Abstract
It is well-known that the Kadec-Klee property is an important property in the geometry of Banach spaces. It is closely connected with the approximation compactness and fixed point property of non-expansive mappings. In this paper, a criterion for Musielak-Orlicz function spaces equipped with the Orlicz norm to have the Kadec-Klee property are given. As a corollary, we obtain that a class of non-reflexive Musielak-Orlicz function spaces have the Fixed Point property.
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07 July 2021
A Correction to this paper has been published: https://doi.org/10.1007/s00010-021-00830-w
References
Wang, T.F., Cui, Y.A., Zhang, T.: Kadec-Klee property in Musielak-Orlicz function spaces equipped with the Luxemburg norm. Sci. Math. 24(3), 339–345 (1998)
Goebel, K., Kirk, W.A.: Topics on metric fixed point theory. Cambridge Univeresity, Cambridge (1990)
Saint Raymond, J.: Kadec-Klee property and fixed point. J. Funct. Anal. 266(8), 5429–5438 (2014)
Saint Raymond, J.: Dual Kadec-Klee property and fixed point. J. Funct. Anal. 272(9), 3825–3844 (2017)
Ciesielski, M., Kolwicz, P., Płuciennik, R.: Local approach to Kadec-Klee properties in symmetric function spaces. J. Math. Anal. Appl. 426(2), 700–726 (2015)
Lukeš, J., Pick, L., Pokorný, D.: On geometric properties of the spaces \(L^{p(x)}\). Revista Matematica Complutense. 24(1), 115–130 (2011)
Khamsi, M.A.: On uniform opial condition and uniform Kadec-Klee property in Banach and metric spaces. Nonlinear Anal. 26(10), 1733–1748 (1996)
Musielak, J.: Orlicz spaces and modular spaces. Springer, Berlin (1983)
Cui, Y.A., Hudzik, H., Ma, H.F.: Geometry of Orlicz spaces equipped with norms generated by some lattice norms in \(R^{2}\). Comptes Rendus de la Academie des Sci. 113(3), 2745–2762 (2019)
Chen, S.T.: Geometry of Orlicz spaces. Diss. Math. 365, 1–204 (1996)
Hanebaly, E.: The fixed point property in Banach spaces via the strict convexity and the Kadec-Klee property[J]. J. Fixed Point Theor. Appl. 22(2), 1–15 (2020)
Hudzik, H., Maligranda, L.: Amemiya norm equals Orlicz norm in general. Indagationes Math. 11(4), 573–585 (2000)
Medzhitov, A., Sukochev, P.: The property \((H)\) in Orlicz spaces. Bull. Polish Acad. Sci. Math. 40(1), 5–11 (1992)
Sims, B.: Examples of fixed point free mappings. In: Kirk, W., Sims, B. (eds.) Handbook of metric fixed point theory, pp. 35–48. Springer-Verlag, Berlin (2001)
Sargsyan, A., Grigoryan, M.: Universal function for a weighted space \(L_{\mu }^{1} [0,1]\)[J]. Positivity 21(4), 1457–1482 (2017)
Lin, P.K.: There is an equivalent norm on \(l_{1}\) that has the fixed point property. Nonlinear Anal. 68(8), 2303–2308 (2008)
Benavides, T.D.: Some questions in metric fixed point theory, by A. W. Kirk, revisited[J]. Arab. J. Math. 1(4), 431–438 (2012)
Hernández-Linares, C.A.H., Japón, M.A., Llorens-Fuster, E.: On the structure of the set of equivalent norms on \(l_{1}\) with the fixed point property. J. Math. Anal. Appl. 387(2), 645–654 (2012)
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This work was supported by the Natural Science Foundation of China(11871181).
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To my supervisor, Henryk Hudzik.
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Cui, Y., Zhao, L. Kadec-Klee property in Musielak-Orlicz function spaces equipped with the Orlicz norm. Aequat. Math. 96, 167–184 (2022). https://doi.org/10.1007/s00010-021-00808-8
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DOI: https://doi.org/10.1007/s00010-021-00808-8
Keywords
- Musielak-Orlicz function spaces
- Orlicz norm
- Kadec-Klee property
- Weak star Kadec-Klee property
- Nonexpansive mapping
- Fixed point