Weak Cyclic Monotonicity and Existence of Solutions of Differential Inclusions

The notion of weak cyclic monotonicity of set-valued maps generalizing the cyclic monotonicity is introduced. The existence of solutions of differential inclusions with compact, upper semi-continuous, not necessarily convex right-hand sides in R^n is proved for weakly cyclic monotone right-hand sides.Comment: 7 pages, 23 reference

Existence of Solutions for Nonconvex Differential Inclusions of Monotone Type

Differential inclusions with compact, upper semi-continuous, not necessarily convex right-hand sides in R^n are studied. Under a weakened monotonicity-type condition the existence of solutions is proved.Comment: 5 pages, 14 reference

Directed Subdifferentiable Functions and the Directed Subdifferential without Delta-Convex Structure

We show that the directed subdifferential introduced for differences of convex (delta-convex, DC) functions by Baier and Farkhi can be constructed from the directional derivative without using any information on the DC structure of the function. The new definition extends to a more general class of functions, which includes Lipschitz functions definable on o-minimal structure and quasidifferentiable functions.Comment: 30 pages, 3 figure

Spline Subdivision Schemes for Compact Sets. A Survey

Dedicated to the memory of our colleague Vasil Popov January 14, 1942 – May 31, 1990 * Partially supported by ISF-Center of Excellence, and by The Hermann Minkowski Center for Geometry at Tel Aviv University, IsraelAttempts at extending spline subdivision schemes to operate on compact sets are reviewed. The aim is to develop a procedure for approximating a set-valued function with compact images from a finite set of its samples. This is motivated by the problem of reconstructing a 3D object from a finite set of its parallel cross sections. The first attempt is limited to the case of convex sets, where the Minkowski sum of sets is successfully applied to replace addition of scalars. Since for nonconvex sets the Minkowski sum is too big and there is no approximation result as in the case of convex sets, a binary operation, called metric average, is used instead. With the metric average, spline subdivision schemes constitute approximating operators for set-valued functions which are Lipschitz continuous in the Hausdorff metric. Yet this result is not completely satisfactory, since 3D objects are not continuous in the Hausdorff metric near points of change of topology, and a special treatment near such points has yet to be designed

Regularity of set-valued maps and their selections through set differences. Part 1: Lipschitz continuity

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