Measures of noncompactness in Köthe spaces
DOI:
https://doi.org/10.12775/TMNA.2024.045Keywords
Measures of noncompactness, fixed point theorems, Orlicz spaces, Lebesgue spaces, r.i. spacesAbstract
In this paper we introduce some measures of noncompactness and establish estimates between them. Such estimates are helpful to apply fixed point theorems of Darbo-Sadovskiĭ type to compositions of operators. Explicit formulas for calculating these measures of noncompactness are also given, with a particular emphasis on Lebesgue and Orlicz spaces.References
R.R. Akhmerov, M.I. Kamenskiı̆, A.S. Potapov, A.E. Rodkina and B.N. Sadovskiı̆,
T. Andô, Weakly compact sets in Orlicz spaces, Canad. Math. J. 14 (1962), 170–176.
J. Appell, Implicit functions, nonlinear integral equations, and the measure of noncompactness of the superposition operator, J. Math. Anal. Appl. 83 (1981), 251–263.
J. Appell and E.M. Semenov, Misure di non compattezza debole in spazi ideali simmetrici, Rend. Ist. Lombardo Sci. Mat. Appl. 122 (1988), 87–104.
J.M. Ayerbe Toledano, T. Domı́nguez Benavides and G. López Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Oper. Theory Adv. Appl., vol. 99, Birkhäuser, Basel, 1997.
J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lect. Notes Pure Appl. Math., vol. 60, Marcel Dekker, New York, 1980.
D.V. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces. Foundations and Harmonic Analysis, Birkhäuser, Basel, 2013.
G. Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Semin. Mat. Univ. Padova 24 (1955), 84–92.
F.S. De Blasi, On a property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R.S. Roumanie 21 (1977), 259–262.
J. Diestel, Uniform integrability: An introduction, Rend. Ist. Mat. Univ. Trieste 23 (1991), 41–80.
G. Emmanuele, Measure of weak noncompactness and fixed point theorems, Bull. Math. Soc. Sci. Math. Roumanie 25 (1981), 353358.
I.T. Gohberg, L.S. Goldenštein and A.S. Markus, Investigations of some properties of bounded linear operators with their q-norms, Učen. Zap. Kishinevsk. Univ. 29 (1957), 29–36.
L.S. Goldenštein and A.S. Markus, On the measure of noncompactness of bounded sets and of linear operators, Studies in Algebra and Math. Anal., Izdat. Karta Moldovenjaske, Kishinev, 1965, 45-54.
M.A. Krasnosel’skiı̆ and Y.B. Rutitskiı̆, Convex Functions and Orlicz Spaces, Fizmatgiz, Moscow, 1958 (in Russian); Engl. transl.: Noordhoff, Groningen, 1961.
K. Kuratowski, Sur les espaces complets, Fund. Math. 15 (1930), 301–309.
J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II: Function Spaces, Springer, Berlin, 1979.
M.M. Rao and Z.D. Ren, Theory of Orlicz Spaces, M. Dekker, New York, 1991.
M.M. Rao and Z.D. Ren, Applications of Orlicz Spaces, M. Dekker, New York, 2002.
B.N. Sadovskiı̆, On a fixed point principle, Funktsional Anal. i Prilozhen. 1 (1967), no. 2, 74–76 (in Russian).
B.N. Sadovskiı̆, Limit-compact and condensing operators, Uspekhi Mat. Nauk 27 (1972), no. 1, 81–146 (in Russian); Engl. transl.: Russian Math. Surveys 27 (1972), no. 1, 85–155.
P.P. Zabreı̆ko, Ideal function spaces, Jaroslav. Gos. Univ. Vestnik 8 (1974), 12–52 (in Russian).
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