##
**Inequalities for differential and integral equations.**
*(English)*
Zbl 1032.26008

Mathematics in Science and Engineering. 197. San Diego, CA: Academic Press, Inc. x, 611 p. (1998).

The present monogragh consists of five chapters. Chapter 1 contains the well-known Gronwall-Bellman linear integral inequality and many one-variable linear generalizations of this inequality. The remarkable results among others given in this chapter are various linear inequalities involving iterated integrals which are initiated by the author in 1973 with a special inequality involving a two-fold iterated integral as follows.

If \(u,f,g\in C(\mathbb{R}_+, \mathbb{R}_+)\), \(\mathbb{R}_+= [0,\infty)\), and \(u_0\geq 0\) is a constant, then \[ u(t)\leq u_0+ \int^t_0 f(s)\Biggl(u(s)+ \int^s_0 g(k) u(k) dk\Biggr) ds,\quad t\in\mathbb{R}_+, \] implies \[ u(t)\leq u_0\Biggl[1+ \int^t_0 f(s)\exp\Biggl(\int^s_0 [f(k)+ g(k)]dk\Biggr) ds\Biggr],\quad t\in\mathbb{R}_+. \] Pachpatte and many other authors including the present reviewer have contributed in this direction in recent years and the general result was proved in the reference [308] of the book by this reviewer [En-Hao Yang, J. Math. Anal. Appl. 103, 184-197 (1984; Zbl 0573.26008)]. These linear integral inequalities cited herein find widespread applications in the study of various differential and integral equations.

Chapter 2 and Chapter 3 contain a number of one-variable nonlinear integral inequalities which can be used to achieve great diversity in both the theory and applications of various types of nonlinear differential and integral equations. Chapter 3 deals with a series of remarkable results due to the author and his Indian co-workers which belong to two different types. Namely, generalizations of the Ou-Iang and Dafermos integral inequality and extensions of the Haraux-Engler integral inequality. The features of these inequalities are: inequalities of the first kind possess a nonlinear power of the unkown function on their left-hand side, and an extra logarithmic factor of the unknown function appears in the integrands of all integral inequalities of the second kind.

Chapters 4 and 5 deal with the linear and nonlinear integral inequalities involving more than one independent variable which plays a vital role in the investigation of various classes of partial differential and integral inequations.

An application section is included in every chapter which indicates some possible applications of the inequalities given in the respective chapter. Each chapter contains also a section on miscellaneous inequalities indicating further sources of information and we find they are very interesting. Just as pointed out by the author: “Taking into account the vast literature, the choice of material for a book devoted to integral inequalities is a difficult task”.

The contents of this monogragh reflects the research interests and results of the author. A large part of the inequalities and their applications are cited directly from over eighty papers (including in press and unpublished papers) of the author. Because the author is a very active investigator and one of the main contributors in the relevant fields in the last two decades, the book can be considered as an important complement to the literature and it presents many very useful inequalities developed recently in some of the main directions of integral inequalities. The reader will find some very interesting inequalities in the book which will motivate new development in the next years. This book is vital reading for mathematicians, physicists, engineers, computer scientists and graduate students in those disciplines.

{Reviewer’s note: A few misprints can be found in the book which may cause problems in possible applications. For example, (i) in the statement of Rodrigues’ inequality on page 28, since the function \(\gamma(t)\) is only defined for \(t\geq\sigma\), the inequalities (1.5.10), (1.5.11) and the relation \(u(t)\leq v(t)\) on page 29 cannot be satisfied for all \(t\in\mathbb{R}_+\); (ii) for all inequalities involving comparison given in Section 3.7 of Chapter 3, the bounds on solutions formulated in terms of some initial value problems of nonlinear differential equations cannot be valid on the whole halfline \(\mathbb{R}_+\) in general. Furthermore, by letting some derivative of the unknown function as a new unknown function, some integro-differential inequalities given herein can be easily reformulated as iterated integral inequalities, which are slightly variants of some results in the book or contained as special cases by some results published in the literature}.

If \(u,f,g\in C(\mathbb{R}_+, \mathbb{R}_+)\), \(\mathbb{R}_+= [0,\infty)\), and \(u_0\geq 0\) is a constant, then \[ u(t)\leq u_0+ \int^t_0 f(s)\Biggl(u(s)+ \int^s_0 g(k) u(k) dk\Biggr) ds,\quad t\in\mathbb{R}_+, \] implies \[ u(t)\leq u_0\Biggl[1+ \int^t_0 f(s)\exp\Biggl(\int^s_0 [f(k)+ g(k)]dk\Biggr) ds\Biggr],\quad t\in\mathbb{R}_+. \] Pachpatte and many other authors including the present reviewer have contributed in this direction in recent years and the general result was proved in the reference [308] of the book by this reviewer [En-Hao Yang, J. Math. Anal. Appl. 103, 184-197 (1984; Zbl 0573.26008)]. These linear integral inequalities cited herein find widespread applications in the study of various differential and integral equations.

Chapter 2 and Chapter 3 contain a number of one-variable nonlinear integral inequalities which can be used to achieve great diversity in both the theory and applications of various types of nonlinear differential and integral equations. Chapter 3 deals with a series of remarkable results due to the author and his Indian co-workers which belong to two different types. Namely, generalizations of the Ou-Iang and Dafermos integral inequality and extensions of the Haraux-Engler integral inequality. The features of these inequalities are: inequalities of the first kind possess a nonlinear power of the unkown function on their left-hand side, and an extra logarithmic factor of the unknown function appears in the integrands of all integral inequalities of the second kind.

Chapters 4 and 5 deal with the linear and nonlinear integral inequalities involving more than one independent variable which plays a vital role in the investigation of various classes of partial differential and integral inequations.

An application section is included in every chapter which indicates some possible applications of the inequalities given in the respective chapter. Each chapter contains also a section on miscellaneous inequalities indicating further sources of information and we find they are very interesting. Just as pointed out by the author: “Taking into account the vast literature, the choice of material for a book devoted to integral inequalities is a difficult task”.

The contents of this monogragh reflects the research interests and results of the author. A large part of the inequalities and their applications are cited directly from over eighty papers (including in press and unpublished papers) of the author. Because the author is a very active investigator and one of the main contributors in the relevant fields in the last two decades, the book can be considered as an important complement to the literature and it presents many very useful inequalities developed recently in some of the main directions of integral inequalities. The reader will find some very interesting inequalities in the book which will motivate new development in the next years. This book is vital reading for mathematicians, physicists, engineers, computer scientists and graduate students in those disciplines.

{Reviewer’s note: A few misprints can be found in the book which may cause problems in possible applications. For example, (i) in the statement of Rodrigues’ inequality on page 28, since the function \(\gamma(t)\) is only defined for \(t\geq\sigma\), the inequalities (1.5.10), (1.5.11) and the relation \(u(t)\leq v(t)\) on page 29 cannot be satisfied for all \(t\in\mathbb{R}_+\); (ii) for all inequalities involving comparison given in Section 3.7 of Chapter 3, the bounds on solutions formulated in terms of some initial value problems of nonlinear differential equations cannot be valid on the whole halfline \(\mathbb{R}_+\) in general. Furthermore, by letting some derivative of the unknown function as a new unknown function, some integro-differential inequalities given herein can be easily reformulated as iterated integral inequalities, which are slightly variants of some results in the book or contained as special cases by some results published in the literature}.

Reviewer: Yang En-Hao (Guangzhou)

### 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 |

34A40 | Differential inequalities involving functions of a single real variable |

35B45 | A priori estimates in context of PDEs |

35R45 | Partial differential inequalities and systems of partial differential inequalities |

45A05 | Linear integral equations |

45G10 | Other nonlinear integral equations |

45J05 | Integro-ordinary differential equations |