On some measures of noncompactness in the space of continuous functions. (English) Zbl 1134.46012

Summary: We consider some quantities in the space of continuous functions on a bounded interval, which are related to monotonicity of functions. Based on those quantities, we construct a few measures of noncompactness in the mentioned function space. Several properties of those measures are established; among others, it is shown that they are regular or “partly” regular measures and equivalent to the Hausdorff measure of noncompactness.


46E15 Banach spaces of continuous, differentiable or analytic functions
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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[1] Akmerov, R.R.; Kamenski, M.I.; Potapov, A.S.; Rodkina, A.E.; Sadovskii, B.N., Measures of noncompactness and condensing operators, (1992), Birkhäuser Verlag Basel · Zbl 0748.47045
[2] Appell, J.; Zabrejko, P.P., ()
[3] Ayerbe Toledano, J.M.; Dominguez Benavides, T.; Lopez Acedo, G., Measures of noncompactness in metric fixed point theory, (1997), Birkhäuser Verlag Basel · Zbl 0885.47021
[4] Banaś, J.; Goebel, K., ()
[5] Banaś, J.; Olszowy, L., Measures of noncompactness related to monotonicity, Comment. math., 41, 13-23, (2001) · Zbl 0999.47041
[6] Banaś, J.; Rodriguez, J.R.; Sadarangani, K., On a nonlinear quadratic integral equation of urysohn – stieltjes type and its applications, Nonlinear anal., 47, 1175-1186, (2001) · Zbl 1042.45502
[7] Danes˘, J., On densifying and related mappings and their application in nonlinear functional analysis, (), 15-56
[8] Darbo, G., Punti uniti in transformazioni a condominio non compatto, Rend. sem. mat. univ. Padova, 24, 84-92, (1955) · Zbl 0064.35704
[9] Goldens˘tein, L.S.; Markus, A.S., On a measure of noncompactness of bounded sets and linear operators, (), 45-54
[10] Kuratowski, K., Sur LES espaces complets, Fund. math., 15, 301-309, (1930) · JFM 56.1124.04
[11] Sadovskii, B.N., Asymptotically compact and condensing operators, Uspekhi mat. nauk, 27, 81-146, (1972) · Zbl 0232.47067
[12] Wiśnicki, A., Hausdorff measure of noncompactness in subspaces of continuous functions of codimension one, Nonlinear anal. TMA, 25, 223-228, (1995) · Zbl 0842.46014
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