##
**Global properties of linear ordinary differential equations.**
*(English)*
Zbl 0784.34009

Mathematics and Its Applications. East European Series. 52. Dordrecht: Kluwer Academic Publishers. xv, 320 p. (1991).

Investigations of linear differential equations (LDEs for short) from the point of view of their transformations, canonical forms and invariants started in the middle of the last century. E. E. Kummer investigated transformations, S. Lie and P. Stäckel derived the most general form of them, A. R. Forsyth, F. Brioschi and G. H. Halphen studied invariants and canonical forms of LDEs. However, all these investigations were of a local character. Local methods and results are not suitable when studying problems of a global nature, like boundedness, periodicity, asymptotic and oscillatory behavior of solutions.

In the fifties O. Borůvka started the systematic study of global properties of the second order LDEs by using methods of algebra and geometry. He summarized his original methods and results in his monograph [Linear differential transformations of the second order, London (1971; Zbl 0222.34002)]. In the last 20 years this approach was extended by the author to LDEs of an arbitrary order \(n\), \(n\geq 2\). He introduced several origin methods and by means of methods and results of algebra, topology, differential geometry, functional analysis, theory of functional equations and theory of LDEs he solved problems which were previously studied only locally. Methods and results are summarized in this excellent and modern monograph. “This book deals with LDEs of the \(n\)th order, \(n\geq 2\), and summarizes results in this field in a unified fashion. However, this monograph is by no means intended to be survey of all results in this area. It contains only a selection of results, which serves to illustrate the unified approach presented here. …The book is self-contained; the notations and results needed in the text are mostly introduced and derived here.”

A short introduction with historical remarks is given in Chapter 1. Notation and auxiliary results of topology, algebraic structures (e.g. Ehresmann and Brandt groupoids), vector spaces, LDEs and functional differential equations (Abel and Euler equations) are introduced. Those are needed throughout the rest of the book and some of them can be found in textbooks on differential equations and algebra.

Chapter 3 deals with global transformations. We say that the LDE \(P_ n(y,x;I): y^{(n)}+ p_{n-1}(x) y^{(n-1)}+ \cdots+ p_ 0(x)y=0\), \(p_ i\in C^ 0(I)\) is globally transformable into the LDE \(Q_ n(z,t;J): z^{(n)}+ q_{n-1}(t) z^{(n-1)}+ \cdots+ q_ 0(t)z=0\), \(q_ i\in C^ 0(J)\), if there exist two functions \(f\) and \(h\) such that (i) \(f\) is of class \(C^ n(J)\), \(f(t)\neq 0\) on \(J\), (ii) \(h\) is a \(C^ n\)-diffeomorphism of interval \(J\) onto interval \(I\), and the function \(z(t)=f(t)y(h(t))\), \(t\in J\) is a solution of equation \(Q_ n(z,t;J)\) whenever \(y\) is a solution of equation \(P_ n(y,x:I)\); and say that \(\langle\langle f,h\rangle\rangle\) is a global transformation of \(P_ n\) into \(Q_ n\). Some algebraic aspects of the global transformations are studied here. The structure of global transformations is originally described in the frame of the theory of categories, by using Brandt and Ehresmann groupoids.

In Chapter 4, analytic, algebraic and geometric aspects of global transformations are introduced. The results presented here serve as a necessary basic for the next chapters.

Chapter 5 deals with criteria of global equivalence. First, the Borůvka’s criterion (by means of terms type and kind of LDE) for the second order LDEs is proved by a new method which is shorter than the origin proof. Next, a new criterion of global equivalence of LDEs the \(n\)th order \((n\geq 3)\) is proved. This criterion uses the \(n\)th order operator of the iterative equation obtained from some second order LDE.

In Chapter 6 stationary groups are considered. The stationary group of an LDE is formed by all global transformations transforming that LDE into itself. A complete list of stationary groups and a characterization of corresponding LDEs is given in the end of this chapter.

Chapter 7 (Canonical forms) deals with finding of representatives, canonical forms in each Brandt groupoid of the Ehresmann groupoid. Two special canonical forms of LDEs have been introduced in the literature, the so-called Laguerre-Forsyth forms and the Halphen forms. These forms are not global in the sense that not every LDE can be transformed in the LDE of the form on its whole interval of definition. Here presented global canonical forms are proved by using a geometrical approach (Cartan’s moving-frame-of-reference method; generalized Frenet formulas) and an analytic approach. A list of canonical forms of the second and third LDEs is introduced.

Invariants and covariants of LDEs with respect to global transformations are studied in Chapter 8. Concrete applications of global transformations of the second order LDEs are given in Chapter 9. Weyl’s limit circle and limit point classification is widely considered and among others a criterion of periodicity of solutions is proved.

Chapter 10 is concerned with zeros of solutions of LDEs. The essence of the author’s approach to questions on distribution of zeros of solutions is based on a certain geometrical representation of zeros, by which a solution has a zero \(x_ 0\) of the multiplicity \(k\) if and only if a hyperplane has a contact of the order \(k-1\) with a curve at a certain point. Some complicated constructions or proofs can be easily understood and hints for possible new results can be obtained by this approach.

The last chapter contains related results and some applications. The titles of the sections give good information about the chapter: Asymptotic properties and zeros of solutions of second order LDEs; Integral inequalities; Affine geometry of plane curves; Isoperimetric theorems; Related results and comments, possible trends of further research.

The author says in the foreword: “This monograph is written for mathematicians working with differential equations and systems and, due to the application of modern aspects of other fields of mathematics, also for those working in algebra, topology, differential geometry, functional-differential equations and functional equations. The description of some situations is quite easy and some answers are obtained without complicated calculation, hence these parts are appropriate for undergraduate students”.

In the fifties O. Borůvka started the systematic study of global properties of the second order LDEs by using methods of algebra and geometry. He summarized his original methods and results in his monograph [Linear differential transformations of the second order, London (1971; Zbl 0222.34002)]. In the last 20 years this approach was extended by the author to LDEs of an arbitrary order \(n\), \(n\geq 2\). He introduced several origin methods and by means of methods and results of algebra, topology, differential geometry, functional analysis, theory of functional equations and theory of LDEs he solved problems which were previously studied only locally. Methods and results are summarized in this excellent and modern monograph. “This book deals with LDEs of the \(n\)th order, \(n\geq 2\), and summarizes results in this field in a unified fashion. However, this monograph is by no means intended to be survey of all results in this area. It contains only a selection of results, which serves to illustrate the unified approach presented here. …The book is self-contained; the notations and results needed in the text are mostly introduced and derived here.”

A short introduction with historical remarks is given in Chapter 1. Notation and auxiliary results of topology, algebraic structures (e.g. Ehresmann and Brandt groupoids), vector spaces, LDEs and functional differential equations (Abel and Euler equations) are introduced. Those are needed throughout the rest of the book and some of them can be found in textbooks on differential equations and algebra.

Chapter 3 deals with global transformations. We say that the LDE \(P_ n(y,x;I): y^{(n)}+ p_{n-1}(x) y^{(n-1)}+ \cdots+ p_ 0(x)y=0\), \(p_ i\in C^ 0(I)\) is globally transformable into the LDE \(Q_ n(z,t;J): z^{(n)}+ q_{n-1}(t) z^{(n-1)}+ \cdots+ q_ 0(t)z=0\), \(q_ i\in C^ 0(J)\), if there exist two functions \(f\) and \(h\) such that (i) \(f\) is of class \(C^ n(J)\), \(f(t)\neq 0\) on \(J\), (ii) \(h\) is a \(C^ n\)-diffeomorphism of interval \(J\) onto interval \(I\), and the function \(z(t)=f(t)y(h(t))\), \(t\in J\) is a solution of equation \(Q_ n(z,t;J)\) whenever \(y\) is a solution of equation \(P_ n(y,x:I)\); and say that \(\langle\langle f,h\rangle\rangle\) is a global transformation of \(P_ n\) into \(Q_ n\). Some algebraic aspects of the global transformations are studied here. The structure of global transformations is originally described in the frame of the theory of categories, by using Brandt and Ehresmann groupoids.

In Chapter 4, analytic, algebraic and geometric aspects of global transformations are introduced. The results presented here serve as a necessary basic for the next chapters.

Chapter 5 deals with criteria of global equivalence. First, the Borůvka’s criterion (by means of terms type and kind of LDE) for the second order LDEs is proved by a new method which is shorter than the origin proof. Next, a new criterion of global equivalence of LDEs the \(n\)th order \((n\geq 3)\) is proved. This criterion uses the \(n\)th order operator of the iterative equation obtained from some second order LDE.

In Chapter 6 stationary groups are considered. The stationary group of an LDE is formed by all global transformations transforming that LDE into itself. A complete list of stationary groups and a characterization of corresponding LDEs is given in the end of this chapter.

Chapter 7 (Canonical forms) deals with finding of representatives, canonical forms in each Brandt groupoid of the Ehresmann groupoid. Two special canonical forms of LDEs have been introduced in the literature, the so-called Laguerre-Forsyth forms and the Halphen forms. These forms are not global in the sense that not every LDE can be transformed in the LDE of the form on its whole interval of definition. Here presented global canonical forms are proved by using a geometrical approach (Cartan’s moving-frame-of-reference method; generalized Frenet formulas) and an analytic approach. A list of canonical forms of the second and third LDEs is introduced.

Invariants and covariants of LDEs with respect to global transformations are studied in Chapter 8. Concrete applications of global transformations of the second order LDEs are given in Chapter 9. Weyl’s limit circle and limit point classification is widely considered and among others a criterion of periodicity of solutions is proved.

Chapter 10 is concerned with zeros of solutions of LDEs. The essence of the author’s approach to questions on distribution of zeros of solutions is based on a certain geometrical representation of zeros, by which a solution has a zero \(x_ 0\) of the multiplicity \(k\) if and only if a hyperplane has a contact of the order \(k-1\) with a curve at a certain point. Some complicated constructions or proofs can be easily understood and hints for possible new results can be obtained by this approach.

The last chapter contains related results and some applications. The titles of the sections give good information about the chapter: Asymptotic properties and zeros of solutions of second order LDEs; Integral inequalities; Affine geometry of plane curves; Isoperimetric theorems; Related results and comments, possible trends of further research.

The author says in the foreword: “This monograph is written for mathematicians working with differential equations and systems and, due to the application of modern aspects of other fields of mathematics, also for those working in algebra, topology, differential geometry, functional-differential equations and functional equations. The description of some situations is quite easy and some answers are obtained without complicated calculation, hence these parts are appropriate for undergraduate students”.

Reviewer: S.Staněk (Olomouc)

### MSC:

34A30 | Linear ordinary differential equations and systems |

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

34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |

34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |

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

34C11 | Growth and boundedness of solutions to ordinary differential equations |