On ranges of set-valued mappings

We derive conditions ensuring that the range of a set-valued 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 convex fibers. Subsequently, in Banach and finite-dimensional spaces, we focus on approximations determined by a convex set of bounded linear mappings such as Pales-Zeidan Jacobian, Clarke's generalized Jacobian, shields by T.H. Sweetser, or Neumaier's interval extensions of the derivative of a smooth mapping. As easy corollaries in Euclidean spaces, we obtain perturbation stability of the property of metric semiregularity under the additive perturbation by a single-valued mapping having sufficiently small calmness modulus; as well as the non-smooth Lyusternik-Graves theorem and Robinson's theorem by A.F. Izmailov. Finally, given two quadratic mappings f and g, a polyhedral convex set Omega, and an ordered interval Theta, we provide conditions guaranteeing that an ordered interval Gamma is such that for each y is an element of Gamma there is an x is an element of Omega with y = f(x) and g(x) is an element of Theta This theorem has direct applications in power network security management such as preventing the electricity blackout. (C) 2022 Elsevier Inc. All rights reserved.