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 $X=(X,\leq)$ to a set $Y$ on which a reflexive binary relation $\vartheta$ is given (this relation is not supposed to be antisymmetric or transitive, i.e., $\vartheta$ 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 $F(x)\ni \tilde{y},$ where $F\colon X \rightrightarrows Y,$ $\tilde{y}\in Y.$ It is assumed that for some given $x_0 \in X,$ there exists $y_{0} \in F(x_{0})$ such that $(\tilde{y},y_{0}) \in \vartheta.$ Conditions for the existence of a solution $x\in X$ satisfying the inequality $x\leq x_0$ 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 $\widetilde{y}$ is also defined and investigated. Namely, the sequence of “perturbed” inclusions $F_i(x)\ni \tilde{y}_i,$ $i\in \mathbb{N},$ is assumed, and the conditions of existence of solutions $x_i \in X$ such that for any increasing sequence of integers $\{i_n\}$ there holds $\sup_{n \in \mathbb{N}}\{x_{i_{n}}\}= x,$ where $x \in X$ is a solution of the initial inclusion, are derived.