Regularity of Set-Valued Maps and Their Selections through Set Differences. Part 1: Lipschitz Continuity
نویسندگان
چکیده
We introduce Lipschitz continuity of set-valued maps with respect to a given set difference. The existence of Lipschitz selections that pass through any point of the graph of the map and inherit its Lipschitz constant is studied. We show that the Lipschitz property of the set-valued map with respect to the Demyanov difference with a given constant is characterized by the same property of its generalized Steiner selections. For a univariate multifunction with only compact values in R, we characterize its Lipschitz continuity in the Hausdorff metric (with respect to the metric difference) by the same property of its metric selections with the same constant.
منابع مشابه
Regularity of Set-Valued Maps and Their Selections through Set Differences. Part 2: One-Sided Lipschitz Properties
We introduce one-sided Lipschitz (OSL) conditions of set-valued maps with respect to given set differences. The existence of selections of such maps that pass through any point of their graphs and inherit uniformly their OSL constants is studied. We show that the OSL property of a convex-valued set-valued map with respect to the Demyanov difference with a given constant is characterized by the ...
متن کاملIntegration and Regularity of Set - Valued Maps Represented
A family of probability measures on the unit ball in R generates a family of generalized Steiner (GS-)points for every convex compact set in R. Such a ”rich” family of probability measures determines a representation of a convex compact set by GS-points. In this way, a representation of a setvalued map with convex compact images is constructed by GS-selections (which are defined by the GS-point...
متن کاملGeneralized Differentiation with Positively Homogeneous Maps: Applications in Set-Valued Analysis and Metric Regularity
We propose a new concept of generalized di erentiation of setvalued maps that captures rst order information. This concept encompasses the standard notions of Fréchet di erentiability, strict di erentiability, calmness and Lipschitz continuity in single-valued maps, and the Aubin property and Lipschitz continuity in set-valued maps. We present calculus rules, sharpen the relationship between th...
متن کاملApproximation theorems for fuzzy set multifunctions in Vietoris topology. Physical implications of regularity
n this paper, we consider continuity properties(especially, regularity, also viewed as an approximation property) for $%mathcal{P}_{0}(X)$-valued set multifunctions ($X$ being a linear,topological space), in order to obtain Egoroff and Lusin type theorems forset multifunctions in the Vietoris hypertopology. Some mathematicalapplications are established and several physical implications of thema...
متن کاملBanach module valued separating maps and automatic continuity
For two algebras $A$ and $B$, a linear map $T:A longrightarrow B$ is called separating, if $xcdot y=0$ implies $Txcdot Ty=0$ for all $x,yin A$. The general form and the automatic continuity of separating maps between various Banach algebras have been studied extensively. In this paper, we first extend the notion of separating map for module case and then we give a description of a linear se...
متن کامل