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Simpliziale Algorithmen zur Berechnung von Fixpunkten mengenwertiger Operatoren. (German) Zbl 0589.65044
Trier: WVT Wissenschaftlicher Verlag Trier. IV, 190 S. DM 34.00 (1986).
The central objective of the book is the development of constructive proofs of fixed point theorems basing only on continuity and compactness requirements (Brouwer’s and Schauder’s fixed point theorems). The first section deals with the computation of fixed points for piecewise linear mappings in \({\mathbb{R}}^ n\). As an example, Merrill’s algorithm is described and known conditions for convergence are generalized. In the next section constructive proofs of fixed point theorems for multivalued mapping in \({\mathbb{R}}^ n\) respectively in normed spaces are developed. This is done by applying the results of section I to appropriate sequences of piecewise linear approximations based on the theory of A- proper resp. collectively compact operators. In particular, versions of Merrill’s algorithm are derived for problems in \({\mathbb{R}}^ n\) and for infinite dimensional problems. In the final section the results of section II are applied to boundary value problems for ordinary differential equations and to control-theoretic problems, which are treated as boundary value problems for differential inclusions, resulting in new numerical methods for solving these problems.
The material is carefully organized and concisely presented. Throughout the book Merrill’s algorithm is chosen as model algorithm to develop the numerical methods resp. the constructive proofs of fixed point theorems. The appendices on convex analysis, functional analysis and Pontryagin’s maximum principle make the book largely self-contained. An ALGOL-program for Merrill’s algorithm in \({\mathbb{R}}^ n\) is also enclosed. The book is highly recommended to those readers who are interested in constructive aspects for fixed point theorems.
Reviewer: W.Zulehner

MSC:
65H10 Numerical computation of solutions to systems of equations
65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
47H10 Fixed-point theorems
47-04 Software, source code, etc. for problems pertaining to operator theory