Geodesic
On the existence theorems of Kantorovich, Miranda and Borsuk
Electronic transactions on numerical analysis, Tome 17 (2004), pp. 102-111
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 -norm. As our main results we then prove that the Kantorovich $\sterling $########

$ 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.$

Classification : 47H10, 47J05, 65H10
Keywords: nonlinear equations, existence theorems, fixed points, Newton-Kantorovich theorem, miranda theorem, borsuk theorem 
@article{ETNA_2004__17__a8,
     author = {Alefeld,  G\"otz and Frommer,  Andreas and Heindl,  Gerhard and Mayer,  Jan},
     title = {On the existence theorems of {Kantorovich,} {Miranda} and {Borsuk}},
     journal = {Electronic transactions on numerical analysis},
     pages = {102--111},
     year = {2004},
     volume = {17},
     zbl = {1080.47042},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2004__17__a8/}
}
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%J Electronic transactions on numerical analysis
%D 2004
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%F ETNA_2004__17__a8
Alefeld,  Götz; Frommer,  Andreas; Heindl,  Gerhard; Mayer,  Jan. On the existence theorems of Kantorovich, Miranda and Borsuk. Electronic transactions on numerical analysis, Tome 17 (2004), pp. 102-111. http://geodesic.mathdoc.fr/item/ETNA_2004__17__a8/