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Article

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

Department of Mathematics, Faculty of Arts and Sciences, Gaziantep University, Gaziantep 27310, Turkey
*
Author to whom correspondence should be addressed.
Symmetry 2018, 10(7), 274; https://doi.org/10.3390/sym10070274
Submission received: 13 June 2018 / Revised: 1 July 2018 / Accepted: 5 July 2018 / Published: 11 July 2018
(This article belongs to the Special Issue Mathematical Physics and Symmetry)

Abstract

:
The main topic in this article is to define and examine new sequence spaces b s ( F ^ ( s , r ) ) and c s ( F ^ ( s , r ) ) ) , where 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 b s ( F ^ ( s , r ) ) and c s ( F ^ ( s , r ) ) are appointed and some matrix transformations of them are given.

1. 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 .
f 0 = 1 , f 1 = 1 and n 2 . The golden ratio is
lim n f n + 1 f n = 1 + 5 2 = φ ( Golden Ratio ) .
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 , b s and c s 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 n k ) , where a n k 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 A x = A n ( x ) is A transform of x and in Y, where
A n ( x ) = k a n k x k
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
λ K = t = ( t k ) w : K t λ .
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 [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22].
Kara [23] recently introduced the F ^ 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:
p ( F ^ ) = x w : F ^ x p , ( 1 p < ) ,
where F ^ = ( f n k ) defined by the sequence ( f n ) as follows:
f n k : = f n + 1 f n , k = n 1 , f n f n + 1 , k = n , 0 , 0 k < n 1 or k > n ,
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 < .
Candan [25] introduced c ( F ^ ( s , r ) ) and c 0 ( F ^ ( s , r ) ) . Later, Candan and Kara [15] have investigated the sequence spaces p ( F ^ ( s , r ) ) in which 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 : a t = ( a k t k ) 1 for all t P , P β = a = ( a k ) w : a t = ( a k t k ) c s for all t P , P γ = a = ( a k ) w : a t = ( a k t k ) b s for all t P ,
respectively.
In Section 2, sequence space b s ( F ^ ) and c s ( F ^ ) are defined and some algebric properties of them are investigated. In the last section, the α -, β - and γ -duals of the spaces b s ( F ^ ) and c s ( F ^ ) are found and some matrix tranformations of them are given.

2. Generalized Fibonacci Difference Spaces of b s and c s Sequences

In this section, spaces b s ( F ^ ( s , r ) ) and c s ( 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 b s ( F ^ ( s , r ) ) and c s ( F ^ ( s , r ) ) as the sets of all sequences whose F ^ ( s , r ) = f n k ( s , r ) transforms are in the sequence space b s and c s ,
b s ( F ^ ( s , r ) ) = x = ( x k ) w : sup n N k = 0 n s f k f k + 1 x k + r f k + 1 f k x k 1 < , c s ( F ^ ( s , r ) ) = x = ( x k ) w : n k = 0 s f k f k + 1 x k + r f k + 1 f k x k 1 c ,
where F ^ ( s , r ) = f n k ( s , r ) is
f n k ( s , r ) : = r f n + 1 f n , k = n 1 , s f n f n + 1 , k = n , 0 , k < n 1 or 0 k > n ,
for all k , n N where s , r R \ 0 . Actually, by using Equation (2), we can get
b s ( F ^ ( s , r ) ) = ( b s ) F ^ ( s , r ) and c s ( F ^ ( s , r ) ) = ( c s ) F ^ ( s , r ) .
With a basic calculation, we can find the inverse matrix of F ^ ( s , r ) = f n k ( s , r ) . The inverse matrix of F ^ ( s , r ) = f n k ( s , r ) is F ^ 1 ( s , r ) = f n k 1 ( s , r ) such that
f n k 1 ( s , r ) = 1 s ( r s ) n k f n + 1 2 f k f k + 1 , 0 k < n , 0 , k > n ,
for all k , n N . If y = ( y n ) is F ^ ( s , r ) -transform of a sequence x = ( x n ) , then the below equality is justified:
y n = ( F ^ ( s , r ) x ) n = s x 0 , n = 0 , s f n f n + 1 x n + r f n + 1 f n x n 1 , n 1 ,
for all n N . In this situation, we see that x n = F ^ 1 ( s , r ) y , i.e.,
x n = k = 0 n 1 s ( r s ) n k f n + 1 2 f k f k + 1 y k
for all n N .
Theorem 1.
b s ( 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.
c s ( 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 b s ( F ^ ( s , r ) ) is a normed space with
x b s ( F ^ ( s , r ) ) = sup n N n k = 0 s f k f k + 1 x k + r f k + 1 f k x k 1 .
Proof. 
It is clear that space b s ( F ^ ( s , r ) ) ensures normed space conditions. ☐
Theorem 4.
The space c s ( F ^ ( s , r ) ) is a normed space with norm Equation (7).
Proof. 
It is clear that normed space conditions are ensured by space c s ( F ^ ( s , r ) ) . ☐
Theorem 5.
b s ( F ^ ( s , r ) ) is linearly isomorphic as isometric to the space b s , that is, b s ( F ^ ( s , r ) ) b s .
Proof. 
For proof, we must demonstrate that bijection and linearly transformation T exist between the space b s ( F ^ ( s , r ) ) and b s . Let us take the transformation T : b s ( F ^ ( s , r ) ) b s mentioned above with the help of Equation (5) by T x = 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 ) b s .
In this case, by using Equations (6) and (7), we find
x b s ( F ^ ( s , r ) ) = sup n N n k = 0 s f k f k + 1 x k + r f k + 1 f k x k 1 = sup n N n k = 0 s f k f k + 1 k i = 0 1 s ( r s ) k i f k + 1 2 f i f i + 1 y i + r f k + 1 f k k 1 i = 0 1 s ( r s ) k i 1 f k 2 f i f i + 1 y i = sup n N n k = 0 y k = y b s .
This result shows that x b s ( 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 b s ( F ^ ( s , r ) ) and b s are linearly isomorphic as isometric.
Theorem 6.
The sequence space c s ( F ^ ( s , r ) ) is linearly isomorphic as isometric to the space c s , that is, c s ( F ^ ( s , r ) ) c s .
Proof. 
If we write c s instead of b s and c s ( F ^ ( s , r ) ) instead of b s ( F ^ ( s , r ) ) in Theorem 5, the proof will be demonstrated. ☐
Theorem 7.
The space b s ( 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 k i ) is a Cauchy sequence in b s ( F ^ ( s , r ) ) for all i N . By using Equation (5), we have
y k i = s f k f k + 1 x k i + r f k + 1 f k x k 1 i
for all i , k N . Since x i = ( x k i ) is a Cauchy sequence, for every ε > 0 , there exists n 0 = n 0 ( ε ) such that
x i x m b s ( F ^ ( s , r ) ) = sup n N n k = 0 s f k f k + 1 ( x k i x k m ) + r f k + 1 f k ( x k 1 i x k 1 m ) = sup n N n k = 0 y k i y k m = y i y m b s < ε
for all i , m n 0 . Since b s is complete, y i y ( i ) such that y b s exist and since the sequence spaces b s ( F ^ ( s , r ) ) and b s are linearly isomorphic as isometric b s ( F ^ ( s , r ) ) is complete. Consequently, b s ( F ^ ( s , r ) ) is a Banach space. ☐
Theorem 8.
The space c s ( 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 k i ) is a Cauchy sequence in c s ( F ^ ( s , r ) ) for all i N . By using Equation (5), we have
y k i = s f k f k + 1 x k i + r f k + 1 f k x k 1 i
for all i , k N . Since x i = ( x k i ) is a Cauchy sequence, for every ε > 0 , there exists n 0 = n 0 ( ε ) such that
x i x m c s ( F ^ ( s , r ) ) = sup n N n k = 0 s f k f k + 1 ( x k i x k m ) + r f k + 1 f k ( x k 1 i x k 1 m ) = sup n N n k = 0 y k i y k m = y i y m c s < ε
for all i , m n 0 . Since c s is complete, y i y ( i ) such that y c s exists and since the sequence spaces c s ( F ^ ( s , r ) ) and c s are linearly isomorphic as isometric c s ( F ^ ( s , r ) ) is complete. Consequently, c s ( F ^ ( s , r ) ) is a Banach space. ☐
Now, let A = ( a n k ) be an arbitrary infinite matrix and list the following:
sup n N k a n k < ,
lim k a n k = 0 for each n N ,
sup m k n = 0 m ( a n k a n , k + 1 ) < ,
lim n k a n k = α for each k N , α C ,
sup n k a n k a n , k + 1 < ,
lim n a n k = α k for each k N , α k C ,
sup N , K F n N k K ( a n k a n , k + 1 ) < ,
sup N , K F n N k K ( a n k a n , k 1 ) < ,
lim n ( a n k a n , k + 1 ) = α for each k N , α C ,
lim n k a n k a n , k + 1 = k lim n ( a n k a n , k + 1 ) ,
sup n lim k a n k < ,
lim n k a n k a n , k + 1 = 0 uniformly in n ,
lim m k m n = 0 ( a n k a n , k + 1 ) = 0 ,
lim m k m n = 0 ( a n k a n , k + 1 ) = k n ( a n k a n , k + 1 ) ,
sup N , K F n N k K [ ( a n k a n , k + 1 ) ( a n 1 , k a n 1 , k + 1 ) ] < ,
sup m N lim k m n = 0 a n k < ,
α k C n a n k = α k for each k N ,
sup N , K F n N k K ( a n k a n 1 , k ) ( a n , k 1 a n 1 , k 1 ) < ,
where F denote the collection of all finite subsets of N .
Now, we can give some matrix transformations in the following Lemma for the next step that we will need in the inclusion Theorems.
Lemma 1.
Let A = ( a n k ) be an arbitrary infinite matrix. Then,
(1) 
A = ( a n k ) ( b s , ) iff Equations (9) and (12) hold (Stieglitz and Tietz [26]),
(2) 
A = ( a n k ) ( c s , c ) iff Equations (12) and (13) hold (Wilansky [27]),
(3) 
A = ( a n k ) ( b s , 1 ) iff Equations (9) and (14) hold (K.-G. Grosse-Erdman [28]).
(4) 
A = ( a n k ) ( c s , 1 ) iff Equation (15) holds (Stieglitz and Tietz [26]).
(5) 
A = ( a n k ) ( b s , c ) iff Equations (9), (16) and (17) hold (K.-G. Grosse-Erdman [28]).
(6) 
A = ( a n k ) ( c s , ) iff Equations (12) and (18) hold (Stieglitz and Tietz [26]).
(7) 
A = ( a n k ) ( b s , c 0 ) iff Equations (9) and (19) hold (Stieglitz and Tietz [26]).
(8) 
A = ( a n k ) ( b s , c s 0 ) iff Equations (9) and (20) hold (Zeller [29]).
(9) 
A = ( a n k ) ( b s , c s ) iff Equations (9) and (21) hold (Zeller [29]).
(10) 
A = ( a n k ) ( b s , b v ) iff Equations (9) and (22) hold (Zeller [29]).
(11) 
A = ( a n k ) ( b s , b s ) iff Equations (9) and (10) hold (Zeller [29]).
(12) 
A = ( a n k ) ( c s , c s ) iff Equations (10) and (11) hold (Hill, [30]).
(13) 
A = ( a n k ) ( b s , b v 0 ) iff Equations (12), (19) and (22) hold (Stieglitz and Tietz [26]).
(14) 
A = ( a n k ) ( c s , c 0 ) iff Equation (12) holds and Equation (13) also holds with α k = 0 for all k N (Dienes [31]).
(15) 
A = ( a n k ) ( c s , b s ) iff Equations (10) and (23) hold (Zeller [29]).
(16) 
A = ( a n k ) ( c s , c s 0 ) iff Equation (10) holds and Equation (24) also holds with α k = 0 for all k N (Zeller [29]).
(17) 
A = ( a n k ) ( c s , b v ) iff Equation (25) holds (Zeller [29]).
(18) 
A = ( a n k ) ( c s , b v 0 ) iff Equation (25) holds and Equation (13) also holds with α k = 0 for all k N (Stieglitz and Tietz [26]).
Theorem 9.
The inclusion b s b s ( F ^ ( s , r ) ) is valid.
Proof. 
Let x b s . We must demonstrate that x b s ( F ^ ( s , r ) ) . It means that F ^ ( s , r ) ( b s , b s ) . For F ^ ( s , r ) ( b s , b s ) , F ^ ( s , r ) must ensure to the conditions of (11) of Lemma 1. We see that
lim k f n k ( s , r ) = 0 for each n N .
The other condition also holds as follows:
sup m k n = 0 m ( f n k ( s , r ) f n , k + 1 ( s , r ) ) = sup m lim p s + r f 1 . f 2 + s + r f 2 . f 3 + + s + r f p + 1 . f p + 2 = 17 10 s + r < .
Consequently, the conditions of (11) of Lemma 1 hold. The proof is complete. ☐
Theorem 10.
If r / s < 1 / 4 , then b s ( F ^ ( s , r ) ) is valid.
Proof. 
Let x b s ( F ^ ( s , r ) ) . Then, y = F ^ ( s , r ) x b s . We must demonstrate that x = F ^ 1 ( s , r ) y . That is, F ^ 1 ( s , r ) ( b s , ) . For F ^ 1 ( s , r ) ( b s , ) , F ^ 1 ( s , r ) must satisfy the conditions of (1) of Lemma 1. It is clear that
lim k f n k 1 ( s , r ) = 0 for each n N .
The other condition is also holds as follows:
sup n k ( f n k 1 ( s , r ) f n , k + 1 1 ( s , r ) ) 2 sup n k ( f n k 1 ( s , r ) r s 4 s k 4 r s k < .
Consequently, the conditions of (1) of Lemma 1 hold. The proof is complete. ☐
Theorem 11.
The inclusion c s c s ( F ^ ( s , r ) ) is valid.
Proof. 
Let x c s . We must demonstrate that x c s ( F ^ ( s , r ) ) . It means that F ^ ( s , r ) ( c s , c s ) . For F ^ ( s , r ) ( c s , c s ) , F ^ ( s , r ) must satisfy the conditions of (12) of Lemma 1. Equation (10) has been satisfied in Theorem 9. Now, we must demonstrate Equation (11). For every k N ,
lim n k f n k ( s , r ) = lim n ( s f n f n + 1 + r f n + 1 f n ) = s φ + r φ =
such that C exist. Consequently, the conditions of (12) of Lemma 1 hold. The proof is complete. ☐
Theorem 12.
If r / s < 1 / 4 , then c s ( F ^ ( s , r ) ) c is valid.
Proof. 
Let x c s ( F ^ ( s , r ) ) . Then, y = F ^ ( s , r ) x c s . We must demonstrate that x = F ^ 1 ( s , r ) y c . That is, F ^ 1 ( s , r ) ( c s , c ) . For F ^ 1 ( s , r ) ( c s , c ) , F ^ 1 ( s , r ) must satisfy the conditions of (2) of Lemma 1. Equation (12) has been satisfied in Theorem 10. Now, we must demonstrate Equation (13). For each k N ,
lim n f n k 1 ( s , r ) lim n f n k 1 ( s , r ) = lim n f n + 1 s f n r s n k f n + 1 f k + 1 f k f n = lim n f n + 1 s f n i = k n 1 r f i + 2 f i + 1 s f i f i + 1 lim n f n + 1 s f n i = k n 1 sup i N r f i + 2 f i + 1 inf i N s f i f i + 1 lim n f n + 1 s f n 4 r s n k = φ s . 0 = 0 .
Thus, Equation (13) is also satisfied. ☐
Theorem 13.
The inclusion c s ( F ^ ( s , r ) ) b s ( F ^ ( s , r ) ) is valid.
Proof. 
Let x c s ( F ^ ( s , r ) ) . Then, y = F ^ ( s , r ) x c s . Hence, k F ^ ( s , r ) x c . c , so it becomes k F ^ ( s , r ) x . That is, F ^ ( s , r ) x b s . Hence, x b s ( F ^ ( s , r ) ) . Consequently, c s ( F ^ ( s , r ) ) b s ( F ^ ( s , r ) ) .
Before giving the corollary about the Schauder basis for the space c s ( 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
lim n x n k = 0 λ k a k = 0 .
Then, ( a k ) is called a Schauder basis for X. ☐
Now, we can give the corollary about Schauder basis.
Corollary 1.
Let us sequence b ( k ) = b n ( k ) n N defined in the c s ( F ^ ( s , r ) ) such that
b n ( k ) = 1 s ( r s ) n k f n + 1 2 f k f k + 1 , n > k , 1 s f k + 1 f k , n = k , 0 , n < k .
Then, sequence b ( k ) n N is a basis of c s ( F ^ ( s , r ) ) and every sequence x c s ( F ^ ( s , r ) ) has a unique representation x = k y k b k , where y k = ( F ^ ( s , r ) x ) k .

3. The α -, β - and γ -Duals of the Spaces b s ( F ^ ( s , r ) ) and c s ( F ^ ( s , r ) ) and Some Matrix Transformations

In this section, the alpha-, beta-, gamma-duals of the spaces b s ( F ^ ( s , r ) ) and c s ( F ^ ( s , r ) ) are determined and characterized the classes of infinite matrices from the space b s ( F ^ ( s , r ) ) and c s ( 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.
Lemma 2.
Suppose that a = ( a n ) w and the infinite matrix B = ( b n k ) is defined by B n = a n ( F ^ 1 ( s , r ) ) n , that is,
b n k = a n f n k 1 ( s , r ) , 0 k < n , 0 , k > n ,
for all k , n N , δ b s , c s . Then, a { δ ( F ^ ( s , r ) ) } α iff B ( δ , 1 ) .
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 = ( B y ) n
for all n N . Hence, we obtain by Equation (5) that a x = ( a n x n ) 1 with x = ( x n ) δ ( F ^ ( s , r ) ) iff B y 1 with y δ . That is, B ( δ , 1 ) . ☐
Lemma 3.
Let [32] C = ( c n k ) be defined via a sequence a = ( a k ) w and the inverse matrix V = ( v n k ) of the triangle matrix U = ( u n k ) by
c n k = j = k n a j v j k , 0 k < n , 0 , k > n ,
for all k , n N . Then, for any sequence space λ,
λ U γ = a = ( a k ) w : C ( λ , ) , λ U β = a = ( a k ) w : C ( λ , c ) .
If we consider Lemmas 1–3 together, the following is obtained.
Corallary 1.
Let B = ( b n k ) and C = ( c n k ) such that
b n k = a n f n k 1 ( s , r ) , 0 k < n 0 , k > n a n d c n k = j = k n 1 s ( r s ) j k f j + 1 2 f k f k + 1 a j .
If we take t 1 , t 2 , t 3 , t 4 , t 5 , t 6 , t 7 and t 8 as follows:
t 1 = a = ( a k ) w : sup N , K F n N k K ( b n k b n , k + 1 ) < , t 2 = a = ( a k ) w : sup N , K F n N k K ( b n k b n , k 1 ) < , t 3 = a = ( a k ) w : lim k c n k = 0 , t 4 = a = ( a k ) w : α C lim n ( c n k c n , k + 1 ) = α , t 5 = a = ( a k ) w : lim n k c n k c n , k + 1 = k lim n ( c n k c n , k + 1 ) , t 6 = a = ( a k ) w : α C lim n c n k = α , f o r a l l k N , t 7 = a = ( a k ) w : sup n N k c n k c n , k + 1 < , t 8 = a = ( a k ) w : sup n N lim k c n k < .
Then, the following statements hold:
(1) 
{ b s ( F ^ ( s , r ) ) } α = t 1 ,
(2) 
{ c s ( F ^ ( s , r ) ) } α = t 2 ,
(3) 
{ b s ( F ^ ( s , r ) ) } β = t 3 t 4 t 5 ,
(4) 
{ c s ( F ^ ( s , r ) ) } β = t 6 t 7 ,
(5) 
{ b s ( F ^ ( s , r ) ) } γ = t 3 t 7 ,
(6) 
{ c s ( F ^ ( s , r ) ) } γ = t 7 t 8 .
Theorem 14.
Let λ b s , c s and μ ⊂ w. Then, A = ( a n k ) ( λ ( F ^ ( s , r ) ) , μ ) iff
D m = ( d n k ( m ) ) ( λ , c ) f o r a l l n N ,
D = ( d n k ) ( λ , μ ) ,
where
d n k ( m ) = j = k m 1 s ( r s ) j k f j + 1 2 f k f k + 1 a n j , 0 k < m , 0 , k > m ,
and
d n k = j = k 1 s ( r s ) j k f j + 1 2 f k f k + 1 a n j
for all k , m , n N .
Proof. 
To prove the necessary part of the theorem, let us suppuse that A = ( a n k ) ( λ ( F ^ ( s , r ) , μ ) and x = ( x k ) λ ( F ^ ( s , r ) ) . By using Equation (6), we find
k = 0 m a n k x k = k = 0 m a n k j = o k 1 s ( r s ) k j f k + 1 2 f j f j + 1 y j = k = 0 m j = k m 1 s ( r s ) j k f j + 1 2 f k f k + 1 a n j y k = k = 0 m d n k ( m ) y k = D n ( m ) ( y )
for all m , n N . For each m N and x = ( x k ) λ ( F ^ ( s , r ) ) , A m ( x ) exists and also lies in c. Then, D n ( m ) also lies in c for each m N . Hence, D ( m ) ( λ , c ) . Now, from Equation (32), we consider for m , and then A x = D y . Consequently, we obtain D = ( d n k ) ( λ , μ ) .
If we want to prove the sufficient part of the theorem, then let us assume that Equations (28) and (29) are satisfied and x = ( x k ) λ ( F ^ ( s , r ) ) . By using Corollary 1 and Equations (28) and (32), we obtain y = F ^ ( s , r ) x λ and D n ( m ) ( y ) = k = 0 m d n k ( m ) y k = k = 0 m a n k x k = A n ( m ) ( x ) c . Hence, A = ( a n k ) k N exists. In addition, in Equation (32), if we consider m . Then, A x = D y . Consequently, we obtain A = ( a n k ) ( λ ( F ^ ( s , r ) ) , μ ) .
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. ☐
Theorem 15.
Let λ b s , c s and μ be an arbitrary subset of w and A = ( a n k ) and B = ( b n k ) be infinite matrices. If we take
b n k : = r f n + 1 f n a n 1 , k + s f n f n + 1 a n k
for all k , n N , then A ( μ , λ ( F ^ ( s , r ) ) ) iff B ( μ , λ ) .
Proof. 
Let us suppose that A ( μ , λ ( F ^ ( s , r ) ) ) and Equation (33) exist. For z = ( z k ) μ , we obtain A z λ ( F ^ ( s , r ) ) from A ( μ , λ ( F ^ ( s , r ) ) ) . Hence, F ^ ( s , r ) ( A z ) λ . On the other hand, we have
k = 0 m b n k z k = k = 0 m r f n + 1 f n a n 1 , k + s f n f n + 1 a n k z k
for all m , n N . If we carry out m to Equation (34), we obtain that
( B z ) n = ( ( F ^ ( s , r ) A ) z ) n = ( F ^ ( s , r ) ( A z ) ) n .
Since F ^ ( s , r ) ( A z ) λ , we find B z = ( B z ) n λ for z = ( z k ) μ from Equation (35). Hence, we obtain that B ( μ , λ ) . This is the desired result. ☐
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 k = 0 m 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 use c ^ and c ^ 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, by c ^ s , we indicate the space of sequences, which is composed of all almost convergent series. The sequences spaces c ^ and c ^ 0 are
c ^ 0 = x = ( x k ) : lim m k = 0 m x n + k m + 1 = 0 uniformly in n , c ^ = x = ( x k ) : C lim m k = 0 m x n + k m + 1 = uniformly in n .
Now, let A = ( a n k ) be an arbitrary infinite matrix and list the following conditions:
α k C f lim a n k = α k for each k N ,
lim q k 1 q + 1 q i = 0 n + i j = 0 ( a j k α k ) = 0 uniformly in n ,
sup n N k n j = 0 a j k < ,
α k C f lim n j = 0 a j k = α k for each k N ,
sup n N k n j = 0 a j k < ,
α k C n k a n k = α k for all k N ,
lim n k n j = 0 ( a j k α k ) = 0 ,
sup n N k n j = 0 a j k q < , q = p p 1 ,
sup m , n N m n = 0 a n k < ,
sup m , l N m n = 0 k = l a n k < ,
sup m , l N m n = 0 l k = 0 a n k < ,
lim m k n = m a n k = 0 ,
n k a n k , convergent ,
lim m m n = 0 ( a n k 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.
Lemma 4.
Let A = ( a n k ) be an infinite matrix for all k , n N . Then,
(1) 
A = ( a n k ) ( c ^ , c s ) iff Equations (24) and (40)–(42) hold (Başar [34]).
(2) 
A = ( a n k ) ( c s , c ^ ) iff Equations (12) and (36) hold (Başar and Çolak [35]).
(3) 
A = ( a n k ) ( b s , c ^ ) iff Equations (9), (12), (36) and (37) hold (Başar and Solak [36]).
(4) 
A = ( a n k ) ( b s , c ^ s ) iff Equations (9) and (37)–(39) hold (Başar and Solak [36]).
(5) 
A = ( a n k ) ( c s , c ^ s ) iff Equations (38) and (39) hold (Başar and Çolak [35]).
(6) 
A = ( a n k ) ( , b s ) = ( c , b s ) = ( c 0 , b s ) iff Equation (40) holds (Zeller [29]).
(7) 
A = ( a n k ) ( p , b s ) iff Equation (43) holds (Jakimovski and Russell [37]).
(8) 
A = ( a n k ) ( , b s ) iff Equation (44) holds (Zeller [29]).
(9) 
A = ( a n k ) ( b v , b s ) iff Equation (45) holds (Zeller [29]).
(10) 
A = ( a n k ) ( b v 0 , b s ) iff Equation (46) holds (Jakimovski and Russell [37]).
(11) 
A = ( a n k ) ( , c s ) iff Equation (47) holds (Zeller [29]).
(12) 
A = ( a n k ) ( c , c s ) iff Equations (11), (40) and (48) hold (Zeller [29]).
(13) 
A = ( a n k ) ( c s 0 , c s ) iff Equations (10) and (49) hold (Zeller [29]).
(14) 
A = ( a n k ) ( p , c s ) iff Equations (11) and (43) hold (Jakimovski and Russell [37]).
(15) 
A = ( a n k ) ( , c s ) iff Equations (11) and (44) hold (Jakimovski and Russell [37]).
(16) 
A = ( a n k ) ( b v , c s ) iff Equations (11), (44) and (46) hold (Zeller [29]).
(17) 
A = ( a n k ) ( b v 0 , c s ) iff Equations (11) and (46) hold (Jakimovski and Russell [37]).
Now, let us list the following condition, where d n k and d n k ( m ) are taken as in Equations (30) and (31):
lim k d n k ( m ) = 0 for all n N ,
d n k C lim n ( d n k ( m ) d n , k + 1 ( m ) ) = d n k for all k , n N ,
lim n k d n k ( m ) d n , k + 1 ( m ) < uniformly in n ,
lim k d n k = 0 for all n N ,
sup n k d n k d n , k + 1 < ,
d k C lim n ( d n k d n , k + 1 ) = d k for all k , n N ,
α C lim n k d n k d n , k + 1 = α uniformly in n ,
sup m N k m n = 0 ( d n k d n , k + 1 ) < ,
lim m k m n = 0 ( d n k d n , k + 1 ) = k n ( d n k d n , k + 1 ) ,
lim m k m n = 0 ( d n k d n , k + 1 ) = 0 ,
sup N , K F n N k K ( d n k d n , k + 1 ) < ,
sup N , K F n N k K ( d n k d n , k + 1 ) ( d n 1 , k d n 1 , k + 1 ) < ,
sup n k d n k ( m ) d n , k + 1 ( m ) < ,
d k C lim n d n k ( m ) = d k for all k , n N ,
sup n N lim k d n k < ,
d k C lim n d n k = d k for all k , n N ,
sup m N lim k m n = 0 d n k < ,
sup m N k m n = 0 ( d n k d n , k 1 ) < ,
d k C n d n k = d k for each k N ,
sup N , K F n N k K ( d n k d n , k 1 ) < ,
d k C f lim d n k = d k for each k N ,
sup N , K F n N k K ( d n k d n 1 , k ) ( d n , k 1 d n 1 , k 1 ) < ,
lim q k 1 q + 1 q i = 0 n + i j = 0 ( d j k α k ) = 0 uniformly in n ,
sup n N k n j = 0 d j k < ,
d k C n k d n k = d k for all k N ,
lim n k n j = 0 ( d j k α k ) = 0 ,
sup n N k n j = 0 d j k < ,
d k C f lim n j = 0 d j k = d k for each k N ,
Now, we can give several conclusions of Theorems 14 and 15, and Lemmas 1 and 4.
Corallary 2.
Let A = ( a n k ) be an infinite matrix for all k , n N . Then,
(1) 
A = ( a n k ) ( b s ( F ^ ( s , r ) , c 0 ) iff Equations (50)–(53) hold and Equation (56) also holds with α = 0 .
(2) 
A = ( a n k ) ( b s ( F ^ ( s , r ) , c s 0 ) iff Equations (50)–(53) and (59) hold.
(3) 
A = ( a n k ) ( b s ( F ^ ( s , r ) , c ) iff Equations (50)–(53), (55) and (56) hold.
(4) 
A = ( a n k ) ( b s ( F ^ ( s , r ) , c s ) iff Equations (50)–(53) and (58) hold.
(5) 
A = ( a n k ) ( b s ( F ^ ( s , r ) , ) iff Equations (50)–(54) hold.
(6) 
A = ( a n k ) ( b s ( F ^ ( s , r ) , b s ) iff Equations (50)–(53) and (57) hold.
(7) 
A = ( a n k ) ( b s ( F ^ ( s , r ) , 1 ) iff Equations (50)–(53) and (60) hold.
(8) 
A = ( a n k ) ( b s ( F ^ ( s , r ) , b v ) iff Equations (50)–(53) and (61) hold.
(9) 
A = ( a n k ) ( b s ( F ^ ( s , r ) , b v 0 ) iff Equations (50)–(52), (54) and (61) hold and Equation (56) also holds with α = 0 .
Corallary 3.
Let A = ( a n k ) be an infinite matrix for all k , n N . Then,
(1) 
A = ( a n k ) ( c s ( F ^ ( s , r ) , c 0 ) iff Equations (54), (62) and (63) hold and Equation (65) also holds with d k = 0 for all k N .
(2) 
A = ( a n k ) ( c s ( F ^ ( s , r ) , c s 0 ) iff Equations (57), (62) and (63) hold and Equation (68) also holds with d k = 0 for all k N .
(3) 
A = ( a n k ) ( c s ( F ^ ( s , r ) , c ) iff Equations (54), (62), (63) and (65) hold.
(4) 
A = ( a n k ) ( c s ( F ^ ( s , r ) , c s ) iff Equations (62), (63), (67) and (68) hold.
(5) 
A = ( a n k ) ( c s ( F ^ ( s , r ) , ) iff Equations (54) and (62)–(64) hold.
(6) 
A = ( a n k ) ( c s ( F ^ ( s , r ) , b s ) iff Equations (57), (62), (63) and (66) hold.
(7) 
A = ( a n k ) ( c s ( F ^ ( s , r ) , 1 ) iff Equations (62), (63) and (69) hold.
(8) 
A = ( a n k ) ( c s ( F ^ ( s , r ) , b v ) iff Equations (62), (63) and (71) hold.
(9) 
A = ( a n k ) ( c s ( F ^ ( s , r ) , b v 0 ) iff Equations (62), (63) and (65) hold and Equation (71) also holds with d k = 0 for all k N .
Corallary 4.
Let A = ( a n k ) be an infinite matrix for all k , n N . Then,
(1) 
A = ( a n k ) ( b s ( F ^ ( s , r ) , c ^ ) iff Equations (50)–(54), (70) and (72) hold.
(2) 
A = ( a n k ) ( b s ( F ^ ( s , r ) , c ^ 0 ) iff Equations (50)–(54) hold and (70) and Equation (72) also hold with α k = 0 in Equation (70) and d k = 0 in (72).
(3) 
A = ( a n k ) ( c s ( F ^ ( s , r ) , c ^ ) iff Equations (54), (62), (63) and (70) hold.
(4) 
A = ( a n k ) ( c s ( F ^ ( s , r ) , c ^ 0 ) iff Equations (62), (63) and (54) hold and Equation (70) also holds with α k = 0 .
(5) 
A = ( a n k ) ( c ^ , c s ( F ^ ( s , r ) ) iff Equations (68) and (73)–(75) hold with b n k instead of d n k , where b n k is defined by Equation (33).
(6) 
A = ( a n k ) ( b s ( F ^ ( s , r ) , c ^ s ) iff Equations (50)–(53), (72), (76) and (77) hold.
(7) 
A = ( a n k ) ( c s ( F ^ ( s , r ) , c ^ s ) iff Equations (62), (63), (76) and (77) hold.
Corallary 5.
Let A = ( a n k ) be an infinite matrix for all k , n N . Then,
(1) 
A = ( a n k ) ( , b s ( F ^ ( s , r ) ) = ( c , b s ) = ( c 0 , b s ) iff Equation (40) holds with b n k instead of a n k , where b n k is defined by (33).
(2) 
A = ( a n k ) ( p , b s ( F ^ ( s , r ) ) iff Equation (43) holds with b n k instead of a n k , where b n k is defined by (33).
(3) 
A = ( a n k ) ( , b s ( F ^ ( s , r ) ) iff Equation (44) holds with b n k instead of a n k , where b n k is defined by Equation (33).
(4) 
A = ( a n k ) ( b v , b s ( F ^ ( s , r ) ) iff Equation (45) holds with b n k instead of a n k , where b n k is defined by Equation (33).
(5) 
A = ( a n k ) ( b v 0 , b s ( F ^ ( s , r ) ) iff Equation (46) holds with b n k instead of a n k , where b n k is defined by Equation (33).
(6) 
A = ( a n k ) ( , c s ( F ^ ( s , r ) ) iff Equation (47) holds with b n k instead of a n k , where b n k is defined by Equation (33).
(7) 
A = ( a n k ) ( c , c s ( F ^ ( s , r ) ) iff Equations (11), (40) and (48) hold with b n k instead of a n k , where b n k is defined by Equation (33).
(8) 
A = ( a n k ) ( c s 0 , c s ( F ^ ( s , r ) ) iff Equations (10) and (49) hold with b n k instead of a n k , where b n k is defined by Equation (33).
(9) 
A = ( a n k ) ( p , c s ( F ^ ( s , r ) ) iff Equations (11) and (43) hold with b n k instead of a n k , where b n k is defined by Equation (33).
(10) 
A = ( a n k ) ( , c s ( F ^ ( s , r ) ) iff Equations (11) and (44) hold with b n k instead of a n k , where b n k is defined by Equation (33).
(11) 
A = ( a n k ) ( b v , c s ( F ^ ( s , r ) ) iff Equations (11), (44) and (46) hold with b n k instead of a n k , where b n k is defined by Equation (33).
(12) 
A = ( a n k ) ( b v 0 , c s ( F ^ ( s , r ) ) iff Equations (11) and (46) hold with b n k instead of a n k , where b n k is defined by Equation (33).

4. Discussion

The difference sequence operator was introduced for the first time in the literature by Kızmaz [38]. Kirişçi and Başar [4] have characterized and investigated generalized difference sequence spaces. The Fibonacci difference matrix F ^ , which is derived from the Fibonacci sequence ( f n ) , was recently introduced by Kara [23] in 2013 and defined the new sequence spaces p ( F ^ ) and ( F ^ ) , which are derived by the matrix domain of F ^ from the sequence spaces p and , respectively, where 1 p < . Candan [25] in 2015 introduced the sequence spaces c ( F ^ ( s , r ) ) and c 0 ( F ^ ( s , r ) ) . Later, Candan and Kara [15] studied the sequence spaces p ( F ^ ( s , r ) ) in which 1 p . In addition, Kara et al. [24] have characterized some class of compact operators in the spaces p ( F ^ ) and ( F ^ ) , where 1 p < .
The study is concerned with matrix domain on a sequences space of a triangle infinite matrix. In this article, we defined spaces b s ( F ^ ( s , r ) ) and c s ( F ^ ( s , r ) ) of Generalized Fibonacci difference of sequences, which constituted bounded and convergence series, respectively. We have demonstrated the sets of b s ( F ^ ( s , r ) ) and c s ( F ^ ( s , r ) ) , which are the linear spaces, and both spaces have the same norm
x = sup n N n k = 0 s f k f k + 1 x k + r f k + 1 f k x k 1 ,
where x b s ( F ^ ( s , r ) ) or x c s ( F ^ ( s , r ) ) . In addition, it was shown that they are normed space and Banach spaces. It was found that b s ( F ^ ( s , r ) ) and b s are linearly isomorphic as isometric. At the same time, c s ( F ^ ( s , r ) ) and c s are linearly isomorphic as isometric. Some inclusions’ theorems were given with respect to b s ( F ^ ( s , r ) ) and c s ( F ^ ( s , r ) ) . According to this, inclusions b s b s ( F ^ ( s , r ) ) , c s c s ( F ^ ( s , r ) ) are valid. In addition, if r / s < 1 / 4 , then b s ( F ^ ( s , r ) ) and c s ( F ^ ( s , r ) ) c are valid. It was concluded that c s ( 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.

5. Conclusions

In this article, we have defined spaces b s ( F ^ ( s , r ) ) and c s ( F ^ ( s , r ) ) of Generalized Fibonacci difference of sequences, which constituted bounded and convergence series, respectively. We have demonstrated that the sets of b s ( F ^ ( s , r ) ) and c s ( 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 b s ( F ^ ( s , r ) ) and c s ( F ^ ( s , r ) ) . It was concluded that c s ( 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.

Acknowledgments

The authors are grateful to the responsible editor and the anonymous reviewers for their valuable comments and suggestions, which have greatly improved this paper.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

The following abbreviations are used in this manuscript:
iffif and only if

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Yaşar, F.; Kayaduman, K. A Different Study on the Spaces of Generalized Fibonacci Difference bs and cs Spaces Sequence. Symmetry 2018, 10, 274. https://doi.org/10.3390/sym10070274

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Yaşar F, Kayaduman K. A Different Study on the Spaces of Generalized Fibonacci Difference bs and cs Spaces Sequence. Symmetry. 2018; 10(7):274. https://doi.org/10.3390/sym10070274

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Yaşar, Fevzi, and Kuddusi Kayaduman. 2018. "A Different Study on the Spaces of Generalized Fibonacci Difference bs and cs Spaces Sequence" Symmetry 10, no. 7: 274. https://doi.org/10.3390/sym10070274

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