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Some properties of the set and ball measures of noncompactness and applications. (English) Zbl 0578.47045

Let X be a metric space. Using the set and ball measures of noncompactness, denoted by \(\alpha\) (.) and \(\beta\) (.) respectively, the following notions are defined in this paper: An infinite bounded set A of X is \(\alpha\)-minimal (resp. \(\beta\)-minimal) if every infinite subset B of A satisfies \(\alpha (B)=\alpha (A)\) (resp. \(\beta (B)=\beta (A))\). It is proved:
(1) Every infinite bounded set A has an \(\alpha\)-minimal (resp.: \(\beta\)- minimal) subset.
(2) If X is separable, every infinite bounded set A has a \(\beta\)-minimal subset B such that \(\beta (B)=\beta (A).\)
These results are applied to prove some relationships between k-contractions, k-set-contractions, and k-ball-contractions. For instance, the main result in the paper is:
Every set-condensing mapping \(T: D\to H\) is ball-condensing, where H is a Hilbert space and D an arbitrary subset of H. The A-properness of several classes of mappings \(T: D\to H\) is also studied, D being an arbitrary subset of H, without any surjectivity or bounded restriction on T.

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J05 Equations involving nonlinear operators (general)
65J15 Numerical solutions to equations with nonlinear operators
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