On stability of solutions of integral equations\\ in the class of measurable functions

W. Merchela

Vestnik rossijskih universitetov. Matematika, Tome 26 (2021) no. 133, pp. 44-54

Résumé

Consider the equation $G(x)=\tilde{y},$ where the mapping $G$ acts from a metric space $X$ into a space $Y,$ on which a distance is defined, $\tilde{y} \in Y.$ The metric in $X$ and the distance in $Y$ can take on the value $\infty,$ the distance satisfies only one property of a metric: the distance between $y, z \in Y$ is zero if and only if $y=z.$ For mappings $X \to Y$ the notions of sets of covering, Lipschitz property, and closedness are defined. In these terms, the assertion is obtained about the stability in the metric space $X$ of solutions of the considered equation to changes of the mapping $G$ and the element $\tilde{y}.$ This assertion is applied to the study of the integral equation $$ f \big(t, \int_0^1 \mathcal{K}(t,s)x(s) ds, x(t)\big)=\tilde{y}(t), \ \ t \in [0.1], $$ with respect to an unknown Lebesgue measurable function $x: [0,1] \to \mathbb {R}.$ Sufficient conditions are obtained for the stability of solutions (in the space of measurable functions with the topology of uniform convergence) to changes of the functions $f, \mathcal{K}, \tilde{y}.$

Détail

Keywords: operator equation; existence of solutions; stability of solutions; covering mapping; distance; space of measurable functions; integral equation.