Multivalued Homotopy on an Ordered Set,Fixed and Coincidence Points of Mappings,and Applications in Game Theory

Mathematical Notes - The article develops results of the authors’ previous papers on the topic. The notion of the homotopy of a multivalued mapping of an ordered set is introduced. We study...     

Common fixed points of a family of commuting mappings of partially ordered sets.

The paper presents conditions providing the existence of a common fixed point of a family of commuting isotone multivalued mappings of a partially ordered set and the existence of the minimal element in the set of common fixed points. Additional conditions that guarantee the existence of the least element in that point set are also presented. Relations of the obtained results to well-known fixed point theorems are discussed.     

Fixed points and coincidences of mappings of partially ordered sets

Possibility of the reduction of a coincidence problem to a fixed point problem is investigated for one-valued and multivalued mappings of partially ordered sets. New fixed point theorems are proved. Connections of the obtained results with well-known fixed point theorems and some recent results on coincidences of two mappings are considered.     

On coincidences of families of mappings on ordered sets

New results on fixed points and coincidences of families of set-valued mappings of partially ordered sets obtained without commutativity assumptions are presented. These results develop theorems on fixed points of an isotone self-mapping of an ordered set (for families of set-valued mappings) and theorems about coincidences of two set-valued mappings one of which is isotone and the other is covering (for finite families of set-valued mappings).     

Preservation of the existence of fixed and coincidence points under homotopy of mappings of ordered sets

In problems of topology and analysis, well-known theorem on the preservation by any continuous homotopy of the property of a mapping to have a fixed point and the property of a pair of mappings to have a coincidence point are extensively applied. Thus, for contraction mappings and some of their generalizations, Frigon’s results on the preservation of the property to have a fixed point by a homotopy of a special type are known. This paper presents theorems on the preservation by order homotopy of the property of a pair of mappings to have a coincidence point. As a corollary, conditions under which such a homotopy preserves the property of a mapping to have a fixed point are obtained.