On the existence theorems of Kantorovich, Miranda and Borsuk (Q1772037)

The theorems of Kantorovich, Miranda and Borsuk all give conditions on the existence of a zero of a nonlinear mapping. The authors concern themselves with relations between these theorems in terms of generality in the case that the mapping is finite-dimensional. To this purpose, they formulate a generalization of Miranda's theorem, holding for arbitrary norms instead of just the \(l_\infty\)-norm. The main results are 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-type theorems are themselves special cases of Borsuk's theorem.