On measures of weak noncompactness. (English) Zbl 0653.47035

The authors give an axiomatic definition of measures of weak noncompactness which is in some sense parallel to B. N. Sadovskij’s definition of measures of (strong) noncompactness [see e.g. Usp. Mat. Nauk 27, No.1, 81-146 (1972; Zbl 0243.47033)]. The first explicit measure of weak noncompactness is due to F. S. de Blasi [Bull. Math. Soc. Sci. Math. R.S.R., n. Sér. 21(69), 259-262 (1977; Zbl 0365.46015)]. The measure \(\gamma\) studied in Section 3 (in the space L 1(a,b)) was previously introduced by the reviewer in Rend. Sci. Mat. Appl. A-119, 157-174 (1985; Zbl 0619.47043). In the last section, a Darbo-type fixed point theorem is proved for operators which diminish a general measure of weak non compactness.
Reviewer: J.Appell


47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
46B10 Duality and reflexivity in normed linear and Banach spaces
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