Preservation of the Coincidence Point Existence Property under Some Discrete Transformations of a Pair of Mappings of Metric Spaces

Abstract

In topology, there are known results on preservation of the fixed point existence property under any homotopy of self-mappings on some spaces in the event that the initial mapping has a nonzero Lefschetz number. For contracting mappings of metric spaces and some of their generalizations, there are known results by M. Frigon on preservation of the contraction property and, consequently, the fixed point existence property under some special homotopies. In 1984, J.W.Walker introduced a discrete counterpart of homotopy for self-mappings of an ordered set, which he called an order isotone homotopy. The naturalness of this notion and its relation to the usual continuous homotopy follow from the work by R. E. Stong (1966). Recently, the author and D. A.Podoprikhin have extended Walker’s notion of an order isotone homotopy and suggested sufficient conditions for preservation of the fixed point (coincidence point) existence property under such discrete homotopy (a pair of homotopies) of a mapping (a pair of mappings) of ordered sets. This paper contains metric counterparts of the obtained results and some corollaries. The method of ordering a metric space proposed by A. Brøndsted in 1974 is used.