On one inclusion with a mapping acting from a partially ordered set to a set with a reflexive binary relation

Abstract

Set-valued mappings acting from a partially ordered space X = (X, ≤) to a set Y on which a reflexive binary relation ϑ is given (this relation is not supposed to be antisymmetric or transitive, i. e., ϑ is not an order in Y), are considered. For such mappings, analogues of the concepts of covering and monotonicity are introduced. These concepts are used to study the inclusion F (x) ∋ y, ∼ where F: X Y, y∼ ∈ Y. It is assumed that for some given x0 ∈ X, there exists y0 ∈ F (x0) such that (∼ y, y0) ∈ ϑ. Conditions for the existence of a solution x ∈ X satisfying the inequality x ≤ x0 are obtained, as well as those for the existence of minimal and least solutions. The property of stability of solutions of the considered inclusion to changes of the set-valued mapping F and of the element ye is also defined and investigated. Namely, the sequence of “perturbed” inclusions Fi(x) ∋ y∼i, i ∈ N, is assumed, and the conditions of existence of solutions xi ∈ X such that for any increasing sequence of integers {in} there holds supn∈N{xin} = x, where x ∈ X is a solution of the initial inclusion, are derived.