##
**Generalized ordinary differential equations. Not absolutely continuous solutions.**
*(English)*
Zbl 1248.34001

Series in Real Analysis 11. Hackensack, NJ: World Scientific (ISBN 978-981-4324-02-1/hbk; 978-981-4324-03-8/ebook). ix, 197 p. (2012).

Generalized ordinary differential equations were introduced by J. Kurzweil in 1957 [Czech. Math. J. 7(82), 418–449 (1957; Zbl 0090.30002)]. The main motivation for this work was the study of continuous dependence of solutions on the right-hand side (as well as the related averaging method, see chapters 2–4 of the present book). Since then, a number of works devoted to generalized ODEs have appeared. Also, it became clear that many familiar types of equations (such as ODEs with impulses, measure differential equations, functional differential equations, dynamic equations on time scales) are special cases of generalized ODEs.

In this book, the author considers equations whose solutions take values in a Banach space \(X\). The notion of a generalized ODE is based on the following concept of an integral: A function \(U:[a,b]\times [a,b]\to X\) is called strongly Kurzweil-Henstock integrable over \([a,b]\) if there exists a function \(u:[a,b]\to X\) (called an SKH-primitive of \(U\)) such that for every \(\varepsilon>0\), there is a function \(\delta:[a,b]\to\mathbb R^+\) such that \[ \sum_{i=1}^k\left\|U(\tau_i,t_i)-U(\tau_i,t_{i-1})-u(t_i)+u(t_{i-1})\right\|<\varepsilon \] for every tagged partition \(a=t_0\leq\tau_1\leq t_1\leq \tau_2\leq\cdots\leq t_{k-1}\leq \tau_k\leq t_k=b\) satisfying \[ [t_{i-1},t_i]\subset(\tau_i-\delta(\tau_i),\tau_i+\delta(\tau_i)),\;\;\;i\in\{1,\ldots,k\}. \] In this case, the strong Kurzweil-Henstock integral is defined as \(\text{(SKH)}\int_a^b {\mathrm D}_t U(\tau ,t)=u(b)-u(a)\).

By changing the definition and requiring the function \(\delta\) to be constant, we obtain the definition of the strong Riemann integral \(\text{(SR)}\int_a^b {\mathrm D}_t U(\tau ,t)\).

A generalized ODE is an integral equation of the form \[ x(s)=x(a)+\int_a^s {\mathrm D}_t F(x(\tau),\tau,t),\;\;\;s\in[a,b], \] where the function \(F\) is defined on a subset of \(X\times\mathbb R^2\). A distinction is made between SR-solutions and SKH-solutions, depending on whether the integral on the right-hand side is interpreted as the SR-integral or SKH-integral, respectively.

The author considers two different sets of assumptions on the right-hand side; for both of them, he obtains theorems concerning the existence and uniqueness of solutions as well as continuous dependence on the right-hand side. The conditions are too technical to be reproduced here; however, the class of generalized ODEs under consideration is large enough to encompass solutions which need not have bounded variation on any interval (cf. the monograph written by Š. Schwabik [Generalized ordinary differential equations. Singapore: World Scientific (1992; Zbl 0781.34003)], which deals primarily with a class of generalized ODEs whose solutions have bounded variation).

The book also discusses a number of related topics, such as integration by parts for the SKH-integral, Gronwall’s inequality, or an elementary derivation of averaging theorems.

After the monograph by Š. Schwabik [loc. cit.], published in the same series, this book represents another major addition to the theory of nonlinear generalized ordinary differential equations.

In this book, the author considers equations whose solutions take values in a Banach space \(X\). The notion of a generalized ODE is based on the following concept of an integral: A function \(U:[a,b]\times [a,b]\to X\) is called strongly Kurzweil-Henstock integrable over \([a,b]\) if there exists a function \(u:[a,b]\to X\) (called an SKH-primitive of \(U\)) such that for every \(\varepsilon>0\), there is a function \(\delta:[a,b]\to\mathbb R^+\) such that \[ \sum_{i=1}^k\left\|U(\tau_i,t_i)-U(\tau_i,t_{i-1})-u(t_i)+u(t_{i-1})\right\|<\varepsilon \] for every tagged partition \(a=t_0\leq\tau_1\leq t_1\leq \tau_2\leq\cdots\leq t_{k-1}\leq \tau_k\leq t_k=b\) satisfying \[ [t_{i-1},t_i]\subset(\tau_i-\delta(\tau_i),\tau_i+\delta(\tau_i)),\;\;\;i\in\{1,\ldots,k\}. \] In this case, the strong Kurzweil-Henstock integral is defined as \(\text{(SKH)}\int_a^b {\mathrm D}_t U(\tau ,t)=u(b)-u(a)\).

By changing the definition and requiring the function \(\delta\) to be constant, we obtain the definition of the strong Riemann integral \(\text{(SR)}\int_a^b {\mathrm D}_t U(\tau ,t)\).

A generalized ODE is an integral equation of the form \[ x(s)=x(a)+\int_a^s {\mathrm D}_t F(x(\tau),\tau,t),\;\;\;s\in[a,b], \] where the function \(F\) is defined on a subset of \(X\times\mathbb R^2\). A distinction is made between SR-solutions and SKH-solutions, depending on whether the integral on the right-hand side is interpreted as the SR-integral or SKH-integral, respectively.

The author considers two different sets of assumptions on the right-hand side; for both of them, he obtains theorems concerning the existence and uniqueness of solutions as well as continuous dependence on the right-hand side. The conditions are too technical to be reproduced here; however, the class of generalized ODEs under consideration is large enough to encompass solutions which need not have bounded variation on any interval (cf. the monograph written by Š. Schwabik [Generalized ordinary differential equations. Singapore: World Scientific (1992; Zbl 0781.34003)], which deals primarily with a class of generalized ODEs whose solutions have bounded variation).

The book also discusses a number of related topics, such as integration by parts for the SKH-integral, Gronwall’s inequality, or an elementary derivation of averaging theorems.

After the monograph by Š. Schwabik [loc. cit.], published in the same series, this book represents another major addition to the theory of nonlinear generalized ordinary differential equations.

Reviewer: Antonín Slavík (Praha)

### 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 |

34G20 | Nonlinear differential equations in abstract spaces |

26A45 | Functions of bounded variation, generalizations |

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

28B05 | Vector-valued set functions, measures and integrals |