Nonlinear functional analysis and its applications. III: Variational methods and optimization. Transl. from the German by Leo F. Boron. (English) Zbl 0583.47051

New York etc.: Springer-Verlag. XXII, 662 p. DM 189.00 (1985).
The third volume (chapters 37-57) was the first that appeared in English translation. In the new form of the opening chapter, delivering an overview of extremal problems, their relationships with approximation methods and critical points are detailed. A new part has been introduced dealing with the compactness and the convexity principles in proving the existence and uniqueness of solutions of extremal problems. The chapter devoted to the Ljusternik-Schnirelman theory and the existence of several eigenfunctions reflects the author’s and other recent investigations while the one about bifurcation for potential operators includes a complete proof and important variants are mentioned. The part on variational inequalities, announced only in the German edition (1978: Zbl 0426.47033 and 1982: Zbl 0482.47022), contains four chapters concerning results about elliptic inequalities and their connections with control problems, evolution inequalities in Hilbert spaces, as well as basic elements of accretive mappings. Finally, the quality of the translation should be emphasized.
Reviewer: D.Pascali


47Hxx Nonlinear operators and their properties
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
49J40 Variational inequalities