Abstract
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.
Highlights
Let N =. {, . . .}; Z =. {. . ., . . .}; R+ =. [, ∞)
Treating the space clos(X) in pair with the Hausdorff metric dist does not lead to substantial results if, for instance, the corresponding maps have images that are infinitely distant from each other
As shown in [ – ], a way to overcome this difficulty and make it possible to use the known methods and standard techniques is to construct in the space clos(X) a metric satisfying the following conditions: ( ) the distance between any closed sets is finite; ( ) if a sequence of closed sets is convergent with respect to the Hausdorff metric, it is convergent with respect to the ‘new’ metric; ( ) the convergence of a sequence {Hi}∞ i= ⊂ clos(X) means the convergence for any r > of the sequence of bounded subsets Hri ⊂ Hi, Hri ⊂ Or, such that i Hri = Hi and Hri ⊂ Hri as soon as r < r
Summary
Since the sequence {Sr(ε)Fi}∞ i= converges (in the Hausdorff metric) to the set Sr(ε)Fr(ε) = Sr(ε)F, for every i, starting with some I, the following inequalities hold: dist Sr(ε)Fi, Sr(ε)F < ε/ , ρo Fi, F < ε/ . If such a set F ⊂ R does exist, taking any of its points x , one gets the following estimates: for every i ≥ |x | + and each r ∈ [|x |, i], dist SrFi, SrF ≥ r – |x |.
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