Preservation of the existence of coincidence points under some discrete transformations of a pair of mappings of metric spaces

In topology there are known results on the preservation under homotopy of the fixed point property of self-mappings in some spaces if the Lefschetz number of the initial mapping is nonzero. For the class of contracting mappings of metric spaces and for some of their generalizations, there are M. Frigon's known results on the preservation of the contraction property and hence of the fixed point property under homotopies of some special type. In 1984 J. W. Walker introduced a discrete counterpart of homotopy for mappings in an ordered set, which he called an order isotone homotopy. R. E. Stong showed the naturalness of this notion and its relation to the usual continuous homotopy. Recently, the author and D. A. Podoprikhin have generalized Walker's notion of order isotone homotopy and suggested sufficient conditions for the preservation under such discrete homotopy (a pair of homotopies) of the property of a mapping (a pair of mappings) of ordered sets to have a fixed point (a coincidence point). 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.