On Ranges of Non-linear Operators

We derive conditions ensuring that the range of a given continuous mapping with a compact convex domain covers a prescribed set. In Frechet spaces, we consider approximations by one single-valued mapping such that the inverse of it has either convex fibers or admits a continuous single-valued selection. Subsequently, in Banach and finite-dimensional spaces, we focus on approximations determined by a convex set of bounded linear mappings. We demonstrate that our approach is highly flexible and provides the unified treatment of various, in general non-local, covering properties of possibly non-smooth mappings. In finite-dimensional spaces, we present sufficient conditions for constrained directional semiregularity, metric regularity and strong metric regularity. These conditions cover, unify, and extend such well-known results as Pourciau's open mapping theorem and Clarke's inverse and implicit function theorems as well as their generalizations established by V. Jeyakumar and D.T. Luc by using upper semi-continuous unbounded pseudo-Jacobians or by A. Neumaier by considering various interval extensions of the derivative of a smooth mapping. Finally, we provide conditions guaranteeing that the non-linear image of a compact convex set contains a prescribed ordered interval which has direct applications in power network security management such as preventing the electricity blackout.