- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2014 (2014), Article ID 427382, 9 pages

http://dx.doi.org/10.1155/2014/427382

## Banach-Saks Type and Gurariǐ Modulus of Convexity of Some Banach Sequence Spaces

^{1}Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland^{2}Department of Mathematical Engineering, Yıldız Technical University, Davutpasa Campus, Esenler, 34750 Istanbul, Turkey^{3}Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India^{4}Department of Mathematics, Istanbul Ticaret University, Üsküdar, Istanbul, Turkey

Received 5 August 2013; Accepted 4 October 2013; Published 26 March 2014

Academic Editor: Marek Wisla

Copyright © 2014 Henryk Hudzik et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- G. Alherk and H. Hudzik, “Uniformly non-${l}_{n}^{\left(1\right)}$ Musielak-Orlicz spaces of Bochner type,”
*Forum Mathematicum*, vol. 1, no. 4, pp. 403–410, 1989. View at Google Scholar - G. Alherk and H. Hudzik, “Copies of ${l}^{1}$ and ${c}_{0}$ in Musielak-Orlicz sequence spaces,”
*Commentationes Mathematicae Universitatis*, vol. 35, no. 1, pp. 9–19, 1994. View at Google Scholar - L. X. Bo, Y. A. Cui, P. Foralewski, and H. Hudzik, “Local uniform rotundity and weak local uniform rotundity of Musielak-Orlicz sequence spaces endowed with the Orlicz norm,”
*Nonlinear Analysis, Theory, Methods and Applications*, vol. 69, no. 5-6, pp. 1559–1569, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - Y. A. Cui, H. Hudzik, and R. Płuciennik, “Banach-Saks property in some Banach sequence spaces,”
*Annales Polonici Mathematici*, vol. 65, no. 2, pp. 193–202, 1997. View at Google Scholar - Y. A. Cui, H. Hudzik, N. Petrot, S. Suantai, and A. Szymaszkiewicz, “Basic topological and geometric properties of Cesàro-Orlicz spaces,”
*Proceedings of the Indian Academy of Sciences*, vol. 115, no. 4, pp. 461–476, 2005. View at Google Scholar - Y. A. Cui, H. Hudzik, M. Wisła, and M. Zou, “Extreme points and strong U-points in Musielak-Orlicz sequence spaces equipped with the Orlicz norm,”
*Zeitschrift fur Analysis und ihre Anwendung*, vol. 26, no. 1, pp. 87–101, 2007. View at Google Scholar · View at Zentralblatt MATH · View at Scopus - M. Denker and H. Hudzik, “Uniformly non-${l}_{n}^{\left(1\right)}$ Musielak-Orlicz sequence spaces,”
*Proceedings of the Indian Academy of Sciences*, vol. 101, no. 2, pp. 71–86, 1991. View at Publisher · View at Google Scholar · View at Scopus - F. Fuentes and F. L. Hernandez, “On weighted Orlicz sequence spaces and their subspaces,”
*Rocky Mountain Journal of Mathematics*, vol. 18, pp. 585–599, 1988. View at Publisher · View at Google Scholar - H. Hudzik, “Uniformly non-${l}_{n}^{\left(1\right)}$ Orlicz spaces with Luxemburg norm,”
*Studia Mathematica*, vol. 81, no. 3, pp. 271–284, 1985. View at Google Scholar - H. Hudzik and D. Pallaschke, “On some convexity properties of Orlicz sequence spaces equipped with the Luxemburg norm,”
*Mathematische Nachrichten*, vol. 186, pp. 167–185, 1997. View at Google Scholar · View at Zentralblatt MATH · View at Scopus - H. Hudzik and Y. N. Ye, “Support functionals and smoothness in Musielak-Orlicz sequence spaces endowed with the Luxemburg norm,”
*Commentationes Mathematicae Universitatis Carolinae*, vol. 031, no. 4, pp. 661–684, 1990. View at Google Scholar - H. Hudzik and L. Szymaszkiewicz, “Basic topological and geometric properties of orlicz spaces over an arbitrary set of atoms,”
*Zeitschrift fur Analysis und ihre Anwendung*, vol. 27, no. 4, pp. 425–449, 2008. View at Google Scholar · View at Scopus - J. E. Jamison, A. Kamińska, and G. Lewicki, “One-complemented subspaces of Musielak-Orlicz sequence spaces,”
*Journal of Approximation Theory*, vol. 130, no. 1, pp. 1–37, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - A. Kamińska, “Strict convexity of sequence Orlicz-Musielak spaces with Orlicz norm,”
*Journal of Functional Analysis*, vol. 50, no. 3, pp. 285–305, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - A. Kamińska, “Uniform rotundity of Musielak-Orlicz sequence spaces,”
*Journal of Approximation Theory*, vol. 47, no. 4, pp. 302–322, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - A. Kamińska, “Flat Orlicz-Musielak sequence spaces,”
*Bulletin de L'Academie Polonaise des Sciences*, vol. 29, pp. 137–144, 1981. View at Google Scholar - A. Kamińska and L. Han Ju, “Banach-Saks properties of Musielak-Orlicz and Nakano sequence spaces,”
*Proceedings of the American Mathematical Society*, vol. 142, pp. 547–558, 2014. View at Google Scholar - A. Kamińska, “Uniform rotundity in every direction of sequence Orlicz spaces,”
*Bulletin de L'Academie Polonaise des Sciences*, vol. 32, pp. 589–594, 1984. View at Google Scholar - A. Kamińska, “Rotundity of sequence Musielak-Orlicz spaces,”
*Bulletin de L'Academie Polonaise des Sciences*, vol. 29, pp. 137–144, 1981. View at Google Scholar - V. Karakaya, “Some geometric properties of sequence spaces involving lacunary sequence,”
*Journal of Inequalities and Applications*, vol. 2007, Article ID 81028, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - E. Katirtzoglou, “Type and Cotype of Musielak-Orlicz Sequence Spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 226, no. 2, pp. 431–455, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - H. Knaust, “Orlicz sequence spaces of Banach-Saks type,”
*Archiv der Mathematik*, vol. 59, no. 6, pp. 562–565, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - P. Kolwicz, “Strictly and uniformly monotone sequential Musielak-Orlicz spaces, Fourth International Conference on Function Spaces (Zielona Gora, 1995),”
*Collectanea Mathematica*, vol. 50, no. 1, pp. 1–17, 1999. View at Google Scholar - W. Kurc, “Strictly and uniformly monotone sequential Musielak-Orlicz spaces,”
*Collectanea Mathematica*, vol. 50, no. 1, pp. 1–17, 1999. View at Google Scholar · View at Scopus - V. Peirats and C. Ruiz, “On ${l}_{p}$-copies in Musielak-Orlicz sequence spaces,”
*Archiv der Mathematik*, vol. 58, no. 2, pp. 164–173, 1992. View at Publisher · View at Google Scholar · View at Scopus - Y. Cui and H. Hudzik, “Packing constant for Cesàro sequence spaces, proceedings of the
third world congress of nonlinear analysis, part 4 (Catania, 2000),”
*Nonlinear Analysis, Theory, Methods and Applications*, vol. 47, no. 4, pp. 2695–2702, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - Y. A. Cui and H. Hudzik, “Some geometric properties related to fixed point theory in Cesàro spaces,”
*Collectanea Mathematica*, vol. 50, no. 3, pp. 277–288, 1999. View at Google Scholar - Y. A. Cui and H. Hudzik, “On the Banach-Saks and weak Banach-Saks properties of some Banach sequence spaces,”
*Acta Scientiarum Mathematicarum*, vol. 65, no. 1-2, pp. 179–187, 1999. View at Google Scholar - Y. A. Cui and H. Hudzik, “Packing constant for Cesàro sequence spaces,”
*Nonlinear Analysis, Theory, Methods and Applications*, vol. 47, no. 4, pp. 2695–2702, 2001. View at Publisher · View at Google Scholar · View at Scopus - Y. A. Cui and C. Meng, “Banach-Saks property and property (
*β*) in Cesàro sequence spaces,”*Southeast Asian Bulletin of Mathematics*, vol. 24, pp. 201–210, 2000. View at Google Scholar - S. Suantai, “In the H-property of some Banach sequence spaces,”
*Archivum Mathematicum*, vol. 39, pp. 309–316, 2003. View at Google Scholar - P. N. Ng and P. Y. Lee, “Cesàro sequences spaces of non-absolute type,”
*Commentationes Mathematicae*, vol. 20, no. 2, pp. 429–433, 1978. View at Google Scholar - P. Foralewski, H. Hudzik, and A. Szymaszkiewicz, “Local rotundity structure of Cesàro-Orlicz sequence spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 345, no. 1, pp. 410–419, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - P. Foralewski, H. Hudzik, and A. Szymaszkiewicz, “Some remarks on Cesàro-Orlicz sequence spaces,”
*Mathematical Inequalities and Applications*, vol. 13, no. 2, pp. 363–386, 2010. View at Google Scholar · View at Zentralblatt MATH · View at Scopus - A. Kamińska and D. Kubiak, “On isometric copies of ${l}_{\infty}$ and James constants in Cesàro-Orlicz sequence spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 372, no. 2, pp. 574–584, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - D. Kubiak, “A note on Cesàro-Orlicz sequence spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 349, no. 1, pp. 291–296, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - P. Foralewski, H. Hudzik, and L. Szymaszkiewicz, “Local rotundity structure of generalized Orlicz-Lorentz sequence spaces,”
*Nonlinear Analysis, Theory, Methods and Applications*, vol. 68, no. 9, pp. 2709–2718, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - P. Foralewski, H. Hudzik, and P. Kolwicz, “Non-squareness properties of Orlicz-Lorentz sequence spaces,”
*Journal of Functional Analysis*, vol. 264, no. 2, pp. 605–629, 2013. View at Publisher · View at Google Scholar - M. Mursaleen, F. Başar, and B. Altay, “On the Euler sequence spaces which include the spaces
*ℓ*p and*ℓ*∞ II,”*Nonlinear Analysis, Theory, Methods and Applications*, vol. 65, no. 3, pp. 707–717, 2006. View at Publisher · View at Google Scholar · View at Scopus - H. Hudzik and A. Narloch, “Relationships between monotonicity and complex rotundity properties with some consequences,”
*Mathematica Scandinavica*, vol. 96, no. 2, pp. 289–306, 2005. View at Google Scholar · View at Zentralblatt MATH · View at Scopus - P. Kolwicz and R. Płuciennik, “Points of upper local uniform monotonicity in Calderón-Lozanovskii spaces,”
*Journal of Convex Analysis*, vol. 17, no. 1, pp. 111–130, 2010. View at Google Scholar · View at Zentralblatt MATH · View at Scopus - P. Kolwicz and K. Leśnik, “Property
*β*of Rolewicz and orthogonal convexities of Calderón-Lozanovskiǐ spaces,”*Nonlinear Analysis, Theory, Methods and Applications*, vol. 74, no. 13, pp. 4352–4368, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - P. Kolwicz, “Kadec-Klee properties of Calderón-Lozanovskiǐ sequence spaces,”
*Collectanea Mathematica*, vol. 63, no. 1, pp. 45–58, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - B. Altay and F. Başar, “On the paranormed Riesz sequence spaces of non-absolute type,”
*Southeast Asian Bulletin of Mathematics*, vol. 26, pp. 701–715, 2002. View at Google Scholar - B. Altay and F. Başar, “Generalization of the sequence space $l\left(p\right)$ derived by weighted mean,”
*Journal of Mathematical Analysis and Applications*, vol. 330, no. 1, pp. 174–185, 2007. View at Publisher · View at Google Scholar · View at Scopus - P. Kolwicz, “Property (
*β*) and orthogonal convexities income class of Köthe sequence spaces,”*Publicationes Mathematicae*, vol. 63, no. 4, pp. 587–609, 2003. View at Google Scholar · View at Scopus - H. Hudzik and P. Kolwicz, “On property (
*β*) of Rolewicz in Köthe-Bochner sequence spaces,”*Studia Mathematica*, vol. 162, no. 3, pp. 195–212, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - I. J. Maddox, “Paranormed sequence spaces generated by infinite matrices,”
*Proceedings-Cambridge Philosophical Society*, vol. 64, pp. 335–340, 1968. View at Google Scholar - E. Malkowsky and M. Mursaleen, “Matrix transformations between
*FK*-spaces and the sequence spaces $m\left(\varphi \right)$ and $n\left(\varphi \right)$,”*Journal of Mathematical Analysis and Applications*, vol. 196, no. 2, pp. 659–665, 1995. View at Publisher · View at Google Scholar · View at Scopus - E. Malkowsky and M. Mursaleen, “Compact matrix operators between the spaces $m\left(\varphi \right)$, $n\left(\varphi \right)$ and ${l}_{p}$,”
*Bulletin of the Korean Mathematical Society*, vol. 48, no. 5, pp. 1093–1103, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - E. Malkowsky and E. Savas, “Matrix transformations between sequence spaces of generalized weighted means,”
*Applied Mathematics and Computation*, vol. 147, no. 2, pp. 333–345, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - M. Mursaleen, “Some geometric proprties of a sequence space related to ${l}^{p}$,”
*Bulletin of the Australian Mathematical Society*, vol. 67, no. 2, pp. 343–347, 2003. View at Publisher · View at Google Scholar · View at Scopus - M. Mursaleen,
*Elements of Metric Spaces*, Anamaya, New Delhi, India, 2005. - M. Mursaleen, R. Çolak, and M. Et, “Some geometric inequalities in a new Banach sequence space,”
*Journal of Inequalities and Applications*, vol. 2007, Article ID 86757, 6 pages, 2007. View at Publisher · View at Google Scholar - W. L. C. Sargent, “Some sequence spaces related to the ${l}^{p}$ spaces,”
*Journal of the London Mathematical Society*, vol. 35, pp. 161–171, 1960. View at Google Scholar - S. Simons, “The sequence spaces $l\left({p}_{v}\right)$ and $m\left({p}_{v}\right)$,”
*Proceedings of the London Mathematical Society*, vol. 15, no. 3, pp. 422–436, 1965. View at Google Scholar - N. Şimşek and V. Karakaya, “On some geometrial properties of generalized modular spaces of Cesàro type defined by weighted means,”
*Journal of Inequalities and Applications*, vol. 2009, Article ID 932734, 13 pages, 2009. View at Publisher · View at Google Scholar - E. Savaş, V. Karakaya, and N. Şimşek, “Some $l\left(p\right)$-Type new sequence spaces and their geometric properties,”
*Abstract and Applied Analysis*, vol. 2009, Article ID 696971, 12 pages, 2009. View at Publisher · View at Google Scholar - N. Şimşek, E. Savas, and V. Karakaya, “On geometrical properties of some banach spaces,”
*Applied Mathematics & Information Sciences*, vol. 7, no. 1, pp. 295–300, 2013. View at Google Scholar - C. S. Wang, “On Nörlund sequence space,”
*Tamkang Journal of Mathematics*, vol. 9, pp. 269–274, 1978. View at Google Scholar - B. C. Tripathy and M. Sen, “On a new class of sequences related to the space ${l}^{p}$,”
*Tamkang Journal of Mathematics*, vol. 33, no. 2, pp. 167–171, 2002. View at Google Scholar - J. Y. T. Woo, “On Modular Sequence spaces,”
*Studia Mathematica*, vol. 48, pp. 271–289, 1973. View at Google Scholar - J. Diestel, “Sequence and series in Banach spaces,” in
*Graduate Texts in Math*, vol. 92, Springer, New York, NY, USA, 1984. View at Google Scholar - J. Garcia-Falset, “Stability and fixed points for nonexpansive mappings,”
*Houston Journal of Mathematics*, vol. 20, pp. 495–505, 1994. View at Google Scholar - J. G. Falset, “The fixed point property in Banach spaces with the NUS-property,”
*Journal of Mathematical Analysis and Applications*, vol. 215, no. 2, pp. 532–542, 1997. View at Publisher · View at Google Scholar · View at Scopus - J. A. Clarkson, “Uniformly convex spaces,”
*Transactions of the American Mathematical Society*, vol. 40, pp. 396–414, 1936. View at Publisher · View at Google Scholar - M. M. Day, “Uniform convexity in factor and conjugate spaces,”
*Annals of Mathematics*, vol. 45, no. 2, pp. 375–385, 1944. View at Google Scholar - V. I. Gurariǐ, “On moduli of convexity and attering of Banach spaces,”
*Soviet Mathematics Doklady*, vol. 161, no. 5, pp. 1003–1006, 1965. View at Google Scholar - V. I. Gurariǐ, “On differential properties of the convexity moduli of Banach spaces,”
*Matematicheskie Issledovaniya*, vol. 2, pp. 141–148, 1969. View at Google Scholar - N. I. Gurariǐ and Y. U. Sozonov, “Normed spaces that do not have distortion of the unit sphere,”
*Matematicheskie Zametki*, vol. 7, pp. 307–310, 1970 (Russian). View at Google Scholar - C. Zanco and A. Zucchi, “Moduli of rotundity and smoothness for convex bodies,”
*Bolletino della Unione Matematica Italiana B*, vol. 7, no. 7, pp. 833–855, 1993. View at Google Scholar