Treffer: On the Proofs of Some Statements Concerning the Theorems of Kantorovich, Moore, and Miranda: On the proofs of some statements concerning the theorems of Kantorovich, Moore, and Miranda

The theorems of Kantorovich, Miranda and Moore are amongst the most important results proving the existence of solutions of nonlinear equations (in \(\mathbb{R}^n\)). Because they are based on assumptions not similar or not directly comparable, it is of interest to establish a hierarchy of these theorems, in several contexts. The main aim of the paper under review is to show that the proof of such a comparison result of Moore's and Miranda's theorems presented by \textit{Z. Shen} and \textit{M. A. Wolfe} [Nonlinear Anal., Theory Methods Appl. 15, 229--232 (1990; Zbl 0727.65045)] is incorrect. This result essentially states that if the assumptions of Moore's theorem are fulfilled, then the assumptions of Miranda's theorem are satisfied, too. The author then provides a correct proof of this statement.