A Different Study on the Spaces of Generalized Fibonacci Difference bs and cs Spaces Sequence

: The main topic in this article is to deﬁne and examine new sequence spaces bs ( ˆ F ( s , r )) and cs ( ˆ F ( s , r ))) , where ˆ F ( s , r ) is generalized difference Fibonacci matrix in which s , r ∈ R \ { 0 } . Some algebric properties including some inclusion relations, linearly isomorphism and norms deﬁned over them are given. In addition, it is shown that they are Banach spaces. Finally, the α -, β - and γ -duals of the spaces bs ( ˆ F ( s , r )) and cs ( ˆ F ( s , r )) are appointed and some matrix transformations of them are given.


Introduction
Italian mathematician Leonardo Fibonacci found the Fibonacci number sequence. The Fibonacci sequence actually originated from a rabbit problem in his first book "Liber Abaci". This sequence is used in many fields. The Fibonacci sequence is as follows: 1, 1, 2, 3, 5, 8, 13, 21, 34, .... The Fibonacci sequence, which is denoted by ( f n ), is defined as the linear reccurence relation f n = f n−1 + f n−2 .
The Golden Ratio, which is also known outside the academic community, is used in many fields of science.
Let w be the set of all real valued sequences. Any subspace of w is called the sequence space. c, c 0 and ∞ are called as sequences space convergent, convergent to zero and bounded, respectively. In addition to these representations, 1 , bs and cs are sequence spaces, which are called absolutely convergent, bounded and convergent series, respectively.
Let us take a two-indexed real valued infinite matrix A = (a nk ), where a nk is real number and k, n ∈ N. A is called a matrix transformation from X to Y if, for every x = (x k ) ∈ X, sequence Ax = {A n (x)} is A transform of x and in Y, where A n (x) = ∑ k a nk x k (1) and Equation (1) converges for each n ∈ N. Let λ be a sequence space and K be an infinite matrix. Then, the matrix domain λ K is introduced by Here, it can be seen that λ K is a sequence space. For calculation of any matrix domain of a sequence, a triangle infinite matrix is used by many authors. So many sequence spaces have been recently defined in this way. For more details, see .
Kara [23] recently introduced theF which is derived from the Fibonacci sequence ( f n ) and defined the new sequence spaces p (F) and ∞ (F) by using sequence spaces p and ∞ , respectively, where 1 ≤ p < ∞. The sequence space p (F) has been defined as: whereF = ( f nk ) defined by the sequence ( f n ) as follows: for all k, n ∈ N. In addition, Kara et al. [24] have characterized some class of compact operators on the spaces p (F) and ∞ (F), where 1 ≤ p < ∞.
The α-, β-and γ-duals P α , P β and P γ of a sequence spaces P are defined, respectively, as P α = {a = (a k ) ∈ w : at = (a k t k ) ∈ 1 for all t ∈ P} , P β = {a = (a k ) ∈ w : at = (a k t k ) ∈ cs for all t ∈ P} , P γ = {a = (a k ) ∈ w : at = (a k t k ) ∈ bs for all t ∈ P} , respectively. In Section 2, sequence space bs(F) and cs(F) are defined and some algebric properties of them are investigated. In the last section, the α-, β-and γ-duals of the spaces bs(F) and cs(F) are found and some matrix tranformations of them are given.

Generalized Fibonacci Difference Spaces of bs and cs Sequences
In this section, spaces bs(F(s, r)) and cs(F(s, r)) of generalized Fibonacci difference of sequences, which constitutes bounded and convergence series, respectively, will be defined. In addition, some algebraic properties of them are investigated. Now, we introduce the sets bs(F(s, r)) and cs(F(s, r)) as the sets of all sequences whosê F(s, r) = { f nk (s, r)} transforms are in the sequence space bs and cs, bs(F(s, r)) = for all k, n ∈ N where s, r ∈ R\ {0}. Actually, by using Equation (2), we can get bs(F(s, r)) = (bs)F (s,r) and cs(F(s, r)) = (cs)F (s,r) .
With a basic calculation, we can find the inverse matrix ofF(s, for all k, n ∈ N. If y = (y n ) isF(s, r)-transform of a sequence x = (x n ), then the below equality is justified: for all n ∈ N. In this situation, we see that x n =F −1 (s, r)y, i.e., for all n ∈ N. Theorem 1. bs(F(s, r)) is the linear space with the co-ordinatewise addition and scalar multiplation.
Proof. We omit the proof because it is clear and easy.
Theorem 2. cs(F(s, r)) is the linear space with the co-ordinatewise addition and scalar multiplation.
Proof. We omit the proof because it is clear and easy.
Theorem 3. The space bs(F(s, r)) is a normed space with Proof. It is clear that space bs(F(s, r)) ensures normed space conditions. Theorem 4. The space cs(F(s, r)) is a normed space with norm Equation (7).
Proof. It is clear that normed space conditions are ensured by space cs(F(s, r)).
Proof. For proof, we must demonstrate that bijection and linearly transformation T exist between the space bs(F(s, r)) and bs. Let us take the transformation T : bs(F(s, r)) → bs mentioned above with the help of Equation (5) by Tx =F(s, r)x. We omit the details that T is both linear and injective because the demonstration is clear.
Let us prove that transformation T is surjective. For this, we get y = (y n ) ∈ bs. In this case, by using Equations (6) and (7), we find This result shows that x ∈ bs(F(s, r)). That is, T is surjective. At the same time, this result also indicates that T is preserving the norm. Therefore, the sequence spaces bs(F(s, r)) and bs are linearly isomorphic as isometric.
Theorem 6. The sequence space cs(F(s, r)) is linearly isomorphic as isometric to the space cs, that is, cs(F(s, r)) ∼ = cs.
Proof. If we write cs instead of bs and cs(F(s, r)) instead of bs(F(s, r)) in Theorem 5, the proof will be demonstrated.
Theorem 7. The space bs(F(s, r)) is a Banach space with the norm, which is given in Equation (7).

Proof.
We can easily see that norm conditions are ensured. Let us take that x i = (x i k ) is a Cauchy sequence in bs(F(s, r)) for all i ∈ N. By using Equation (5), we have for all i, k ∈ N. Since x i = (x i k ) is a Cauchy sequence, for every ε > 0, there exists n 0 = n 0 (ε) such that Since bs is complete, y i → y (i → ∞) such that y ∈ bs exist and since the sequence spaces bs(F(s, r)) and bs are linearly isomorphic as isometric bs(F(s, r)) is complete. Consequently, bs(F(s, r)) is a Banach space.
Theorem 8. The space cs(F(s, r)) is a Banach space with the norm, which is given in Equation (7).
Proof. Let x ∈ bs. We must demonstrate that x ∈ bs(F(s, r)). It means thatF(s, r) ∈ (bs, bs). ForF(s, r) ∈ (bs, bs),F(s, r) must ensure to the conditions of (11) of Lemma 1. We see that The other condition also holds as follows: Consequently, the conditions of (11) of Lemma 1 hold. The proof is complete.
Before giving the corollary about the Schauder basis for the space cs(F(r, s)), let us define the Schauder basis which was introduced by J. Schauder in 1927. Let (X, . ) be normed space and be a sequence (a k ) ∈ X . There exists a unique sequence (λ k ) of scalars such that x = ∞ ∑ k=0 λ k a k , and Then, sequence b (k) n∈N is a basis of cs(F(s, r)) and every sequence x ∈ cs(F(s, r)) has a unique representation x = ∑ k y k b k , where y k = (F(s, r)x) k .

The α-, β-and γ-Duals of the Spaces bs(F(s, r)) and cs(F(s, r)) and Some Matrix Transformations
In this section, the alpha-, beta-, gamma-duals of the spaces bs(F(s, r)) and cs(F(s, r)) are determined and characterized the classes of infinite matrices from the space bs(F(s, r)) and cs(F(s, r)) to some other sequence spaces. Now, we give the two lemmas to prove the theorems that will be given in the next stage.
Proof. Let a = (a n ) and x = (x n ) be an arbitrary subset of w. y = (y n ) such that y =F(s, r)x, which is defined by Equation (5). Then, a n x n = a n (F −1 (s, r)y) n = (By) n for all n ∈ N. Hence, we obtain by Equation (5) that ax = (a n x n ) ∈ 1 with x = (x n ) ∈ δ(F(s, r)) iff By ∈ 1 with y ∈ δ. That is, B ∈ (δ, 1 ).
Lemma 3. Let [32] C = (c nk ) be defined via a sequence a = (a k ) ∈ w and the inverse matrix V = (v nk ) of the triangle matrix U = (u nk ) by c nk = ∑ n j=k a j v jk , 0 ≤ k < n, 0, k > n, for all k, n ∈ N. Then, for any sequence space λ, If we consider Lemmas 1-3 together, the following is obtained. Corallary 1. Let B = (b nk ) and C = (c nk ) such that b nk = a n f −1 nk (s, r), 0 ≤ k < n 0, k > n and c nk = If we take t 1 , t 2 , t 3 , t 4 , t 5 , t 6 , t 7 and t 8 as follows: Then, the following statements hold: {cs(F(s, r))} γ = t 7 ∩ t 8 .
In Theorem 14, we take λ (F(s, r)) instead of µ and µ instead of λ (F(s, r)), and then we get the following theorem.
At this stage, let us consider almost convergent sequences spaces, which were given by Lorentz [33]. This is because they will help in calculating some of the results of Theorems 14 and 15. Let a sequence x = (x k ) ∈ ∞ . x is said to be almost convergent to the generalized limit iff lim m→∞ ∑ m k=0 x n+k m+1 = uniformly in n and is denoted by f − lim x = . By f and f 0 , we indicate the space of all almost convergent and almost null sequences, respectively. However, in this article, we useĉ andĉ 0 instead of f and f 0 , respectively, in order to avoid any confusion. This is because the Fibonacci sequence is also denoted by f . In addition, byĉs, we indicate the space of sequences, which is composed of all almost convergent series. The sequences spacesĉ andĉ 0 arê x n+k m + 1 = uniformly in n . Now, let A = (a nk ) be an arbitrary infinite matrix and list the following conditions: (a nk − a n,k+1 ) = α, for each k ∈ N, α ∈ C.
Let us give some matrix transformations in the following Lemma for use in the next step.
(17) A = (a nk ) ∈ (bv 0 , cs) iff Equations (11) and (46) hold (Jakimovski and Russell [37]). Now, let us list the following condition, where d nk and d (m) nk are taken as in Equations (30) and (31): Now, we can give several conclusions of Theorems 14 and 15, and Lemmas 1 and 4.  where x ∈ bs(F(s, r)) or x ∈ cs(F(s, r)). In addition, it was shown that they are normed space and Banach spaces. It was found that bs(F(s, r)) and bs are linearly isomorphic as isometric. At the same time, cs(F(s, r)) and cs are linearly isomorphic as isometric. Some inclusions' theorems were given with respect to bs(F(s, r)) and cs(F(s, r)). According to this, inclusions bs ⊂ bs(F(s, r)), cs ⊂ cs(F(s, r)) are valid. In addition, if |r/s| < 1/4, then bs(F(s, r)) ⊂ ∞ and cs(F(s, r)) ⊂ c are valid. It was concluded that cs(F(s, r)) has a Schauder basis. Finally, the α-, β-and γ-duals of the both spaces are calculated and some matrix transformations of them were given.

Conclusions
In this article, we have defined spaces bs(F(s, r)) and cs(F(s, r)) of Generalized Fibonacci difference of sequences, which constituted bounded and convergence series, respectively. We have demonstrated that the sets of bs(F(s, r)) and cs(F(s, r)) are the linear spaces and both spaces have the same norm. In addition, it was shown that they are Banach spaces. Some inclusions theorems were given with respect to bs(F(s, r)) and cs(F(s, r)). It was concluded that cs(F(s, r)) has a Schauder basis. Finally, the α-, β-and γ-duals of the both spaces were calculated and some matrix transformations of them were given.
Author Contributions: Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.
Funding: This research received no external funding