Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences
Serdica Mathematical Journal, Vol. 39, No 3-4, (2013), 365p-390p
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^n, 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. 2010 Mathematics Subject Classification: 54C65, 54C60, 26E25.