On Strong Approximations of USC Nonconvex-Valued Mappings

AbstractFor any upper semicontinuous and compact-valued (usco) mapping F : X→Y from a metric space X without isolated points into a normed space Y we prove the existence of a single-valued continuous mapping f : X→Y such that the Hausdorff distance between graphs ΓF and Γf is arbitrarily small, whenever “measure of nonconvexity” of values of F admits an appropriate common upper estimate. Hence, we prove a version of the Beer–Cellina theorem, under controlled withdrawal of convexity of values of multifunctions. We also give conditions for such strong approximability of star-shaped-valued upper'semicontinuous (usc) multifunctions in comparison with Beer's result for Hausdorff continuous star-shaped-valued multifunctions