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**Oscillation theory for neutral differential equations with delay.**
*(English)*
Zbl 0747.34037

Bristol etc.: Adam Hilger. vi, 280 p. (1991).

The book under review consists of results concerning the oscillatory properties of solutions of delay differential equations of neutral type, i.e. differential equations which contain the leading derivative at different times.

Chapter I is introductory. Chapter II deals with first order neutral delay differential equations (NDDE). The typical object under discussion is the NDDE \((d/dt)[x(t)+h(t)x(\tau(t))]=f(t,x(g(t)))\). Oscillation and nonoscillation criteria are given for the first order NDDEs both linear and nonlinear, autonomous and nonautonomous. NDDEs with distributed delay are also considered. Asymptotic behavior of nonoscillating solutions is described. Section 2.5 is devoted to applications for the delay logistic equation arising in population growth models. Chapter III is devoted to oscillation and asymptotic properties of second order NDDEs of the form \((d^ 2/dt^ 2)[y(t)+P(t)y(t-\tau)]+f(y(t-\sigma))=0\) and their particular cases and generalizations. The results analogous to those stated in Chapter II are described. In Chapter IV nth order NDDEs \((d^ n/dt^ n)[y(t)+P(t)y(t-\tau)]+Q(t)y(t-\sigma)=0\) are considered. In Chapter V criteria for the existence of nonoscillating solutions for linear systems of NDDEs with constant coefficient are given. Chapter VI deals with partial NDDEs. Parabolic and hyperbolic NDDEs, both linear and nonlinear, and also parabolic and hyperbolic equations with “maxima” are considered. The book contains many illustrative examples. Each chapter is supplied with bibliographical notes and comments. The bibliography consists of 163 entries.

Chapter I is introductory. Chapter II deals with first order neutral delay differential equations (NDDE). The typical object under discussion is the NDDE \((d/dt)[x(t)+h(t)x(\tau(t))]=f(t,x(g(t)))\). Oscillation and nonoscillation criteria are given for the first order NDDEs both linear and nonlinear, autonomous and nonautonomous. NDDEs with distributed delay are also considered. Asymptotic behavior of nonoscillating solutions is described. Section 2.5 is devoted to applications for the delay logistic equation arising in population growth models. Chapter III is devoted to oscillation and asymptotic properties of second order NDDEs of the form \((d^ 2/dt^ 2)[y(t)+P(t)y(t-\tau)]+f(y(t-\sigma))=0\) and their particular cases and generalizations. The results analogous to those stated in Chapter II are described. In Chapter IV nth order NDDEs \((d^ n/dt^ n)[y(t)+P(t)y(t-\tau)]+Q(t)y(t-\sigma)=0\) are considered. In Chapter V criteria for the existence of nonoscillating solutions for linear systems of NDDEs with constant coefficient are given. Chapter VI deals with partial NDDEs. Parabolic and hyperbolic NDDEs, both linear and nonlinear, and also parabolic and hyperbolic equations with “maxima” are considered. The book contains many illustrative examples. Each chapter is supplied with bibliographical notes and comments. The bibliography consists of 163 entries.

Reviewer: R.R.Akhmerov (Novosibirsk)

### MSC:

34K99 | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |

35R10 | Partial functional-differential equations |

34K40 | Neutral functional-differential equations |

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |

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

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