Using the Rådström-Hörmander theorem on embedding of the hyperspace of closed convex sets in a Banach space, we prove multivalued versions of some results known for real functions.
This paper provides a convergent numerical approximation of the Pareto optimal set for finite-horizon multiobjective optimal control problems in which the objective space is not necessarily convex. Our approach is based on Viability Theory. We first introduce a set-valued return function V and show that the epigraph of V equals the viability kernel of a certain related augmented dynamical system. We then introduce an approximate set-valued return function with finite set-values as the solution of...
2000 Mathematics Subject Classification: 26E25, 41A35, 41A36, 47H04, 54C65.The paper is an updated survey of our work on the approximation of univariate set-valued functions by samples-based linear approximation operators, beyond the results reported in our previous overview. Our approach is to adapt operators for real-valued functions to set-valued functions, by replacing operations between numbers by operations between sets. For set-valued functions with compact convex images we use Minkowski...
We investigate absolute retracts for hereditarily unicoherent continua, and also the continua that have the arc property of Kelley (i.e., the continua that satisfy both the property of Kelley and the arc approximation property). Among other results we prove that each absolute retract for hereditarily unicoherent continua (for tree-like continua, for λ-dendroids, for dendroids) has the arc property of Kelley.
Relations between homomorphisms on a real function algebra and different properties (such as being inverse-closed and closed under bounded inversion) are studied.
Let be a locally connected, -compact metric space and a closed subset of . Let be the space of all continuous real-valued functions defined on some closed subsets of . We prove the equivalence of the and topologies on , where is the so called Attouch-Wets topology, defined in terms of uniform convergence of distance functionals, and is the topology of Kuratowski convergence on compacta.