An extension of the Borsuk antipodal fixed point theory with essential maps

In this paper we first present new essential map type results for Kakutani maps satisfying certain antipodal type conditions on the boundary of a set. This enables us to obtain some new fixed point results for maps with an inclusion property satisfying some antipodal type conditions on the boundary of a set. In particular we use new antipodal type conditions to discuss essentiality and the existence of fixed points for general classes of maps which have a continuous or upper semicontinuous selection property.