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

Abstract

The main topic in this article is to define and examine new sequence spaces $bs(\widehat{F}(s,r))$ and $cs(\widehat{F}(s,r)))$, where $\widehat{F}(s,r)$ is generalized difference Fibonacci matrix in which $s,r\in \mathbb{R}\backslash \left\{0\right\}$. 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 $\alpha$-, $\beta$- and $\gamma$-duals of the spaces $bs(\widehat{F}(s,r))$ and $cs(\widehat{F}(s,r))$ 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:

$$1,1,2,3,5,8,13,21,34,\dots .$$

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 $n\ge 2$. The golden ratio is

$$\underset{n\to \infty }{lim}\frac{f_{n+1}}{f_n}=\frac{1+\sqrt{5}}{2}=\phi (\mathrm{Golden}\mathrm{Ratio}).$$

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 ${\ell }_{\infty }$ are called as sequences space convergent, convergent to zero and bounded, respectively. In addition to these representations, ${\ell }_1$, $bs$ 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 $k,n\in \mathbb{N}$. $A$ is called a matrix transformation from X to Y if, for every $x=(x_k)\in X$, sequence $Ax=\left\{A_n(x)\right\}$ is A transform of $x$ and in Y, where

$$A_n(x)=\sum _k{a}_{nk}{x}_k$$

(1)

and Equation (1) converges for each $n\in \mathbb{N}$.

Let $\lambda$ be a sequence space and $K$ be an infinite matrix. Then, the matrix domain ${\lambda }_K$ is introduced by

$${\lambda }_K=\left\{t=(t_k)\in w:Kt\in \lambda \right\}.$$

(2)

Here, it can be seen that ${\lambda }_K$ 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 $\widehat{F}$ which is derived from the Fibonacci sequence $(f_n)$ and defined the new sequence spaces ${\ell }_p(\widehat{F})$ and ${\ell }_{\infty }(\widehat{F})$ by using sequence spaces ${\ell }_p$ and ${\ell }_{\infty }$, respectively, where $1\le p<\infty$. The sequence space ${\ell }_p(\widehat{F})$ has been defined as:

$${\ell }_p(\hat{F})=\left\{x\in w:\widehat{F}x\in {\ell }_p\right\},(1\le p<\infty ),$$

where $\widehat{F}=(f_{nk})$ defined by the sequence $(f_n)$ as follows:

$$f_{nk}:=\left\{\begin{array}{cc}-\frac{f_{n+1}}{f_n},& k=n-1,\hfill \\ \frac{f_n}{f_{n+1}},& k=n,\hfill \\ 0,& 0\le k<n-1\mathrm{or}k>n,\hfill \end{array}\right.$$

for all $k,n\in \mathbb{N}$. In addition, Kara et al. [24] have characterized some class of compact operators on the spaces ${\ell }_p(\widehat{F})$ and ${\ell }_{\infty }(\widehat{F})$, where $1\le p<\infty$.

Candan [25] introduced $c(\widehat{F}(s,r))$ and $c_0(\widehat{F}(s,r))$. Later, Candan and Kara [15] have investigated the sequence spaces ${\ell }_p(\widehat{F}(s,r))$ in which $1\le p\le \infty$.

The $\alpha$-, $\beta$- and $\gamma$-duals $P^{\alpha },P^{\beta }$ and $P^{\gamma }$ of a sequence spaces P are defined, respectively, as

$$\begin{array}{ccc}\hfill P^{\alpha }& =& \left\{a=(a_k)\in w:at=(a_k{t}_k)\in {\ell }_1\mathrm{for}\mathrm{all}t\in P\right\},\hfill \\ \hfill P^{\beta }& =& \left\{a=(a_k)\in w:at=(a_k{t}_k)\in cs\mathrm{for}\mathrm{all}t\in P\right\},\hfill \\ \hfill P^{\gamma }& =& \left\{a=(a_k)\in w:at=(a_k{t}_k)\in bs\mathrm{for}\mathrm{all}t\in P\right\},\hfill \end{array}$$

respectively.

In Section 2, sequence space $bs(\widehat{F})$ and $cs(\widehat{F})$ are defined and some algebric properties of them are investigated. In the last section, the $\alpha$-, $\beta$- and $\gamma$-duals of the spaces $bs(\widehat{F})$ and $cs(\widehat{F})$ are found and some matrix tranformations of them are given.

2. Generalized Fibonacci Difference Spaces of $bs$ and $cs$ Sequences

In this section, spaces $bs(\widehat{F}(s,r))$ and $cs(\widehat{F}(s,r))$ 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 $bs(\widehat{F}(s,r))$ and $cs(\widehat{F}(s,r))$ as the sets of all sequences whose $\widehat{F}(s,r)=\left\{f_{nk}(s,r)\right\}$ transforms are in the sequence space $bs$ and $cs$,

$$\begin{array}{ccc}\hfill bs(\widehat{F}(s,r))& =& \left\{x=(x_k)\in w:\underset{n\in \mathbb{N}}{sup}\left|{\displaystyle \sum _{k=0}^n}\left(s\frac{f_k}{f_{k+1}}{x}_k+r\frac{f_{k+1}}{f_k}{x}_{k-1}\right)\right|<\infty \right\},\hfill \\ \hfill cs(\widehat{F}(s,r))& =& \left\{x=(x_k)\in w:\underset{k=0}{\sum ^n}\left(s\frac{f_k}{f_{k+1}}{x}_k+r\frac{f_{k+1}}{f_k}{x}_{k-1}\right)\in c\right\},\hfill \end{array}$$

where $\widehat{F}(s,r)=\left\{f_{nk}(s,r)\right\}$ is

$$f_{nk}(s,r):=\left\{\begin{array}{cc}r\frac{f_{n+1}}{f_n},\hfill & k=n-1,\hfill \\ s\frac{f_n}{f_{n+1}},\hfill & k=n,\hfill \\ 0,\hfill & k<n-1\mathrm{or}0\le k>n,\hfill \end{array}\right.$$

(3)

for all $k,n\in \mathbb{N}$ where $s,r\in \mathbb{R}\backslash \left\{0\right\}$. Actually, by using Equation (2), we can get

$$bs(\widehat{F}(s,r))={(bs)}_{\widehat{F}(s,r)}\mathrm{and}cs(\widehat{F}(s,r))={(cs)}_{\widehat{F}(s,r)}.$$

With a basic calculation, we can find the inverse matrix of $\widehat{F}(s,r)=\left\{f_{nk}(s,r)\right\}$. The inverse matrix of $\widehat{F}(s,r)=\left\{f_{nk}(s,r)\right\}$ is ${\widehat{F}}^{-1}(s,r)=\left(f_{nk}^{-1}(s,r)\right)$ such that

$$f_{nk}^{-1}(s,r)=\left\{\begin{array}{cc}\frac{1}{s}{(-\frac{r}{s})}^{n-k}\frac{f_{n+1}^2}{f_k{f}_{k+1}},& 0\le k<n,\\ 0,& k>n,\end{array}\right.$$

(4)

for all $k,n\in \mathbb{N}$. If $y=(y_n)$ is $\widehat{F}(s,r)$-transform of a sequence $x=(x_n)$, then the below equality is justified:

$$y_n={(\widehat{F}(s,r)x)}_n=\left\{\begin{array}{cc}s{x}_0,\hfill & n=0,\hfill \\ s\frac{f_n}{f_{n+1}}{x}_n+r\frac{f_{n+1}}{f_n}{x}_{n-1},\hfill & n\ge 1,\hfill \end{array}\right.$$

(5)

for all $n\in \mathbb{N}.$ In this situation, we see that $x_n={\widehat{F}}^{-1}(s,r)y$, i.e.,

$$x_n=\stackrel{n}{\sum _{k=0}}\frac{1}{s}{(-\frac{r}{s})}^{n-k}\frac{f_{n+1}^2}{f_k{f}_{k+1}}{y}_k$$

(6)

for all $n\in \mathbb{N}$.

Theorem 1.

$bs(\widehat{F}(s,r))$ 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.

$cs(\widehat{F}(s,r))$ 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 $bs(\widehat{F}(s,r))$ is a normed space with

$${∥x∥}_{bs(\widehat{F}(s,r))}=\underset{n\in \mathbb{N}}{sup}\left|\underset{k=0}{\sum ^n}\left(s\frac{f_k}{f_{k+1}}{x}_k+r\frac{f_{k+1}}{f_k}{x}_{k-1}\right)\right|.$$

(7)

Proof. 

It is clear that space $bs(\widehat{F}(s,r))$ ensures normed space conditions. ☐

Theorem 4.

The space $cs(\widehat{F}(s,r))$ is a normed space with norm Equation (7).

Proof. 

It is clear that normed space conditions are ensured by space $cs(\widehat{F}(s,r))$. ☐

Theorem 5.

$bs(\widehat{F}(s,r))$ is linearly isomorphic as isometric to the space $bs,$ that is, $bs(\widehat{F}(s,r))\cong bs$.

Proof. 

For proof, we must demonstrate that bijection and linearly transformation T exist between the space $bs(\widehat{F}(s,r))$ and $bs$. Let us take the transformation $T:bs(\widehat{F}(s,r))\to bs$ mentioned above with the help of Equation (5) by $Tx=\widehat{F}(s,r)x$. 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 $y=(y_n)\in bs$.

In this case, by using Equations (6) and (7), we find

$$\begin{array}{ccc}\hfill {∥x∥}_{bs(\widehat{F}(s,r))}& =& \underset{n\in \mathbb{N}}{sup}\left|\underset{k=0}{\sum ^n}\left(s\frac{f_k}{f_{k+1}}{x}_k+r\frac{f_{k+1}}{f_k}{x}_{k-1}\right)\right|\hfill \\ & =& \underset{n\in \mathbb{N}}{sup}\left|\underset{k=0}{\sum ^n}\left[s\frac{f_k}{f_{k+1}}\left(\underset{i=0}{\sum ^k}-\frac{1}{s}{(-\frac{r}{s})}^{k-i}\frac{f_{k+1}^2}{f_i{f}_{i+1}}{y}_i\right)\right.\right.\hfill \\ & & \left.\left.+r\frac{f_{k+1}}{f_k}\left(\underset{i=0}{\sum ^{k-1}}-\frac{1}{s}{(-\frac{r}{s})}^{k-i-1}\frac{f_k^2}{f_i{f}_{i+1}}{y}_i\right)\right]\right|\hfill \\ & =& \underset{n\in \mathbb{N}}{sup}\left|\underset{k=0}{\sum ^n}{y}_k\right|={∥y∥}_{bs}.\hfill \end{array}$$

This result shows that $x\in bs(\widehat{F}(s,r))$. That is, T is surjective. At the same time, this result also indicates that T is preserving the norm. Therefore, the sequence spaces $bs(\widehat{F}(s,r))$ and $bs$ are linearly isomorphic as isometric.

Theorem 6.

The sequence space $cs(\widehat{F}(s,r))$ is linearly isomorphic as isometric to the space $cs$, that is, $cs(\widehat{F}(s,r))\cong cs$.

Proof. 

If we write $cs$ instead of $bs$ and $cs(\widehat{F}(s,r))$ instead of $bs(\widehat{F}(s,r))$ in Theorem 5, the proof will be demonstrated. ☐

Theorem 7.

The space $bs(\widehat{F}(s,r))$ 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 $bs(\widehat{F}(s,r))$ for all $i\in \mathbb{N}$. By using Equation (5), we have

$$y_k^i=s\frac{f_k}{f_{k+1}}{x}_k^i+r\frac{f_{k+1}}{f_k}{x}_{k-1}^i$$

for all $i,k\in \mathbb{N}$. Since $x^i=(x_k^i)$ is a Cauchy sequence, for every $\epsilon >0$, there exists $n_0=n_0(\epsilon )$ such that

$$\begin{array}{ccc}\hfill {∥x^i-x^m∥}_{bs(\widehat{F}(s,r))}& =& \underset{n\in \mathbb{N}}{sup}\left|\underset{k=0}{\sum ^n}\left(s\frac{f_k}{f_{k+1}}(x_k^i-x_k^m)+r\frac{f_{k+1}}{f_k}(x_{k-1}^i-x_{k-1}^m)\right)\right|\hfill \\ & =& \underset{n\in \mathbb{N}}{sup}\left|\underset{k=0}{\sum ^n}\left(y_k^i-y_k^m\right)\right|={∥y^i-y^m∥}_{bs}<\epsilon \hfill \end{array}$$

for all $i,m\ge n_0$. Since $bs$ is complete, $y^i\to y(i\to \infty )$ such that $y\in bs$ exist and since the sequence spaces $bs(\widehat{F}(s,r))$ and $bs$ are linearly isomorphic as isometric $bs(\widehat{F}(s,r))$ is complete. Consequently, $bs(\widehat{F}(s,r))$ is a Banach space. ☐

Theorem 8.

The space $cs(\widehat{F}(s,r))$ 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 $cs(\widehat{F}(s,r))$ for all $i\in \mathbb{N}$. By using Equation (5), we have

$$y_k^i=s\frac{f_k}{f_{k+1}}{x}_k^i+r\frac{f_{k+1}}{f_k}{x}_{k-1}^i$$

for all $i,k\in \mathbb{N}$. Since $x^i=(x_k^i)$ is a Cauchy sequence, for every $\epsilon >0$, there exists $n_0=n_0(\epsilon )$ such that

$$\begin{array}{ccc}\hfill {∥x^i-x^m∥}_{cs(\widehat{F}(s,r))}& =& \underset{n\in \mathbb{N}}{sup}\left|\underset{k=0}{\sum ^n}\left(s\frac{f_k}{f_{k+1}}(x_k^i-x_k^m)+r\frac{f_{k+1}}{f_k}(x_{k-1}^i-x_{k-1}^m)\right)\right|\hfill \\ & =& \underset{n\in \mathbb{N}}{sup}\left|\underset{k=0}{\sum ^n}\left(y_k^i-y_k^m\right)\right|={∥y^i-y^m∥}_{cs}<\epsilon \hfill \end{array}$$

for all $i,m\ge n_0$. Since $cs$ is complete, $y^i\to y(i\to \infty )$ such that $y\in cs$ exists and since the sequence spaces $cs(\widehat{F}(s,r))$ and $cs$ are linearly isomorphic as isometric $cs(\widehat{F}(s,r))$ is complete. Consequently, $cs(\widehat{F}(s,r))$ is a Banach space. ☐

Now, let $A=(a_{nk})$ be an arbitrary infinite matrix and list the following:

$$\underset{n\in \mathbb{N}}{sup}\sum _k\left|a_{nk}\right|<\infty ,$$

(8)

$$\underset{k}{lim}{a}_{nk}=0\mathrm{for}\mathrm{each}n\in \mathbb{N},$$

(9)

$$\underset{m}{sup}\sum _k\left|\stackrel{m}{\sum _{n=0}}(a_{nk}-a_{n,k+1})\right|<\infty ,$$

(10)

$$\underset{n}{lim}\sum _k{a}_{nk}=\alpha \mathrm{for}\mathrm{each}k\in \mathbb{N},\alpha \in \mathbb{C},$$

(11)

$$\underset{n}{sup}\sum _k\left|a_{nk}-a_{n,k+1}\right|<\infty ,$$

(12)

$$\underset{n}{lim}{a}_{nk}={\alpha }_k\mathrm{for}\mathrm{each}k\in \mathbb{N},{\alpha }_k\in \mathbb{C},$$

(13)

$$\underset{N,K\in \mathcal{F}}{sup}\left|\sum _{n\in N}\sum _{k\in K}(a_{nk}-a_{n,k+1})\right|<\infty ,$$

(14)

$$\underset{N,K\in \mathcal{F}}{sup}\left|\sum _{n\in N}\sum _{k\in K}(a_{nk}-a_{n,k-1})\right|<\infty ,$$

(15)

$$\underset{n}{lim}(a_{nk}-a_{n,k+1})=\alpha \mathrm{for}\mathrm{each}k\in \mathbb{N},\alpha \in \mathbb{C},$$

(16)

$$\underset{n\to \infty }{lim}\sum _k\left|a_{nk}-a_{n,k+1}\right|=\sum _k\left|\underset{n\to \infty }{lim}(a_{nk}-a_{n,k+1})\right|,$$

(17)

$$\underset{n}{sup}\left|\underset{k}{lim}{a}_{nk}\right|<\infty ,$$

(18)

$$\underset{n\to \infty }{lim}\sum _k\left|a_{nk}-a_{n,k+1}\right|=0\mathrm{uniformly}\mathrm{in}n,$$

(19)

$$\underset{m}{lim}\sum _k\left|\underset{n=0}{\sum ^m}(a_{nk}-a_{n,k+1})\right|=0,$$

(20)

$$\underset{m}{lim}\sum _k\left|\underset{n=0}{\sum ^m}(a_{nk}-a_{n,k+1})\right|=\sum _k\left|\sum _n(a_{nk}-a_{n,k+1})\right|,$$

(21)

$$\underset{N,K\in \mathcal{F}}{sup}\left|\sum _{n\in N}\sum _{k\in K}[(a_{nk}-a_{n,k+1})-(a_{n-1,k}-a_{n-1,k+1})]\right|<\infty ,$$

(22)

$$\underset{m\in \mathbb{N}}{sup}\left|\underset{k}{lim}\underset{n=0}{\sum ^m}{a}_{nk}\right|<\infty ,$$

(23)

$$\exists {\alpha }_k\in \mathbb{C}\ni \sum _n{a}_{nk}={\alpha }_k\mathrm{for}\mathrm{each}k\in \mathbb{N},$$

(24)

$$\underset{N,K\in \mathcal{F}}{sup}\left|\sum _{n\in N}\sum _{k\in K}\left[(a_{nk}-a_{n-1,k})-(a_{n,k-1}-a_{n-1,k-1})\right]\right|<\infty ,$$

(25)

where $\mathcal{F}$ denote the collection of all finite subsets of $\mathbb{N}$.

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 $A=(a_{nk})$ be an arbitrary infinite matrix. Then,

(1) 

$A=(a_{nk})\in (bs,{\ell }_{\infty })$ iff Equations (9) and (12) hold (Stieglitz and Tietz [26]),

(2) 

$A=(a_{nk})\in (cs,c)$ iff Equations (12) and (13) hold (Wilansky [27]),

(3) 

$A=(a_{nk})\in (bs,{\ell }_1)$ iff Equations (9) and (14) hold (K.-G. Grosse-Erdman [28]).

(4) 

$A=(a_{nk})\in (cs,{\ell }_1)$ iff Equation (15) holds (Stieglitz and Tietz [26]).

(5) 

$A=(a_{nk})\in (bs,c)$ iff Equations (9), (16) and (17) hold (K.-G. Grosse-Erdman [28]).

(6) 

$A=(a_{nk})\in (cs,{\ell }_{\infty })$ iff Equations (12) and (18) hold (Stieglitz and Tietz [26]).

(7) 

$A=(a_{nk})\in (bs,c_0)$ iff Equations (9) and (19) hold (Stieglitz and Tietz [26]).

(8) 

$A=(a_{nk})\in (bs,c{s}_0)$ iff Equations (9) and (20) hold (Zeller [29]).

(9) 

$A=(a_{nk})\in (bs,cs)$ iff Equations (9) and (21) hold (Zeller [29]).

(10) 

$A=(a_{nk})\in (bs,bv)$ iff Equations (9) and (22) hold (Zeller [29]).

(11) 

$A=(a_{nk})\in (bs,bs)$ iff Equations (9) and (10) hold (Zeller [29]).

(12) 

$A=(a_{nk})\in (cs,cs)$ iff Equations (10) and (11) hold (Hill, [30]).

(13) 

$A=(a_{nk})\in (bs,b{v}_0)$ iff Equations (12), (19) and (22) hold (Stieglitz and Tietz [26]).

(14) 

$A=(a_{nk})\in (cs,c_0)$ iff Equation (12) holds and Equation (13) also holds with ${\alpha }_k=0$ for all $k\in \mathbb{N}$ (Dienes [31]).

(15) 

$A=(a_{nk})\in (cs,bs)$ iff Equations (10) and (23) hold (Zeller [29]).

(16) 

$A=(a_{nk})\in (cs,c{s}_0)$ iff Equation (10) holds and Equation (24) also holds with ${\alpha }_k=0$ for all $k\in \mathbb{N}$ (Zeller [29]).

(17) 

$A=(a_{nk})\in (cs,bv)$ iff Equation (25) holds (Zeller [29]).

(18) 

$A=(a_{nk})\in (cs,b{v}_0)$ iff Equation (25) holds and Equation (13) also holds with ${\alpha }_k=0$ for all $k\in \mathbb{N}$ (Stieglitz and Tietz [26]).

Theorem 9.

The inclusion $bs\subset bs(\widehat{F}(s,r))$ is valid.

Proof. 

Let $x\in bs$. We must demonstrate that $x\in bs(\widehat{F}(s,r))$. It means that $\widehat{F}(s,r)\in (bs,bs)$. For $\widehat{F}(s,r)\in (bs,bs)$, $\widehat{F}(s,r)$ must ensure to the conditions of (11) of Lemma 1. We see that

$$\underset{k}{lim}{f}_{nk}(s,r)=0\mathrm{for}\mathrm{each}n\in \mathbb{N}.$$

The other condition also holds as follows:

$$\begin{array}{ccc}\hfill \underset{m}{sup}\sum _k\left|\stackrel{m}{\sum _{n=0}}(f_{nk}(s,r)-f_{n,k+1}(s,r))\right|& =& \underset{m}{sup}\underset{p}{lim}\left(\frac{\left|s+r\right|}{f_{1.}{f}_2}+\frac{\left|s+r\right|}{f_{2.}{f}_3}+\dots +\frac{\left|s+r\right|}{f_{p+1.}{f}_{p+2}}\right)\hfill \\ & =& \frac{17}{10}\left|s+r\right|<\infty .\hfill \end{array}$$

Consequently, the conditions of (11) of Lemma 1 hold. The proof is complete. ☐

Theorem 10.

If $\left|r/s\right|<1/4$, then $bs(\widehat{F}(s,r))\subset {\ell }_{\infty }$ is valid.

Proof. 

Let $x\in bs(\widehat{F}(s,r))$. Then, $y=\widehat{F}(s,r)x\in bs$. We must demonstrate that $x={\widehat{F}}^{-1}(s,r)y\in {\ell }_{\infty }$. That is, ${\widehat{F}}^{-1}(s,r)\in (bs,{\ell }_{\infty })$. For ${\widehat{F}}^{-1}(s,r)\in (bs,{\ell }_{\infty })$, ${\widehat{F}}^{-1}(s,r)$ must satisfy the conditions of (1) of Lemma 1. It is clear that

$$\underset{k}{lim}{f}_{nk}^{-1}(s,r)=0\mathrm{for}\mathrm{each}n\in \mathbb{N}.$$

The other condition is also holds as follows:

$$\begin{array}{ccc}\hfill \underset{n}{sup}\sum _k\left|(f_{nk}^{-1}(s,r)-f_{n,k+1}^{-1}(s,r))\right|& \le & 2\underset{n}{sup}\sum _k\left|(f_{nk}^{-1}(s,r)\right|-\left|\frac{r}{s}\right|\hfill \\ & \le & \frac{4}{s}\sum _k{\left(\frac{4r}{s}\right)}^k<\infty .\hfill \end{array}$$

(26)

Consequently, the conditions of (1) of Lemma 1 hold. The proof is complete. ☐

Theorem 11.

The inclusion $cs\subset cs(\widehat{F}(s,r))$ is valid.

Proof. 

Let $x\in cs$. We must demonstrate that $x\in cs(\widehat{F}(s,r))$. It means that $\widehat{F}(s,r)\in (cs,cs)$. For $\widehat{F}(s,r)\in (cs,cs)$, $\widehat{F}(s,r)$ 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 $k\in \mathbb{N}$,

$$\underset{n\to \infty }{lim}\sum _k{f}_{nk}(s,r)=\underset{n}{lim}(s\frac{f_n}{f_{n+1}}+r\frac{f_{n+1}}{f_n})=\frac{s}{\phi }+r\phi =\ell$$

such that $\ell \in \mathbb{C}$ exist. Consequently, the conditions of (12) of Lemma 1 hold. The proof is complete. ☐

Theorem 12.

If $\left|r/s\right|<1/4$, then $cs(\widehat{F}(s,r))\subset c$ is valid.

Proof. 

Let $x\in cs(\widehat{F}(s,r))$. Then, $y=\widehat{F}(s,r)x\in cs$. We must demonstrate that $x={\widehat{F}}^{-1}(s,r)y\in c$. That is, ${\widehat{F}}^{-1}(s,r)\in (cs,c)$. For ${\widehat{F}}^{-1}(s,r)\in (cs,c)$, ${\widehat{F}}^{-1}(s,r)$ 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 $k\in \mathbb{N},$

$$\begin{array}{ccc}\hfill \underset{n}{lim}{f}_{nk}^{-1}(s,r)& \le & \underset{n}{lim}\left|f_{nk}^{-1}(s,r)\right|=\underset{n}{lim}\left|\frac{f_{n+1}}{s{f}_n}{\left(-\frac{r}{s}\right)}^{n-k}\frac{\frac{f_{n+1}}{f_{k+1}}}{\frac{f_k}{f_n}}\right|=\underset{n}{lim}\left|\frac{f_{n+1}}{s{f}_n}\stackrel{n-1}{\prod _{i=k}}\frac{r\frac{f_{i+2}}{f_{i+1}}}{s\frac{f_i}{f_{i+1}}}\right|\hfill \\ & \le & \underset{n}{lim}\frac{f_{n+1}}{\left|s\right|f_n}\stackrel{n-1}{\prod _{i=k}}\left|\frac{{sup}_{i\in \mathbb{N}}r\frac{f_{i+2}}{f_{i+1}}}{{inf}_{i\in \mathbb{N}}s\frac{f_i}{f_{i+1}}}\right|\le \underset{n}{lim}\frac{f_{n+1}}{\left|s\right|f_n}{\left(\frac{4r}{s}\right)}^{n-k}=\frac{\phi }{\left|s\right|}.0=0.\hfill \end{array}$$

Thus, Equation (13) is also satisfied. ☐

Theorem 13.

The inclusion $cs(\widehat{F}(s,r))\subset bs(\widehat{F}(s,r))$ is valid.

Proof. 

Let $x\in cs(\widehat{F}(s,r))$. Then, $y=\widehat{F}(s,r)x\in cs$. Hence, ${\sum }_k\widehat{F}(s,r)x\in c$. $c\subset {\ell }_{\infty }$, so it becomes ${\sum }_k\widehat{F}(s,r)x\in {\ell }_{\infty }$. That is, $\widehat{F}(s,r)x\in bs$. Hence, $x\in bs(\widehat{F}(s,r))$. Consequently, $cs(\widehat{F}(s,r))\subset bs(\widehat{F}(s,r))$.

Before giving the corollary about the Schauder basis for the space $cs(\widehat{F}(r,s))$, let us define the Schauder basis which was introduced by J. Schauder in 1927. Let $(X,∥.∥)$ be normed space and be a sequence $(a_k)\in X$. There exists a unique sequence $({\lambda }_k)$ of scalars such that $x={\displaystyle \sum _{k=0}^{\infty }}{\lambda }_k{a}_k$, and

$$\underset{n\to \infty }{lim}∥x-\underset{k=0}{\sum ^n}{\lambda }_k{a}_k∥=0.$$

Then, $(a_k)$ is called a Schauder basis for X. ☐

Now, we can give the corollary about Schauder basis.

Corollary 1.

Let us sequence $b^{(k)}={\left\{b_n^{(k)}\right\}}_{n\in \mathbb{N}}$ defined in the $cs(\widehat{F}(s,r))$ such that

$$b_n^{(k)}=\left\{\begin{array}{cc}\frac{1}{s}{(-\frac{r}{s})}^{n-k}\frac{f_{n+1}^2}{f_k{f}_{k+1}},\hfill & n>k,\hfill \\ \frac{1}{s}\frac{f_{k+1}}{f_k},\hfill & n=k,\hfill \\ 0,\hfill & n<k.\hfill \end{array}\right.$$

Then, sequence ${\left\{b^{(k)}\right\}}_{n\in \mathbb{N}}$ is a basis of $cs(\widehat{F}(s,r))$ and every sequence $x\in cs(\widehat{F}(s,r))$ has a unique representation $x={\displaystyle \sum _k}{y}_k{b}^k$, where $y_k={(\widehat{F}(s,r)x)}_k$.

3. The $\alpha$-, $\beta$- and $\gamma$-Duals of the Spaces $bs(\widehat{F}(s,r))$ and $cs(\widehat{F}(s,r))$ and Some Matrix Transformations

In this section, the alpha-, beta-, gamma-duals of the spaces $bs(\widehat{F}(s,r))$ and $cs(\widehat{F}(s,r))$ are determined and characterized the classes of infinite matrices from the space $bs(\widehat{F}(s,r))$ and $cs(\widehat{F}(s,r))$ 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 $a=(a_n)\in w$ and the infinite matrix $B=(b_{nk})$ is defined by $B_n=a_n{({\widehat{F}}^{-1}(s,r))}_n$, that is,

$$b_{nk}=\left\{\begin{array}{cc}{a}_n{f}_{nk}^{-1}(s,r),\hfill & 0\le k<n,\hfill \\ 0,\hfill & k>n,\hfill \end{array}\right.$$

for all $k,n\in \mathbb{N},\delta \in \left\{bs,cs\right\}$. Then, $a\in {\{\delta (\widehat{F}(s,r))\}}^{\alpha }$ iff $B\in (\delta ,{\ell }_1)$.

Proof. 

Let $a=(a_n)$ and $x=(x_n)$ be an arbitrary subset of w. $y=(y_n)$ such that $y=\widehat{F}(s,r)x,$ which is defined by Equation (5). Then,

$$a_n{x}_n=a_n{({\widehat{F}}^{-1}(s,r)y)}_n={(By)}_n$$

(27)

for all $n\in \mathbb{N}$. Hence, we obtain by Equation (5) that $ax=(a_n{x}_n)\in {\ell }_1$ with $x=(x_n)\in \delta (\widehat{F}(s,r))$ iff $By\in {\ell }_1$ with $y\in \delta$. That is, $B\in (\delta ,{\ell }_1)$. ☐

Lemma 3.

Let [32] $C=(c_{nk})$ be defined via a sequence $a=(a_k)\in w$ and the inverse matrix $V=(v_{nk})$ of the triangle matrix $U=(u_{nk})$ by

$$c_{nk}=\left\{\begin{array}{cc}\sum _{j=k}^n{a}_j{v}_{jk},& 0\le k<n,\\ 0,& k>n,\end{array}\right.$$

for all $k,n\in \mathbb{N}$. Then, for any sequence space λ,

$$\begin{array}{ccc}\hfill {\lambda }_U^{\gamma }& =& \left\{a=(a_k)\in w:C\in (\lambda ,{\ell }_{\infty })\right\},\hfill \\ \hfill {\lambda }_U^{\beta }& =& \left\{a=(a_k)\in w:C\in (\lambda ,c)\right\}.\hfill \end{array}$$

If we consider Lemmas 1–3 together, the following is obtained.

Corallary 1.

Let $B=(b_{nk})$ and $C=(c_{nk})$ such that

$$b_{nk}=\left\{\begin{array}{cc}{a}_n{f}_{nk}^{-1}(s,r),& 0\le k<n\\ 0,& k>n\end{array}\right.and{c}_{nk}=\sum _{j=k}^n\frac{1}{s}{(-\frac{r}{s})}^{j-k}\frac{f_{j+1}^2}{f_k{f}_{k+1}}{a}_j.$$

If we take $t_1,t_2,t_3,t_4,t_5,t_6,t_7$ and $t_8$ as follows:

$$\begin{array}{ccc}\hfill t_1& =& \left\{a=(a_k)\in w:\underset{N,K\in \mathcal{F}}{sup}\left|\sum _{n\in N}\sum _{k\in K}(b_{nk}-b_{n,k+1})\right|<\infty \right\},\hfill \\ \hfill t_2& =& \left\{a=(a_k)\in w:\underset{N,K\in \mathcal{F}}{sup}\left|\sum _{n\in N}\sum _{k\in K}(b_{nk}-b_{n,k-1})\right|<\infty \right\},\hfill \\ \hfill t_3& =& \left\{a=(a_k)\in w:\underset{k\to \infty }{lim}{c}_{nk}=0\right\},\hfill \\ \hfill t_4& =& \left\{a=(a_k)\in w:\exists \alpha \in \mathbb{C}\ni \underset{n\to \infty }{lim}(c_{nk}-c_{n,k+1})=\alpha \right\},\hfill \\ \hfill t_5& =& \left\{a=(a_k)\in w:\underset{n\to \infty }{lim}\sum _k\left|c_{nk}-c_{n,k+1}\right|=\sum _k\left|\underset{n\to \infty }{lim}(c_{nk}-c_{n,k+1})\right|\right\},\hfill \\ \hfill t_6& =& \left\{a=(a_k)\in w:\exists \alpha \in \mathbb{C}\underset{n\to \infty }{lim}{c}_{nk}=\alpha ,forallk\in \mathbb{N}\right\},\hfill \\ \hfill t_7& =& \left\{a=(a_k)\in w:\underset{n\in \mathbb{N}}{sup}\sum _k\left|c_{nk}-c_{n,k+1}\right|<\infty \right\},\hfill \\ \hfill t_8& =& \left\{a=(a_k)\in w:\underset{n\in \mathbb{N}}{sup}\left|\underset{k}{lim}{c}_{nk}\right|<\infty \right\}.\hfill \end{array}$$

Then, the following statements hold:

(1) 

${\{bs(\widehat{F}(s,r))\}}^{\alpha }=t_1,$

(2) 

${\{cs(\widehat{F}(s,r))\}}^{\alpha }=t_2,$

(3) 

${\{bs(\widehat{F}(s,r))\}}^{\beta }=t_3\cap t_4\cap t_5,$

(4) 

${\{cs(\widehat{F}(s,r))\}}^{\beta }=t_6\cap t_7,$

(5) 

${\{bs(\widehat{F}(s,r))\}}^{\gamma }=t_3\cap t_7,$

(6) 

${\{cs(\widehat{F}(s,r))\}}^{\gamma }=t_7\cap t_8.$

Theorem 14.

Let $\lambda \in \left\{bs,cs\right\}$ and μ ⊂ w. Then, $A=(a_{nk})\in (\lambda (\widehat{F}(s,r)),\mu )$ iff

$$D^m=(d_{nk}^{(m)})\in (\lambda ,c)foralln\in \mathbb{N},$$

(28)

$$D=(d_{nk})\in (\lambda ,\mu ),$$

(29)

where

$$d_{nk}^{(m)}=\left\{\begin{array}{cc}\sum _{j=k}^m\frac{1}{s}{(-\frac{r}{s})}^{j-k}\frac{f_{j+1}^2}{f_k{f}_{k+1}}{a}_{nj},\hfill & 0\le k<m,\hfill \\ 0,\hfill & k>m,\hfill \end{array}\right.$$

(30)

and

$$d_{nk}=\sum _{j=k}^{\infty }\frac{1}{s}{(-\frac{r}{s})}^{j-k}\frac{f_{j+1}^2}{f_k{f}_{k+1}}{a}_{nj}$$

(31)

for all $k,m,n\in \mathbb{N}$.

Proof. 

To prove the necessary part of the theorem, let us suppuse that $A=(a_{nk})\in (\lambda (\widehat{F}(s,r),\mu )$ and $x=(x_k)\in \lambda (\widehat{F}(s,r))$. By using Equation (6), we find

$$\begin{array}{ccc}\hfill \sum _{k=0}^m{a}_{nk}{x}_k& =& \sum _{k=0}^m{a}_{nk}\sum _{j=o}^k\frac{1}{s}{(-\frac{r}{s})}^{k-j}\frac{f_{k+1}^2}{f_j{f}_{j+1}}{y}_j\hfill \\ & =& \sum _{k=0}^m\sum _{j=k}^m\frac{1}{s}{(-\frac{r}{s})}^{j-k}\frac{f_{j+1}^2}{f_k{f}_{k+1}}{a}_{nj}{y}_k=\sum _{k=0}^m{d}_{nk}^{(m)}{y}_k=D_n^{(m)}(y)\hfill \end{array}$$

(32)

for all $m,n\in \mathbb{N}$. For each $m\in \mathbb{N}$ and $x=(x_k)\in \lambda (\widehat{F}(s,r))$, $A_m(x)$ exists and also lies in c. Then, $D_n^{(m)}$ also lies in c for each $m\in \mathbb{N}$. Hence, $D^{(m)}\in (\lambda ,c)$. Now, from Equation (32), we consider for $m\to \infty$, and then $Ax=Dy$. Consequently, we obtain $D=(d_{nk})\in (\lambda ,\mu )$.

If we want to prove the sufficient part of the theorem, then let us assume that Equations (28) and (29) are satisfied and $x=(x_k)\in \lambda (\widehat{F}(s,r))$. By using Corollary 1 and Equations (28) and (32), we obtain $y=\widehat{F}(s,r)x\in \lambda$ and $D_n^{(m)}(y)={\sum }_{k=0}^m{d}_{nk}^{(m)}{y}_k={\sum }_{k=0}^m{a}_{nk}{x}_k=A_n^{(m)}(x)\in c$. Hence, $A={(a_{nk})}_{k\in \mathbb{N}}$ exists. In addition, in Equation (32), if we consider $m\to \infty$. Then, $Ax=Dy$. Consequently, we obtain $A=(a_{nk})\in (\lambda (\widehat{F}(s,r)),\mu )$.

In Theorem 14, we take $\lambda (\widehat{F}(s,r))$ instead of $\mu$ and $\mu$ instead of $\lambda (\widehat{F}(s,r))$, and then we get the following theorem. ☐

Theorem 15.

Let $\lambda \in \left\{bs,cs\right\}$ and μ be an arbitrary subset of $w$ and $A=(a_{nk})$ and $B=(b_{nk})$ be infinite matrices. If we take

$$b_{nk}:=r\frac{f_{n+1}}{f_n}{a}_{n-1,k}+s\frac{f_n}{f_{n+1}}{a}_{nk}$$

(33)

for all $k,n\in \mathbb{N}$, then $A\in (\mu ,\lambda (\widehat{F}(s,r)))$ iff $B\in (\mu ,\lambda )$.

Proof. 

Let us suppose that $A\in (\mu ,\lambda (\widehat{F}(s,r)))$ and Equation (33) exist. For $z=(z_k)\in \mu ,$ we obtain $Az\in \lambda (\widehat{F}(s,r))$ from $A\in (\mu ,\lambda (\widehat{F}(s,r)))$. Hence, $\widehat{F}(s,r)(Az)\in \lambda$. On the other hand, we have

$$\sum _{k=0}^m{b}_{nk}{z}_k=\sum _{k=0}^m\left(r\frac{f_{n+1}}{f_n}{a}_{n-1,k}+s\frac{f_n}{f_{n+1}}{a}_{nk}\right)z_k$$

(34)

for all $m,n\in \mathbb{N}$. If we carry out $m\to \infty$ to Equation (34), we obtain that

$${(Bz)}_n={((\widehat{F}(s,r)A)z)}_n={(\widehat{F}(s,r)(Az))}_n.$$

(35)

Since $\widehat{F}(s,r)(Az)\in \lambda$, we find $Bz={(Bz)}_n\in \lambda$ for $z=(z_k)\in \mu$ from Equation (35). Hence, we obtain that $B\in (\mu ,\lambda )$. 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 $x=(x_k)\in {\ell }_{\infty }$. x is said to be almost convergent to the generalized limit ℓ iff $\underset{m\to \infty }{lim}{\sum }_{k=0}^m\frac{x_{n+k}}{m+1}=\ell$ uniformly in n and is denoted by $f-limx=\ell$. By f and $f_0$, we indicate the space of all almost convergent and almost null sequences, respectively. However, in this article, we use $\widehat{c}$ and ${\widehat{c}}_0$ 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 $\widehat{c}s$, we indicate the space of sequences, which is composed of all almost convergent series. The sequences spaces $\widehat{c}$ and ${\widehat{c}}_0$ are

$$\begin{array}{ccc}\hfill {\widehat{c}}_0& =& \left\{x=(x_k)\in {\ell }_{\infty }:\underset{m\to \infty }{lim}\sum _{k=0}^m\frac{x_{n+k}}{m+1}=0\mathrm{uniformly}\mathrm{in}n\right\},\hfill \\ \hfill \widehat{c}& =& \left\{x=(x_k)\in {\ell }_{\infty }:\exists \ell \in \mathbb{C}\ni \underset{m\to \infty }{lim}\sum _{k=0}^m\frac{x_{n+k}}{m+1}=\ell \mathrm{uniformly}\mathrm{in}n\right\}.\hfill \end{array}$$

Now, let $A=(a_{nk})$ be an arbitrary infinite matrix and list the following conditions:

$$\exists {\alpha }_k\in \mathbb{C}\ni f-lim{a}_{nk}={\alpha }_k\mathrm{for}\mathrm{each}k\in \mathbb{N},$$

(36)

$$\underset{q}{lim}\sum _k\frac{1}{q+1}\left|\underset{i=0}{\sum ^q}△\left[\underset{j=0}{\sum ^{n+i}}(a_{jk}-{\alpha }_k)\right]\right|=0\mathrm{uniformly}\mathrm{in}n,$$

(37)

$$\underset{n\in \mathbb{N}}{sup}\sum _k\left|△\left[\underset{j=0}{\sum ^n}{a}_{jk}\right]\right|<\infty ,$$

(38)

$$\exists {\alpha }_k\in \mathbb{C}\ni f-lim\underset{j=0}{\sum ^n}{a}_{jk}={\alpha }_k\mathrm{for}\mathrm{each}k\in \mathbb{N},$$

(39)

$$\underset{n\in \mathbb{N}}{sup}\sum _k\left|\underset{j=0}{\sum ^n}{a}_{jk}\right|<\infty ,$$

(40)

$$\exists {\alpha }_k\in \mathbb{C}\ni \sum _n\sum _k{a}_{nk}={\alpha }_k\mathrm{for}\mathrm{all}k\in \mathbb{N},$$

(41)

$$\underset{n}{lim}\sum _k\left|△\left[\underset{j=0}{\sum ^n}(a_{jk}-{\alpha }_k)\right]\right|=0,$$

(42)

$$\underset{n\in \mathbb{N}}{sup}\sum _k{\left|\underset{j=0}{\sum ^n}{a}_{jk}\right|}^q<\infty ,q=\frac{p}{p-1},$$

(43)

$$\underset{m,n\in \mathbb{N}}{sup}\left|\underset{n=0}{\sum ^m}{a}_{nk}\right|<\infty ,$$

(44)

$$\underset{m,l\in \mathbb{N}}{sup}\left|\underset{n=0}{\sum ^m}\underset{k=l}{\sum ^{\infty }}{a}_{nk}\right|<\infty ,$$

(45)

$$\underset{m,l\in \mathbb{N}}{sup}\left|\underset{n=0}{\sum ^m}\underset{k=0}{\sum ^l}{a}_{nk}\right|<\infty ,$$

(46)

$$\underset{m}{lim}\sum _k\left|\underset{n=m}{\sum ^{\infty }}{a}_{nk}\right|=0,$$

(47)

$$\sum _n\sum _k{a}_{nk},\mathrm{convergent},$$

(48)

$$\underset{m\to \infty }{lim}\underset{n=0}{\sum ^m}(a_{nk}-a_{n,k+1})=\alpha ,\mathrm{for}\mathrm{each}k\in \mathbb{N},\alpha \in \mathbb{C}.$$

(49)

Let us give some matrix transformations in the following Lemma for use in the next step.

Lemma 4.

Let $A=(a_{nk})$ be an infinite matrix for all $k,n\in \mathbb{N}$. Then,

(1) 

$A=(a_{nk})\in (\widehat{c},cs)$ iff Equations (24) and (40)–(42) hold (Başar [34]).

(2) 

$A=(a_{nk})\in (cs,\widehat{c})$ iff Equations (12) and (36) hold (Başar and Çolak [35]).

(3) 

$A=(a_{nk})\in (bs,\widehat{c})$ iff Equations (9), (12), (36) and (37) hold (Başar and Solak [36]).

(4) 

$A=(a_{nk})\in (bs,\widehat{c}s)$ iff Equations (9) and (37)–(39) hold (Başar and Solak [36]).

(5) 

$A=(a_{nk})\in (cs,\widehat{c}s)$ iff Equations (38) and (39) hold (Başar and Çolak [35]).

(6) 

$A=(a_{nk})\in ({\ell }_{\infty },bs)=(c,bs)=(c_0,bs)$ iff Equation (40) holds (Zeller [29]).

(7) 

$A=(a_{nk})\in ({\ell }_p,bs)$ iff Equation (43) holds (Jakimovski and Russell [37]).

(8) 

$A=(a_{nk})\in (\ell ,bs)$ iff Equation (44) holds (Zeller [29]).

(9) 

$A=(a_{nk})\in (bv,bs)$ iff Equation (45) holds (Zeller [29]).

(10) 

$A=(a_{nk})\in (b{v}_0,bs)$ iff Equation (46) holds (Jakimovski and Russell [37]).

(11) 

$A=(a_{nk})\in ({\ell }_{\infty },cs)$ iff Equation (47) holds (Zeller [29]).

(12) 

$A=(a_{nk})\in (c,cs)$ iff Equations (11), (40) and (48) hold (Zeller [29]).

(13) 

$A=(a_{nk})\in (c{s}_0,cs)$ iff Equations (10) and (49) hold (Zeller [29]).

(14) 

$A=(a_{nk})\in ({\ell }_p,cs)$ iff Equations (11) and (43) hold (Jakimovski and Russell [37]).

(15) 

$A=(a_{nk})\in (\ell ,cs)$ iff Equations (11) and (44) hold (Jakimovski and Russell [37]).

(16) 

$A=(a_{nk})\in (bv,cs)$ iff Equations (11), (44) and (46) hold (Zeller [29]).

(17) 

$A=(a_{nk})\in (b{v}_0,cs)$ iff Equations (11) and (46) hold (Jakimovski and Russell [37]).

Now, let us list the following condition, where $d_{nk}$ and $d_{nk}^{(m)}$ are taken as in Equations (30) and (31):

$$\underset{k}{lim}{d}_{nk}^{(m)}=0\mathrm{for}\mathrm{all}n\in \mathbb{N},$$

(50)

$$\exists d_{nk}\in \mathbb{C}\ni \underset{n\to \infty }{lim}(d_{nk}^{(m)}-d_{n,k+1}^{(m)})=d_{nk}\mathrm{for}\mathrm{all}k,n\in \mathbb{N},$$

(51)

$$\underset{n\to \infty }{lim}\sum _k\left|d_{nk}^{(m)}-d_{n,k+1}^{(m)}\right|<\infty \mathrm{uniformly}\mathrm{in}n,$$

(52)

$$\underset{k}{lim}{d}_{nk}=0\mathrm{for}\mathrm{all}n\in \mathbb{N},$$

(53)

$$\underset{n}{sup}\sum _k\left|d_{nk}-d_{n,k+1}\right|<\infty ,$$

(54)

$$\exists d_k\in \mathbb{C}\ni \underset{n\to \infty }{lim}(d_{nk}-d_{n,k+1})=d_k\mathrm{for}\mathrm{all}k,n\in \mathbb{N},$$

(55)

$$\exists \alpha \in \mathbb{C}\ni \underset{n\to \infty }{lim}\sum _k\left|d_{nk}-d_{n,k+1}\right|=\alpha \mathrm{uniformly}\mathrm{in}n,$$

(56)

$$\underset{m\in \mathbb{N}}{sup}\sum _k\left|\underset{n=0}{\sum ^m}(d_{nk}-d_{n,k+1})\right|<\infty ,$$

(57)

$$\underset{m}{lim}\sum _k\left|\underset{n=0}{\sum ^m}(d_{nk}-d_{n,k+1})\right|=\sum _k\left|\sum _n(d_{nk}-d_{n,k+1})\right|,$$

(58)

$$\underset{m}{lim}\sum _k\left|\underset{n=0}{\sum ^m}(d_{nk}-d_{n,k+1})\right|=0,$$

(59)

$$\underset{N,K\in \mathcal{F}}{sup}\left|\sum _{n\in N}\sum _{k\in K}(d_{nk}-d_{n,k+1})\right|<\infty ,$$

(60)

$$\underset{N,K\in \mathcal{F}}{sup}\left|\sum _{n\in N}\sum _{k\in K}(d_{nk}-d_{n,k+1})-(d_{n-1,k}-d_{n-1,k+1})\right|<\infty ,$$

(61)

$$\underset{n}{sup}\sum _k\left|d_{nk}^{(m)}-d_{n,k+1}^{(m)}\right|<\infty ,$$

(62)

$$\exists d_k\in \mathbb{C}\ni \underset{n}{lim}{d}_{nk}^{(m)}=d_k\mathrm{for}\mathrm{all}k,n\in \mathbb{N},$$

(63)

$$\underset{n\in \mathbb{N}}{sup}\left|\underset{k}{lim}{d}_{nk}\right|<\infty ,$$

(64)

$$\exists d_k\in \mathbb{C}\ni \underset{n}{lim}{d}_{nk}=d_k\mathrm{for}\mathrm{all}k,n\in \mathbb{N},$$

(65)

$$\underset{m\in \mathbb{N}}{sup}\left|\underset{k}{lim}\underset{n=0}{\sum ^m}{d}_{nk}\right|<\infty ,$$

(66)

$$\underset{m\in \mathbb{N}}{sup}\sum _k\left|\underset{n=0}{\sum ^m}(d_{nk}-d_{n,k-1})\right|<\infty ,$$

(67)

$$\exists d_k\in \mathbb{C}\ni \sum _n{d}_{nk}=d_k\mathrm{for}\mathrm{each}k\in \mathbb{N},$$

(68)

$$\underset{N,K\in \mathcal{F}}{sup}\sum _{n\in N}\left|\sum _{k\in K}(d_{nk}-d_{n,k-1})\right|<\infty ,$$

(69)

$$\exists d_k\in \mathbb{C}\ni f-lim{d}_{nk}=d_k\mathrm{for}\mathrm{each}k\in \mathbb{N},$$

(70)

$$\underset{N,K\in \mathcal{F}}{sup}\left|\sum _{n\in N}\sum _{k\in K}(d_{nk}-d_{n-1,k})-(d_{n,k-1}-d_{n-1,k-1})\right|<\infty ,$$

(71)

$$\underset{q}{lim}\sum _k\frac{1}{q+1}\left|\underset{i=0}{\sum ^q}△\left[\underset{j=0}{\sum ^{n+i}}(d_{jk}-{\alpha }_k)\right]\right|=0\mathrm{uniformly}\mathrm{in}n,$$

(72)

$$\underset{n\in \mathbb{N}}{sup}\sum _k\left|\underset{j=0}{\sum ^n}{d}_{jk}\right|<\infty ,$$

(73)

$$\exists d_k\in \mathbb{C}\ni \sum _n\sum _k{d}_{nk}=d_k\mathrm{for}\mathrm{all}k\in \mathbb{N},$$

(74)

$$\underset{n}{lim}\sum _k\left|△\left[\underset{j=0}{\sum ^n}(d_{jk}-{\alpha }_k)\right]\right|=0,$$

(75)

$$\underset{k}{\underset{n\in \mathbb{N}}{\sup }}\left|△\left[\underset{j=0}{\sum ^n}{d}_{jk}\right]\right|<\infty ,$$

(76)

$$\exists d_k\in \mathbb{C}\ni f-lim\underset{j=0}{\sum ^n}{d}_{jk}=d_k\mathrm{for}\mathrm{each}k\in \mathbb{N},$$

(77)

Now, we can give several conclusions of Theorems 14 and 15, and Lemmas 1 and 4.

Corallary 2.

Let $A=(a_{nk})$ be an infinite matrix for all $k,n\in \mathbb{N}$. Then,

(1) 

$A=(a_{nk})\in (bs(\widehat{F}(s,r),c_0)$ iff Equations (50)–(53) hold and Equation (56) also holds with $\alpha =0$.

(2) 

$A=(a_{nk})\in (bs(\widehat{F}(s,r),c{s}_0)$ iff Equations (50)–(53) and (59) hold.

(3) 

$A=(a_{nk})\in (bs(\widehat{F}(s,r),c)$ iff Equations (50)–(53), (55) and (56) hold.

(4) 

$A=(a_{nk})\in (bs(\widehat{F}(s,r),cs)$ iff Equations (50)–(53) and (58) hold.

(5) 

$A=(a_{nk})\in (bs(\widehat{F}(s,r),{\ell }_{\infty })$ iff Equations (50)–(54) hold.

(6) 

$A=(a_{nk})\in (bs(\widehat{F}(s,r),bs)$ iff Equations (50)–(53) and (57) hold.

(7) 

$A=(a_{nk})\in (bs(\widehat{F}(s,r),{\ell }_1)$ iff Equations (50)–(53) and (60) hold.

(8) 

$A=(a_{nk})\in (bs(\widehat{F}(s,r),bv)$ iff Equations (50)–(53) and (61) hold.

(9) 

$A=(a_{nk})\in (bs(\widehat{F}(s,r),b{v}_0)$ iff Equations (50)–(52), (54) and (61) hold and Equation (56) also holds with $\alpha =0$.

Corallary 3.

Let $A=(a_{nk})$ be an infinite matrix for all $k,n\in \mathbb{N}.$ Then,

(1) 

$A=(a_{nk})\in (cs(\widehat{F}(s,r),c_0)$ iff Equations (54), (62) and (63) hold and Equation (65) also holds with $d_k=0$ for all $k\in \mathbb{N}$.

(2) 

$A=(a_{nk})\in (cs(\widehat{F}(s,r),c{s}_0)$ iff Equations (57), (62) and (63) hold and Equation (68) also holds with $d_k=0$ for all $k\in \mathbb{N}$.

(3) 

$A=(a_{nk})\in (cs(\widehat{F}(s,r),c)$ iff Equations (54), (62), (63) and (65) hold.

(4) 

$A=(a_{nk})\in (cs(\widehat{F}(s,r),cs)$ iff Equations (62), (63), (67) and (68) hold.

(5) 

$A=(a_{nk})\in (cs(\widehat{F}(s,r),{\ell }_{\infty })$ iff Equations (54) and (62)–(64) hold.

(6) 

$A=(a_{nk})\in (cs(\widehat{F}(s,r),bs)$ iff Equations (57), (62), (63) and (66) hold.

(7) 

$A=(a_{nk})\in (cs(\widehat{F}(s,r),{\ell }_1)$ iff Equations (62), (63) and (69) hold.

(8) 

$A=(a_{nk})\in (cs(\widehat{F}(s,r),bv)$ iff Equations (62), (63) and (71) hold.

(9) 

$A=(a_{nk})\in (cs(\widehat{F}(s,r),b{v}_0)$ iff Equations (62), (63) and (65) hold and Equation (71) also holds with $d_k=0$ for all $k\in \mathbb{N}$.

Corallary 4.

Let $A=(a_{nk})$ be an infinite matrix for all $k,n\in \mathbb{N}.$ Then,

(1) 

$A=(a_{nk})\in (bs(\widehat{F}(s,r),\widehat{c})$ iff Equations (50)–(54), (70) and (72) hold.

(2) 

$A=(a_{nk})\in (bs(\widehat{F}(s,r),{\widehat{c}}_0)$ iff Equations (50)–(54) hold and (70) and Equation (72) also hold with ${\alpha }_k=0$ in Equation (70) and $d_k=0$ in (72).

(3) 

$A=(a_{nk})\in (cs(\widehat{F}(s,r),\widehat{c})$ iff Equations (54), (62), (63) and (70) hold.

(4) 

$A=(a_{nk})\in (cs(\widehat{F}(s,r),{\widehat{c}}_0)$ iff Equations (62), (63) and (54) hold and Equation (70) also holds with ${\alpha }_k=0$.

(5) 

$A=(a_{nk})\in (\widehat{c},cs(\widehat{F}(s,r))$ iff Equations (68) and (73)–(75) hold with $b_{nk}$ instead of $d_{nk}$, where $b_{nk}$ is defined by Equation (33).

(6) 

$A=(a_{nk})\in (bs(\widehat{F}(s,r),\widehat{c}s)$ iff Equations (50)–(53), (72), (76) and (77) hold.

(7) 

$A=(a_{nk})\in (cs(\widehat{F}(s,r),\widehat{c}s)$ iff Equations (62), (63), (76) and (77) hold.

Corallary 5.

Let $A=(a_{nk})$ be an infinite matrix for all $k,n\in \mathbb{N}.$ Then,

(1) 

$A=(a_{nk})\in ({\ell }_{\infty },bs(\widehat{F}(s,r))=(c,bs)=(c_0,bs)$ iff Equation (40) holds with $b_{nk}$ instead of $a_{nk}$, where $b_{nk}$ is defined by (33).

(2) 

$A=(a_{nk})\in ({\ell }_p,bs(\widehat{F}(s,r))$ iff Equation (43) holds with $b_{nk}$ instead of $a_{nk}$, where $b_{nk}$ is defined by (33).

(3) 

$A=(a_{nk})\in (\ell ,bs(\widehat{F}(s,r))$ iff Equation (44) holds with $b_{nk}$ instead of $a_{nk}$, where $b_{nk}$ is defined by Equation (33).

(4) 

$A=(a_{nk})\in (bv,bs(\widehat{F}(s,r))$ iff Equation (45) holds with $b_{nk}$ instead of $a_{nk}$, where $b_{nk}$ is defined by Equation (33).

(5) 

$A=(a_{nk})\in (b{v}_0,bs(\widehat{F}(s,r))$ iff Equation (46) holds with $b_{nk}$ instead of $a_{nk}$, where $b_{nk}$ is defined by Equation (33).

(6) 

$A=(a_{nk})\in ({\ell }_{\infty },cs(\widehat{F}(s,r))$ iff Equation (47) holds with $b_{nk}$ instead of $a_{nk}$, where $b_{nk}$ is defined by Equation (33).

(7) 

$A=(a_{nk})\in (c,cs(\widehat{F}(s,r))$ iff Equations (11), (40) and (48) hold with $b_{nk}$ instead of $a_{nk}$, where $b_{nk}$ is defined by Equation (33).

(8) 

$A=(a_{nk})\in (c{s}_0,cs(\widehat{F}(s,r))$ iff Equations (10) and (49) hold with $b_{nk}$ instead of $a_{nk}$, where $b_{nk}$ is defined by Equation (33).

(9) 

$A=(a_{nk})\in ({\ell }_p,cs(\widehat{F}(s,r))$ iff Equations (11) and (43) hold with $b_{nk}$ instead of $a_{nk}$, where $b_{nk}$ is defined by Equation (33).

(10) 

$A=(a_{nk})\in (\ell ,cs(\widehat{F}(s,r))$ iff Equations (11) and (44) hold with $b_{nk}$ instead of $a_{nk}$, where $b_{nk}$ is defined by Equation (33).

(11) 

$A=(a_{nk})\in (bv,cs(\widehat{F}(s,r))$ iff Equations (11), (44) and (46) hold with $b_{nk}$ instead of $a_{nk}$, where $b_{nk}$ is defined by Equation (33).

(12) 

$A=(a_{nk})\in (b{v}_0,cs(\widehat{F}(s,r))$ iff Equations (11) and (46) hold with $b_{nk}$ instead of $a_{nk}$, where $b_{nk}$ 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 $\widehat{F}$ , which is derived from the Fibonacci sequence $(f_n),$ was recently introduced by Kara [23] in 2013 and defined the new sequence spaces ${\ell }_p(\widehat{F})$ and ${\ell }_{\infty }(\widehat{F}),$ which are derived by the matrix domain of $\widehat{F}$ from the sequence spaces ${\ell }_p$ and ${\ell }_{\infty }$, respectively, where $1\le p<\infty$. Candan [25] in 2015 introduced the sequence spaces $c(\widehat{F}(s,r))$ and $c_0(\widehat{F}(s,r))$. Later, Candan and Kara [15] studied the sequence spaces ${\ell }_p(\widehat{F}(s,r))$ in which $1\le p\le \infty$. In addition, Kara et al. [24] have characterized some class of compact operators in the spaces ${\ell }_p(\widehat{F})$ and ${\ell }_{\infty }(\widehat{F})$, where $1\le p<\infty$.

The study is concerned with matrix domain on a sequences space of a triangle infinite matrix. In this article, we defined spaces $bs(\widehat{F}(s,r))$ and $cs(\widehat{F}(s,r))$ of Generalized Fibonacci difference of sequences, which constituted bounded and convergence series, respectively. We have demonstrated the sets of $bs(\widehat{F}(s,r))$ and $cs(\widehat{F}(s,r)),$ which are the linear spaces, and both spaces have the same norm

$$∥x∥=\underset{n\in \mathbb{N}}{sup}\left|\underset{k=0}{\sum ^n}\left(s\frac{f_k}{f_{k+1}}{x}_k+r\frac{f_{k+1}}{f_k}{x}_{k-1}\right)\right|,$$

where $x\in bs(\widehat{F}(s,r))$ or $x\in cs(\widehat{F}(s,r))$. In addition, it was shown that they are normed space and Banach spaces. It was found that $bs(\widehat{F}(s,r))$ and $bs$ are linearly isomorphic as isometric. At the same time, $cs(\widehat{F}(s,r))$ and $cs$ are linearly isomorphic as isometric. Some inclusions’ theorems were given with respect to $bs(\widehat{F}(s,r))$ and $cs(\widehat{F}(s,r))$. According to this, inclusions $bs\subset bs(\widehat{F}(s,r))$, $cs\subset cs(\widehat{F}(s,r))$ are valid. In addition, if $\left|r/s\right|<1/4$, then $bs(\widehat{F}(s,r))\subset {\ell }_{\infty }$ and $cs(\widehat{F}(s,r))\subset c$ are valid. It was concluded that $cs(\widehat{F}(s,r))$ has a Schauder basis.

Finally, the $\alpha$-, $\beta$- and $\gamma$-duals of the both spaces are calculated and some matrix transformations of them were given.

5. Conclusions

In this article, we have defined spaces $bs(\widehat{F}(s,r))$ and $cs(\widehat{F}(s,r))$ of Generalized Fibonacci difference of sequences, which constituted bounded and convergence series, respectively. We have demonstrated that the sets of $bs(\widehat{F}(s,r))$ and $cs(\widehat{F}(s,r))$ 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 $bs(\widehat{F}(s,r))$ and $cs(\widehat{F}(s,r))$. It was concluded that $cs(\widehat{F}(s,r))$ has a Schauder basis. Finally, the $\alpha$-, $\beta$- and $\gamma$-duals of the both spaces were calculated and some matrix transformations of them were given.