## Oscillation theory of two-term differential equations.(English)Zbl 0878.34022

Mathematics and its Applications (Dordrecht). 396. Dordrecht: Kluwer Academic Publishers. vii, 217 p. (1997).
The book contains a comprehensive treatment of oscillation properties of linear differential equations of the form $L_ny+p(x)y=0,\tag{1}$ where $L_ny:=\varrho_n(x)(\varrho_{n-1}(x)\dots(\varrho_1(x)(\varrho_0(x)y)')'\dots)' \tag{2}$ is the $$n$$-th order disconjugate differential operator with $$\varrho_i>0,$$ $$\varrho_i\in C^{n-i}$$, $$i=0,\dots,n$$ and with $$p$$ of constant sign. Equation (1) includes as a special case the classical two-term equation $y^{(n)}+p(x)y=0.\tag{3}$
Investigation of oscillation properties of linear differential equations has a long history and as its origin it is usually regarded the pioneering work of Sturm from 1836. Since that time a tremendous number of papers devoted to this topic has been published and hence any work presenting a unified approach to the problem (or at least to a part of it) is of great value. This monograph represents a successful attempt along this line.
Here is the list of chapters of the book: 0. Introduction, 1. The basic lemma, 2. Boundary value functions, 3. Bases of solutions, 4. Comparison of boundary value problems, 5. Comparison theorems for two equations, 6. Disfocality and its characterization, 7. Various types of disfocality, 8. Solutions on an infinite interval, 9. Disconjugacy and its characterization, 10. Eigenvalue problems, 11. More extremal points, 12. Minors of the Wronskians, 13. The dominance property of solutions.
The author starts with basic concepts of oscillation theory of linear differential equations, like canonical factorization of disconjugate differential operators, solvability of boundary value problems (BVP’s) associated with (1), special bases of the solution space of (1) and of its adjoint equation, conjugate, focal and extremal points. It is shown that the disconjugate differential operator (2) and quasiderivatives defined by $L_0y=\varrho_0y,\quad L_ky=\varrho_k(L_{k-1}y)',\quad k=1,\dots,n,$ behave essentially like the operator $${\text{d}^n\over \text{d}x^n}$$ and the usual derivatives. A particular attention is also devoted to the investigation of the relation between the existence of conjugate (focal, extremal) points and solvability of certain corresponding BVP’s associated with (1).
Chapters 4 and 5 deal with various comparison theorems for equation (1). In Chapter 4 two different BVP’s for one equation are compared and as a particular case inequalities and existence criteria for focal and conjugate points are obtained. Chapter 5 presents comparison theorems for two equations, both of pointwise and integral type. In case $$n=2$$ these statements reduce to the well-known Sturm and Hille-Wintner type theorems for second order equations.
The next four chapters, Chapter 6-9, may be regarded as the most important part of the book. Here the very voluminous material on disconjugacy, oscillation and asymptotic properties of solutions of (1) is elaborated. The fact that the author concentrates his attention to a special class of linear equations of the form (1) enables him to present a large variety of problems concerning oscillation theory of linear equations in a very consistent form. In addition to the characterization of disconjugacy and disfocality of (1) and relations between these concepts, the reviewer particularly enjoyed the treatment of properties A,B of solutions of this equation.
Chapter 10 deals with distribution of eigenvalues of BVP $L_ny+\lambda p(x)y=0,\quad +\text{\;boundary conditions}$ and related problems, like the number of zeros of eigenfunctions and dependence of eigenvalues on boundary conditions. Chapter 11 presents some additional properties of conjugate, focal and extremal points of solutions of (1). The last two chapters contain essentially the author’s recent results on asymptotic properties of some Wronskians and linear combinations of solutions of (1). Let us mention that in these chapters the author also formulated several open problems concerning dominance properties and uniqueness of the so-called Kneser solution of (1).
The book is self-contained and it is determined to researchers and postgraduate students interested in qualitative theory of differential equations. The people who are familiar with the problems treated in the book will particularly enjoy the unified and well-arranged approach to the problem. In many cases completely new proofs are given and in no case the original proof is copied verbatim. The theoretical material is illustrated by a number of examples and corollaries dealing with explicitly solvable equations or equations of the form (3) which facilitate the reading of the book.
Reviewer: O.Došlý (Brno)

### MSC:

 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations 34A30 Linear ordinary differential equations and systems 34B05 Linear boundary value problems for ordinary differential equations 34B24 Sturm-Liouville theory