A new metric in the space clos(X) of all closed subsets of a metric space X is proposed. This metric, unlike the generalized Hausdorff metric, takes finite values only, and the convergence of a sequence of closed sets H i , i = 1, 2, . . . , with respect to this metric is equivalent to the convergence (in the sense of Hausdorff ) for any r ≥ 0 of the unions of H i with a closed 'exterior ball' of radius r. Using this metric allows one to investigate multi-valued maps that have images in clos(X) and are not continuous in the Hausdorff metric. In the work, the necessary and sufficient conditions for a multi-valued map to be continuous and Lipschitz with respect to the metric presented are studied, a connection of these properties with their analogues in the Hausdorff metric is derived, and a generalization of the Nadler fixed point theorem is obtained. MSC: 47H04; 47H10; 54E35

Получены теоремы о существовании и единственности неподвижной точки отображения полного метрического пространства в себя, обобщающие теоремы Л.В. Канторовича для гладких отображений банаховых пространств. Эти результаты распространены на точки совпадения как обычных, так и многозначных отображений, действующих в метрических пространствах.

No abstract

No abstract