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Electronic Transactions on Numerica1 Analysis. Volume 17, pp. 102.111,2004.
 

Summary: - -- -- - . -- -. - - --. - - -- - --.-
Electronic Transactions on Numerica1 Analysis.
Volume 17, pp. 102.111,2004.
Copyright @ 2004, Kent State University.
ISSN 1068.%13.
ETNA
Kent State University
etna@mcs.kent.edu
ON THE EXISTENCE THEOREMS OF KANTOROVICH, MIRANDA AND
BORSUK*
0012 ALEFELOt, ANOREAS FROMMER~, GERHARD HEINDO, AND JAN MAYERt
Abstract. Tbe theorems of Kantorovich, Miranda and Borsuk all give conditions on the existence of a zero of a
nonIinear mapping. In this paper we are concerned with relations between these theorems in terms of generality in
the case that the mapping is finite-dimensional. To this purpose we formulate a generalization of Miranda's theorem,
holding for arbitrary norms instead of just the 100-norm. As our main results we then prove that the Kantorovich
theorem reduces to a special case of this generalized Miranda theorem as weil as to a special case of Borsuk's
theorem. Moreover, it turns out that, essentially, the Miranda theorems are themselves special cases of Borsuk's
theorem.
Key words. nonlinear equations, existence theorems, fixed points, Newton-Kantorovich theorem, Miranda
theorem, Borsuk theorem.

  

Source: Alefeld, Götz - Institut für Angewandte und Numerische Mathematik & Fakultät für Mathematik, Universität Karlsruhe

 

Collections: Mathematics