On a consequence of the Schauder theorem

Abstract

A “topological digression”, which is a consequence of the Brouwer fixed-point theorem, appears in Sec. 4.1 in the book of Lee and Markus [1]. Under some restrictions on the continuous mapping, this consequence states that the interior of the image of the unit ball in Rn is nonempty under such a mapping. This consequence was widely used by these authors (e.g., in the proof of Theorem 3 in Sec. 4.1 [1]). This consequence was also used by the author of this paper in [2], [3]. In the present paper, we prove a generalization of this result to the case of Banach spaces.