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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.

#### Abstract

Banach-Saks type is calculated for two types of Banach sequence spaces and Gurariǐ modulus of convexity is estimated from above for the spaces of one type among them.

#### 1. Introduction

Recently, there has been a lot of interest in investigating geometric properties of sequence spaces besides topological and some other usual properties. In the literature, there are many papers concerning geometric properties of various Banach sequence spaces. For example, geometry of Orlicz spaces and of Musielak-Orlicz spaces has been studied in [1–25]. Several authors including Cui and Hudzik [26–29], Cui and Meng [30], Suantai [31], and Lee [32] investigated the geometric properties of Cesàro sequence space . Also Cesàro-Orlicz sequence spaces equipped with Luxemburg norm have been studied in [5, 33–36]. Additionally, geometry of Orlicz-Lorentz sequence spaces and of generalized Orlicz-Lorentz sequence space were studied in [37, 38]. Furthermore, Mursaleen et al. [39] studied some geometric properties of normed Euler sequence space. Additionally, Hudzik and Narloch [40] have studied relationships between monotonicity and complex rotundity properties with some consequences. Besides, some geometrical properties of Calderon-Lozanovskii sequence spaces have been investigated in [41–43].

Quite recently, Karakaya [20] defined a new sequence space involving lacunary sequence space equipped with the Luxemburg norm and studied Kadec-Klee and rotundity properties of these spaces. Further information on topological and geometric properties of sequence spaces can be found in [39, 44–62].

Let be a real Banach space and and be the unit sphere and the unit ball of , respectively. Let , , , , and be the spaces of all real sequences, null, convergent, and bounded sequences and absolutely convergent series, respectively, and let be the space of those real sequences which have only a finite number of nonzero coordinates and .

Note that , , and are Banach spaces with the sup-norm and are Banach spaces with the norm , while is not a Banach space with respect to any norm.

Let us recall that a sequence in a Banach space is called of (or for short) if for each there exists a unique sequence of scalars such that , that is, .

A sequence space with a linear topology is called a - if each of the projection maps defined by for is continuous for each natural . A is a complete metric linear space and the metric is generated by an -norm and a Fréchet space which is a -space is called an -; that is, a -space is called an -space if is a complete linear metric space. In other words, is an -space if is a Fréchet space with continuous coordinate projections. All the sequence spaces mentioned above are spaces except the space .

An -space which contains the space is said to have the if, for every sequence , , where . The spaces equipped with the sup-norm equipped with the norm and endowed with the metric have property , while and do not have property . Banach spaces with continuous coordinate projections are called -spaces.

A Banach space is said to be a (see [28, 63]) if is a subspace of such that(i)if , and , for all , then and ;(ii)there exists an element such that for all .

We say that is order continuous if, for any sequence in (the positive cone in ) such that and for each , we have holds.

A Köthe sequence space is said to be order continuous that if all sequences in are order continuous. It is easy to see that is order continuous if and only if as .

A Köthe sequence space is said to have the if, for any real sequence and any in such that coordinatewisely and , we have the fact that and .

A Banach space is said to have the - if every bounded sequence in admits a subsequence such that the sequence is convergent in with respect to the norm, where A Banach space is said to have the whenever given any weakly null sequence in there exists its subsequence such that the sequence converges to zero strongly.

Given any , we say that a Banach space has the if there exists a constant such that every weakly null sequence has a subsequence such that (see [22]) The Banach-Saks property of type and the weak Banach-Saks property for Cesro sequence spaces have been considered in [28]. These properties and stronger property for Musielak-Orlicz and Nakano sequence spaces have been studied in [17].

We say that a Banach space has the if every nonexpansive self-mapping defined on a nonempty weakly compact convex subset of has a fixed point in .

In [64], Garcia-Falset introduced the following coefficient for a Banach space : and he proved (see [64, 65]) that a Banach space with has the weak fixed point property.

Clarkson of a normed space is defined (see Clarkson [66] and Day [67]) by the formula for any . The inequality for all characterizes the uniform convexity of and the equality characterizes strict convexity (=rotundity) of .

The Gurariǐ modulus of convexity of a normed space is defined (see [68, 69]) by for any . It is obvious that for any Banach space and any . It is also known that for any and that is rotund if and only if and as well as that is uniformly convex if and only if for any . Gurariǐ [68] proved that if is renormed by the norm then for any and .

Gurariǐ and Sozonov [70] proved that a normed linear space is an inner product space if and only if, for every Zanco and Zucchi [71] showed an example of a normed space with .

Now, we will define Köthe sequence spaces and that will be considered in this paper.

Let denote the set whose elements are finite sets of distinct positive integers. Given any element of , we denote by the sequence such that for , and otherwise. Further, we define that is, is the set of those whose support has cardinality at most . Let us define where .

Given any , we define the following sequence space, introduced in [55]: Sargent [55] established the relationship of this space to the space and characterized some matrix transformations. In [49], matrix classes have been characterized, where is assumed to be any -space.

Recently in [52], some of the geometric properties of have been investigated. In [61], Tripathy and Sen extended the space to as follows: for and .

It has been proved in [61] that, for , is a Banach space if it is endowed with the norm and that one has the following.(i)If , for all , then . Moreover, .(ii)If , then . Also, for any .(iii) if and only if .

It is easy to see that is a Köthe sequence space, indeed a -space with respect to its natural norm (see [55]). Note that throughout the present paper we will study the space except the case , for which it is reduced to the space .

Now we will introduce the space . Let and let be arbitrary real sequences with all coordinates and different from zero and let, for any , It is obvious that this is a linear space. It is known (see [48]) that the functional is a norm in and that the couple is a Banach space. The space is a generalization of three spaces. Namely, one has the following.(i)If and , then is the Cesáro sequence space of nonabsolute type (see [32]) and .(ii)Let be a real sequence with and for all . If and , where for any . Then we obtain that is the Riesz sequence space of nonabsolute type denoted by and that (see [44]).(iii)Let be a real sequence with and , for all , . If and , where for any . Then reduces to the Nörlund sequence space of nonabsolute type denoted by and (see [60]).

#### 2. Banach-Saks Type of Sequence Space

In this section, we investigate some properties of the space such as the Fatou property, the Banach-Saks property of type , and the weak fixed point property. Let us start with the following lemma.

Lemma 1. *If a -space containing has the , then it is order continuous, that is, as for any .*

*Proof. *From the definition of property , we have that every has the unique representation as . Hence as , that is, as , which means that is order continuous.

Corollary 2. *The space is order continuous.*

*Proof. *It is easy to see that contains and that every has the unique representation as , which means that has the property . Hence, from the above lemma, is order continuous.

Theorem 3. *The space has the Fatou property.*

*Proof. *Let x be any real sequence from and be any non-decreasing sequence of non-negative elements from such that as coordinatewisely and .

Let us denote . Then, since the supremum is homogeneous, we have
Moreover, by the assumptions that is non-decreasing and convergent to coordinatewisely and by the Beppo-Levi theorem, we have
whence
Therefore, . On the other hand, since for any natural number and the sequence is non-decreasing, we obtain that the sequence is bounded from above by . In consequence , which together with the opposite inequality proved already, yields that .

Theorem 4. *The space has the Banach-Saks property of the type .*

*Proof. *Let be a sequence of positive numbers with . Let be a weakly null sequence in . Let us set and . Then, there exists such that
where consist of the elements of which exceed . Since implies that coordinatewise, there is such that
for all . Set . Then there exists such that
where consist of all elements of which exceed . Using again the fact that coordinatewise, there exists such that
for all .

Continuing this process, we can find two increasing sequences and such that
where and consist of the elements of which exceed . Since , we have
for all and . Hence
By using the norm of the space , we have
Therefore,
This completes the proof of the theorem.

Theorem 5. *For , the space has the weak fixed point property, if , where .*

*Proof. *If , for all , it follows that
Hence
where stands for the Garcia-Falset coefficient of . Therefore, has in this case the weak fixed point property.

#### 3. Banach-Saks Type and Gurariǐ Modulus of Sequence Spaces

Theorem 6. *The space has the Banach-Saks property of the type .*

*Proof. *Let be a sequence of positive numbers for which . Let be a weakly null sequence in . Set and . Then there exists such that
Since the fact that is a weakly null sequence implies that , coordinatewise, there is an such that
for all . Set . Then there exists an such that
By using the fact that coordinatewise, there exists an such that
for all . Continuing this process, we can find by induction two increasing subsequences and of natural numbers such that
for all and
where . Hence,
On the other hand, since , it can be easily seen that . Therefore, and
Hence we obtain
By using the inequality for all and , we have
Therefore, the space has the Banach-Saks type , which completes the proof of the theorem.

Let us define the matrix by for all , where depends only on and depends only on . The matrix is called generalized weighted mean or factorable matrix. By , we denote the inverse of the matrix as follows:

Theorem 7. *For , by (39), one has the fact that the Gurariǐ modulus of convexity for the normed space satisfies the inequality
**
for any .*

*Proof. *Let . By using (39), we have
Let . Then using (40), let us consider the following sequences:
Since and , we have
By using the sequences given above, we obtain the following equalities:
To complete the upper estimate of the Gurariǐ modulus of convexity, it remains to calculate the infimum of for . We have
Consequently, we get for the inequality
which is the desired result.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### 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. - 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. - 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. - 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. - 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 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. - 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 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. - 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 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. - 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. - A. Kamińska, “Uniform rotundity in every direction of sequence Orlicz spaces,”
*Bulletin de L'Academie Polonaise des Sciences*, vol. 32, pp. 589–594, 1984. - A. Kamińska, “Rotundity of sequence Musielak-Orlicz spaces,”
*Bulletin de L'Academie Polonaise des Sciences*, vol. 29, pp. 137–144, 1981. - 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. - W. Kurc, “Strictly and uniformly monotone sequential Musielak-Orlicz spaces,”
*Collectanea Mathematica*, vol. 50, no. 1, pp. 1–17, 1999. 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. - 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. - 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. - S. Suantai, “In the H-property of some Banach sequence spaces,”
*Archivum Mathematicum*, vol. 39, pp. 309–316, 2003. - P. N. Ng and P. Y. Lee, “Cesàro sequences spaces of non-absolute type,”
*Commentationes Mathematicae*, vol. 20, no. 2, pp. 429–433, 1978. - 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 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 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 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. - 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 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. - 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. - 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. - 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. - C. S. Wang, “On Nörlund sequence space,”
*Tamkang Journal of Mathematics*, vol. 9, pp. 269–274, 1978. - 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. - J. Y. T. Woo, “On Modular Sequence spaces,”
*Studia Mathematica*, vol. 48, pp. 271–289, 1973. - J. Diestel, “Sequence and series in Banach spaces,” in
*Graduate Texts in Math*, vol. 92, Springer, New York, NY, USA, 1984. - J. Garcia-Falset, “Stability and fixed points for nonexpansive mappings,”
*Houston Journal of Mathematics*, vol. 20, pp. 495–505, 1994. - 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. - V. I. Gurariǐ, “On moduli of convexity and attering of Banach spaces,”
*Soviet Mathematics Doklady*, vol. 161, no. 5, pp. 1003–1006, 1965. - V. I. Gurariǐ, “On differential properties of the convexity moduli of Banach spaces,”
*Matematicheskie Issledovaniya*, vol. 2, pp. 141–148, 1969. - 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). - 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.