Volume 17, pp. 102-111, 2004.

On the existence theorems of Kantorovich, Miranda and Borsuk

Götz Alefeld, Andreas Frommer, Gerhard Heindl, and Jan Mayer

Abstract

The theorems of Kantorovich, Miranda and Borsuk all give conditions on the existence of a zero of a nonlinear 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 $l_{\mathrm{\infty }}$-norm. As our main results we then prove that the Kantorovich theorem reduces to a special case of this generalized Miranda theorem as well 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.

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Key words

nonlinear equations, existence theorems, fixed points, Newton-Kantorovich theorem, Miranda theorem, Borsuk theorem.

AMS subject classifications

47H10, 47J05, 65H10.