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On ideal convergence Fibonacci difference sequence spaces
Advances in Difference Equations volume 2018, Article number: 199 (2018)
Abstract
The Fibonacci sequence was firstly used in the theory of sequence spaces by Kara and Başarir (Casp. J. Math. Sci. 1(1):43–47, 2012). Afterward, Kara (J. Inequal. Appl. 2013(1):38, 2013) defined the Fibonacci difference matrix F̂ by using the Fibonacci sequence \((f_{n})\) for \(n\in{\{0, 1, \ldots\}}\) and introduced new sequence spaces related to the matrix domain of F̂. In this paper, by using the Fibonacci difference matrix F̂ defined by the Fibonacci sequence and the notion of ideal convergence, we introduce the Fibonacci difference sequence spaces \(c^{I}_{0}(\hat {F})\), \(c^{I}(\hat{F})\), and \(\ell^{I}_{\infty}(\hat{F})\). Further, we study some inclusion relations concerning these spaces. In addition, we discuss some properties on these spaces such as monotonicity and solidity.
1 Introduction
Let \(\mathbb{N}\) and \(\mathbb{R}\) denote the sets of natural and real numbers, respectively. By ω we denote the vector space of all real sequences. Any vector subspace of ω is called a sequence space. Throughout the paper, \(\ell_{\infty}\), c, and \(c_{0}\) are the classes of bounded, convergent, and null sequences, respectively, with norm \(\x\_{\infty}=\sup_{k\in\mathbb{N}}x_{k}\). Let λ and μ be two sequence spaces, and let \(A =(a_{nk})\) be an infinite matrix of real numbers \(a_{nk}\), \(n,k\in{\mathbb{N}}\). Then we say that A defines a matrix transformation from λ into μ, and we denote it by writing \(A:\lambda\longrightarrow\mu\) if for every sequence \(x=(x_{k})\in{\lambda}\), the sequence \(Ax=\{A_{n}(x)\}\), the Atransform of x, is in μ, where
By \((\lambda,\mu)\) we denote the class of all matrices A. Thus \(A\in {(\lambda,\mu)}\) if and only if the series on the right side of (1.1) converges for each \(n\in{\mathbb{N}}\) and every \(x\in{\lambda}\) and \(Ax\in{\mu}\) for all \(x\in{\lambda}\), where \(A_{n}=(a_{nk})_{k\in {\mathbb{N}}}\) denotes the sequence in the nth row of A. The concept of matrix domain has fundamental importance for this study. So, the matrix domain of an infinite matrix A in a sequence space λ is defined by
which is a sequence space. If \(A=\triangle\), where △ is the backward difference matrix defined by
for all \(n,k\in{\mathbb{N}}\), then \(\lambda_{\triangle}\) is called the difference sequence space defined by the domain of a triangle matrix A whenever λ is a normed linear space or paranormed sequence space. The notion of difference sequence spaces was introduced by Kizmaz [22] as follows:
for \(\lambda\in{\{\ell_{\infty},c,c_{0}\}}\). In recent years, some researchers have addressed the approach to constructing a new sequence space by means of the matrix domain of a particular limitation method; see, for instance, [2–4, 10, 11, 15, 20, 26] and the references therein. Quite recently, Kara [12] has introduced the difference sequence space
which is derived by the Fibonacci difference matrix \(\hat{F}=(\hat {f}_{nk})\) defined as follows:
for all \(n,k\in{\mathbb{N}}\), where \(\{f_{n}\}_{n=0}^{\infty}\) is the sequence of Fibonacci numbers defined by the linear recurrence equalities \(f_{0}= f_{1}=1\) and \(f_{n}=f_{n1}+f_{n2}\), \(n\geq2\), with the following fundamental properties (see Koshy [23]):
which yields \(f_{n1}^{2}+f_{n}f_{n1}f_{n}^{2}=(1)^{n+1}\) by substituting for \(f_{n+1}\) in Cassini’s formula.
For a more detailed information about Fibonacci sequence spaces, we refer to [5–7, 18, 25]. By using the same infinite Fibonacci matrix F̂ and the same technique, Başarir et al. [1] have introduced the Fibonacci difference sequence spaces \(c_{0}(\hat{F})\) and \(c(\hat {F})\) as the sets of all sequences whose F̂transforms are in the spaces \(c_{0}\) and c, respectively, that is,
and
where the sequence \(\hat{F}_{n}(x)\) is the F̂transform of a sequence \(x=(x_{n})\in{\omega}\), defined as follows:
By an ideal we mean a family of sets \(I\subset P(X)\) (where \(P(X)\) is the power set of X) such that (i) \(\emptyset\in{I}\), (ii) \(A\cup B \in {I}\) for all \(A,B\in{I}\), and (iii) for each \(A\in{I}\) and \(B\subset A\), we have \(B\in{I}\); I is called admissible in X if it contains all singletons, that is, if \(I\supset\{\{x\}:x\in{X}\}\). A filter on X is a nonempty family of sets \(\mathcal{F}\subset P(X)\) satisfying (i) \(\emptyset\notin\mathcal{F}\), (ii) \(A, B\in{\mathcal{F}} \) implies that \(A\cap B \in\mathcal{F}\), and (iii) for any \(A\in {\mathcal{F}}\) and \(B\supset A \), we have \(B\in{\mathcal{F}}\). For each ideal I, there is a filter \(\mathcal{F}(I)\) corresponding to I (a filter associated with ideal I), that is, \(\mathcal{F}(I)=\{ K\subseteq X: K^{c} \in{I}\}\), where \(K^{c}=X\setminus K\). In 1999, Kostyrko et al. [24] defined the notion of Iconvergence, which depends on the structure of ideals of subsets of \(\mathbb{N}\) as a generalization of statistical convergence introduced by Fast [9] and Steinhaus [29] in 1951. Later on, the notion of Iconvergence was further investigated from the sequence space point of view and linked with the summability theory by Šalát et al. [27], Tripathy and Hazarika [30–32], Khan and Ebadullah [19], Das et al. [8], and many other authors. Šalát et al. [28] extended the notion of summability fields of an infinite matrix of operators A with the help of the notion of Iconvergence, that is, the notion of Isummability and introduced new sequence spaces \(c_{A}^{I}\) and \(m_{A}^{I}\), the Iconvergence field and bounded Iconvergence field of an infinite matrix A, respectively. For further details on ideal convergence, we refer to [14, 16, 17].
Throughout the paper, \(c^{I}_{0}\), \(c^{I}\), and \(\ell^{I}_{\infty}\) denote the Inull, Iconvergent, and Ibounded sequence spaces, respectively. In this paper, by combining the definitions of Fibonacci difference matrix F̂ and ideal convergence we introduce the sequence spaces \(c_{0}^{I}(\hat{F})\), \(c^{I}(\hat{F})\), and \(\ell _{\infty}^{I}(\hat{F})\). Further, we study some topological and algebraic properties of these spaces. Also, we study some inclusion relations concerning these spaces.
Now, we recall some definitions and lemmas, which will be used throughout the paper.
Definition 1.1
A sequence \(x=(x_{n})\in{\omega}\) is said to be statistically convergent to a number \(\ell\in{\mathbb{R}}\) if, for every \(\epsilon>0\), the natural density of the set \(\{n\in{\mathbb{N}}:x_{n}\ell\geq\epsilon\}\) equals zero, and we write \(\mathit{st}\mbox{}\lim x_{n}=\ell\). If \(\ell=0\), then \(x=(x_{n})\in{\omega}\) is said to be stnull.
Definition 1.2
([27])
A sequence \(x=(x_{n})\in{\omega} \) is said to be ICauchy if, for every \(\epsilon>0\), there exists a number \(N=N(\epsilon)\) such that the set \(\{n\in{\mathbb {N}}:x_{n}x_{N}\geq\epsilon \}\in{I}\).
Definition 1.3
([24])
A sequence \(x=(x_{n})\in{\omega} \) is said to be Iconvergent to a number \(\ell\in{\mathbb{R}}\) if, for every \(\epsilon>0\), the set \(\{n\in{\mathbb{N}}:x_{n}\ell\geq\epsilon \}\in{I}\), and we write \(I\mbox{}\lim x_{n}=\ell\). If \(\ell=0\), then \((x_{n})\in{\omega}\) is said to be Inull.
Definition 1.4
([19])
A sequence \(x=(x_{n})\in{\omega}\) is said to be Ibounded if there exists \(K>0\) such that the set \(\{n\in{\mathbb {N}}:x_{n}\geq K\}\in{I}\).
Definition 1.5
([27])
Let \(x=(x_{n})\) and \(z=(z_{n})\) be two sequences. We say that \(x_{n} = z_{n}\) for almost all n relative to I (in short, a.a.n.r.I) if the set \(\{n\in{\mathbb{N}}: x_{n}\neq z_{n}\}\in{I} \).
Definition 1.6
([27])
A sequence space E is said to be solid or normal if \((\alpha_{n}x_{n})\in{E}\) for any sequence \((x_{n})\in{E} \) and any sequence of scalars \((\alpha_{n})\in{\omega}\) with \(\alpha_{n}<1\) for all \(n\in{\mathbb{N}}\).
Lemma 1.1
([27])
Every solid space is monotone.
Definition 1.7
([27])
A sequence space E is said to be a sequence algebra if \((x_{n})*(z_{n})=(x_{n}\cdot z_{n})\in{E}\) for all \((x_{n}),(z_{n})\in{E}\).
Definition 1.8
([27])
Let \(K =\{n_{i}\in{\mathbb{N}}: n_{1}< n_{2}<\cdots \}\subseteq\mathbb{N}\), and let E be a sequence space. The Kstep space of E is the sequence space
A canonical preimage of a sequence \((x_{n_{i}})\in{\lambda}^{E}_{K}\) is the sequence \((y_{n})\in{\omega} \) defined as
A canonical preimage of the step space \(\lambda_{K}^{E}\) is the set of canonical preimages of all elements in \(\lambda_{K}^{E}\), that is, y is in the canonical preimage of \(\lambda_{K}^{E} \) iff y is the canonical preimage of some element \(x\in{\lambda}_{K}^{E}\).
Definition 1.9
([27])
A sequence space E is said to be monotone if it contains the canonical preimages of its step space (i.e., if for all infinite \(K\subseteq\mathbb{N}\) and \((x_{n})\in{E}\), the sequence \((\alpha_{n}x_{n})\) with \(\alpha_{n}=1\) for \(n\in{K}\) and \(\alpha _{n}=0\) otherwise belongs to E).
Definition 1.10
A map h defined on a domain \(D\subset X\) (i.e., \(h: D\subset X\longrightarrow\mathbb{R}\)) is said to satisfy the Lipschitz condition if \(h(x)h(y)\leq Kxy\), where K is called the Lipschitz constant.
Remark 1.1
([27])
The convergence field of Iconvergence is the set
Definition 1.11
([24])
The convergence field \(\mathcal{F}(I)\) is a closed linear subspace of \(\ell_{\infty}\) with respect to the supremum norm, \(\mathcal{F}(I)=\ell_{\infty}\cap c^{I}\).
Lemma 1.2
([28])
Let \(K\in{\mathcal{F}}(I) \) and \(M \subseteq\mathbb{N}\). If \(M \notin{I}\), then \(M\cap K \notin{I}\).
Definition 1.12
([27])
The function \(h: D\subset X\longrightarrow\mathbb {R}\) defined by \(h(x)=I\mbox{}\lim x\) for all \(x\in{\mathcal{F}(I)}\) is a Lipschitz function.
2 IConvergence Fibonacci difference sequence spaces
In this section, we introduce the sequence spaces as the sets of sequences whose F̂transforms are in the spaces \(c^{I}_{0}\), \(c^{I}\), and \(\ell^{I}_{\infty}\). Further, we present some inclusion theorems and study some topological and algebraic properties on these resulting. Throughout the paper, we suppose that a sequence \(x=(x_{n})\in{\omega}\) and \(\hat{F}_{n}(x)\) are connected by relation (1.5) and I is an admissible ideal of subset of \(\mathbb{N}\). We define
We write
and
With notation (1.2), the spaces \(c_{0}^{I}(\hat{F})\), \(c^{I}(\hat{F})\), \(\ell_{\infty}^{I}(\hat{F})\), \(m^{I}(\hat{F})\), and \(m_{0}^{I}(\hat{F})\) can be redefined as follows:
Definition 2.1
Let I be an admissible ideal of subsets of \(\mathbb{N}\). A sequence \(x=(x_{n})\in{\omega} \) is called Fibonacci ICauchy if for each \(\epsilon>0\), there exists a number \(N=N(\epsilon)\in{\mathbb{N}}\) such that \(\{ n\in{\mathbb{N}}:\hat{F}_{n}(x)\hat{F}_{N}(x)\geq\epsilon \}\in{I}\).
Example 2.1
Define \(I_{f}=\{A\subseteq\mathbb{N}: A \mbox{ is finite}\}\). Then \(I_{f}\) is an admissible ideal in \(\mathbb{N}\), and \(c^{I_{f}}(\hat {F})=c(\hat{F})\).
Example 2.2
Define the nontrivial ideal \(I_{d}=\{A\subseteq\mathbb{N}: d(A)=0\}\), where \(d(A)\) is the natural density of a set A. In this case, \(c^{I_{d}}(\hat{F})=S(\hat{F})\), where \(S(\hat{F})\) is the space of Fibonacci difference statistically convergent sequence defined as
Theorem 2.1
The sequence spaces \(c^{I}(\hat{F})\), \(c_{0}^{I}(\hat{F})\), \(\ell _{\infty}^{I}(\hat{F})\), \(m_{0}^{I}(\hat{F})\), and \(m^{I}(\hat{F})\) are linear over \(\mathbb{R}\).
Proof
Let \(x=(x_{n})\) and \(y=(y_{n})\) be two arbitrary elements of the space \(c^{I}(\hat{F})\), and let α, β are scalars. Then, for given \(\epsilon>0\), there exist \(L_{1},L_{2}\in{\mathbb{R}}\) such that
and
Now, let
and
be such that \(A_{1}^{c},A_{2}^{c}\in{I}\). Then
Thus, the sets on the righthand side of (2.7) belong to \(\mathcal {F}(I)\). By the definition of the filter associated with an ideal the complement of the set on the lefthand side of (2.7) belongs to I. This implies that \((\alpha x+\beta y )\in{c^{I}}(\hat {F})\). Hence \(c^{I}(\hat{F})\) is a linear space. The proof of the remaining results is similar. □
Theorem 2.2
The spaces \(X(\hat{F})\) are normed spaces with the norm
Proof
The proof of the result is easy by existing techniques and hence is omitted. □
Theorem 2.3
Let \(I \subseteq2^{\mathbb{N}}\) be a nontrivial ideal. Then the inclusion \(c(\hat{F})\subset c^{I}(\hat{F})\) is strict.
Proof
We know that \(c\subseteq c^{I}\) and, for any X and Y spaces, \(X\subseteq Y\) implies \(X(\hat{F})\subseteq Y(\hat{F})\) (see [21], Lemma 2.1). Hence it is easy to see that \(c(\hat {F})\subset c^{I}(\hat{F})\). The following example shows the strictness of the inclusion.
Example 2.3
Define the sequence \(x=(x_{n})\in{\omega}\) such that
Then \(x\in{c^{I_{d}}(\hat{F})}\), but \(x\notin{c(\hat{F})}\).
Example 2.4
Define the ideal I such that
Then I is a nontrivial ideal in \(\mathbb{N}\). When
we have
and \((x_{n})\in{c_{0}^{I}(\hat{F})}\). Hence \(A_{\epsilon}\in{I}\) and \(\hat{F}_{n}(x)\in{c^{I}}\). Now let us look at the statistical convergence of the sequence:
where B is a finite number, and \(A_{\epsilon}\) is the cardinality of \(A_{\epsilon}\). Hence \(\hat{F}_{n}(x)\notin{S}\). □
Theorem 2.4
A sequence \(x=(x_{n})\in{\omega}\) Fibonacci Iconverges if and only if for every \(\epsilon>0 \), there exists \(N=N(\epsilon)\in{\mathbb {N}}\) such that
Proof
Suppose that a sequence \(x=(x_{n})\in{\omega}\) is Fibonacci Iconvergent to some number \(L\in{\mathbb{R}}\). Then, for given \(\epsilon>0\), the set
Fix an integer \(N=N(\epsilon)\in{B_{\epsilon}}\). Then we have
for all \(n\in{B_{\epsilon}}\). Hence (2.9) holds.
Conversely, suppose that (2.9) holds for all \(\epsilon>0\). Then
Let \(J_{\epsilon}= [\hat{F}_{n}(x)\epsilon, \hat {F}_{n}(x)+\epsilon ]\). Fixing \(\epsilon>0\), we have \(C_{\epsilon}\in{\mathcal{F}(I)} \) and \(C_{\frac{\epsilon}{2}}\in {\mathcal{F}(I)}\). Hence \(C_{\epsilon}\cap C_{\frac{\epsilon}{2}}\in{\mathcal{F}(I)}\). This implies that
that is,
and thus
where \(\operatorname{diam} (J)\) denotes the length of an interval J. Proceeding in this way, by induction we get a sequence of closed intervals
such that
and
Then there exists a number \(L\in{\bigcap_{n\in{\mathbb{N}}}{I_{n}}}\), and it is a routine work to verify that \(L=I\mbox{}\lim{\hat{F}_{n}(x)}\), showing that \(x=(x_{n})\in{\omega}\) Fibonacci Iconverges. Hence the result. □
Theorem 2.5
Let I be an admissible ideal. Then the following are equivalent:

(a)
\((x_{n})\in{c^{I}(\hat{F})}\);

(b)
There exists \((y_{n})\in{c(\hat{F})}\) such that \(x_{n}= y_{n}\) for a.a.n.r.I;

(c)
There exist \((y_{n})\in{c(\hat{F})}\) and \((z_{n})\in {c_{0}^{I}(\hat{F})}\) such that \(x_{n}=y_{n}+z_{n}\) for all \(n\in {\mathbb{N}}\) and \(\{n\in{{\mathbb{N}}}: \vert \hat {F}_{n}(x)L \vert \geq\epsilon \}\in{I}\);

(d)
There exists a subset \(K= \{n_{i}:i\in{\mathbb{N}}, n_{1}< n_{2}< n_{3}<\cdots \}\) of \(\mathbb{N}\) such that \(K\in {\mathcal{F}(I)}\) and \(\lim_{n\to\infty} \vert \hat {F}_{n_{i}}(x)L \vert =0\).
Proof
(a) implies (b). Let \(x=(x_{n})\in{c^{I}(\hat{F})}\). Then, for any \(\epsilon>0\), there exists \(L\in{\mathbb{R}}\) such that
Let \((m_{t})\) be an increasing sequence with \(m_{t}\in{\mathbb{N}}\) such that
Define the sequence \(y=(y_{n})\) as \(y_{n}=z_{n}\) for all \(n\leq m_{1}\) and, for \(m_{t}< n < m_{t+1}\), \(t\in{\mathbb{N}}\), as
Then \(y_{n}\in{c(\hat{F})}\), and from the inclusion
we get \(x_{n}=y_{n} \) for a.a.n.r.I.
(b) implies (c). For \(x=(x_{n})\in{c^{I}(\hat{F})}\), there exists \(y=(y_{n})\in{c(\hat{F})}\) such that \(x_{n}=y_{n}\) for a.a.n.r.I. Let \(K=\{n\in{\mathbb{N}}:x_{n}\neq y_{n}\}\). Then \(K\in{I}\). Define the sequence \(z=(z_{n})\) as
Then \((z_{n})\in{c_{0}^{I}(\hat{F})}\) and \((y_{n})\in{c(\hat{F})}\).
(c) implies (d). Let \(P= \{n\in{\mathbb{N}}:\hat {F}_{n}(x)\geq\epsilon \}\in{I}\) and
Then we have
(d) implies (a). Let \(\epsilon>0\) be given and suppose that (c) holds. Then, for any \(\epsilon>0\), by Lemma 1.2 we have
Thus \((x_{n})\in{c^{I}(\hat{F})}\). □
Theorem 2.6
The inclusions \({c_{0}^{I}(\hat{F})}\subset{c^{I}(\hat{F})}\subset {\ell_{\infty}^{I}(\hat{F})}\) are strict.
Proof
The inclusion \({c_{0}^{I}(\hat{F})}\subset{c^{I}(\hat{F})}\) is obvious. Now, to show its strictness, consider the sequence \(x=(x_{n})\in{\omega }\) such that \(\hat{F}_{n}(x)=1\). It easy to see that \(\hat{F}_{n}(x)\in {c^{I}}\) but \(\hat{F}_{n}(x)\notin{c_{0}^{I}}\), that is, \(x\in {c^{I}(\hat{F})\setminus c_{0}^{I}(\hat{F})}\). Next, let \(x=(x_{n})\in {c^{I}(\hat{F})}\). Then there exists \(L\in{\mathbb{R}}\) such that \(I\mbox{}\lim\hat{F}_{n}(x)L=0\), that is,
We have
From this it easily follows that the sequence \((x_{n})\) must belong to \(\ell_{\infty}^{I}(\hat{F})\). Further, we show the strictness of the inclusion \({c^{I}(\hat{F})}\subset{\ell_{\infty}^{I}(\hat{F})}\) by constructing the following example.
Example 2.5
Consider the sequence \(x=(x_{n})\in{\omega}\) such that
Then \(\hat{F}_{n}(x)\in{\ell_{\infty}^{I}}\), but \(\hat{F}_{n}(x)\notin {c^{I}}\), which implies that \(x\in{\ell_{\infty}^{I}(\hat{F})\setminus c^{I}(\hat{F})}\).
Thus the inclusion \({c_{0}^{I}(\hat{F})}\subset{c^{I}(\hat{F})}\subset {\ell_{\infty}^{I}(\hat{F})}\) is strict. □
Remark 2.1
A Fibonacci bounded sequence is obviously Fibonacci Ibounded as the empty set belongs to the ideal I. However, the converse is not true. For example, consider the sequence
Clearly, \(\hat{F}_{n}(x)\) is not a bounded sequence. However, \(\{n\in {\mathbb{N}}: \vert \hat{F}_{n}(x) \vert \geq\frac{1}{2}\}\in{I}\). Hence \(x=(x_{n})\) is Fibonacci Ibounded.
Theorem 2.7
The spaces \(m^{I}(\hat{F})\) and \(m_{0}^{I}(\hat{F})\) are Banach spaces normed by (2.8).
Proof
Let \((x_{n}^{(i)})\) be a Cauchy sequence in \({m^{I}}(\hat{F})\subset {\ell_{\infty}(\hat{F})}\). Then \((x_{n}^{(i)})\) converges in \({\ell _{\infty}(\hat{F})}\), and \(\lim_{i\to\infty}\hat {F}_{n}^{(i)}(x)=\hat{F}_{n}(x)\). Let \(I\mbox{}\lim \hat{F}_{n}^{(j)}(x)=L_{i}\) for \(i\in {\mathbb{N}}\). Then we have to show that

(i)
\((L_{i})\) is convergent say to L and

(ii)
\(I\mbox{}\lim\hat{F}_{n}(x)=L\).
(i) Since \((x_{n}^{(i)})\) is a Cauchy sequence, for each \(\epsilon>0\), there exists \(n_{0}\in{\mathbb{N}}\) such that
Now let \(E_{i}\) and \(E_{j}\) be the following sets in I:
and
Consider \(i,j\geq n_{0}\) and \(n\notin E_{i}\cap E_{j}\). Then we have
Thus \((L_{i})\) is a Cauchy sequence in \(\mathbb{R}\) and thus converges, say to L, that is, \(\lim_{i\to\infty}L_{i}=L\).
(ii) Let \(\delta>0\) be given. Then we can find \(m_{0}\) such that
We have \((x_{n}^{(i)})\rightarrow x_{n}\) as \(i\to\infty\). Thus
Since \((\hat{F}_{n}^{(j)})\) is Iconverges to \(L_{j}\), there exists \(D\in{I}\) such that, for each \(n\notin D\), we have
Without loss of generality, let \(j>m_{0}\). Then, for all \(n\notin D\), we have by (2.13), (2.14), and (2.15) that
Hence \((x_{n})\) is Fibonacci Iconvergent to L. Thus \(m^{I}(\hat{F})\) is a Banach space. The other cases can be similarly established. □
The following results are consequences of Theorem 2.7.
Theorem 2.8
The spaces \(m^{I}(\hat{F})\) and \(m_{0}^{I}(\hat{F})\) are Kspaces.
Theorem 2.9
The set \({m^{I}}(\hat{F})\) is a closed subspace of \({\ell_{\infty}(\hat{F})}\).
Since the inclusions \(m^{I}(\hat{F})\subset{\ell_{ \infty}}(\hat{F})\) and \(m_{0}^{I}(\hat{F})\subset{\ell_{\infty}(\hat{F})}\) are strict, in view of Theorem 2.9, we have the following result.
Theorem 2.10
The spaces \(m^{I}(\hat{F})\) and \(m^{I}_{0}(\hat{F})\) are nowhere dense subsets of \(\ell_{\infty}(\hat{F})\).
Theorem 2.11
The spaces \({c^{I}_{0}(\hat{F})}\) and \({m_{0}^{I}(\hat{F})} \) are solid and monotone.
Proof
We will prove the result for \({c^{I}_{0}(\hat{F})}\); for \({m_{0}^{I}(\hat{F})}\), the result can be established similarly. Let \(x=(x_{n})\in{c^{I}_{0}(\hat{F})}\). For \(\epsilon>0\), the set
Let \(\alpha=(\alpha_{n})\) be a sequence of scalars with \(\alpha\leq 1\) for all \(n\in{\mathbb{N}}\). Then
From this inequality and from (2.16) we have that
implies
Therefore \((\alpha x_{n})\in{c_{0}^{I}(\hat{F})}\). Hence the space \({c_{0}^{I}(\hat{F})}\) is solid, and hence by Lemma 1.1 the space \({c_{0}^{I}(\hat{F})}\) is monotone. □
Theorem 2.12
The spaces \(c_{0}^{I}(\hat{F})\) and \(c^{I}(\hat{F})\) are sequence algebras.
Proof
Let \(x=(x_{n}),y=(y_{n})\in{c_{0}^{I}(\hat{F})}\). Then
Therefore, from (2.17) we have \(I\mbox{}\lim\hat {F}_{n}(x\cdot y)=0\). This implies that \(\{n\in{\mathbb{N}}:\hat {F}_{n}(x\cdot y)\geq\epsilon \}\in{I}\). Thus, \((x\cdot y)\in {c_{0}^{I}(\hat{F})}\). Hence \({c_{0}^{I}(\hat{F})}\) is sequence algebra. Similarly, we can prove that \({c^{I}(\hat{F})}\), is a sequence algebra. □
Theorem 2.13
The function \(h:{m^{I}(\hat{F})}\rightarrow\mathbb{R}\) defined by , where \(m^{I}(\hat {F})={\ell_{\infty}(\hat{F})}\cap{c^{I}(\hat{F})}\), is a Lipschitz function and hence uniformly continuous.
Proof
First of all, we show that the function is well defined. Let \(x,y\in {m^{I}}(\hat{F})\) be such that
Thus h is well defined. Next, let \(x=(x_{n}), y=(y_{n})\in{m^{I}(\hat {F})}\), \(x\neq y\). Then
and
where \(xy_{*}= \sup_{n} \vert \hat{F}_{n}(x)\hat{F}_{n}(y) \vert \). Thus
and
Hence \(B=B_{x}\cap B_{y}\in{\mathcal{F}(I)}\), so that B is a nonempty set. Therefore, choosing \(n\in{B}\), we have
Thus, h is a Lipschitz function and hence uniformly continuous. □
Theorem 2.14
If \(x=(x_{n}),y=(y_{n})\in{m^{I}(\hat{F})}\) with \(\hat{F}_{n}(x\cdot y)=\hat {F}_{n}(x)\cdot \hat{F}_{n}(y)\), then \((x\cdot y)\in{m^{I}(\hat{F})}\) and \(h(x\cdot y) = h(x)\cdot h(y)\), where \(h:{m^{I}(\hat{F})}\rightarrow\mathbb{R}\) is defined by \(h(x)= \vert I\mbox{}\lim\hat{F}_{n}(x) \vert \).
Proof
For \(\epsilon>0\),
and
where \(\epsilon=xy_{*}= \sup_{n} \vert \hat{F}_{n}(x)\hat {F}_{n}(y) \vert \). Now, we have
As \({m^{I}(\hat{F})}\subseteq{\ell_{\infty}(\hat{F})}\), there exists \(M\in{\mathbb{R}}\) such that \(\hat{F}_{n}(x)< M\). Therefore, from equations (2.18), (2.19), and (2.20) we have
for all \(n\in{B_{x}\cap B_{y}}\in{\mathcal{F}(I)}\). Hence \((x\cdot y)\in {m^{I}(\hat{F})}\) and \(h(x\cdot y)= h(x)\cdot h(y)\). □
3 Conclusion
In this paper, we have introduced and studied new difference sequence spaces \(c_{0}^{I}(\hat{F})\), \(c^{I}(\hat{F})\), and \(\ell_{\infty }^{I}(\hat{F})\). We investigated the general type of convergence, that is, Fibonacci Iconvergence for sequences related to the Fibonacci difference matrix F̂ derived by the sequence of Fibonacci numbers. We studied some inclusion relations concerning these spaces. Further, we investigated some topological and algebraic properties of these spaces. These definitions and results provide new tools to deal with the convergence problems of sequences occurring in many branches of science and engineering.
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Acknowledgements
The authors would like to thank the referees for a careful reading and several constructive comments and making some useful corrections that have improved the presentation of the paper.
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Vakeel A. Khan received the M.Phil. and Ph.D., degrees in Mathematics from Aligarh Muslim University, Aligarh, India. Currently, he is an Associate Professor at Aligarh Muslim University, Aligarh, India. A vigorous researcher in the area of sequence spaces, he has published a number of research papers in reputed national and international journals, including Numerical Functional Analysis and Optimization (Taylors and Francis), Information Sciences (Elsevier), Applied Mathematics Letters (Elsevier), A Journal of Chinese Universities (SpringerVerlag, China). Rami K.A. Rababah is working as an Assistant Professor in the Department of Mathematics, Amman Arab University, Jordan. Kamal M.A.S. Alshlool received M.Sc., from Aligarh Muslim University and is currently a Ph.D., scholar at Aligarh Muslim University. Sameera A.A. Abdullah received M.Sc., from Aligarh Muslim University and is currently a Ph.D., scholar at Aligarh Muslim University. Ayaz Ahmad is working as an Assistant Professor in the National Institute Technology, Patna, India.
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This work was supported by Department of Mathematics, Amman Arab University, Amman, Jordan.
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Khan, V.A., Rababah, R.K.A., Alshlool, K.M.A.S. et al. On ideal convergence Fibonacci difference sequence spaces. Adv Differ Equ 2018, 199 (2018). https://doi.org/10.1186/s1366201816392
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DOI: https://doi.org/10.1186/s1366201816392