Continuity of fuzzy multivalued mappings

Continuity of multivalued mappings is characterized by two types of semi-continuity: lower and upper semi-continuity. In this paper, different equivalent ways in which these concepts can be expressed are investigated and then used to define lower and upper semi-continuous fuzzy multivalued mappings. Some shortcomings in existing definitions of upper semi-continuous multivalued mappings are addressed. Several relationships between the lower semi-continuity of fuzzy multivalued mappings and the classical lower semi-continuity of their cuts, or the lower semi-continuity w.r.t. the level topologies are established in the setting of stratified fuzzy topological spaces. Two different types of upper semi-continuity of fuzzy multivalued mappings are introduced, inspired by the two equivalent ones in the crisp case. The relationships between these two types are studied completely. Many interesting properties of lower and upper semi-continuous fuzzy multivalued mappings are presented, including the composition and union of lower and upper semi-continuous fuzzy multivalued mappings and the compactness preserving property of upper semi-continuous and compact-valued fuzzy multivalued mappings. (C) 1998 Elsevier Science B.V.