Summary
In this paper an axiomatic approach to the notion of a measure of weak noncompactness is presented. Several properties of the defined measures are given. Moreover, we provide a few concrete realizations of the accepeted axiomatic system in some Banach spaces.
References
J. Banaś -K. Goebel,Measures of noncompactness in Banach spaces, Lect. Notes in Pure and Appl. Math., Marcel Dekker,60 (1980), New York and Basel.
J. Banaś -A. Hajnosz -S. Wedrychowicz,On the equation x′=f(t, x) in Banach spaces, Comment. Math. Univ. Carolinae,23 (1982), pp. 233–247.
E. Cramer -V. Lakshmikantham -A. R. Mitchell,On the existence of weak solutions of differential equations in nonreflexive Banach spaces, Nonlinear Anal. T.M.A.,2 (1978), pp. 169–177.
J. Daneš,On densifying and related mappings and their applications in nonlinear functional analysis, Theory of Nonlinear Operators, Akademie-Verlag, Berlin (1984), pp. 15–56.
G. Darbo,Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Math. Univ. Padova,24 (1955), pp. 84–92.
F. S. De Blasi,On a property of the unit sphere in Banach spaces, Bull. Math. Soc. Math. Roum.,21 (1977), pp. 259–262.
N.Dunford - J. T.Schwartz,Linear Operators, New York, 1958.
G. Emmanuele,Measures of weak noncompactness and fixed points theorems, Bull. Math. Soc. Sci. Math. Roum.,25 (1981), pp. 353–358.
M. Furi -M. Martelli,On the minimal displacement of points under α-Lipschitz maps in normed spaces, Boll. Un. Mat. Ital.,4 (1974), pp. 791–799.
M. Furi -A. Vignoli,On a property of the unit sphere in a linear normed space, Bull. Acad. Polon. Sci., Ser. Sci. Math. Astron. Phys.,18 (1970), pp. 333–334.
G.Köthe,Topological Vector Spaces I, Springer (1969).
I. Kubiaczyk,A functional differential equation in Banach space, Demonstr. Math.,15 (1982), pp. 113–130.
I. Kubiaczyk,Kneser type theorems for ordinary differential equations in Banach spaces, J. Differ. Equations,45 (1982), pp. 133–146.
B. N. Sadovskii,Asymptotically compact and densifying operators, Uspehi Mat. Nauk,27 (1972), pp. 81–146.
Author information
Authors and Affiliations
Additional information
This paper was done while the first author visited the Universidad de los Andes (Venezuela).
Rights and permissions
About this article
Cite this article
Banaś, J., Rivero, J. On measures of weak noncompactness. Annali di Matematica pura ed applicata 151, 213–224 (1988). https://doi.org/10.1007/BF01762795
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01762795
Keywords
- Banach Space
- Axiomatic System
- Axiomatic Approach
- Concrete Realization
- Weak Noncompactness