In any Banach space a monotone operator with a norm-to-weak upper semicontinuous multivalued selection on an open set D is singlevalued and norm-to-norm upper semicontinuous at the points of a dense G subset of D. Monotone operators | and especially a special case of them, subdiierentials of convex functions | play an important role in various parts of nonlinear analysis. One of the often inves...

We formulate a multi-valued version of the Tietze-Urysohn extension theorem. Precisely, we prove that any upper semicontinuous multi-valued map with nonempty closed convex values defined on a closed subset (resp. closed perfectly normal subset) of a completely normal (resp. of a normal) space X into the unit interval [0, 1] can be extended to the whole space X. The extension is upper semicontin...

We develop a framework for studying dynamical properties of hybrid systems based on considering the space of all possible trajectories of the system. In this way it is possible to treat both nondeterministic systems and systems for which the evolution does not depend continuously on the initial conditions. We show that the trajectory space is compact for the class of upper-semicontinuous hybrid...

for all x, y ∈ X and all natural numbers n, where φn : [0,∞)→ [0,∞) and limn→∞φn = φ, uniformly on any bounded interval [0,b]. Suppose that φ is upper semicontinuous and that φ(t) < t for all t > 0. Furthermore, suppose that there exists a positive integer n∗ such that φn∗ is upper semicontinuous and φn∗(0)= 0. If there exists x0 ∈ X which has a bounded orbit O(x0)= {x0,Tx0,T2x0, . . .}, then T...

In this paper, the relationships between semicontinuity and preinvexity of fuzzy mappings are investigated. The concepts of intermediate-point preinvex fuzzy mappings and weakly preinvex fuzzy mappings are first introduced, and the authors establish a characterization for a weakly preinvex fuzzy mapping in terms of its epigraph. In this context, a characterization of a preinvex fuzzy mapping in...

We prove that any locally bounded from below, upper semicontinuous v-convex function in any Carnot group is h-convex. §