On stability and continuous dependence on parameter of the set of coincidence points of two mappings acting in a space with a distance

T. V. Zhukovskaya ; W. Merchela

Vestnik rossijskih universitetov. Matematika, Tome 27 (2022) no. 139, pp. 247-260

Résumé

We consider the problem of coincidence points of two mappings $\psi,\varphi$ acting from a metric space $(X,\rho)$ into a space $(Y,d)$ in which a distance $d$ has only one of the properties of the metric: $ d(y_1,y_2)=0$ $\Leftrightarrow$ $y_1=y_2,$ and is assumed to be neither symmetric nor satisfying the triangle inequality. The question of well-posedness of the equation $$\psi(x)=\varphi(x)$$ which determines the coincidence point, is investigated. It is shown that if $x=\xi$ is a solution to this equation, then for any sequence of $\alpha_i$-covering mappings $\psi_i :X\to Y$ and any sequence of $\beta_i$-Lipschitz mappings $\varphi_i :X\to Y,$ $\alpha_i> \beta_i \geq 0,$ in the case of convergence {${d(\varphi_i(\xi),\psi_i(\xi))\to 0}$}, equation $\psi_i(x)= \varphi_i(x)$ has, for any $i,$ a solution $x=\xi_i$ such that $\rho(\xi_i,\xi)\to 0.$ Further in the article, the dependence of the set $\mathrm{Coin}(t)$ of coincidence points of mappings $\psi(\cdot,t),\varphi(\cdot,t ):X\to Y$ on a parameter $t,$ an element of the topological space $T,$ is investigated. Assuming that the first of these mappings is $\alpha$-covering and the second one is $\beta$-Lipschitz, we obtain an assertion on upper semicontinuity, lower semicontinuity, and continuity of the set-valued mapping $\mathrm {Coin}:T\rightrightarrows X.$

Détail

Keywords: well-posedness of equation, continuous dependence on parameter, coincidence point of two mappings, covering mapping.
Mots-clés : distance