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On a generalization of statistical cluster and limit points
Advances in Difference Equations volume 2015, Article number: 55 (2015)
Abstract
This paper is concerned with the notions of statistical limit and cluster points defined by Fridy. Following the concept of a Δdensity for a subset of a time scale, we established a generalization of these notions which are called Δlimit and Δcluster points for a function defined on a time scale \(\mathbb{T}\).
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 LebesgueStieltjes Δ and ∇measures and by using these measures, they defined an integral which is adaptable to a time scale, specifically the LebesgueStieltjes Δintegral, 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.
We define the forward jump operator \(\sigma:\mathbb{T\rightarrow T}\), for each \(t\in\mathbb{T}\), by
For \(a,b\in\mathbb{T}\) with \(a\leq b\) we define the interval \([ a,b]\) in \(\mathbb{T}\) by
Open intervals and halfopen intervals are defined similarly.
Let \(\mathcal{S}\) be a semiring of leftclosed and rightopen intervals and \(m^{\ast}\) be a Carathéodory extension of the Lebesgue set function m which is defined by \(m([a,b))=ba\), associated with the family \(\mathcal{S}\) in the time scale \(\mathbb{T}\) as in the real case. Also let \(\mathcal{M}(m^{*})\) be the σalgebra of all \(m^{\ast}\)measurable sets. Recall that \(\mathcal{M}(m^{*})\) consisting of such a subset E has the property that \(m^{\ast}(A)=m^{\ast}(A\cap E)+m^{\ast}(A\cap E^{c})\) for all \(A\subset\mathbb{T}\). It is well known that the restriction of \(m^{\ast}\) to \(\mathcal{M}(m^{\ast})\), which we denote by \(\mu_{\Delta}\), is a countably additive measure on \(\mathcal {M}(m^{\ast})\). This measure is called a Lebesgue Δmeasure. The measurable subsets of \(\mathbb{T}\) is called Δmeasurable and a function \(f:\mathbb{T}\rightarrow\mathbb{R}\) is called a measurable function, if \(f^{1}(O)\in\mathcal{M}(m^{\ast})\) for every open subset O of ℝ. From [2] we know that if \(a,b\in\mathbb{T}\) and \(a\leq b\), then
If \(a,b\in\mathbb{T}\setminus\{\max\mathbb{T}\}\) and \(a\leq b\), then
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 ‘Δdensity’, which is defined in [9]. Let us recall the definitions of Δdensity, Δconvergence, and the ΔCauchy 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.
Let A be a Δmeasurable subset of \(\mathbb{T}\) and \(a=\min \mathbb{T}\), the Δdensity of A in \(\mathbb{T}\) is defined by
(if this limit exists) where \(A(s)= \{ t\in A:t\leq s \} \). The Δdensity function can be consider as a probabilistic finite additive measure on the algebra of a subset of \(\mathbb{T}\) which has a Δdensity. We will denote this space by \(\mathcal{M}_{d}\). Obviously a subset of \(\mathbb{T}\) which has a zero Δdensity is an element of \(\mathcal{M}_{d}\). This collection is denoted by \(\mathcal{M}_{d}^{0}\) and it has the ring structure. By using the Δdensity we obtained the following new type of convergence, which is a generalization of the definitions of natural statistical convergence and statistical Cauchy sequences. Let us recall the Δconvergence and ΔCauchy definitions for functions:
A measurable function f is Δconvergent to the number L provided that for each \(\varepsilon>0\)
A measurable function f is ΔCauchy provided that for every \(\varepsilon>0\) there exists a number \(t_{0}\in\mathbb{T}\) such that
Note that these definitions coincide with the notions of statistical convergence and of statistical Cauchy definitions for the real sequences whenever \(\mathbb{T}\) is taken as the natural numbers. The reason of this fact is that the notions of Δdensity and natural density coincide in the case \(\mathbb{T}=\mathbb{N}\). In [9] we showed that these definitions are equivalent for a measurable function. Recall the main result in [9].
Theorem 1.1
Let \(f:\mathbb{T}\rightarrow\mathbb{R}\) be a measurable function. The following statements are equivalent:

(i)
A function f is Δconvergent,

(ii)
a function f is ΔCauchy,

(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 Δdensity zero.
For further information as regards statistically convergence in a time scale, see [10, 11], and [12].
2 ΔLimit point, Δcluster point
In the present section we investigate the Δlimit point and Δcluster 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
(ΔLimit point)
A real number L is called a Δlimit point of a function \(f:\mathbb{T}\rightarrow\mathbb{R}\) if there exists a subset K of \(\mathbb{T}\) with a nonzero Δdensity or if it does not have a Δdensity 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 Δdensity or may not have even a Δdensity. For describing this situation we will use the Δnonthin subset notation. This notation can be considered as a modified Fridy nonthin term defined for subsequences. Detailed information as regards the classical thin or nonthin concepts can be found in [8]. We proceed with the next definition.
Definition 2.2
(ΔCluster point)
A real number L is called a Δcluster 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 Δnonthin set.
We denote the set of Δlimit points and Δcluster points of f by \(\Lambda_{f}\) and \(\Gamma_{f}\), respectively.
Definition 2.3
(ΔBoundedness)
A measurable function \(f:\mathbb{T}\rightarrow \mathbb{R}\) is called Δbounded 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
(ΔMonotone increasing)
A function \(f:\mathbb{T}\rightarrow \mathbb{R}\) is called Δmonotone 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
Let \(L\in\Lambda_{f}\). Then there exists a Δnonthin set \(K\subset\mathbb{T}\) such that
and
Let an arbitrary \(\varepsilon>0\) be given. The set \(\{t\in K:f(t)L\geq \varepsilon\}\) is measurable and bounded. From
we have
Finally we have
which means the set \(\{ t\in\mathbb{T}:f(t)L<\varepsilon \} \) is a Δnonthin set; \(L\in\Gamma_{f}\). □
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.
(ii) Let \(q>1\) be a fixed integer and \(\mathbb{T}=\{q^{m}:m\in\mathbb {N}\}\). Consider the sequence of natural numbers \((k_{n})\) such that \(k_{n+1}k_{n}>1\). If we take the subset \(K=\{q^{k_{n}}:\in\mathbb{N}\}\) then we can easily show that K is a Δnonthin set. Let \(k_{n}\leq k< k_{n+1}\) and \(t=q^{k}\), then we have
If one can take the limit \(t\rightarrow\infty\) in the above equality we can conclude that the limit does not exist. That means K can be considered as a Δnonthin set.
(iii) Let us consider the continuous case \(\mathbb{T}=[0,\infty)\) and the function \(f:\mathbb{T}=[0,\infty)\rightarrow\mathbb{R}\) defined by
for \(n\in\mathbb{N}\). Since for each \(n=0,1,2,\ldots \) , we have
and we have \(1/(n+1)\in\Lambda_{f}\). Moreover,
implies that
This means that \(0\in\Gamma_{f}\). The set of Δcluster points is exactly the set \(\{1,1/2,1/3,\ldots\}\cup\{0\}\). Now we shall show that \(0\notin\Lambda_{f}\). For our purpose, we can consider a measurable subset \(A\subset\mathbb{T}\) with
We claim that \(\delta_{\Delta}(A)=0\). Let \(\varepsilon>0\) be given. There exists a natural number m such that \(2^{m1}<\varepsilon/3\). Then there exists \(s_{1}\in\mathbb{T}\) such that
holds for all \(s>s_{1}\). On the other hand, from (2.1), \(s_{2}\in \mathbb{T}\) can be chosen such that
holds for all \(s>s_{2}\). Let us define \(s_{0}=\max\{s_{1},s_{2}\}\). From (2.2) and (2.3) we have
for all \(s>s_{0}\).
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
Assume that \(\Delta\mbox{}\!\lim_{t\rightarrow\infty}f(t)=L\). From Theorem 1.1 there exists a measurable function \(g:\mathbb{T}\rightarrow \mathbb{R}\) such that \(\lim_{t\rightarrow\infty}g(t)=L\), and \(f(t)=g(t)\) holds for \(\Delta\mbox{}a.a.~t.\) Let us define \(K= \{ t\in\mathbb {T}:f(t)=g(t) \} \). From the definition of \(\Delta\mbox{}a.a.~t.\), \(\delta_{\Delta}(K)=1\) and
That is, \(L\in\Lambda_{f}\). By Proposition 2.5, \(L\in\Gamma_{f}\).
Now we will show that L is a unique element of \(\Gamma_{f}\). Let \(\varepsilon >0\) be given. Since \(\Delta\mbox{}\!\lim_{t\rightarrow\infty}f(t)=L\), we have \(K_{1}=\{t\in\mathbb{T}:f(t)L<\varepsilon/2\}\) with \(\delta_{\Delta }(K_{1})=1\). If \(L^{\prime}\) is another element of \(\Gamma_{f}\) then we can define \(K_{2}=\{t\in \mathbb{T}:f(t)L^{\prime}<\varepsilon/2\}\); then
We claim that \(K_{1} \cap K_{2}\neq\emptyset\). Assume that \(K_{1}\cap K_{2}= \emptyset\) then \(K_{1}\subset ( K_{2} ) ^{c}\) and from \(\delta_{\Delta}(K_{1})=1\) we have \(\delta _{\Delta}( ( K_{2} ) ^{c})=1\) and so \(\delta_{\Delta }(K_{2})=0\). But this contradicts (2.4). For each \(t_{0}\in K_{1}\cap K_{2}\) we have
Since \(\varepsilon>0\) is arbitrary, we have \(L=L^{{\prime}}\). □
Proposition 2.7
Let \(f:\mathbb{T}\rightarrow\mathbb{R}\) be a measurable function; then the set \(\Gamma_{f}\) is closed.
Proof
Let \((L_{n})\) be a sequence in \(\Gamma_{f}\) such that \(L_{n}\rightarrow L\) whenever \(n\rightarrow\infty\). For a given \(\varepsilon>0\), choose \(n_{0}\) large enough to make \(L\varepsilon< L_{n_{0}}<L+\varepsilon\) and choose \(\varepsilon^{\prime}>0\) such that \((L_{n_{0}}\varepsilon ^{\prime }, L_{n_{0}}+\varepsilon^{\prime})\subset(L\varepsilon ,L+\varepsilon)\). In this case, from
and
we have
That means the set \(\{ t\in\mathbb{T}:\vert f(t)L\vert <\varepsilon \} \) is not a set that has zero Δdensity; \(L\in\Gamma_{f}\). □
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
Let \(L\in\Lambda_{f}\). Then there exists a Δnonthin set K such that
and
Since \(\delta_{\Delta}( \{ t\in\mathbb{T}:f(t)\neq g(t) \} )=0\) we have \(\delta_{\Delta}( \{ t\in K:f(t)\neq g(t) \} )=0\). This means that the set \(\{ t\in K:f(t)=g(t) \} \) is not a set that has zeroΔdensity and so \(L\in\Lambda_{g}\). Then we have \(\Lambda _{f}\subset\Lambda_{g}\). It is easy to see that \(\Lambda_{g}\subset\Lambda_{f}\) from symmetry. Finally we have \(\Lambda_{f}=\Lambda_{g}\). The equality \(\Gamma_{f}=\Gamma_{g}\) can be shown in a similar way. □
Proposition 2.9
Let \(f:\mathbb{T}\rightarrow\mathbb{R}\) be a measurable function. If f is bounded on a Δnonthin set then \(\Gamma_{f}\neq\emptyset\).
Proof
Assume that f is bounded on K, which is a Δnonthin subset of \(\mathbb{T} \) and \(\Gamma_{f}=\emptyset\). Then for each \(t\in K\) there exists a neighborhood of \(f(t)\)
such that \(\delta_{\Delta} ( f^{1} ( \mathcal{N} ( f(t) ) ) ) =0\). Moreover, it is clear that
If \(f(K)\) is not closed then we can consider \(\overline{f(K)}\) by adding limit points \(\beta_{i}\) (\(i\in I\)) of \(f(K)\) and we have
We have
and note that positive real numbers \(\varepsilon(i)\) can be chosen such that \(\delta_{\Delta} ( f^{1} ( \mathcal{N} ( \beta_{i} ) ) ) =0\). Since \(\overline{f(K)}\) is compact, we can find \(\mathcal{B}_{1},\mathcal {B}_{2},\ldots,\mathcal{B}_{n}\) which is a finite subcover of \(\{ \mathcal{N} ( f(t) ) \} _{t\in K}\cup \{ \mathcal{N} ( \beta_{i} ) \} _{i\in I}\) such that
Then we have
Since \(\delta_{\Delta} ( \bigcup_{k=1}^{n}f^{1}(\mathcal {B}_{k}) ) =0\) we can conclude that \(\delta_{\Delta}(K)=0\). But this conclusion contradicts the Δnonthin property of K. One can prove everything in a similar way if \(f(K)\) is closed. □
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 Δbounded, then \(\Gamma_{f}\neq\emptyset\).
Proposition 2.11
Let \(f:\mathbb{T}\rightarrow\mathbb{R}\) be a measurable function. If f is Δmonotone increasing and Δbounded, then it is Δconvergent.
Proof
Since f is Δmonotone increasing, there exists a subset \(K_{1}\subset\mathbb{T}\) with Δdensity equal to one, and for every \(t_{1},t_{2}\in K_{1}\) and \(t_{1}< t_{2}\) implies \(f(t_{1})\leq f(t_{2})\). Since f is Δbounded there exists a real number A and \(K_{2}= \{ t\in\mathbb{T}:\vert f(t)\vert \leq A \} \) with \(\delta_{\Delta} ( K_{2} ) =1\). Setting \(K=K_{1}\cap K_{2}\), one has \(\delta_{\Delta} ( K ) =1\). We now consider the following set:
and let β be the supremum of \(f(K)\) over K. For an arbitrary \(\varepsilon>0\) there is a point \(t_{0}\in K\) that satisfies \(\beta\varepsilon< f(t_{0})\leq\beta\). From the monotonicity we can write \(\beta\varepsilon< f(t)<\beta+\varepsilon\) for every \(t\in K \) satisfying \(t\geq t_{0}\). The last statement implies that \(\Delta\mbox{}\!\lim_{t\rightarrow\infty}f(t)=\beta\). □
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The authors thank the editor and referees for making possible a significant revision of the manuscript.
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NÖT carried out the measure theoretic studies, participated in the sequence alignment, and drafted the manuscript. MSS carried out the subjects of time scale and statistically convergence. NÖT and MSS conceived of the study, participated in its design and coordination, and helped to draft the manuscript. All authors read and approved the final manuscript.
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Seyyit Seyyidog̃lu, M., Özkan Tan, N. On a generalization of statistical cluster and limit points. Adv Differ Equ 2015, 55 (2015). https://doi.org/10.1186/s1366201503959
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DOI: https://doi.org/10.1186/s1366201503959