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Fixed point theory in topological vector spaces. (English) Zbl 0576.47030
Novi Sad: University of Novi Sad, Institute of Mathematics. V, 337 p. $ 35.00 (1984).
The literature on the theory and applications of a fixed point principles in normed or locally convex spaces is vast. Some spaces which occur in applications, however, are topological vector spaces which are not locally convex. For instance, the space S(\(\Omega)\) of all measurable functions over some domain \(\Omega\), the Lebesgue space \(L^ p(\Omega)\) with \(0<p<1\), and the Hardy space \(H^ p\) with \(0<p<1\) are all linear metric spaces without locally convex topology [many facts about such spaces can be found in S. Rolewicz’ book ”Metric linear spaces” (1972; Zbl 0226.46001), which is missing in the bibliography of the book under review].
Fixed point theory in general topological vector spaces has become the subject of several papers in the last years; all these results, however, are scattered over the literature, and hence the aim of the present book is, according to the author, to ”represent some of the results of this theory together with some applications”. As far as the theory is concerned, the author provides in fact a rather complete and self- contained reference text. On the other hand, the applications and examples given by the author are mostly rather artificial. This is, of course, not her fault, since the theory is highly abstract; nevertheless, she had better illustrate at least the numerous definitions by means of some examples in known function spaces (The only part dealing with concrete function spaces is the section on locally convex subsets of the Orlicz spaces \(KL_{\phi}\) and \(SL_{\phi}.)\)
To give an idea of the topics covered by the book, let us recall the headings of the 15 sections: Admissible subsets of topological vector spaces and fixed point theorems for F-approachable mappings.
Some classes of admissible subsets of topological vector spaces.
Examples of locally convex subsets in admissible topological vector spaces \(KL_{\phi}\) and \(SL_{\phi}.\)
Some further fixed point theorems in topological vector spaces.
Degree theory in topological vector spaces.
The eigenvalue problem in topological vector spaces.
Stability of solutions of operator equations in topological vector spaces.
Fixed point theorems for mulitvalued mappings.
Some applications of fixed point theorems from II.1.
The almost continuous selection property.Homotopy extension theorems and applications on the fixed point theorems and eigenvalue problem.
Leray-Schauder principles for condensing multivalued mappings. Fixed point theorems in probabilistic metric and random normed spaces.
Random operators on complete separable metric spaces. Fixed point theorems for multivalued mappings in uniform and probabilistic metric spaces.
The book is clearly written and well organized. It contains a list of symbols, a subject index, and a useful bibliography containing 374 items, and will certainly be a standard reference for specialists in abstract functional analysis, nonlinear operator theory, and fixed point principles. Moreover, it would be interesting to study the known concrete function spaces (or sequence spaces) under the viewpoint of the concepts developped in this book; in the referee’s opinion, there is a deplorable lack of papers like that of T. Riedrich on the admissibility of S(0,1) and \(L^ p(0,1)\) [Wiss. Z. TU Dresden 12, 1149-1152 (1963; Zbl 0135.352), and ibid. 13, 1-6 (1964; Zbl 0158.134)].
Reviewer: J.Appell

MSC:
47H10 Fixed-point theorems
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
47J05 Equations involving nonlinear operators (general)
46A04 Locally convex Fréchet spaces and (DF)-spaces