the structure of the chapter. The first chapter deals with oscillations in difference equations and seems to present one of the first attempts of the systematic presentation of the subject which has attracted recently efforts of many researchers. For scalar difference equations, the authors introduce such basic concepts as oscillation (strict oscillation) around \(a,\) oscillation (strict oscillation) around a sequence, regular oscillation and periodic oscillation. Related results and examples are discussed. In the Section 1.3 the oscillation of some classes of orthogonal polynomials (Chebyshev polynomials, Hermite polynomials, and Legendre polynomials) in the point-wise sense is proved. In the next section a concept of oscillation in the global sense is introduced and studied. The oscillation in ordered sets, linear spaces, and Archimedian spaces is discussed Sections 1.5-1.7. Partial difference equations and their oscillatory properties are considered in Section 1.8. The next section deals with the oscillation of systems of equations. In Section 1.10 another generalization of the concept of oscillation, namely, oscillation between sets, is introduced and examined. The remaining part of Chapter 1 is devoted to the study of oscillation for various classes of difference equations including but not limited to even/odd order difference equations, neutral/mixed difference equations, difference equations involving quasi-differences, difference equations with distributed deviating arguments, partial difference equations, etc. It should be noted that the authors have carefully selected the most interesting results in their opinion on the oscillation of difference equations and provided, whenever possible, illustrative examples which in some cases are far from being trivial. Most theorems are supplied with detailed proofs and references to the literature.

Chapter 2 is devoted to the oscillation of functional-differential equations with deviating arguments and functional-differential equations of neutral type. The authors attempt to present the results on oscillation of \(n\)th-order equations from the unified point of view limiting themselves mostly to integral oscillation criteria and some comparison theorems. Due to a large number of results collected in this chapter, the proofs of only those criteria which the authors thought would be best to illustrate the main techniques and ideas involved have been selected with the references to the literature provided for the rest of theorems. The second chapter starts with the introduction of the main concepts like oscillation, nonoscillation and almost oscillation of solutions to functional-differential equations and a presentation of a number of auxiliary results extensively used in the following exposition. Some interesting recent oscillation results for ordinary differential equations are collected in Section 2.3. In Section 2.4 a total of 28 key results providing sufficient or necessary and sufficient conditions for the oscillation of certain special classes of functional-differential equations are given. Sections 2.5-2.7 deal with comparison results which enable one to deduce oscillatory/nonoscillatory behaviour of a given equation comparing it with the one whose oscillation properties are known or can be easily established. In Sections 2.8-2.13 the authors collect oscillation results for equations with middle term with and without forcing term, forced differential equations, superlinear and sublinear forced differential equations. Two types of comparison results for the neutral equations are discussed in Sections 2.14 and 2.15 they are compared with nonneutral equations and equations of the same form. In the remainder of Chapter 2, oscillation criteria for various classes of neutral and functional-differential equations involving quasi-derivatives and some results for neutral differential equations of mixed type and systems of higher-order functional-differential equations are presented.

An extensive bibliography contains 332 references to the papers and monographs published mainly during the last two decades not limited only to personal contribution to the subject by the authors themselves, but covering the most significant results due to other researchers. The exposition is clear and almost self-contained.

This monograph serves as a reference for specialists in difference and functional-differential equations and can be also used as a valuable source of material for graduate students. Unfortunately, many researchers and students may never open this book because of its quite elevated price.

The monograph is divided into two chapters, each containing 20 sections, dealing with difference and functional-differential equations. The first section in each chapter introduces the reader to the subject and explains the structure of the chapter. The first chapter deals with oscillations in difference equations and seems to present one of the first attempts of the systematic presentation of the subject which has attracted recently efforts of many researchers. For scalar difference equations, the authors introduce such basic concepts as oscillation (strict oscillation) around \(a,\) oscillation (strict oscillation) around a sequence, regular oscillation and periodic oscillation. Related results and examples are discussed. In the Section 1.3 the oscillation of some classes of orthogonal polynomials (Chebyshev polynomials, Hermite polynomials, and Legendre polynomials) in the point-wise sense is proved. In the next section a concept of oscillation in the global sense is introduced and studied. The oscillation in ordered sets, linear spaces, and Archimedian spaces is discussed Sections 1.5-1.7. Partial difference equations and their oscillatory properties are considered in Section 1.8. The next section deals with the oscillation of systems of equations. In Section 1.10 another generalization of the concept of oscillation, namely, oscillation between sets, is introduced and examined. The remaining part of Chapter 1 is devoted to the study of oscillation for various classes of difference equations including but not limited to even/odd order difference equations, neutral/mixed difference equations, difference equations involving quasi-differences, difference equations with distributed deviating arguments, partial difference equations, etc. It should be noted that the authors have carefully selected the most interesting results in their opinion on the oscillation of difference equations and provided, whenever possible, illustrative examples which in some cases are far from being trivial. Most theorems are supplied with detailed proofs and references to the literature.

Chapter 2 is devoted to the oscillation of functional-differential equations with deviating arguments and functional-differential equations of neutral type. The authors attempt to present the results on oscillation of \(n\)th-order equations from the unified point of view limiting themselves mostly to integral oscillation criteria and some comparison theorems. Due to a large number of results collected in this chapter, the proofs of only those criteria which the authors thought would be best to illustrate the main techniques and ideas involved have been selected with the references to the literature provided for the rest of theorems. The second chapter starts with the introduction of the main concepts like oscillation, nonoscillation and almost oscillation of solutions to functional-differential equations and a presentation of a number of auxiliary results extensively used in the following exposition. Some interesting recent oscillation results for ordinary differential equations are collected in Section 2.3. In Section 2.4 a total of 28 key results providing sufficient or necessary and sufficient conditions for the oscillation of certain special classes of functional-differential equations are given. Sections 2.5-2.7 deal with comparison results which enable one to deduce oscillatory/nonoscillatory behaviour of a given equation comparing it with the one whose oscillation properties are known or can be easily established. In Sections 2.8-2.13 the authors collect oscillation results for equations with middle term with and without forcing term, forced differential equations, superlinear and sublinear forced differential equations. Two types of comparison results for the neutral equations are discussed in Sections 2.14 and 2.15 they are compared with nonneutral equations and equations of the same form. In the remainder of Chapter 2, oscillation criteria for various classes of neutral and functional-differential equations involving quasi-derivatives and some results for neutral differential equations of mixed type and systems of higher-order functional-differential equations are presented.

An extensive bibliography contains 332 references to the papers and monographs published mainly during the last two decades not limited only to personal contribution to the subject by the authors themselves, but covering the most significant results due to other researchers. The exposition is clear and almost self-contained.

This monograph serves as a reference for specialists in difference and functional-differential equations and can be also used as a valuable source of material for graduate students. Unfortunately, many researchers and students may never open this book because of its quite elevated price.

Reviewer: Yuri V.Rogovchenko (Mersin)

### MSC:

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

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

34K10 | Boundary value problems for functional-differential equations |

39A11 | Stability of difference equations (MSC2000) |

34K40 | Neutral functional-differential equations |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |