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Some topological and geometrical properties of new Banach sequence spaces
Journal of Inequalities and Applicationsvolume 2013, Article number: 38 (2013)
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.
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.
Let Δ denote the matrix defined by
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 . In 1981, Kızmaz  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  and in the case by Başar and Altay  and Çolak et al. . The paranormed difference sequence space
Recently, Başar et al.  have defined the sequence spaces and by
where is an arbitrary fixed sequence and for all . These spaces are generalization of the space for . Quite recently, Kirişçi and Başar  have introduced and studied the generalized difference sequence spaces
for , , c and , where and (). Following Kirişçi and Başar , Sönmez  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
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  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
where cs and bs are the sequence spaces of all convergent and bounded series, respectively .
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.,
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,
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 [, 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
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),
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
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  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
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
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 [, 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 for all .
Now, we list the following conditions:
Then, by combining Lemma 5.1 with the results in , we immediately derive the following results.
if and only (5.2), (5.9) and (5.10) hold.
if and only if (5.2), (5.6), (5.9) and (5.10) hold.
if and only if (5.2), (5.6) with , (5.9) and (5.10) hold.
if and only (5.2), (5.10) and (5.11) hold.
Theorem 5.3 Let . Then we have
if and only if (5.1), (5.2) and (5.4) hold.
if and only if (5.1), (5.2), (5.4) and (5.6) hold.
if and only if (5.1), (5.2), (5.4) and (5.6) with hold.
if and only if (5.1), (5.2) and (5.5) hold.
if and only (5.2), (5.3) and (5.4) with hold.
if and only (5.2), (5.3), (5.6) and (5.7) hold.
if and only (5.2), (5.3) and (5.8) hold.
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 .
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 , 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 , 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 .
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 ).
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
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 .
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The authors wish to thank the referee for his/her valuable suggestions, which improved the paper considerably.
The author declares that they have no competing interests.
About this article
- sequence spaces
- Fibonacci numbers
- difference matrix
- α-, β-, γ-duals
- matrix transformations
- fixed point property
- Banach-Saks type p