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Approximation methods for expanding operators

Conference paper
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Part of the Lecture Notes in Mathematics book series (LNM, volume 506)

Abstract

An attempt is made in this report to give a very rough survey on expanding operators. The phenomenon of expanding operators T seems to appear very often. Some classical fixed point theorems cover cases with expanding operators; numerical examples for these are given. Furthermore, there exists a fixed point theorem of Krasnoselskii, which is applicable to nonlinear integral equations of Hammerstein-type under certain conditions: for this numerical examples are given. But usually it is not yet possible to get exact inclusion theorems for solutions u of u=Tu.

A general numerical procedure, working in the last mentioned cases also for not well posed problems, and problems with several solutions, is described and applied in concrete cases. It was not the intention of this paper to give the greatest possible generality but to illustrate the situation by many examples. It is hoped that more mathematicians than hitherto will deal with expanding operators and that there will be much success in this new field of research in the future.

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© Springer-Verlag 1976

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