On ideal convergence Fibonacci difference sequence spaces
- Vakeel A. Khan^{1}Email author,
- Rami K. A. Rababah^{2},
- Kamal M. A. S. Alshlool^{1},
- Sameera A. A. Abdullah^{1} and
- Ayaz Ahmad^{3}
https://doi.org/10.1186/s13662-018-1639-2
© The Author(s) 2018
Received: 23 January 2018
Accepted: 10 May 2018
Published: 25 May 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.
Keywords
1 Introduction
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 I-convergence, 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 I-convergence 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 I-convergence, that is, the notion of I-summability and introduced new sequence spaces \(c_{A}^{I}\) and \(m_{A}^{I}\), the I-convergence field and bounded I-convergence 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 I-null, I-convergent, and I-bounded 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 st-null.
Definition 1.2
([27])
A sequence \(x=(x_{n})\in{\omega} \) is said to be I-Cauchy 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 I-convergent 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 I-null.
Definition 1.4
([19])
A sequence \(x=(x_{n})\in{\omega}\) is said to be I-bounded 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 \(n\in{\mathbb{N}}\).
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])
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 \(|h(x)-h(y)|\leq K|x-y|\), where K is called the Lipschitz constant.
Remark 1.1
([27])
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 I-Convergence Fibonacci difference sequence spaces
Definition 2.1
Let I be an admissible ideal of subsets of \(\mathbb{N}\). A sequence \(x=(x_{n})\in{\omega} \) is called Fibonacci I-Cauchy 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
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
Theorem 2.2
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
Example 2.4
Theorem 2.4
Proof
Theorem 2.5
- (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
Theorem 2.6
The inclusions \({c_{0}^{I}(\hat{F})}\subset{c^{I}(\hat{F})}\subset {\ell_{\infty}^{I}(\hat{F})}\) are strict.
Proof
Example 2.5
Thus the inclusion \({c_{0}^{I}(\hat{F})}\subset{c^{I}(\hat{F})}\subset {\ell_{\infty}^{I}(\hat{F})}\) is strict. □
Remark 2.1
Theorem 2.7
The spaces \(m^{I}(\hat{F})\) and \(m_{0}^{I}(\hat{F})\) are Banach spaces normed by (2.8).
Proof
- (i)
\((L_{i})\) is convergent say to L and
- (ii)
\(I\mbox{-}\lim\hat{F}_{n}(x)=L\).
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 K-spaces.
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
Theorem 2.12
The spaces \(c_{0}^{I}(\hat{F})\) and \(c^{I}(\hat{F})\) are sequence algebras.
Proof
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
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
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 I-convergence 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.
Declarations
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.
Authors’ information
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 (Springer-Verlag, 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.
Funding
This work was supported by Department of Mathematics, Amman Arab University, Amman, Jordan.
Authors’ contributions
All authors of the manuscript have read and agreed to its content and are accountable for all aspects of the accuracy and integrity of the manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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