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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 finitedimensional. To this purpose we formulate a generalization of Miranda's theorem,
holding for arbitrary norms instead of just the 100norm. 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, NewtonKantorovich theorem, Miranda
theorem, Borsuk theorem.
