Approximation methods for expanding operators
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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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- Bredendiek, E. and L. Collatz : Simultanapproximation bei Randwertaufgaben, to appear in Internat. Ser. Num. Math., Birkhäuser, 1975.Google Scholar
- Collatz, L. : Functional Analysis and Numerical Analysis, Academic Press 1966, 473 p.Google Scholar
- Flügge, H. : Zur Tschebyscheff-Approximation mit Funktionen mehrerer Variablen, Dissertation Universität Hamburg 1975, 71 p.Google Scholar
- Krasnoselskii, M.A. : Positive solutions of operator equations, P. Noordhoff, Groningen 1964.Google Scholar
- Reissig, R., G. Sansone and R. Conti ,: Nichtlineare Differentialgleichungen höherer Ordnung, Roma 1969, 738 p.Google Scholar
- Smart, D.R. : Fixed point theorems, Cambridge Univ. Press (1974) 93 p.Google Scholar
- Sprekels, J. : Einige Existenz-und Einschließungssätze für Fixpunkte expandierender Integraloperatoren, erscheint demnächst.Google Scholar