Coincidence points of two mappings acting from a partially ordered space to an arbitrary set

A coincidence point of a pair of mappings is an element these mappings take the same values at. Coincidence points of mappings of partially ordered spaces are investigated by A.V. Arutyunov, E.S. Zhukovsky, S.E. Zhukovsky (see Topology and its Applications, 2015, V. 179, No. 1, p. 13–33); in particular, it was proved that an orderly covering mapping and a monotone mapping both acting from a partially ordered space into a partially ordered space possess a coincidence point. We consider the problem of existence of a coincidence point of a pair of mappings acting from a partially ordered space into a set, where no binary relation is defined, and, thus, it is impossible to determine the properties of covering and monotonicity of maps. In order to study such a problem, we define the notion of “quasi-coincidence” point, i.e. such element for which there exists an element not greater than the initial element such that the value of the first mapping at it is equal to the value of the second mapping at the initial element. It turns out that it is sufficient to require the following condition to be fulfilled for the existence of a coincidence point: any chain of “quasi-coincidence” points is bounded and has a lower boundary, which is also a point of “quasi-coincidence”. An example of mappings that satisfy the proposed conditions, but to which the results on coincidence points of the orderly covering and monotone mappings cannot be applied, is given in the article. An interpretation of the stability concept of a coincidence point of mappings with respect to their small perturbations in partially ordered space was proposed, and the conditions for such stability were obtained in the article.