On the existence theorems of Kantorovich, Miranda and Borsuk.

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 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.