##
**Regular variation and differential equations.**
*(English)*
Zbl 0946.34001

Lecture Notes in Mathematics. 1726. Berlin: Springer. x, 127 p. (2000).

This little monograph deals with the precise asymptotic behavior of solutions to second-order differential equations of the form \(y''= f(x)\phi(y)\), both for the linear and for some nonlinear cases. By that as usual one means to find a continuous, positive function \(g\) such that \(y(x)/g(x)\to 1\) as \(x\to\infty\). Equations of the form \(y''+ g(x) y'+ h(x)y= 0\) and \(y'''- yy''+ \lambda(1+ y^{\prime 2})= 0\) are also studied. The novelty of the approach in this book consists of using the notion of regular variation of J. Karamata [Mathematica 4, 38-53 (1930; JFM 56.0907.01)] and its extensions. A positive measurable function \(\rho\) defined on some neighbourhood \([a,\infty)\) of infinity is called regularly varying at infinity of index \(\alpha\) \((\alpha\in\mathbb{R})\) if for each \(\lambda> 0\), \(\lim_{x\to\infty} \rho(\lambda x)/\rho(x)= \lambda^\alpha\). The real number \(\alpha\) is called the index of regular variation.

The book has two parts. Part 1 deals with the existence and asymptotic behavior of regular solutions for linear equations. Necessary and sufficient conditions for the solutions to belong to the Karamata class of functions are presented, here. Part 2 is devoted to nonlinear equations of Thomas-Fermi type and to an equation arising in boundary-layer theory. An Appendix summarizing basic properties of regularly varying and related functions, a list of references (67 titles) and a general index conclude the book.

The book is a wonderful addition to the literature on asymptotic behavior of solutions to ordinary differential equations. The writing is concise, precise, elegant and enjoyable. These recommend it to graduate students and to all interested in this topic.

The book has two parts. Part 1 deals with the existence and asymptotic behavior of regular solutions for linear equations. Necessary and sufficient conditions for the solutions to belong to the Karamata class of functions are presented, here. Part 2 is devoted to nonlinear equations of Thomas-Fermi type and to an equation arising in boundary-layer theory. An Appendix summarizing basic properties of regularly varying and related functions, a list of references (67 titles) and a general index conclude the book.

The book is a wonderful addition to the literature on asymptotic behavior of solutions to ordinary differential equations. The writing is concise, precise, elegant and enjoyable. These recommend it to graduate students and to all interested in this topic.

Reviewer: R.Precup (Cluj-Napoca)

### 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 |

34E05 | Asymptotic expansions of solutions to ordinary differential equations |

26A12 | Rate of growth of functions, orders of infinity, slowly varying functions |