The second volume (chapters 18-36) of Zeidler’s treatise is a broad and up-to-date survey of the monotone operator theory, a popular and successful research area of nonlinear analysis, over the past 25 years. In comparison with the German original (1979; Zbl 0426.47032 and 1981; Zbl 0481.47041) the number of pages of its English augmented translation has been increased five times; this compelled the publisher to divide the content into two subvolumes, which form a unit. The first subvolume, devoted to linear positive (monotone) operators, is a competent introduction to Hilbert space methods in PDE theory. As a prelude to the subject, Hilbert orthogonality and the Ritz method, as well as an elegant functional analytic foundation of quadratic variational problems are presented. More attention is paid to the Friedrichs extension for unbounded linear operators and its consequences for eigenvalue problems and the semigroup approach of evolution problems (mild solutions). Completion and smoothing principles and relationships between stability, consistency, and convergence in difference methods have been added.
The second part of the first subvolume deals with the approximation-solvability of linear operator equations and the convergence of the Galerkin method. Hilbert space methods are successively implemented to elliptic, parabolic and hyperbolic problems, whose peculiarities are emphasized. New topics on functional and interpolation inequalities, and regularity of generalized solutions are developed. Sufficient rules for strongly positive bilinear forms, existence of eigenvalues, and the Fredholm alternative are interacted to investigate the boundary value problems for elliptic equations. Along this perspective, the monotone mappings appear as a natural tool, extending Hilbert space methods to nonlinear problems. Moreover, monotone operator machinery yields a rigorous justification of the Dirichlet principle, based on the fundamental concept of energy.
After a digest of the basic ideas in the theory of monotone operators, the second subvolume develops by progressive stages the implementation of monotone operators to stationary and nonstationary problems. Upon general hypotheses, quasilinear elliptic or nonlinear integral Hammerstein equations can be rewritten as functional equations involving monotone-like operators. Furthermore, the pseudomonotone operator theory unifies compactness and monotonicity arguments. With nonlinear Fredholm alternative are now associated noncoercive problems, the Cesàri method and the Landesman-Lazer principle, a main bifurcation theorem, as well as continuation and multiplicity results. The core of the second subvolume emphasizes the efficiency of monotonicity arguments, nonlinear semigroups and the Galerkin method in the study of nonlinear evolution equations. Kato’s new results on semibounded evolutions, generalized Korteweg-de Vries equation, or Amann’s look at quasilinear parabolic problems regarded as dynamical systems are mentioned. A more constituent additional attribute of monotone operators is their maximality. Maximal monotone (set-valued) mappings have significant properties and applications to variational problems and inequalities or to Hammerstein equations. Their structure is based on both the Galerkin method and the regularization procedure via duality map. Special monotone mapping subclasses with nice features, such as angle-bounded and 3-monotone operators, are described. Key theorems about pseudomonotone perturbations of maximal monotone mappings and basic geometric elements concerning ranges of sums of monotone operators are inserted. Recent progress in semilinear wave equations and generalized viscosity solutions are interconnected with the Galerkin method for second-order evolution equations. Discretization methods, inner and external approximation schemes, for nonlinear equations are theorized in the last part of the volume. This leads naturally to the introduction of A-proper mappings, for which a multivalued topological degree is defined. For the convenience of the reader, an appendix summarizes the frequently used notion in measure and integration theory, discrete Sobolev spaces, functional calculus of selfadjoint operators, interpolation theory of function spaces, fractional powers of sectorial operators, Hausdorff measure and fractals, etc. Besides, many suggestive schematic overviews explain the interrelations between fundamental ideas, concepts and results in nonlinear operator theory. Since the proposed subject is too ample, the author has often chosen an intermediate way between monograph and directory. Each chapter closes with complementary problems and references, orienting the reader to current research trends. Moreover, a comprehensive bibliography of more than 40 pages concludes the book. Like in other previously published parts of this five-volume encyclopaedic series (Part I: 1986; Zbl 0583.47050; Part III: 1985; Zbl 0583.47051 and Part IV: 1988; Zbl 0648.47036), the lecturer enjoys reading the many quotations and remarks made by outstanding scientists or writers on the topics covered in the various chapters. It should also be emphasized that the author’s approach goes straight to a numerical functional analysis with applications to finite elements methods in Sobolev spaces.
All in all, this book is an excellent readable introduction in the prevalent classes of nonlinear operators. A large audience of mathematicians and natural scientists will find it a stimulating guide through the contemporary operator treatment of nonlinear partial differential and integral equations.