On a generalization of statistical cluster and limit points
- Mustafa Seyyit Seyyidog̃lu^{1} and
- Neşet Özkan Tan^{1}Email author
https://doi.org/10.1186/s13662-015-0395-9
© Seyyit Seyyidog̃lu and Özkan Tan; licensee Springer. 2015
Received: 31 October 2014
Accepted: 30 January 2015
Published: 24 February 2015
Abstract
This paper is concerned with the notions of statistical limit and cluster points defined by Fridy. Following the concept of a Δ-density for a subset of a time scale, we established a generalization of these notions which are called Δ-limit and Δ-cluster points for a function defined on a time scale \(\mathbb{T}\).
Keywords
MSC
1 Introduction
The theory of time scales was first constructed by Hilger in his PhD thesis in [1]. Measure theory on time scales has been introduced in [2], then further studies were made in [3] and [4]. Deniz and Ufuktepe defined the Lebesgue-Stieltjes Δ and ∇-measures and by using these measures, they defined an integral which is adaptable to a time scale, specifically the Lebesgue-Stieltjes Δ-integral, in [5]. In the light of these studies, let us introduce some time scale and measure theoretic notations.
The time scale \(\mathbb{T}\) is an arbitrary nonempty closed subset of the real numbers ℝ. In fact \(\mathbb{T}\) is a complete metric space with the usual metric. Throughout this paper we consider a time scale \(\mathbb{T}\) with the topology inherited from the real numbers with the standard topology.
Open intervals and half-open intervals are defined similarly.
The theory of statistical convergence has been introduced in [6]. Fridy made progress with the concept of a statistically Cauchy sequence in [7] and proved that it is equivalent to statistical convergence. Besides, in [8] the notion of the statistical limit point is defined. The central purpose of the present paper is to extend the notions of statistical limit point and statistical cluster point for functions defined on a time scale and valued on real numbers. It is natural to attempt to do this by adapting the ‘Δ-density’, which is defined in [9]. Let us recall the definitions of Δ-density, Δ-convergence, and the Δ-Cauchy property, and some results necessary for our purpose. Throughout this paper let us take all time scales unbounded from above and having a minimum point.
Theorem 1.1
- (i)
A function f is Δ-convergent,
- (ii)
a function f is Δ-Cauchy,
- (iii)
for a function f there exists a measurable and convergent function \(g:\mathbb{T}\rightarrow\mathbb{R}\) such that \(f(t)=g(t)\) for \(\Delta\mbox{-}a.a.~t.\)
The notation of \(\Delta\mbox{-}a.a.~t.\) for a property means that the set of elements for which the property does not hold is a set of Δ-density zero.
For further information as regards statistically convergence in a time scale, see [10, 11], and [12].
2 Δ-Limit point, Δ-cluster point
In the present section we investigate the Δ-limit point and Δ-cluster point concepts for a function defined on a time scale \(\mathbb{T}\). The results of this section coincide with the statistical limit point and the statistical cluster point in the case \(\mathbb{T}=\mathbb{N}\). In other words, these new notions give us a progression of generalizations of results for statistical limit points and statistical cluster points, introduced by Fridy in [8].
Definition 2.1
(Δ-Limit point)
A real number L is called a Δ-limit point of a function \(f:\mathbb{T}\rightarrow\mathbb{R}\) if there exists a subset K of \(\mathbb{T}\) with a non-zero Δ-density or if it does not have a Δ-density such that \(f(t)\rightarrow L\) whenever \(t\rightarrow\infty\) in K.
Note that in Definition 2.1 the measurable set K may have a positive Δ-density or may not have even a Δ-density. For describing this situation we will use the Δ-non-thin subset notation. This notation can be considered as a modified Fridy non-thin term defined for subsequences. Detailed information as regards the classical thin or non-thin concepts can be found in [8]. We proceed with the next definition.
Definition 2.2
(Δ-Cluster point)
A real number L is called a Δ-cluster point of a measurable function \(f:\mathbb{T}\rightarrow\mathbb{R}\) if for all \(\varepsilon>0\) the set \(\{t\in\mathbb{T}:|f(t)-L|<\varepsilon \}\) is a Δ-non-thin set.
We denote the set of Δ-limit points and Δ-cluster points of f by \(\Lambda_{f}\) and \(\Gamma_{f}\), respectively.
Definition 2.3
(Δ-Boundedness)
A measurable function \(f:\mathbb{T}\rightarrow \mathbb{R}\) is called Δ-bounded if there exists a real number r such that \(\delta_{\Delta}(\{ t\in\mathbb{T}: \vert f(t)\vert\leq r\}) =1\).
Definition 2.4
(Δ-Monotone increasing)
A function \(f:\mathbb{T}\rightarrow \mathbb{R}\) is called Δ-monotone increasing if there exists a subset K of \(\mathbb{T}\) with \(\delta_{\Delta}(K)=1\) such that f is monotone on K. That is, for each pair \(t_{1},t_{2}\in K\), \(t_{1}< t_{2}\) implies \(f(t_{1})\leq f(t_{2})\).
Proposition 2.5
Let \(f:\mathbb{T}\rightarrow\mathbb{R}\) be a measurable function, then \(\Lambda_{f}\subset\Gamma_{f}\).
Proof
We will proceed some special cases of the above concepts. We also should emphasize that the sets \(\Lambda_{f}\) and \(\Gamma_{f}\) are not equal in general. Details are in the following example.
Example
(i) Let the time scale be \(\mathbb{T}=\mathbb{N}\). This case is called the discrete case and it is easy to see that all definitions above coincide with the definition of a limit point and cluster point in the classical statistically convergence theory.
Proposition 2.6
Let \(f:\mathbb{T}\rightarrow\mathbb{R}\) be a measurable function with \(\Delta\mbox{-}\!\lim_{t\rightarrow\infty}f(t)=L\), then \(\Lambda _{f}=\Gamma_{f}= \{ L \} \).
Proof
Proposition 2.7
Let \(f:\mathbb{T}\rightarrow\mathbb{R}\) be a measurable function; then the set \(\Gamma_{f}\) is closed.
Proof
Theorem 2.8
Let \(f,g:\mathbb{T}\rightarrow\mathbb{R}\) be measurable functions. If \(f(t)=g(t)\) for \(\Delta\mbox{-}a.a.~t.\) then \(\Lambda_{f}=\Lambda_{g}\) and \(\Gamma _{f}=\Gamma_{g}\).
Proof
Proposition 2.9
Let \(f:\mathbb{T}\rightarrow\mathbb{R}\) be a measurable function. If f is bounded on a Δ-non-thin set then \(\Gamma_{f}\neq\emptyset\).
Proof
The following corollary can be obtained immediately from Proposition 2.9.
Corollary 2.10
If a measurable function \(f:\mathbb{T}\rightarrow\mathbb{R}\) is Δ-bounded, then \(\Gamma_{f}\neq\emptyset\).
Proposition 2.11
Let \(f:\mathbb{T}\rightarrow\mathbb{R}\) be a measurable function. If f is Δ-monotone increasing and Δ-bounded, then it is Δ-convergent.
Proof
Declarations
Acknowledgements
The authors thank the editor and referees for making possible a significant revision of the manuscript.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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