UK Nonlinear News, February 1997

New Books

(We would like to carry reviews of any of these books in future issues of UK Nonlinear News.)

New from American Mathematical Society

Monotone Operators in Banach Space and Nonlinear Partial Differential Equations

R.E. Showalter

Mathematical Surveys and Monographs, Volume 49; 1997;
278 pages; ISBN 0-8218-0500-2

"The objectives of this monograph are to present some topics from the the theory of monotone operators and nonlinear semigroup theory which are directly applicable to the existence and uniqueness theory of initial-boundary-value problems for partial differential equations and to construct such operators as realisations of those problems in appropriate function spaces.

A highlight of this presentation is the large number and variety of examples introduced to illustrate the connection between the theory of nonlinear operators and partial differential equations. These include primarily semilinear ot quasilinear equations of elliptic or of parabolic type, degenerate cases with change of type, related systems and variational inequalities, and spatial boundary conditions of the usual Dirichlet, Neumann, Robin or dynamic type."

New Books from Springer

Springer America.
Springer Europe.

Yuri Gliklikh.
1996 240 pages ISBN 0-387-94867-8
Applied Mathematical Sciences Volume 122

"This book gives a common treatment to three areas of application of global analysis to mathematical physics previously considered quite distant from each other. These areas are the geometry of manifolds applied to classical mechanics, stochastic differential geometry, used in quantum and statistical mechanics, and infinite-dimensional differential geometry, fundamental for hydrodynamics. The unification of these topics is made possible by considering Newton's Equation and its natural generalisations and analogues as a fundamental equation of motion. New geometric and stochastic methods of investigation are developed and new results on existence, uniqueness and qualitative behaviour of solutions are obtained in this book".

Vy Khoi Le and Klaus Schmitt.
1997 app 240 pages ISBN 0-387-94886-4
Applied Mathematical Science, Volume 123

"This book presents an up-to-date and unified treatment of bifurcation theory for variational inequalities in reflexive spaces and the use of the theory in a variety of applications, such as obstacle problems from elasticity theory, unilateral problems, torsion problems, equations from fluid dynamics and quasilinear elliptic partial differential equations. The tools employed are the tools of modern nonlinear analysis. The contents are accessible to graduate students and researchers who work in nonlinear analysis, nonlinear partial differential equations, as well as research disciplines that use nonlinear mathematics extensively".

CONTENTS: Introduction; Some Auxiliary Results; Variational Inequalities Defined on Convex Sets in Hilbert Spaces: Homogenisation Procedures; Degree Calculations - The Hilbert Space Case; Bifurcation in Hilbert Spaces; Bifurcation from Infinity in Hilbert Spaces; Bibliography; Index.

Roger Temam.
1997 app 600 pages ISBN 0-387-94866-X
Applied Mathematical Sciences, Volume 68

"The study of dynamical systems of infinite dimensions has been a very active area of research. This book, written by a well-known researcher, presents a systematic study of infinite dimensional dynamical systems generated by dissipative evolution partial differential equations arising in mechanics and physics and in other areas of sciences and technology. This second edition has been completely updated and extended, including many of the latest results in the subject".

David R. Merkin, Fred Afagh, and Andrei L. Smirnov
1996 344pages ISBN 0-387-94761-2
Texts in Applied Mathematics, Volume 24

"This books deals with the issues of stability of motion which most often are encountered in the analysis of scientific and technical problems. There are many comprehensive monographs on the theory of stability of motion, with each one devoted to a separate complicated issue of the theory. The main advantage of this book, however, is its simple yet simultaneous rigorous presentation of the concepts of the theory, which often are presented in the context of applied problems with detailed examples, demonstrating effective methods of solving practical problems".

New from Cambridge University Press

Dynamical Systems and Numerical Analysis


A.M. Stuart and A.R. Humphries

[Cambridge Monographs on Applied and Computational Mathematics 2]

"Dynamical systems are pervasive in the modelling of naturally occurring phenomena. Most of the models arising in practice cannot be completely solved by analytic techniques; thus, numerical simulations are of fundamental importance in gaining an understanding of dynamical systems. It is therefore crucial to understand the behaviour of numerical simulations of dynamical systems in order to interpret the data obtained from such simulations and to facilitate the design of algorithms which provide correct qualitative information without being unduly expensive. These two concerns lead to the study of the convergence and stability properties of numerical methods for dynamical systems.

The first three chapters of this book contain the elements of the theory of dynamical systems and the numerical solution of initial-value problems. In the remaining chapters, numerical methods are formulated as dynamical systems, and the convergence and stability properties of the methods are examined. Topics studied include the stability of numerical methods for contractive, dissipative, gradient, and Hamiltonian systems together with the convergence properties of equilibria, phase portraits, periodic solutions, and strange attractors under numerical approximation.

This book will be an invaluable tool for graduate students and researchers in the fields of numerical analysis and dynamical systems."

1. Finite Dimensional Maps
2. Ordinary Differential Equations
3. Numerical Methods for Initial Value Problems
4. Numerical Methods as Dynamical Systems
5. Global Stability
6. Convergence of Invariant Sets
7. Global Properties and Attractors Under Discretisation
8. Hamiltonian and Conservative Systems

Published October 1996 700 pages 28 line diagrams 258 exercises ISBN 0-521-49672-1 (Hardback)

For more information, consult the online catalogues at and, or send enquiries to

Source: Alan Harvey (

New from Oxford University Press


Patters and Waves (second edition)

Peter Grindrod

"This newly edited textbook introduces students and mathematicians to the theory of reaction-diffusion equations; it provides useful techniques for their analysis and shows how they can be applied in a variety of settings."

(Oxford Applied Mathematics and Computing Science Series)
1996 288pp

Oxford University Press WWW sites:

<< Move to UK Nonlinear News Issue 7 Index Page (February 1997).
Last Updated: 28th January 1997.