##
**Generalized ordinary differential equations.**
*(English)*
Zbl 0781.34003

Series in Real Analysis. 5. Singapore: World Scientific. ix, 382 p. (1992).

The book is devoted to the theory of generalized ordinary differential equations which was developed mainly by the Czech school (Jarnik, Kurzweil, Schwabik). The material of the book is devided into ten chapters.

Chapter I has an introductory character and contains basic notions of the theory of generalized integral in the Henstock-Kurzweil sense. For example, the author describes the notions of the tagged interval, a gauge function, an \(o\)-fine system and so on. It is pointed out that the classical Riemann, Stieltjes and Perron integral can be obtained as special cases of the Henstock-Kurzweil integral. Moreover, several properties of the integral in question are proved. For example, integral inequalities of Bihari-Gronwall type are derived.

Chapter II discusses in details the theorem on the existence of solutions of the Cauchy problem (1) \(x'=f(t,x)\), \(x(0)=x_ 0\), where the solution is understood in the sense of Perron-Henstock. It is worthwhile mentioning that the assumptions imposed on the right hand side \(f=f(t,x)\) imply that it may be represented as the sum of a Carathéodory function \(h(t,x)\) and a Perron integrable function \(g(t)\).

The next two Chapters III and IV are devoted to the study of the generalized ordinary differential equations of the form (2) \(dx/d\tau=DF(t,x)\), where the letter \(D\) indicates that the equation is treated in generalized sense. A function \(x: [a,b]\to \mathbb{R}^ n\) is said to be a solution of the equation (2) if \(x(u)-x(v)= \int^ u_ v DF(t,x(\tau))\) for every \(u,v\in [a,b]\), where the integral is taken as the generalized Perron integral. It is shown that the solutions of (2) are functions being locally of bounded variation.

In Chapter V a comparison of the generalized ordinary differential equations with other important types is made. Particularly, it is shown that Carathéodory equations, the so-called measure differential equations as well as differential equations with impulses can be treated from the unified point of view as special cases of generalized ordinary differential equations.

In Chapter VI a special attention is focused on generalized linear differential equations of the form (3) \(dx/d\tau= D[A(t)x+ g(t)]\), where the matrix \(A\) is assumed to be of locally bounded variation while \(g(t)\) is a function of the same type. A lot of results concerning the equation (3) are proved which show the analogy with the classical theory of linear differential equations.

Chapter VII describes fundamental properties of integral products. This notion replaces the Riemann integral sums by partial integral products. The approach to the generalized linear differential equations via product integral turns out to be a very fruitful idea what is shown in this chapter.

The last three chapters treat the problem of continuous dependence of solutions on the right hand sides of generalized ordinary differential equations and variational stability of solutions of equations of such a type. Moreover, in Chapter IX there is considered the situation when the right hand sided converge to the Dirac delta function.

The considerations in the book are correct and clear, in general. But some aspects seem to be unclear and their presentation is not satisfactory enough. For example, the concepts of the Henstock-Kurzweil integral (in the general setting) and of generalized ordinary differential equation are not sufficiently explained. Nevertheless, these minor failings do not diminish the value of this good book.

The topics described in this book are very hard and the author was forced to use a very complicated machinery. Despite of this the book is well- written and organized. The theory is illustrated by well-chosen examples. The list of references contains 173 items devoted mainly to the subject studied in the book. This volume covers a large range of ideas and techniques exploited in the theory of generalized ordinary differential equations. It can be warmly recommended to specialists working in differential equations. On the other hand every reader may find the book helpful as an introduction to the theory of generalized integrals and as a guide to the literature for further details.

Chapter I has an introductory character and contains basic notions of the theory of generalized integral in the Henstock-Kurzweil sense. For example, the author describes the notions of the tagged interval, a gauge function, an \(o\)-fine system and so on. It is pointed out that the classical Riemann, Stieltjes and Perron integral can be obtained as special cases of the Henstock-Kurzweil integral. Moreover, several properties of the integral in question are proved. For example, integral inequalities of Bihari-Gronwall type are derived.

Chapter II discusses in details the theorem on the existence of solutions of the Cauchy problem (1) \(x'=f(t,x)\), \(x(0)=x_ 0\), where the solution is understood in the sense of Perron-Henstock. It is worthwhile mentioning that the assumptions imposed on the right hand side \(f=f(t,x)\) imply that it may be represented as the sum of a Carathéodory function \(h(t,x)\) and a Perron integrable function \(g(t)\).

The next two Chapters III and IV are devoted to the study of the generalized ordinary differential equations of the form (2) \(dx/d\tau=DF(t,x)\), where the letter \(D\) indicates that the equation is treated in generalized sense. A function \(x: [a,b]\to \mathbb{R}^ n\) is said to be a solution of the equation (2) if \(x(u)-x(v)= \int^ u_ v DF(t,x(\tau))\) for every \(u,v\in [a,b]\), where the integral is taken as the generalized Perron integral. It is shown that the solutions of (2) are functions being locally of bounded variation.

In Chapter V a comparison of the generalized ordinary differential equations with other important types is made. Particularly, it is shown that Carathéodory equations, the so-called measure differential equations as well as differential equations with impulses can be treated from the unified point of view as special cases of generalized ordinary differential equations.

In Chapter VI a special attention is focused on generalized linear differential equations of the form (3) \(dx/d\tau= D[A(t)x+ g(t)]\), where the matrix \(A\) is assumed to be of locally bounded variation while \(g(t)\) is a function of the same type. A lot of results concerning the equation (3) are proved which show the analogy with the classical theory of linear differential equations.

Chapter VII describes fundamental properties of integral products. This notion replaces the Riemann integral sums by partial integral products. The approach to the generalized linear differential equations via product integral turns out to be a very fruitful idea what is shown in this chapter.

The last three chapters treat the problem of continuous dependence of solutions on the right hand sides of generalized ordinary differential equations and variational stability of solutions of equations of such a type. Moreover, in Chapter IX there is considered the situation when the right hand sided converge to the Dirac delta function.

The considerations in the book are correct and clear, in general. But some aspects seem to be unclear and their presentation is not satisfactory enough. For example, the concepts of the Henstock-Kurzweil integral (in the general setting) and of generalized ordinary differential equation are not sufficiently explained. Nevertheless, these minor failings do not diminish the value of this good book.

The topics described in this book are very hard and the author was forced to use a very complicated machinery. Despite of this the book is well- written and organized. The theory is illustrated by well-chosen examples. The list of references contains 173 items devoted mainly to the subject studied in the book. This volume covers a large range of ideas and techniques exploited in the theory of generalized ordinary differential equations. It can be warmly recommended to specialists working in differential equations. On the other hand every reader may find the book helpful as an introduction to the theory of generalized integrals and as a guide to the literature for further details.

Reviewer: J.Banaś (Rzeszów)

### MSC:

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

34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |

34A99 | General theory for ordinary differential equations |

26A39 | Denjoy and Perron integrals, other special integrals |