Summary: - -- -- - . -- -. - - --. - - -- - --.-
Electronic Transactions on Numerica1 Analysis.
Volume 17, pp. 102.111,2004.
Copyright @ 2004, Kent State University.
Kent State University
ON THE EXISTENCE THEOREMS OF KANTOROVICH, MIRANDA AND
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
Key words. nonlinear equations, existence theorems, fixed points, Newton-Kantorovich theorem, Miranda
theorem, Borsuk theorem.