Weakly Convergent Sequence Coefficient in Musielak-Orlicz Sequence Spaces

Liying Fan; Yunan Cui; Henryk Hudzik

doi:10.68381/jca16007

Abstract

A formula for the weakly convergent sequence coefficient of the Musielak-Orlicz sequence space

l_\Phi

equipped with the Luxemburg norm is calculated. As a consequence of this result the cofficient for Nakano sequence spaces

l^{(p_i)}

is found. Criteria for weakly uniformly normal structure of Musielak-Orlicz sequence spaces equipped with the Luxemburg are also given.

Author information

Contact details are reproduced from the original publication and may be historical.

Liying Fan

Dept. of Mathematics, Univ. of Science and Technology, Harbin 150080, P. R. China

Frankying@yahoo.com.cn

Yunan Cui

Dept. of Mathematics, Univ. of Science and Technology, Harbin 150080, P. R. China

yunan_cui@yahoo.com.cn

Henryk Hudzik

Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznan, Poland

hudzik@amu.edu.pl

Keywords

Fixed point property
Musielak-Orlicz sequence spaces
weak convergent sequence coefficient

Mathematics Subject Classification

46B20, 47H10

References

J. M. Ayerbe, T. Domíngues Benavides: Connections between some Banach space coefficients concerning normal structure, J. Math. Anal. Appl. 172(1) (1993) 53–61.https://doi.org/10.1006/jmaa.1993.1006 Google Scholar
T. Domíngues Benavides: Weak uniform normal structure in direction sum spaces, Stud. Math. 103(3) (1992) 293–299.https://doi.org/10.4064/sm-103-3-283-290 Google Scholar
W. L. Bynum: Normal structure coefficient for Banach spaces, Pacific J. Math. 86 (1980) 427–436.https://doi.org/10.2140/pjm.1980.86.427 Google Scholar
K. Deimling: Nonlinear Functional Analysis, Springer, Berlin (1985).https://doi.org/10.1007/978-3-662-00547-7 Google Scholar
J. Diestel: Sequences and Series in a Banach Space, Graduate Texts in Math. 92, Springer, New York (1984).https://doi.org/10.1007/978-1-4612-5200-9 Google Scholar
P. Foralewski, H. Hudzik: On some geometrical and topological properties of generalized Calderón-Lozanovskiǐ sequence spaces, Houston J. Math. 25(3) (1999) 523–543.Google Scholar
H. Hudzik: Banach lattices with order isometric copies of ℓ∞, Indag. Math., New Ser. 9(4) (1998) 521–527.https://doi.org/10.1016/S0019-3577(98)80031-1 Google Scholar
H. Hudzik, A. Kamińska: On uniformly convexitiable and B-convex Musielak-Orlicz spaces, Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 25 (1985) 59–75.https://doi.org/10.14492/hokmj/1381757691 Google Scholar
H. Kamińska: Uniform rotundity of Musielak-Orlicz sequence spaces, J. Approximation Theory 47 (1986) 302–322.https://doi.org/10.1016/0021-9045(86)90020-1 Google Scholar
L. V. Kantorovich, G. P. Akilov. Functional Analysis, Nauka, Moscow (1977) (in Russian).Google Scholar
W. A. Kirk: A fixed point theorem for mappings which do not increase distances, Amer. Math. Mon. 72 (1965) 1004–1006.https://doi.org/10.2307/2313345 Google Scholar
W. A. Kirk, B. Sims (eds.): Handbook of Metric Fixed Point Theory, Kluwer, Dordrecht (2001).https://doi.org/10.1007/978-94-017-1748-9 Google Scholar
M. A. Krasnoselskiǐ, Ya. B. Rutitskiǐ: Convex Functions and Orlicz Spaces, Noordhoff, Groningen (1961).Google Scholar
L. Maligranda: Orlicz Spaces and Interpolation, Seminars in Mathematics 5, Departamento de Matemática, Universidade Estadual de Campinas, Campinas (1989).Google Scholar
J. Musielak: Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer, Berlin (1983).https://doi.org/10.1007/BFb0072210 Google Scholar
E. Maluta: Uniformly normal structure and related coefficients, Pacific J. Math. 111 (1984) 357–369.https://doi.org/10.2140/pjm.1984.111.357 Google Scholar
M. M. Rao, Z. D. Ren: Theory of Orlicz Spaces, Marcel Dekker, New York (1991).Google Scholar
G. L. Zhang: Weakly convergent sequence coefficient of product space, Proc. Amer. Math. Soc. 117(3) (1992) 637–643.https://doi.org/10.1090/S0002-9939-1993-1152993-1 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2009. Rights now held by Banach Press LLC.

[1] J. M. Ayerbe, T. Domíngues Benavides: Connections between some Banach space coefficients concerning normal structure, J. Math. Anal. Appl. 172(1) (1993) 53–61.https://doi.org/10.1006/jmaa.1993.1006 Google Scholar

[2] T. Domíngues Benavides: Weak uniform normal structure in direction sum spaces, Stud. Math. 103(3) (1992) 293–299.https://doi.org/10.4064/sm-103-3-283-290 Google Scholar

[3] W. L. Bynum: Normal structure coefficient for Banach spaces, Pacific J. Math. 86 (1980) 427–436.https://doi.org/10.2140/pjm.1980.86.427 Google Scholar

[4] K. Deimling: Nonlinear Functional Analysis, Springer, Berlin (1985).https://doi.org/10.1007/978-3-662-00547-7 Google Scholar

[5] J. Diestel: Sequences and Series in a Banach Space, Graduate Texts in Math. 92, Springer, New York (1984).https://doi.org/10.1007/978-1-4612-5200-9 Google Scholar

[6] P. Foralewski, H. Hudzik: On some geometrical and topological properties of generalized Calderón-Lozanovskiǐ sequence spaces, Houston J. Math. 25(3) (1999) 523–543.Google Scholar

[7] H. Hudzik: Banach lattices with order isometric copies of ℓ∞, Indag. Math., New Ser. 9(4) (1998) 521–527.https://doi.org/10.1016/S0019-3577(98)80031-1 Google Scholar

[8] H. Hudzik, A. Kamińska: On uniformly convexitiable and B-convex Musielak-Orlicz spaces, Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 25 (1985) 59–75.https://doi.org/10.14492/hokmj/1381757691 Google Scholar

[9] H. Kamińska: Uniform rotundity of Musielak-Orlicz sequence spaces, J. Approximation Theory 47 (1986) 302–322.https://doi.org/10.1016/0021-9045(86)90020-1 Google Scholar

[10] L. V. Kantorovich, G. P. Akilov. Functional Analysis, Nauka, Moscow (1977) (in Russian).Google Scholar

[11] W. A. Kirk: A fixed point theorem for mappings which do not increase distances, Amer. Math. Mon. 72 (1965) 1004–1006.https://doi.org/10.2307/2313345 Google Scholar

[12] W. A. Kirk, B. Sims (eds.): Handbook of Metric Fixed Point Theory, Kluwer, Dordrecht (2001).https://doi.org/10.1007/978-94-017-1748-9 Google Scholar

[13] M. A. Krasnoselskiǐ, Ya. B. Rutitskiǐ: Convex Functions and Orlicz Spaces, Noordhoff, Groningen (1961).Google Scholar

[14] L. Maligranda: Orlicz Spaces and Interpolation, Seminars in Mathematics 5, Departamento de Matemática, Universidade Estadual de Campinas, Campinas (1989).Google Scholar

[15] J. Musielak: Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer, Berlin (1983).https://doi.org/10.1007/BFb0072210 Google Scholar

[16] E. Maluta: Uniformly normal structure and related coefficients, Pacific J. Math. 111 (1984) 357–369.https://doi.org/10.2140/pjm.1984.111.357 Google Scholar

[17] M. M. Rao, Z. D. Ren: Theory of Orlicz Spaces, Marcel Dekker, New York (1991).Google Scholar

[18] G. L. Zhang: Weakly convergent sequence coefficient of product space, Proc. Amer. Math. Soc. 117(3) (1992) 637–643.https://doi.org/10.1090/S0002-9939-1993-1152993-1 Google Scholar