Measures of noncompactness and some applications. (English) Zbl 1009.47047

The author gives a thorough review (mostly without proofs) of various notions with the common name “measure of noncompactness”. So, the Kuratowski’s, the Hausdorff’s, the Istrǎţescu’s and other measures of noncompactness, both of subsets of metric spaces and of operators between such spaces, are considered. The basic properties of these notions are listed and main applications explained, often referring to the long bibliography given at the end of the paper. Particularly, applications to the Fredholm and the semi-Fredholm operators are explained, that contain most of the author’s earlier results mentioned in the paper.


47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
47A53 (Semi-) Fredholm operators; index theories
47A55 Perturbation theory of linear operators