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

Abstract

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.