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
Full Text: DOI


[1] Banaś, J.; Goebel, K., Measures of noncompactness in Banach spaces, Lect. Notes in Pure and Appl. Math. (1980), New York and Basel: Marcel Dekker, New York and Basel · Zbl 0441.47056
[2] Banaś, J.; Hajnosz, A.; Wedrychowicz, S., On the equation x′=f(t, x) in Banach spaces, Comment. Math. Univ. Carolinae, 23, 233-247 (1982) · Zbl 0502.34050
[3] Cramer, E.; Lakshmikantham, V.; Mitchell, A. R., On the existence of weak solutions of differential equations in nonreflexive Banach spaces, Nonlinear Anal. T.M.A., 2, 169-177 (1978) · Zbl 0379.34041
[4] Daneš, J., On densifying and related mappings and their applications in nonlinear functional analysis, Theory of Nonlinear Operators, 15-56 (1984), Berlin: Akademie-Verlag, Berlin
[5] Darbo, G., Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Math. Univ. Padova, 24, 84-92 (1955) · Zbl 0064.35704
[6] De Blasi, F. S., On a property of the unit sphere in Banach spaces, Bull. Math. Soc. Math. Roum., 21, 259-262 (1977) · Zbl 0365.46015
[7] N.Dunford - J. T.Schwartz,Linear Operators, New York, 1958. · Zbl 0084.10402
[8] Emmanuele, G., Measures of weak noncompactness and fixed points theorems, Bull. Math. Soc. Sci. Math. Roum., 25, 353-358 (1981) · Zbl 0482.47027
[9] Furi, M.; Martelli, M., On the minimal displacement of points under α-Lipschitz maps in normed spaces, Boll. Un. Mat. Ital., 4, 791-799 (1974) · Zbl 0304.47050
[10] Furi, M.; Vignoli, A., On a property of the unit sphere in a linear normed space, Bull. Acad. Polon. Sci., Ser. Sci. Math. Astron. Phys., 18, 333-334 (1970) · Zbl 0194.43501
[11] G.Köthe,Topological Vector Spaces I, Springer (1969). · Zbl 0179.17001
[12] Kubiaczyk, I., A functional differential equation in Banach space, Demonstr. Math., 15, 113-130 (1982) · Zbl 0509.34063
[13] Kubiaczyk, I., Kneser type theorems for ordinary differential equations in Banach spaces, J. Differ. Equations, 45, 133-146 (1982) · Zbl 0505.34048
[14] Sadovskii, B. N., Asymptotically compact and densifying operators, Uspehi Mat. Nauk, 27, 81-146 (1972) · Zbl 0232.47067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.