##
**Half-linear differential equations.**
*(English)*
Zbl 1090.34001

North-Holland Mathematics Studies 202. Amsterdam: Elsevier (ISBN 0-444-52039-2/hbk). xiv, 517 p. (2005).

The book under review is an excellent survey on recent results (up to 2005) and progress in theory of half-linear differential equations. The simplest type of this equation, the equation
\[
\Bigl(r(t)\Phi(x')\Bigr)'+c(t)\Phi(x)=0, \qquad \Phi(x)=| x| ^{p-1}\text{sgn}\, x, \;p>1,\tag{1}
\]
has been studied since 1976 by Mirzov, Elbert and others and attracted a broad interest. It has been observed already by Mirzov and Elbert that many results established in the theory of linear ordinary differential equations (which can be obtained from (1) by letting \(p=2\)) can be extended to (1). This book is a systematic study which extends many important results from the linear theory to (1) and provides also results which are different (such as nonexistence of Wronskian, Fredholm alternative) or have no counterparts in the linear theory (such as comparison theorems with respect to \(p\)) and thus strongly depend on the nonlinearity of (1).

The book is not only a unified presentation and collection of published papers scattered in many journals, but also a clearly written comprehensive text which includes new results, as well as new proofs of existing results. This approach makes the text more compact and understandable to the reader. An important part of the book is devoted also to several important extensions of equation (1). These extensions cover partial differential equations with \(p\)-Laplacian, half-linear difference equations, differential inequalities, higher-order equations, equations on time scales and others.

The book is divided into 9 chapters: Basic theory (46 pages), Methods of oscillation theory (36 pages), Oscillation and nonoscillation criteria (40 pages), Nonoscillatory solutions (66 pages), Various oscillation problems (122 pages), BVPs for half-linear differential equations (42 pages), partial differential equations with \(p\)-Laplacian (30 pages), Half-linear difference equations (34 pages) and Related differential equations and inequalities (54 pages). The book is very useful as an up-to-date source of references for researchers working as well as a textbook for graduate students.

The book is not only a unified presentation and collection of published papers scattered in many journals, but also a clearly written comprehensive text which includes new results, as well as new proofs of existing results. This approach makes the text more compact and understandable to the reader. An important part of the book is devoted also to several important extensions of equation (1). These extensions cover partial differential equations with \(p\)-Laplacian, half-linear difference equations, differential inequalities, higher-order equations, equations on time scales and others.

The book is divided into 9 chapters: Basic theory (46 pages), Methods of oscillation theory (36 pages), Oscillation and nonoscillation criteria (40 pages), Nonoscillatory solutions (66 pages), Various oscillation problems (122 pages), BVPs for half-linear differential equations (42 pages), partial differential equations with \(p\)-Laplacian (30 pages), Half-linear difference equations (34 pages) and Related differential equations and inequalities (54 pages). The book is very useful as an up-to-date source of references for researchers working as well as a textbook for graduate students.

Reviewer: Robert Mařík (Brno)

### MSC:

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

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

39A10 | Additive difference equations |

35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |

34N05 | Dynamic equations on time scales or measure chains |