Real-valued non compactness measures in topological vector spaces and a pplications

Abstract

A non compactness measure with values in the lattice of extended non negative real numbers $[0, +\infty ]$ is introduced in the general setting of a Hausdorff topological vector space E. This generalizes the classical Kuratowski and Hausdorff non compactness measures. In order to achieve this, we introduce the notions of basic and sufficient collections of zero neighborhoods. We then show that our measure satisfies most of the properties of the classical non compactness measures. We particularly show that if E is locally p-convex for some $0 < p \le 1$, our measure is stable by the transition to the closed p-convex hull. This allows us to obtain, as applications, generalizations of the well-known three fixed point theorems, namely Schauder, Darbo, and Sadovskii’s ones in the setting of locally p-convex spaces. As another application, we establish a quantification of Ascoli theorem in the space C(X, E) of vector-valued continuous functions on a Hausdorff completely regular space with values in a topological vector space E, giving an alternative of Ambrosetti theorem initially stated in the metric spaces setting.