Geodesic
On the existence theorems of Kantorovich, Miranda and Borsuk
Electronic transactions on numerical analysis, Tome 17 (2004), pp. 102-111.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 
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     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},
     publisher = {mathdoc},
     volume = {17},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2004__17__a8/}
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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/