Inclusions with mappings acting from a metric space to a space with distance

W. Merchela

Vestnik rossijskih universitetov. Matematika, Tome 27 (2022) no. 137, pp. 27-36

Résumé

The article deals with an inclusion in which a multivalued mapping acts from a metric space $ (X, \rho) $ into a set $Y$ with distance $d.$ This distance satisfies only the first axiom of the metric: $d(y_1, y_2)$ is equal to zero if and only if $y_1= y_2.$ The distance does not have to be symmetric or to satisfy the triangle inequality. For the space $(Y, d),$ the simplest concepts (of a ball, convergence, the distance from a point to a set) are defined, and for a multivalued map $G: X \rightrightarrows Y,$ the sets of covering, Lipschitz and closedness are introduced. In these terms (allowing us to adapt the classical conditions of covering, Lipschitz property and closedness of mappings of metric spaces to the maps with values in $(Y,d)$ and to weaken such conditions), a theorem on solvability of the inclusion $F(x,x) \ni \widehat{y} $ is formulated, and an estimate for the deviation in the space $(X,\rho)$ of the set of solutions from a given element $x_0 \in X$ is given. The main conditions of the obtained statement are the following: for any $x$ from some ball, the pair $(x,\widehat{y})$ belongs to the $\alpha$-covering set of the mapping $F(\cdot, x)$ and to the $\beta$-Lipschitz set of the mapping $F(x,\cdot),$ where $\alpha> \beta.$ The proof of the corresponding statement is based on the construction of the sequences $\{x_n\} \subset X$ and $\{y_n\} \subset Y$ satisfying the relations \begin{equation*} y_n \in F(x_{n}, x_{n}), \ \ \widehat{y} \in F(x_{n+1}, x_{n}), \ \alpha \rho(x_{n+1}, x_n) \leq d(\widehat{y}, y_n) \leq \beta \rho(x_{n}, x_{n-1}). \end{equation*} Also, in the paper, we obtain sufficient conditions for the stability of solutions of the considered inclusion to changes in the multivalued mapping $F$ and in the element $\widehat{y}.$

Détail

Keywords: metric, covering multivalued mapping.
Mots-clés : distance, inclusion, existence of solution