Theorems on perturbations of covering mappings in spaces with a distance and in spaces with a binary relation

S. Benarab ; E. S. Zhukovskiy ; W. Merchela

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 4, pp. 52-63

Résumé

Statements on the existence of solutions of special-type equations in spaces with a distance and in spaces with a binary relation are derived. The results obtained generalize the well-known theorems on coincidence points of a covering and a Lipschitz mappings and on Lipschitz perturbations of covering mappings in metric spaces as well as the theorems on coincidence points of a covering and an isotonic mappings and on antitone perturbations of covering mappings in partially ordered spaces. In the first part of the paper, we consider a mapping $F\colon X\times X \to Y$, where $X$ is a metric space and $Y$ is equipped with a distance satisfying only the identity axiom. “Weakened analogs” of the notions of covering and Lipschitz mappings from $X$ to $Y$ are defined. Under the assumption that $F$ is covering in the first argument and Lipschitz in the second argument (in the sense of the definitions of these properties given in the paper), the existence of a solution $x$ to the equation $F(x,x)=y$ is established. It is shown that this statement yields conditions for the existence of a coincidence point of a covering and a Lipschitz mappings acting from $X$ to $Y$. In the second part of the paper, similar results are obtained in the case when $X$ is a partially ordered space and $Y$ is equipped with a reflexive binary relation (which is neither transitive nor antisymmetric). “Weakened analogs” of the notions of ordered covering and monotonicity of mappings from $X$ to $Y$ are defined. Under the assumption that $F$ is covering in the first argument and antitone in the second argument (in the sense of the definitions of these properties given in the paper), the existence of a solution $x$ to the equation $F(x,x)=y$ is established and conditions for the existence of a coincidence point of a covering and an isotone mappings acting from $X$ to $Y$ are deduced from this statement. In the third part, a connection between the obtained statements is established. Namely, it is proved that the theorem on the solvability of an operator equation in spaces with a binary relation implies a similar theorem in spaces with a distance and, accordingly, the statements on coincidence points.

Détail

Keywords: metric space, ordered space, covering mapping, Lipschitz mapping, monotone mapping.