- Research
- Open Access
- Published:
Some topological and geometrical properties of new Banach sequence spaces
Journal of Inequalities and Applications volume 2013, Article number: 38 (2013)
Abstract
In the present paper, we introduce a new band matrix and define the sequence space
where is the k th Fibonacci number for every . We also establish some inclusion relations concerning this space and determine its α-, β-, γ-duals. Further, we characterize some matrix classes on the space and examine some geometric properties of this space.
MSC:11B39, 46A45, 46B45, 46B20.
1 Introduction
Let ω be the space of all real-valued sequences. Any vector subspace of ω is called a sequence space. By , c, and (), we denote the sets of all bounded, convergent, null sequences and p-absolutely convergent series, respectively. Also, we use the conventions that and is the sequence whose only non-zero term is 1 in the n th place for each , where .
Let X and Y be two sequence spaces and be an infinite matrix of real numbers , where . We write instead of . Then we say that A defines a matrix mapping from X into Y and we denote it by writing if for every sequence , the sequence , the A-transform of x, is in Y, where
For simplicity in notation, here and in what follows, the summation without limits runs from 0 to ∞. Also, if , then we write instead of .
By , we denote the class of all matrices A such that . Thus, if and only if the series on the right-hand side of (1.1) converges for each and every and we have for all .
The matrix domain of an infinite matrix A in a sequence space X is defined by
which is a sequence space.
The approach constructing a new sequence space by means of the matrix domain of a particular limitation method has recently been employed by several authors; see, for instance, [1–12].
Let Δ denote the matrix defined by
or
In the literature, the matrix domain is called the difference sequence space whenever λ is a normed or paranormed sequence space. The idea of difference sequence spaces was introduced by Kızmaz [13]. In 1981, Kızmaz [13] defined the sequence spaces
for , c and . The difference space , consisting of all sequences such that is in the sequence space , was studied in the case by Altay and Başar [14] and in the case by Başar and Altay [4] and Çolak et al. [15]. The paranormed difference sequence space
was examined by Ahmad and Mursaleen [16] and Malkowsky [17], where is any of the paranormed spaces , and defined by Simons [18] and Maddox [19].
Recently, Başar et al. [20] have defined the sequence spaces and by
and
where is an arbitrary fixed sequence and for all . These spaces are generalization of the space for . Quite recently, Kirişçi and Başar [21] have introduced and studied the generalized difference sequence spaces
for , , c and , where and (). Following Kirişçi and Başar [21], Sönmez [22] has examined the sequence space as the set of all sequences whose -transforms are in the space , where denotes the triple band matrix defined by
for all and . Also in [23–34], the authors studied some difference sequence spaces.
In this paper, we define the Fibonacci difference matrix by using the Fibonacci sequence and introduce new sequence spaces and related to the matrix domain of in the sequence spaces and , respectively, where . This study is organized as follows.
In Section 2, we give some notations and basic concepts including the Fibonacci sequence and a BK-space. In Section 3, we define a new band matrix with Fibonacci numbers and introduce the sequence spaces and . Also, we establish some inclusion relations concerning these spaces and construct the basis of the space for . In Section 4, we determine the α-, β-, γ-duals of the spaces and . In Section 5, we characterize the classes and , where and X is any of the spaces , , c and . In the final section of the paper, we investigate some geometric properties of the space for .
2 The Fibonacci difference sequence space
Define the sequence of Fibonacci numbers given by the linear recurrence relations
Fibonacci numbers have many interesting properties and applications in arts, sciences and architecture. For example, the ratio sequences of Fibonacci numbers converges to the golden ratio which is important in sciences and arts. Also, some basic properties of Fibonacci numbers [35] are given as follows:

Substituting for in Cassini’s formula yields .
A sequence space X is called a FK-space if it is a complete linear metric space with continuous coordinates (), where ℝ denotes the real field and for all and every . A BK space is a normed FK space, that is, a BK-space is a Banach space with continuous coordinates. The space () is a BK-space with and , c and are BK-spaces with .
A sequence in a normed space X is called a Schauder basis for X if for every , there is a unique sequence of scalars such that , i.e., .
The α-, β- and γ-duals of the sequence space X are respectively defined by

and
where cs and bs are the sequence spaces of all convergent and bounded series, respectively [36].
We assume throughout that with and denote the collection of all finite subsets of ℕ by ℱ.
3 The Fibonacci difference sequence spaces and
In this section, we define the Fibonacci band matrix and introduce the sequence spaces and , where . Also, we present some inclusion theorems and construct the Schauder basis of the space for.
Let be the n th Fibonacci number for every . Then we define the infinite matrix by
Now, we introduce the Fibonacci difference sequence spaces and as the set of all sequences such that their -transforms are in the space and , respectively, i.e.,

and
With the notation of (1.2), the sequence spaces and may be redefined by
Define the sequence , which will be frequently used, by the -transform of a sequence , i.e.,
Now, we may begin with the following theorem which is essential in the text.
Theorem 3.1 Let . Then is a BK-space with the norm , that is,
and
Proof Since (3.1) holds, and are BK-spaces with respect to their natural norms and the matrix is a triangle; Theorem 4.3.12 of Wilansky [[37], p.63] gives the fact that the spaces and are BK-spaces with the given norms, where . This completes the proof. □
Remark 3.2 One can easily check that the absolute property does not hold on the spaces and , that is, and for at least one sequence in the spaces and , and this shows that and are the sequence spaces of non-absolute type, where and .
Theorem 3.3 The Fibonacci difference sequence space of non-absolute type is linearly isomorphic to the space , that is, for .
Proof To prove this, we should show the existence of a linear bijection between the spaces and for . Consider the transformation T defined, with the notation of (3.2), from to by . Then for every . Also, the linearity of T is clear. Further, it is trivial that whenever and hence T is injective.
Furthermore, let for and define the sequence by
Then, in the cases and , we get

and
respectively. Thus, we have (). Hence, T is surjective and norm preserving. Consequently, T is a linear bijection which shows that the spaces and are linearly isomorphic for . This concludes the proof. □
Now, we give some inclusion relations concerning the space .
Theorem 3.4 The inclusion strictly holds for .
Proof To prove the validity of the inclusion for , it suffices to show the existence of a number such that for every .
Let and . Since the inequalities and hold for every , we obtain with the notation of (3.2),
and
which together yield, as expected,

for . Further, since the sequence is in , the inclusion is strict for . Similarly, one can easily prove that inequality (3.4) also holds in the case , and so we omit the details. This completes the proof. □
Theorem 3.5 Neither of the spaces and includes the other one, where .
Proof Let and . Then, since and , we conclude that x is in but not in . Now, consider the equation
Then whenever k is odd, which implies that the series is not convergent, where . Thus, is not in for . Additionally, since , the sequence e is in . Hence, the sequence spaces and overlap but neither contains the other, as asserted. □
Theorem 3.6 If , then .
Proof Let and . Then we obtain from Theorem 3.1 that , where y is the sequence given by (3.2). Thus, the well-known inclusion yields . This means that and hence, the inclusion holds. This completes the proof. □
Now, we give a sequence of the points of the space which forms a basis for the space ().
Theorem 3.7 Let and define the sequence for every fixed by
Then the sequence is a basis for the space , and every has a unique representation of the form
Proof Let . Then it is obvious by (3.5) that () and hence for all .
Further, let be given. For every non-negative integer m, we put
Then we have that
and hence
Now, for any given , there is a non-negative integer such that
Therefore, we have for every that
which shows that and hence x is represented as in (3.6).
Finally, let us show the uniqueness of the representation (3.6) of . For this, suppose that . Since the linear transformation T defined from to in the proof of Theorem 3.3 is continuous, we have
Hence, the representation (3.6) of is unique. This concludes the proof. □
4 The α-, β- and γ-duals of the space
In this section, we determine the α-, β- and γ-duals of the sequence space of non-absolute type. Since the case can be proved by analogy, we omit the proof of that case and consider only the case in the proof of Theorems 4.5 and 4.6, respectively.
The following known results [38] are fundamental for our investigation.
Lemma 4.1 if and only if
Lemma 4.2 if and only if


Lemma 4.3 if and only if (4.1) holds and
Lemma 4.4 if and only if (4.2) holds with .
Theorem 4.5 The α-dual of the space is the set
where .
Proof Let . For any fixed sequence , we define the matrix by
for all . Also, for every , we put . Then it follows by (3.2) that
Thus, we observe by (4.4) that whenever if and only if whenever . Therefore, we derive by using Lemma 4.1 that
which implies that . □
Theorem 4.6 Define the sets , and by

and
Then and , where .
Proof Let and consider the equality
where is defined by
Then we deduce from Lemma 4.2 with (3.2) that whenever if and only if whenever . Thus, if and only if and by (4.1) and (4.2), respectively. Hence, . It is clear that one can also prove the case by the technique used in the proof of the case with Lemma 4.3 instead of Lemma 4.2. So, we leave the detailed proof to the reader. □
Theorem 4.7 , where .
Proof This result can be obtained from Lemma 4.4 by using (4.5). □
5 Some matrix transformations related to the sequence space
In this section, we characterize the classes , where and X is any of the spaces , , c and .
For simplicity in notation, we write
for all .
The following lemma is essential for our results.
Lemma 5.1 (see [[21], Theorem 4.1])
Let λ be an FK-space, U be a triangle, V be its inverse and μ be an arbitrary subset of ω. Then we have if and only if
and
where
and for all .
Now, we list the following conditions:












Then, by combining Lemma 5.1 with the results in [38], we immediately derive the following results.
Theorem 5.2
-
(a)
if and only (5.2), (5.9) and (5.10) hold.
-
(b)
if and only if (5.2), (5.6), (5.9) and (5.10) hold.
-
(c)
if and only if (5.2), (5.6) with , (5.9) and (5.10) hold.
-
(d)
if and only (5.2), (5.10) and (5.11) hold.
Theorem 5.3 Let . Then we have
-
(a)
if and only if (5.1), (5.2) and (5.4) hold.
-
(b)
if and only if (5.1), (5.2), (5.4) and (5.6) hold.
-
(c)
if and only if (5.1), (5.2), (5.4) and (5.6) with hold.
-
(d)
if and only if (5.1), (5.2) and (5.5) hold.
Theorem 5.4
-
(a)
if and only (5.2), (5.3) and (5.4) with hold.
-
(b)
if and only (5.2), (5.3), (5.6) and (5.7) hold.
-
(c)
if and only (5.2), (5.3) and (5.8) hold.
-
(d)
if and only (5.2), (5.3) and (5.12) hold.
6 Some geometric properties of the space ()
In this section, we study some geometric properties of the space for .
For these properties, we refer to [3, 39–47].
A Banach space X is said to have the Banach-Saks property if every bounded sequence in X admits a subsequence such that the sequence is convergent in the norm in X [40], where
A Banach space X is said to have the weak Banach-Saks property whenever, given any weakly null sequence , there exists a subsequence of such that the sequence is strongly convergent to zero.
In [43], García-Falset introduces the following coefficient:
where denotes the unit ball of X.
Remark 6.1 A Banach space X with has the weak fixed point property [44].
Let . A Banach space is said to have the Banach-Saks type p or the property if every weakly null sequence has a subsequence such that for some ,
for all ( see [45]).
Now, we may give the following results related to some geometric properties, mentioned above, of the space , where .
Theorem 6.2 Let . Then the space has the Banach-Saks type p.
Proof Let be a sequence of positive numbers for which , and also let be a weakly null sequence in . Set and . Then there exists such that
Since being a weakly null sequence implies coordinatewise, there is an such that
when . Set . Then there exists an such that
Again using the fact that coordinatewise, there exists an such that
when .
If we continue this process, we can find two increasing subsequences and such that
for each and
where . Hence,
On the other hand, it can be seen that . Therefore, we have that
Hence, we obtain
By using the fact that for all and , we have
Hence, has the Banach-Saks type p. This concludes the proof. □
Remark 6.3 Note that since is linearly isomorphic to .
Hence, by Remarks 6.1 and 6.3, we have the following theorem.
Theorem 6.4 The space has the weak fixed point property, where .
References
Aydın C, Başar F:Some new sequence spaces which include the spaces and . Demonstr. Math. 2005, 38(3):641–656.
Mursaleen M, Başar F, Altay B:On the Euler sequence spaces which include the spaces and II. Nonlinear Anal. TMA 2006, 65(3):707–717. 10.1016/j.na.2005.09.038
Savaş E, Karakaya V, Şimşek N:Some -type new sequence spaces and their geometric properties. Abstr. Appl. Anal. 2009., 2009: Article ID 696971
Başar F, Altay B: On the space of sequences of p -bounded variation and related matrix mappings. Ukr. Math. J. 2003, 55: 136–147. 10.1023/A:1025080820961
Altay B, Başar F:Generalization of the sequence space derived by weighted mean. J. Math. Anal. Appl. 2007, 330: 174–185. 10.1016/j.jmaa.2006.07.050
Mursaleen M, Noman AK:On some new sequence spaces of non-absolute type related to the spaces and I. Filomat 2011, 25(2):33–51. 10.2298/FIL1102033M
Aydın C, Başar F:Some generalizations of the sequence spaces . Iran. J. Sci. Technol., Trans. A, Sci. 2006, 30(A2):175–190.
Savaş E: Matrix transformations of some generalized sequence spaces. J. Orissa Math. Soc. 1985, 4(1):37–51.
Savaş E: Matrix transformations and absolute almost convergence. Bull. Inst. Math. Acad. Sin. 1987, 15(3):345–355.
Savaş E: Matrix transformations between some new sequence spaces. Tamkang J. Math. 1988, 19(4):75–80.
Savaş E: Matrix transformations and almost convergence. Math. Stud. 1991, 59(1–4):170–176.
Savaş E, Mursaleen M: Matrix transformations in some sequence spaces. Istanb. Üniv. Fen Fak. Mat. Derg. 1993, 52: 1–5.
Kızmaz H: On certain sequence spaces. Can. Math. Bull. 1981, 24(2):169–176. 10.4153/CMB-1981-027-5
Altay B, Başar F:The matrix domain and the fine spectrum of the difference operator Δ on the sequence space , (). Commun. Math. Anal. 2007, 2(2):1–11.
Çolak R, Et M, Malkowsky E: Some Topics of Sequence Spaces. Fırat Univ. Press, Elazığ; 2004:1–63. ISBN:975–394–0386–6
Ahmad ZU, Mursaleen M: Köthe-Toeplitz duals of some new sequence spaces and their matrix maps. Publ. Inst. Math. (Belgr.) 1987, 42: 57–61.
Malkowsky E: Absolute and ordinary Köthe-Toeplitz duals of some sets of sequences and matrix transformations. Publ. Inst. Math. (Belgr.) 1989, 46(60):97–103.
Simons S:The sequence spaces and . Proc. Lond. Math. Soc. 1965, 3(15):422–436.
Maddox IJ: Continuous and Köthe-Toeplitz duals of certain sequence spaces. Proc. Camb. Philos. Soc. 1965, 65: 431–435.
Altay B, Başar F, Mursaleen M: Some generalizations of the space of p -bounded variation sequences. Nonlinear Anal. TMA 2008, 68: 273–287. 10.1016/j.na.2006.10.047
Kirişçi M, Başar F: Some new sequence spaces derived by the domain of generalized difference matrix. Comput. Math. Appl. 2010, 60: 1299–1309. 10.1016/j.camwa.2010.06.010
Sönmez A: Some new sequence spaces derived by the domain of the triple band matrix. Comput. Math. Appl. 2011, 62(2):641–650. 10.1016/j.camwa.2011.05.045
Choudhary B, Mishra SK: A note on Köthe-Toeplitz duals of certain sequence spaces and their matrix transformations. Int. J. Math. Math. Sci. 1995, 18(4):681–688. 10.1155/S0161171295000871
Sarıgöl MA: On difference sequence spaces. J. Karadeniz Tech. Univ., Fac. Arts Sci., Ser. Math.-Phys. 1987, 10: 63–71.
Et M: On some difference sequence spaces. Turk. J. Math. 1993, 17: 18–24.
Mursaleen M: Generalized spaces of difference sequences. J. Math. Anal. Appl. 1996, 203(3):738–745. 10.1006/jmaa.1996.0409
Mishra SK: Matrix maps involving certain sequence spaces. Indian J. Pure Appl. Math. 1993, 24(2):125–132.
Gaur AK, Mursaleen M: Difference sequence spaces. Int. J. Math. Math. Sci. 1998, 21(4):701–706. 10.1155/S0161171298000970
Malkowsky E, Mursaleen M:Some matrix transformations between the difference sequence spaces , and . Filomat 2001, 15: 353–363.
Mursaleen M, Gaur AK, Saifi AH: Some new sequence spaces and their duals and matrix transformations. Bull. Calcutta Math. Soc. 1996, 88(3):207–212.
Sönmez A: Almost convergence and triple band matrix. Math. Comput. Model. 2012. doi:10.1016/j.mcm.2011.11.079
Başar F, Kirişçi M: Almost convergence and generalized difference matrix. Comput. Math. Appl. 2011, 61(3):602–611. 10.1016/j.camwa.2010.12.006
Et M, Çolak R: On some generalized difference sequence spaces. Soochow J. Math. 1995, 21: 377–386.
Et M, Esi A: On Köthe-Toeplitz duals of generalized difference sequence spaces. Bull. Malays. Math. Soc. 2000, 23: 25–32.
Koshy T: Fibonacci and Lucas Numbers with Applications. Wiley, New York; 2001.
Başar F: Summability Theory and Its Applications. Bentham Science Publishers, İstanbul; 2012. (e-books, Monographs, to appear)
Wilansky A North-Holland Mathematics Studies 85. In Summability Through Functional Analysis. Elsevier, Amsterdam; 1984.
Stieglitz M, Tietz H: Matrix transformationen von folgenräumen eine ergebnisübersicht. Math. Z. 1977, 154: 1–16. 10.1007/BF01215107
Demiriz S, Çakan C: Some topological and geometrical properties of a new difference sequence space. Abstr. Appl. Anal. 2011., 2011: Article ID 213878
Mursaleen M, Başar F, Altay B:On the Euler sequence spaces which include the spaces and II. Nonlinear Anal. TMA 2006, 65(3):707–717. 10.1016/j.na.2005.09.038
Mursaleen M, Çolak R, Et M: Some geometric inequalities in a new Banach sequence space. J. Inequal. Appl. 2007., 2007: Article ID 86757
Diestel J Graduate Texts in Mathematics 92. In Sequence and Series in Banach Spaces. Springer, New York; 1984.
García-Falset J: Stability and fixed points for nonexpansive mappings. Houst. J. Math. 1994, 20(3):495–506.
García-Falset J: The fixed point property in Banach spaces with the NUS-property. J. Math. Anal. Appl. 1997, 215(2):532–542. 10.1006/jmaa.1997.5657
Knaust H: Orlicz sequence spaces of Banach-Saks type. Arch. Math. 1992, 59(6):562–565. 10.1007/BF01194848
Kananthai A, Mursaleen M, Sanhan W, Suantai S: On property (H) and rotundity of difference sequence spaces. J. Nonlinear Convex Anal. 2002, 3(3):401–409.
Mursaleen M:On some geometric properties of a sequence space related to . Bull. Aust. Math. Soc. 2003, 67: 343–347. 10.1017/S0004972700033803
Acknowledgements
The authors wish to thank the referee for his/her valuable suggestions, which improved the paper considerably.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Kara, E.E. Some topological and geometrical properties of new Banach sequence spaces. J Inequal Appl 2013, 38 (2013). https://doi.org/10.1186/1029-242X-2013-38
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2013-38
Keywords
- sequence spaces
- Fibonacci numbers
- difference matrix
- α-, β-, γ-duals
- matrix transformations
- fixed point property
- Banach-Saks type p