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**Integral inequalities and applications. Transl. by R. A. M. Hoksbergen.**
*(English)*
Zbl 0759.26012

Mathematics and Its Applications. East European Series. 57. Dordrecht etc.: Kluwer Academic Publishers. xi, 245 p. (1992).

This important, very well written monograph is devoted to one of the most useful tools applied in equations (differential, integral, difference) theories, namely Gronwall type inequalities, their generalizations and analogues.

The book is divided into three chapters. The first two chapters deal with scalar linear and nonlinear integral inequalities. Chapter 3 contains: integral inequalities (i.i.) with iterated integrals, i.i. for functions of many variables, i.i. involving several integrals, integro-differential inequalities, i.i. with delay, discrete inequalities, impulse i.i., i.i. for Stieltjes integrals, systems of i.i., inequalities in partially ordered spaces.

The monograph surveys, besides the inequalities, also the methods used in Gronwall inequalities and possible applications. Each chapter ends with an Applications section. These applications concern such problems as: uniqueness and differentiability, estimate for the Cauchy matrix, exponential and ordinary dichotomy, stability, boundedness, two-sided estimates, asymptotic equivalence, asymptotic behavior, terminal value problems, perturbed equations; and refer to: ordinary, partial, integral, integro- differential, functional-differential, difference, and impulse equations. Many of the recently obtained results particularly for the continuous cases are contained and discussed as well as the methods which lead to the final bounds. The Bibliography contains 250 items.

The book is divided into three chapters. The first two chapters deal with scalar linear and nonlinear integral inequalities. Chapter 3 contains: integral inequalities (i.i.) with iterated integrals, i.i. for functions of many variables, i.i. involving several integrals, integro-differential inequalities, i.i. with delay, discrete inequalities, impulse i.i., i.i. for Stieltjes integrals, systems of i.i., inequalities in partially ordered spaces.

The monograph surveys, besides the inequalities, also the methods used in Gronwall inequalities and possible applications. Each chapter ends with an Applications section. These applications concern such problems as: uniqueness and differentiability, estimate for the Cauchy matrix, exponential and ordinary dichotomy, stability, boundedness, two-sided estimates, asymptotic equivalence, asymptotic behavior, terminal value problems, perturbed equations; and refer to: ordinary, partial, integral, integro- differential, functional-differential, difference, and impulse equations. Many of the recently obtained results particularly for the continuous cases are contained and discussed as well as the methods which lead to the final bounds. The Bibliography contains 250 items.

Reviewer: J.Popenda (Poznań)

### MSC:

26D10 | Inequalities involving derivatives and differential and integral operators |

26-02 | Research exposition (monographs, survey articles) pertaining to real functions |

26D15 | Inequalities for sums, series and integrals |