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**On oscillatory properties of solutions of functional differential equations.**
*(English)*
Zbl 0843.34070

This book is dedicated to the study of oscillatory properties of solutions of a functional differential equation of the form (1) \(u^{(n)} (t)+F(u)(t)=0\), where \(n \geq 1\) and \(F \in V (\tau)\) \((F \in V (\tau, \sigma))\); \(\tau, \sigma \in C(\mathbb{R}_+; \mathbb{R}_+)\), \(\lim_{t \to+\infty} \tau (t)=+ \infty\), \(\tau (t) \leq \sigma (t)\) for \(t \in \mathbb{R}_+\) and \(V (\tau)\) \((V (\tau, \sigma))\) denotes the set of continuous mappings \(F:C (\mathbb{R}_+; \mathbb{R}) \to L_{\text{loc}} (\mathbb{R}_+; \mathbb{R})\) satisfying the condition: \(F(x) (t)=F(y) (t)\) holds for any \(t \in \mathbb{R}_+\) and \(x,y \in C (\mathbb{R}_+; \mathbb{R})\) provided that \(x(s)=y(s)\) for \(s \geq \tau (t)\) \((\tau (t) \leq s \leq \sigma (t))\). For any \(t_0 \in \mathbb{R}_+\) by \(H_{t_0, \tau}\) is denoted the set of all functions \(u \in C (\mathbb{R}_+; \mathbb{R})\) satisfying \(u(t) \neq 0\) for \(t \geq t_*\), where \(t_*=\min \{t_0, \tau_* (t_0)\}\), \(\tau_* (t)=\inf \{\tau (s):s \geq t\}\). There is assumed that either the condition \(F(u) (t)u(t) \geq 0\) for \(t \geq t_0\), \(u \in H {t_0, \tau}\) or the condition \(F(u ) (t)u(t) \leq 0\) for \(t \geq t_0\), \(u \in H_{t_0, \tau}\) is fulfilled throughout the work. The following definitions are used in the work.

Let \(t_0 \in \mathbb{R}_+\). A function \(u:\langle t_0,+\infty) \to \mathbb{R}\) is said to be a proper solution of equation (1) if it is locally absolutely continuous together with its derivatives up to order \(n-1\) inclusive, \(\sup \{|u(s) |:s \in\langle t,+\infty)\} > 0\) for \(t \geq t_0\), and there exists a function \(\overline u \in C (\mathbb{R}_+; \mathbb{R})\) such that \(\overline u(t) \equiv u(t)\) on \( \langle t_0,+\infty)\) and the equality \(\overline u^{(n)} (t)+F (\overline u(t))=0\) holds for \(t \in \langle t_0,+\infty)\).

A proper solution \(u:\langle t_0,+\infty) \to \mathbb{R}\) of equation (1) is said to be oscillatory if it has a sequence of zeros tending to \(+ \infty\). Otherwise the solution \(u\) is said to be nonoscillatory.

We say that equation (1) has property \(\mathbb{A}\) if any of its proper solutions is oscillatory when \(n\) is even and either is oscillatory or satisfies (2) \(|u^{ (i)} (t) |\to 0\) as \(t \to+\infty\) \((i=0, \dots, n-1)\) when \(n\) is odd. We say that equation (1) has property \(\mathbb{B}\) if any of its proper solutions either is oscillatory or satisfies either (2) or (3) \(|u^{(i)} (t) |\to+\infty\) as \(t \to+\infty\) \((i=0, \dots, n-1)\) when \(n\) is even, and either is oscillatory or satisfies (3) when \(n\) is odd. We say that equation (1) has property \(\widehat \mathbb{A}\) if any of its proper solution is oscillatory when \(n\) is odd, and either is oscillatory or satisfies (3) when \(n\) is even. We say that equation (1) has property \(\widetilde \mathbb{B}\) if any of its proper solutions is oscillatory when \(n\) is odd, and either is oscillatory or satisfies (2) when \(n\) is even.

The book is divided into six chapters. Chapter 1 is concerned with equations having property \(\mathbb{A}\) or \(\mathbb{B}\). Comparison theorems are proved in \(\S 2\), thereby making it possible to derive property \(\mathbb{A}\) or \(\mathbb{B}\) of the considered equations. Based on these theorems, sufficient (necessary and sufficient) conditions are established in §3 (in §4) for an essentially nonlinear functional differential equation to have property \(\mathbb{A}\) or \(\mathbb{B}\).

Chapter 2 deals with analogous problems for equation (1) with the operator \(F\) admitting a linear minorant. The new results for equation (1) obtained in this chapter improve some of the previous well-known results for equation \(u^{(n)} (t) + p(t) u(t) = 0\). The chapter concludes with some sufficient conditions for equation (1) not to have property \(\mathbb{A} (\mathbb{B})\).

Chapter 3 and 4 are concerned with solutions satisfying conditions (2) and (3). The reader finds here some auxiliary lemmas which enable one to establish the asymptotic behaviour near \(+ \infty\) of solutions satisfying (2) of differential equations and inequalities with a delayed argument.

In Chapter 5 the previously obtained results are used to find the sufficient or necessary and sufficient conditions for equation (1) to have property \(\widetilde \mathbb{A}\) or \(\widetilde \mathbb{B}\) and also the sufficient or necessary and sufficient conditions are established for any solution of (1) to be oscillatory. The results presented in this chapter are specific of functional differential equations and have no analoguous for ordinary differential equations.

Chapter 6 is dedicated to second order differential equations with a delayed argument.

The results of this work make it possible to extend a number of the earlier results concerning the oscillatory behaviour of differential equations with deviating arguments to the case of general functional differential equations. Besides, the work presents new results specific of functional differential equations.

Let \(t_0 \in \mathbb{R}_+\). A function \(u:\langle t_0,+\infty) \to \mathbb{R}\) is said to be a proper solution of equation (1) if it is locally absolutely continuous together with its derivatives up to order \(n-1\) inclusive, \(\sup \{|u(s) |:s \in\langle t,+\infty)\} > 0\) for \(t \geq t_0\), and there exists a function \(\overline u \in C (\mathbb{R}_+; \mathbb{R})\) such that \(\overline u(t) \equiv u(t)\) on \( \langle t_0,+\infty)\) and the equality \(\overline u^{(n)} (t)+F (\overline u(t))=0\) holds for \(t \in \langle t_0,+\infty)\).

A proper solution \(u:\langle t_0,+\infty) \to \mathbb{R}\) of equation (1) is said to be oscillatory if it has a sequence of zeros tending to \(+ \infty\). Otherwise the solution \(u\) is said to be nonoscillatory.

We say that equation (1) has property \(\mathbb{A}\) if any of its proper solutions is oscillatory when \(n\) is even and either is oscillatory or satisfies (2) \(|u^{ (i)} (t) |\to 0\) as \(t \to+\infty\) \((i=0, \dots, n-1)\) when \(n\) is odd. We say that equation (1) has property \(\mathbb{B}\) if any of its proper solutions either is oscillatory or satisfies either (2) or (3) \(|u^{(i)} (t) |\to+\infty\) as \(t \to+\infty\) \((i=0, \dots, n-1)\) when \(n\) is even, and either is oscillatory or satisfies (3) when \(n\) is odd. We say that equation (1) has property \(\widehat \mathbb{A}\) if any of its proper solution is oscillatory when \(n\) is odd, and either is oscillatory or satisfies (3) when \(n\) is even. We say that equation (1) has property \(\widetilde \mathbb{B}\) if any of its proper solutions is oscillatory when \(n\) is odd, and either is oscillatory or satisfies (2) when \(n\) is even.

The book is divided into six chapters. Chapter 1 is concerned with equations having property \(\mathbb{A}\) or \(\mathbb{B}\). Comparison theorems are proved in \(\S 2\), thereby making it possible to derive property \(\mathbb{A}\) or \(\mathbb{B}\) of the considered equations. Based on these theorems, sufficient (necessary and sufficient) conditions are established in §3 (in §4) for an essentially nonlinear functional differential equation to have property \(\mathbb{A}\) or \(\mathbb{B}\).

Chapter 2 deals with analogous problems for equation (1) with the operator \(F\) admitting a linear minorant. The new results for equation (1) obtained in this chapter improve some of the previous well-known results for equation \(u^{(n)} (t) + p(t) u(t) = 0\). The chapter concludes with some sufficient conditions for equation (1) not to have property \(\mathbb{A} (\mathbb{B})\).

Chapter 3 and 4 are concerned with solutions satisfying conditions (2) and (3). The reader finds here some auxiliary lemmas which enable one to establish the asymptotic behaviour near \(+ \infty\) of solutions satisfying (2) of differential equations and inequalities with a delayed argument.

In Chapter 5 the previously obtained results are used to find the sufficient or necessary and sufficient conditions for equation (1) to have property \(\widetilde \mathbb{A}\) or \(\widetilde \mathbb{B}\) and also the sufficient or necessary and sufficient conditions are established for any solution of (1) to be oscillatory. The results presented in this chapter are specific of functional differential equations and have no analoguous for ordinary differential equations.

Chapter 6 is dedicated to second order differential equations with a delayed argument.

The results of this work make it possible to extend a number of the earlier results concerning the oscillatory behaviour of differential equations with deviating arguments to the case of general functional differential equations. Besides, the work presents new results specific of functional differential equations.

Reviewer: J.Ohriska (Košice)