Upper Namioka property of compact-valued mappings

We introduce and study the notions of upper Namioka property, upper Namioka space and upper co-Namioka space which are development of the notions of Namioka property, Namioka space and co-Namioka space on the case of compact-valued mappings. We obtain the following results: the class of upper Namioka spaces consists of Baire spaces with everywhere dense set of isolated points; any subset of a upper co-Namioka compact space is separable; every well-ordered upper co-Namioka compact and every upper co-Namioka compact Valdivia are metrizable; the double arrow space is not upper co-Namioka; there exist a compact-valued mapping F∈LU(X,Y) defined on the product of Namioka and co-Namioka spaces such that F has not upper Namioka property; if there exists a non-metrizable linearly ordered upper co-Namioka space, then the set of its non-isolated neighbor points contains a subset always of the first category subset; every compact-valued mapping F∈LU(X,Y) defined on the product of a β-σ′-unfavorable space X and a separable linearly ordered compact space Y has the upper Namioka property.