On the continuous dependence on the parameter of the set of solutions of the operator equation

E. S. Zhukovskiy ; W. Merchela

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 54 (2019), pp. 27-37

Résumé

For mappings acting from a metric space $ (X, \rho_X) $ to a space $ Y, $ on which a distance is defined (i.e., a function $ d: X \times X \to \mathbb{R}_+ $ such that $ d (x, u) = 0 \Leftrightarrow x=u $), the following analogue of the covering property is defined. The set $$ \mathrm{Cov}_{\alpha} [f] = \{(x, \tilde{y}) \in X \times Y: \, \exists \tilde{x} \in X \ f(\tilde{x}) =\tilde{y}, \ \rho_{X}(\tilde{x}, x) \leq {\alpha}^{-1} d_{Y} \bigl(\tilde{y}, f(x) \bigr)\}$$ is called the set of $\alpha$-covering of the mapping $f:X \to Y.$ For given $ \tilde{y} \in Y, $ $ \Phi: X \times X \to Y $ the equation $\Phi(x,x)=\tilde{y}$ is considered. A theorem on the existence of a solution is formulated. The problem of the stability of solutions on small perturbations of the mapping $\Phi$ is investigated. Namely, we consider a sequence of mappings $\Phi_{n}: X\times X\to Y, $ $n = 1,2,\ldots,$ such that for all $x \in X$ the following holds: $(x,\tilde{y}) \in \mathrm{Cov}_{\alpha}\big[\Phi_n(\cdot,x)\big],$ the mapping $\Phi_n( x,\cdot)$ is $\beta$-Lipschitz and for the solution $x^{*}$ of the initial equation $d_{Y} \big (\tilde{y}, \Phi_{n} (x^{*}, x^{*}) \big) \to 0.$ Under these conditions, it is proved that for any $n$ there exists $x^{*}_{n}$ such that $\Phi_{n} (x^{*}_{n}, x^{*}_{n}) = \tilde{y}$ and $\{x^{*}_{n} \} $ converges to $x^{*}$ in the metric space $X.$  Moreover, we consider the equation $\Phi(x,x,t)=\tilde{y}$ with the parameter $t$ which is an element of a topological space. It is assumed that $(x, \tilde{y})\in \mathrm{Cov}_{\alpha} \big [\Phi_n (\cdot, x, t) \big], $ the mapping $\Phi_n (x, \cdot, t) $ is $\beta$-Lipschitz, and the mapping $ \Phi_n (x, x, \cdot) $ is continuous. Statements on the upper and lower semicontinuous dependence of the solutions set on the parameter $t$ are proved.

Détail

Keywords: operator equation, continuous dependence of a solution on parameters, metric space, covering mapping, Lipschitz mapping.
Mots-clés : existence of solutions, estimation of solutions