Nonlinear problems in abstract cones. (English) Zbl 0661.47045

Notes and Reports in Mathematics in Science and Engineering, 5. Boston, MA: Academic Press, Inc. viii, 275 p. $ 34.95 (1988).
Many problems in mathematical analysis lead quite naturally to operator equations in spaces with cones. Beginning with M. A. Krasnosel’skii’s classical monograph [“Positive solutions of operator equations” (1964; Zbl 0121.10604)], the theory and applications of ordered Banach spaces and positive operators have become an important and well-developed field of both linear and nonlinear functional analysis. There are two excellent recent books concerned with these topics, both written by leading experts in functional analysis and operator theory, and both addressed to a wide readership of non-specialists. The first one is the book on “Positive linear systems” by M. A. Krasnosel’skii, E. A. Lifshits, and A. V. Sobolev [Nauka, Moscow (1985; Zbl 0578.47030); English translation Heldermann-Verlag, Berlin (1989; Zbl 0674.47036)], which essentially deals with linear operator equations in spaces with cones, as well as applications to linear spectral problems and iteration procedures. The second one is the book under review, which is meant as an introduction to the nonlinear theory and its applications to differential and integral equations. Each of these books is a perfect complementation of the other one.
The present book consists of 4 chapters. The first chapter introduces the basic notions and important special classes of cones (solid, reproducing, normal, regular, fully regular, minihedral, strongly minihedral, etc.), as well as various types of “deviations” and “oscillations” between elements and subspaces [e.g. Hilbert’s projective metric or Scheffer’s deviation). The second chapter is the main part of the book and deals with fixed points of positive operators. Here the authors provide a great deal of results and methods, building on several tools of nonlinear analysis (including degree theory). Interestingly, these tools do not only give existence theorems, but also multiplicity results for fixed points.
In the reviewer’s opinion, the most interesting part of the book is the final couple of chapters on applications to nonlinear equations. These two chapters contain a wealth of examples and applications which show that the notions developed in the first part are not just of theoretical interest, but have quite a natural meaning and applicability. The applications refer, on the one hand, to nonlinear integral equations (nonlinearities of polynomial type, eigenvalues and eigenvectors, special equations arising in physics and engineering, variational methods for Hammerstein-type equations, etc.) and, on the other hand, to nonlinear differential equations (differential inequalities, upper and lower solutions, comparison principles, cone-valued Lyapunov functions, etc.).
The book is carefully written and both informative and clear. Some results have been published before only in journal papers. It should be mentioned that the results due to the first author seem to be widely unknown (with a few exceptions, all his papers cited in the bibliography are written in Chinese); moreover, he seems to be unaware of parallel results due to Krasnosel’skij and his pupils which may be found, for instance, in the above-mentioned book.
In summary, this book can be recommended to anyone who wants not only to get a readable introduction to the theory of positive nonlinear operators, but is also interested in significant examples and applications.
Reviewer: J.Appell


47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
47H10 Fixed-point theorems
46A40 Ordered topological linear spaces, vector lattices
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
45-02 Research exposition (monographs, survey articles) pertaining to integral equations