Fatou's lemma for multifunctions with unbounded values

For multifunctions having unbounded sets as values we give Fatou-type inclusions for the Kuratowski limes superior, in finite as well as infinite dimensions. These are derived from similar, known Fatou-type inequalities for single-valued multi-functions (i.e., ordinary functions), that is, from Balder's unifying Fatou lemma in case the image set is finite-dimensional, and from an update of related results by Balder in the infinite dimensional case. For this extension from the single-valued to the multivalued situation a lemma due to Hess, used to prove earlier Fatou-type inclusions, is of critical importance. Also, an asymptotic correction term, introduced here, plays an important role. The two main results thus obtained subsume and extend an entire class of comparable Fatou lemmas.