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**Oscillation theory of differential equations with deviating arguments.**
*(English)*
Zbl 0832.34071

Pure and Applied Mathematics, Marcel Dekker. 110. New York, NY: Marcel Dekker, Inc. vi, 308 p. (1987).

This book offers a systematic treatment of oscillation and nonoscillation theory of differential equations with deviating arguments. The book is divided into six chapters.

The first chapter introduces some basic concepts from the theory of differential equations with deviating arguments and demonstrates some new problems in oscillation theory caused by deviating arguments. This begins with the statement of the basic initial value problems and classification of equations with deviating arguments. Then it summarizes certain main topics in oscillation theory and, finally, presents some fixed point theorems which are important tools in oscillation theory.

The second chapter deals with the first-order linear differential equations with deviating arguments. The authors begin with first-order equations with a deviating argument and then discuss the case with several deviating arguments presenting various available techniques.

In chapter three, the oscillatory and nonoscillatory behaviour of nonlinear first-order ordinary differential equations with deviating arguments is studied. We find here oscillation and nonoscillation results for a single as well as several deviating arguments, various results on differential inequalities with deviating arguments, investigation of differential equations with mixed type of deviating arguments, oscillatory results for general nonlinear equations with and without forcing terms, and others.

Chapter Four deals with second-order differential equations. This begins with classification of solutions of linear equations of the type \((r(t) y'(t))'= p(t) y(g(t))\). Then the authors state certain important properties for second-order linear unstable equations and extend some of these results to a class of nonlinear equations. Moreover, oscillation results for equations with deviating arguments of distributed type are studied in this chapter.

Chapter Five is devoted to the study of the oscillation of high order differential equations and inequalities with deviating arguments. The authors begin with the study of third- and fourth-order differential equations. Then they discuss even order differential equations and present several results which are important in studying differential equations and inequalities with oscillation of high order. Also, differential equations with deviating arguments of mixed type and differential equations with forcing terms are studied here.

In chapter Six two-dimensional linear and nonlinear as well as higher order linear systems are discussed.

The book concludes with a complete set of references (of 309 items) and an index.

The first chapter introduces some basic concepts from the theory of differential equations with deviating arguments and demonstrates some new problems in oscillation theory caused by deviating arguments. This begins with the statement of the basic initial value problems and classification of equations with deviating arguments. Then it summarizes certain main topics in oscillation theory and, finally, presents some fixed point theorems which are important tools in oscillation theory.

The second chapter deals with the first-order linear differential equations with deviating arguments. The authors begin with first-order equations with a deviating argument and then discuss the case with several deviating arguments presenting various available techniques.

In chapter three, the oscillatory and nonoscillatory behaviour of nonlinear first-order ordinary differential equations with deviating arguments is studied. We find here oscillation and nonoscillation results for a single as well as several deviating arguments, various results on differential inequalities with deviating arguments, investigation of differential equations with mixed type of deviating arguments, oscillatory results for general nonlinear equations with and without forcing terms, and others.

Chapter Four deals with second-order differential equations. This begins with classification of solutions of linear equations of the type \((r(t) y'(t))'= p(t) y(g(t))\). Then the authors state certain important properties for second-order linear unstable equations and extend some of these results to a class of nonlinear equations. Moreover, oscillation results for equations with deviating arguments of distributed type are studied in this chapter.

Chapter Five is devoted to the study of the oscillation of high order differential equations and inequalities with deviating arguments. The authors begin with the study of third- and fourth-order differential equations. Then they discuss even order differential equations and present several results which are important in studying differential equations and inequalities with oscillation of high order. Also, differential equations with deviating arguments of mixed type and differential equations with forcing terms are studied here.

In chapter Six two-dimensional linear and nonlinear as well as higher order linear systems are discussed.

The book concludes with a complete set of references (of 309 items) and an index.

Reviewer: J.Ohriska (Košice)

### MSC:

34K11 | Oscillation theory of functional-differential equations |

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

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