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Topological Methods in Nonlinear Analysis

Measures of noncompactness in Köthe spaces
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Measures of noncompactness in Köthe spaces

Authors

  • Jürgen Appell
  • Agnieszka Chlebowicz https://orcid.org/0000-0002-7079-901X
  • Tomás Domínguez Benavides https://orcid.org/0000-0003-0281-3745
  • Simon Reinwand https://orcid.org/0000-0003-1447-711X
  • Beata Rzepka https://orcid.org/0000-0002-6460-4749

DOI:

https://doi.org/10.12775/TMNA.2024.045

Keywords

Measures of noncompactness, fixed point theorems, Orlicz spaces, Lebesgue spaces, r.i. spaces

Abstract

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,

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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.

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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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Published

2025-06-14

How to Cite

1.
APPELL, Jürgen, CHLEBOWICZ, Agnieszka, BENAVIDES, Tomás Domínguez, REINWAND, Simon and RZEPKA, Beata. Measures of noncompactness in Köthe spaces. Topological Methods in Nonlinear Analysis. Online. 14 June 2025. pp. 1 - 19. [Accessed 5 July 2025]. DOI 10.12775/TMNA.2024.045.
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