On Essential Fixed Points of Compact Mappings on Arbitrary Absolute Neighborhood Retracts and Their Application to Multivalued Fractals

Abstract

The existence of essential fixed points is proved for compact self-maps of arbitrary absolute neighborhood retracts, provided the generalized Lefschetz number is nontrivial and the topological dimension of a fixed point set is equal to zero. Furthermore, continuous self-maps of some special compact absolute neighborhood retracts, whose Lefschetz number is nontrivial, are shown to possess pseudo-essential fixed points even without the zero dimensionality assumption. Both results are applied to the existence of essential and pseudo-essential multivalued fractals. An illustrative example of this application is supplied.