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Some Properties of the Set and Ball Measures of Non‐Compactness and Applications
Author(s) -
Benavides Tomás Domínguez
Publication year - 1986
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-34.1.120
Subject(s) - compact space , ball (mathematics) , set (abstract data type) , mathematics , computer science , pure mathematics , mathematical analysis , programming language
Let X be a metric space. Using the set and ball measures of non‐compactness, we define the notions of α‐minimal and β‐minimal sets, and prove that X has an α‐minimal subset. If X is separable, and B is a bounded set of X , we prove that B has a β‐minimal subset A such that β( A ) = β( B ). These results are applied to prove that, if Y is another metric space, T : A → Y is condensing and α( A ) > 0, then for some k < 1 there exists a non‐precompact subset B of A such that T : B → Y is k ‐contractive. If X is a separable Hilbert space we prove that, if T : D → X is set‐condensing, then T is ball‐condensing, where D is an arbitrary subset of X . Some other relations are proved. We also study the A ‐properness of several classes of mappings T : D → X , where D is an arbitrary subset of a Hilbert space, without any surjectivity or boundary restriction on T .