Generalization of Banach’s theorem for cones and covering along curves

A. V. Arutyunov ; S. E. Zhukovskiy

Vestnik rossijskih universitetov. Matematika, Tome 28 (2023) no. 144, pp. 361-370

Résumé

This work is devoted to the study of the covering property of linear and nonlinear mappings of Banach spaces. We consider linear continuous operators acting from one Banach space to another. For a given operator, it is shown that for any point $y_0$ from the relative interior of the image of a given convex closed cone there exists a conical neighborhood of $y_0$, with respect to which the given operator has the covering property at zero with a covering constant depending on the point $y_0.$ We provide an example showing that for a linear continuous operator the covering property with respect to the image of a given cone at zero may fail, i. e. the statement of Banach's theorem on an open mapping may not hold for restrictions of linear continuous operators to closed convex cones. We obtain a corollary of the obtained theorem for the case when the target space is finite-dimensional. Moreover, nonlinear twice differentiable mappings of Banach spaces are considered. For them, conditions for local covering along a certain curve with respect to a given cone are presented. The corresponding sufficient conditions are formulated in terms of $2$-regular directions. They remain meaningful even in the case of degeneracy of the first derivative of the mapping under consideration at a given point.

Détail

Keywords: Banach's open mapping theorem, convex cone, abnormal point, $2$-regularity, covering along a curve