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Lacunary statistical boundedness on time scales
Advances in Difference Equations volume 2021, Article number: 513 (2021)
Abstract
In this paper, we introduce the concept of lacunary statistical boundedness of Δ-measurable real-valued functions on an arbitrary time scale. We also give the relations between statistical boundedness and lacunary statistical boundedness on time scales.
1 Introduction
The idea of statistical convergence was formally introduced by Fast [1] and Steinhaus [2], independently. This concept is a generalization of the classical convergence, and it depends on the density of subsets of the natural numbers \(\mathbb{N}\). Let \(K \subseteq \mathbb{N} \) and \({K_{n}} = \{ {k \le n:k \in K} \} \). Then the natural density of K is defined by \(\delta ( K ) = {\lim_{n}}{n^{ - 1}} \vert {{K_{n}}} \vert \) if the limit exists, where \(\vert {{K_{n}}} \vert \) indicates the cardinality of \({K_{n}}\). A sequence \(x = ( {{x_{k}}} )\) is said to be statistically convergent to L if, for every \(\varepsilon > 0\), the set \({K_{\varepsilon }}: = \{ {k \in \mathbb{N}: \vert {{x_{k}} - L} \vert \ge \varepsilon } \} \) has natural density zero, i.e., for each \(\varepsilon > 0\),
and written as \(st - \lim x = L\).
Statistical convergence has become very active in different fields of mathematics over the years and has been studied by many researchers, see [3–21].
Before moving on to the main results of the study, we need to give a brief introduction to time scale theory. A time scale is an arbitrary closed subset of the real numbers \(\mathbb{R}\) in the usual topology which is denoted by \(\mathbb{T}\). The calculus of time scales has been constructed by Hilger [22]. This theory is an efficient tool to unify continuous and discrete analyses in one theory as it allows integration and differentiation of the independent domain used. Therefore, it has received much attention in different branches of science and engineering. The readers can find basic calculus of time scales in [23–25]. Moreover, the idea of statistical convergence was first studied on time scales in [26] and [27], independently. Since then, many concepts related to statistical convergence and summability theory have been applied to time scales by various researchers in the literature [28–36].
The forward jump operator \(\sigma :\mathbb{T} \to \mathbb{T}\) is defined by
for \(t \in \mathbb{T}\), and also the graininess function \(\mu :\mathbb{T} \to [ {0,\infty } )\) is defined by \(\mu ( t ) = \sigma ( t ) - t\). A closed interval on a time scale \(\mathbb{T}\) is given by \({ [ {a,b} ]_{\mathbb{T}}} = \{ {t \in \mathbb{T}:a \le t \le b} \} \). Open intervals or half-open intervals are defined accordingly.
In this paper, we use the Lebesgue Δ-measure by \({\mu _{\Delta }}\) introduced by Guseinov [24]. In this case, it is known that if \(a \in \mathbb{T} \backslash \{ {\max \mathbb{T}} \} \), then the single point set \(\{ a \} \) is Δ-measurable and \({\mu _{\Delta }} ( { \{ a \} } ) = \sigma ( a ) - a\). If \(a,b \in \mathbb{T}\) and \(a \le b\), then \({\mu _{\Delta }} ( {{{ [ {a,b} )}_{\mathbb{T}}}} ) = b - a\) and \({\mu _{\Delta }} ( {{{ ( {a,b} )}_{\mathbb{T}}}} ) = b - \sigma ( a )\); if \(a,b \in \mathbb{T}\backslash \{ {\max \mathbb{T}} \} \) and \(a \le b\), then \({\mu _{\Delta }} ( {{{ ( {a,b} ]}_{\mathbb{T}}}} ) = \sigma ( b ) - \sigma ( a )\) and \({\mu _{\Delta }} ( {{{ [ {a,b} ]}_{\mathbb{T}}}} ) = \sigma ( b ) - a\), see [24].
We now recall some of the concepts defined using the time scale calculus on the summability theory:
Throughout this paper, we consider that \(\mathbb{T}\) is a time scale satisfying \(\inf \mathbb{T} = {t_{0}} > 0\) and \(\sup \mathbb{T} = \infty \).
Definition 1.1
([27])
A Δ-measurable function \(f: \mathbb{T}\to \mathbb{R}\) is statistically convergent to a number L on \(\mathbb{T}\) if, for every \(\varepsilon > 0\),
which is denoted by \(st{_{\mathbb{T}}} - \lim_{t \to \infty } f ( t ) = L\).
Let \(\theta = ( {{k_{r}}} )\) be an increasing sequence of nonnegative integers with \({k_{0}} = 0\) and \(\sigma ( {{k_{r}}} ) - \sigma ( {{k_{r - 1}}} ) \to \infty \) as \(r \to \infty \). Then θ is called a lacunary sequence with respect to \(\mathbb{T}\) [29].
Definition 1.2
([29])
Let θ be a lacunary sequence on \(\mathbb{T}\). A Δ-measurable function \(f: \mathbb{T}\to \mathbb{R}\) is said to be lacunary statistically convergent to a number L if, for every \(\varepsilon > 0\),
which is denoted by \({st_{ \theta - \mathbb{T}}} - \lim_{t \to \infty } f ( t ) = L\).
If \(f: \mathbb{T}\to \mathbb{R}\) is a function such that \(f(t)\) satisfies a property P for all t except a set which has zero lacunary density on time scale, then it is said that \(f ( t )\) has the property P almost all t with respect to θ.
Definition 1.3
([28])
Let \(f: \mathbb{T}\to \mathbb{R}\) be a Δ-measurable function. Then f is said to be statistically bounded on \(\mathbb{T}\) if there exists a number \(M > 0\) such that
The set of all statistically bounded functions on \(\mathbb{T}\) is denoted by \({S_{\mathbb{T}}} ( B )\).
The aim of this study is to introduce and examine the concept of lacunary statistical boundedness on time scales.
2 Main results
In this part, we begin by presenting a new definition, namely lacunary statistical boundedness on time scale. We then give some results related to this concept.
Definition 2.1
Let \(\theta = ( {{k_{r}}} )\) be a lacunary sequence on \(\mathbb{T}\) and \(f:\mathbb{T} \to \mathbb{R}\) be a Δ-measurable function. Then f is said to be lacunary statistically bounded on \(\mathbb{T}\) if there exists a number \(M>0\) such that
i.e., \(\vert {f ( s )} \vert \le M\) (almost all s with respect to θ). The set of all lacunary statistically bounded functions on \(\mathbb{T}\) is denoted by \({S_{\theta - \mathbb{T}}} ( B )\).
Remark 2.1
-
(i)
If we take \(\mathbb{T} = \mathbb{N}\) in Definition 2.1, then lacunary statistical boundedness on \(\mathbb{T}\) reduces to lacunary statistical boundedness of sequences which is introduced in [19].
-
(ii)
If we choose \(\mathbb{T} = [ {a,\infty } )\) (\(a > 1\)) in Definition 2.1, then lacunary statistical boundedness on \(\mathbb{T}\) reduces to lacunary statistical boundedness of measurable functions which is introduced in [21].
Theorem 2.1
Every lacunary statistically convergent function on \(\mathbb{T}\) is lacunary statistically bounded on \(\mathbb{T}\), but the converse does not need to be true.
Proof
Let \(f:\mathbb{T} \to \mathbb{R}\) be lacunary statistically convergent to M. Then, for each \(\varepsilon > 0\), we have
From the fact that
we obtain that \(\vert {f ( s )} \vert \le \varepsilon + M\) (almost all s with respect to θ) which is the desired result. For the converse, we consider the following example: Let \(f ( s ) = { ( { - 1} )^{s}}\) be a function where \(s \in \mathbb{T} = \mathbb{N}\). Then f is lacunary statistically bounded, but it is not a lacunary statistically convergent function. □
Theorem 2.2
Let \(\theta = ( {{k_{r}}} )\) be a lacunary sequence on \(\mathbb{T}\) and \(f:\mathbb{T} \to \mathbb{R}\) be a Δ-measurable function. Then f is lacunary statistically bounded if and only if there exists a bounded function \(g:\mathbb{T} \to \mathbb{R}\) such that \(f ( s ) = g ( s )\) almost all s with respect to θ.
Proof
First assume that f is lacunary statistically bounded. Then there exists \(M \ge 0\) such that \({\delta _{\theta - \mathbb{T}}} ( K ) = 0\), where \(K = \{ {s \in \mathbb{T}: \vert {f ( s )} \vert > M} \} \). Let us consider the function \(g:\mathbb{T} \to \mathbb{R}\) defined as follows:
It is clear that g is a Δ-measurable bounded function, and \(f ( s ) = g ( s )\) almost all s with respect to θ. Conversely, since g is bounded, there exists \(L \ge 0\) such that \(\vert {g ( s )} \vert \le L\) for all \(s \in \mathbb{T}\). Let \(D = \{ {s \in \mathbb{T}:f ( s ) \ne g ( s )} \} \). As \({\delta _{\theta - \mathbb{T}}} ( D ) = 0\), so \(\vert {f ( s )} \vert \le L\) almost all s with respect to θ. This means that f is lacunary statistically bounded. □
Theorem 2.3
Let \(\theta = ( {{k_{r}}} )\) be a lacunary sequence on \(\mathbb{T}\). Then we have
Proof
Sufficiency. The sufficiency part of this theorem can be proved using a similar technique to Lemma 3.1 of [30].
Necessity. Conversely, assume that \(\mathop{\lim \inf }_{r \to \infty } \frac{{\sigma ( {{k_{r}}} )}}{{\sigma ( {{k_{r - 1}}} )}} = 1\). Let us now select a subsequence \(( {{k_{r ( j )}}} )\) of \(\theta = ( {{k_{r}}} )\) satisfying
where \(r ( j ) > r ( {j - 1} ) + 1\).
Let us define Δ-measurable function \(f:\mathbb{T} \to \mathbb{R}\) by
Now, for any \(M > 0\), there exists \({j_{0}} \in \mathbb{N}\) such that \({k_{r ( {{j_{0}}} ) - 1}} > M\). If \(r = r ( j )\), we have
and therefore
for all \(j \ge {j_{0}}\). Also, if \(r \ne r ( j )\), then we get
Hence, \(f \notin {S_{\theta - \mathbb{T}}} ( B )\).
Indeed, for any sufficiently \(t \in \mathbb{T}\), we can find a unique \(j \in \mathbb{N}\) for which \({k_{r ( j ) - 1}} < t \le {k_{r ( {j + 1} ) - 1}}\), and we can write
Since \(t \to \infty \) implies \(j \to \infty \), we have \(f \in {S_{\mathbb{T}}} ( B )\). Therefore, we find \({S_{\mathbb{T}}} ( B ) \not \subset {S_{\theta - \mathbb{T}}} ( B )\), which is a contradiction. □
Remark 2.2
The function \(f:\mathbb{T} \to \mathbb{R}\) given in the necessity part of Theorem 2.3 is an example of a statistically bounded function which is not lacunary statistically bounded.
Theorem 2.4
Let \(\theta = ( {{k_{r}}} )\) be a lacunary sequence on \(\mathbb{T}\) such that \(\mu ( t ) \le Kt\) for some \(K > 0\) and for all \(t \in \mathbb{T}\). Then we have
Proof
Sufficiency. The sufficiency part of this theorem can be proved using a similar technique to Lemma 3.2 of [30].
Necessity. Conversely, assume that \(\mathop{\lim \sup }_{r \to \infty } \frac{{\sigma ( {{k_{r}}} )}}{{\sigma ( {{k_{r - 1}}} )}} = \infty \). By the hypothesis, we can get
and so
Let us select a subsequence \(( {{k_{r ( j )}}} )\) of \(\theta = ( {{k_{r}}} )\) such that \(\frac{{{k_{r ( j )}}}}{{\sigma ( {{k_{r ( j ) - 1}}} )}} > j\).
Now define Δ-measurable function \(f:\mathbb{T} \to \mathbb{R}\) by
Letting
If \(r \ne r ( j )\), then we can easily see that \({\tau _{r}} = 0\). If \(r = r ( j )\), then we get
Here, there are two possible cases: \(2\sigma ( {{k_{r ( j ) - 1}}} ) \in \mathbb{T}\) and \(2\sigma ( {{k_{r ( j ) - 1}}} ) \notin \mathbb{T}\). Let us now examine these: If \(2\sigma ( {{k_{r ( j ) - 1}}} ) \in \mathbb{T}\), then we find
and so
If \(2\sigma ( {{k_{r ( j ) - 1}}} ) \notin \mathbb{T}\), then we can write
where \({\alpha _{j}}: = \max \{ {s \in \mathbb{T}:s < 2\sigma ( {{k_{r ( j ) - 1}}} )} \} \). Therefore, using the hypothesis, we get
and so
Thus, we get that \(f \in {S_{\theta - \mathbb{T}}} ( B )\).
On the other hand, for any real \(M > 0\), there exists some \({j_{0}} \in \mathbb{N}\) such that \({k_{r ( j ) - 1}} > M\) for all \(j \ge {j_{0}}\). Then we have
for all \(j \ge {j_{0}}\). Here, if \(2\sigma ( {{k_{r ( j ) - 1}}} ) \in \mathbb{T}\), then
If \(2\sigma ( {{k_{r ( j ) - 1}}} ) \notin \mathbb{T}\), then we can write
and
where \({\alpha _{j}}\) is the same as in the above and \({\beta _{j}}: = \min \{ {s \in \mathbb{T}:s > 2\sigma ( {{k_{r ( j ) - 1}}} )} \} \). Hence, if \(2\sigma ( {{k_{r ( j ) - 1}}} ) \notin \mathbb{T}\), then we have
Therefore, we get that \(f \notin {S_{\mathbb{T}}} ( B )\). Consequently, we find that \({S_{\theta - \mathbb{T}}} ( B ) \not \subset {S_{\mathbb{T}}} ( B )\), which is a contradiction. □
Remark 2.3
The function \(f:\mathbb{T} \to \mathbb{R}\) given in the necessity part of Theorem 2.4 is an example of a lacunary statistically bounded function which is not statistically bounded.
Theorem 2.5
Let \(\theta = ( {{k_{r}}} )\) and \(\theta ' = ( {{l_{r}}} )\) be two lacunary sequences on \(\mathbb{T}\) such that \({ ( {{k_{r - 1}},{k_{r}}} ]_{\mathbb{T}}} \subset { ( {{l_{r - 1}},{l_{r}}} ]_{\mathbb{T}}}\) for all \(r \in \mathbb{N}\). Then we have the following:
-
(i)
If \(\mathop{\lim \inf }_{r \to \infty } \frac{{{\mu _{\Delta }} ( {{{ ( {{k_{r - 1}},{k_{r}}} ]}_{\mathbb{T}}}} )}}{{{\mu _{\Delta }} ( {{{ ( {{l_{r - 1}},{l_{r}}} ]}_{\mathbb{T}}}} )}} > 0\), then \({S_{\theta ' - \mathbb{T}}} ( B ) \subseteq {S_{\theta - \mathbb{T}}} ( B )\).
-
(ii)
If \(\lim_{r \to \infty } \frac{{{\mu _{\Delta }} ( {{{ ( {{k_{r - 1}},{k_{r}}} ]}_{\mathbb{T}}}} )}}{{{\mu _{\Delta }} ( {{{ ( {{l_{r - 1}},{l_{r}}} ]}_{\mathbb{T}}}} )}} = 1\), then \({S_{\theta - \mathbb{T}}} ( B ) \subseteq {S_{\theta ' - \mathbb{T}}} ( B )\).
Proof
(i) Suppose that \({ ( {{k_{r - 1}},{k_{r}}} ]_{\mathbb{T}}} \subset { ( {{l_{r - 1}},{l_{r}}} ]_{\mathbb{T}}}\) for all \(r \in \mathbb{N}\) and \(\mathop{\lim \inf }_{r \to \infty } \frac{{{\mu _{\Delta }} ( {{{ ( {{k_{r - 1}},{k_{r}}} ]}_{\mathbb{T}}}} )}}{{{\mu _{\Delta }} ( {{{ ( {{l_{r - 1}},{l_{r}}} ]}_{\mathbb{T}}}} )}} > 0\). For \(M > 0\), we have
and so
for all \(r \in \mathbb{N}\). Now, taking the limit as \(r \to \infty \), we get \({S_{\theta ' - \mathbb{T}}} ( B ) \subseteq {S_{\theta - \mathbb{T}}} ( B )\).
(ii) Suppose that \({ ( {{k_{r - 1}},{k_{r}}} ]_{\mathbb{T}}} \subset { ( {{l_{r - 1}},{l_{r}}} ]_{\mathbb{T}}}\) for all \(r \in \mathbb{N}\) and \(\lim_{r \to \infty } \frac{{{\mu _{\Delta }} ( {{{ ( {{k_{r - 1}},{k_{r}}} ]}_{\mathbb{T}}}} )}}{{{\mu _{\Delta }} ( {{{ ( {{l_{r - 1}},{l_{r}}} ]}_{\mathbb{T}}}} )}} = 1\). For \(M > 0\), we may write
for all \(r \in \mathbb{N}\). Since \(\lim_{r \to \infty } \frac{{{\mu _{\Delta }} ( {{{ ( {{k_{r - 1}},{k_{r}}} ]}_{\mathbb{T}}}} )}}{{{\mu _{\Delta }} ( {{{ ( {{l_{r - 1}},{l_{r}}} ]}_{\mathbb{T}}}} )}} = 1\), if \(f \in {S_{\theta - \mathbb{T}}} ( B )\), then we get \(f \in {S_{\theta ' - \mathbb{T}}} ( B )\). Thus, this implies that \({S_{\theta - \mathbb{T}}} ( B ) \subseteq {S_{\theta ' - \mathbb{T}}} ( B )\). □
References
Fast, H.: Sur la convergence statistique. Colloq. Math. 2, 241–244 (1951)
Steinhaus, H.: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 2(1), 73–74 (1951)
Schoenberg, I.J.: The integrability of certain functions and related summability methods. Am. Math. Mon. 66, 361–375 (1959)
Freedman, A.R., Sember, J.J., Raphael, M.: Some Cesàro-type summability spaces. Proc. Lond. Math. Soc. 37(3), 508–520 (1978)
Fridy, J.A.: On statistical convergence. Analysis 5, 301–313 (1985)
Fridy, J.A., Orhan, C.: Lacunary statistical convergence. Pac. J. Math. 160, 43–51 (1993)
Connor, J.S.: The statistical and strong p-Cesàro convergence of sequences. Analysis 8, 47–63 (1988)
Mursaleen, M.: λ-statistical convergence. Math. Slovaca 50(1), 111–115 (2000)
Móricz, F.: Statistical limits of measurable functions. Analysis 24(1), 1–18 (2004)
Srivastava, H.M., Et, M.: Lacunary statistical convergence and strongly lacunary summable functions of order α. Filomat 31(6), 1573–1582 (2017)
Belen, C., Mohiuddine, S.A.: Generalized weighted statistical convergence and application. Appl. Math. Comput. 219(18), 9821–9826 (2013)
Braha, N.L., Srivastava, H.M., Mohiuddine, S.A.: A Korovkin’s type approximation theorem for periodic functions via the statistical summability of the generalized de la Vallée Poussin mean. Appl. Math. Comput. 228, 162–169 (2014)
Kadak, U., Mohiuddine, S.A.: Generalized statistically almost convergence based on the difference operator which includes the \(({p,q} )\)-gamma function and related approximation theorems. Results Math. 73(9), 1–31 (2018)
Mohiuddine, S.A., Asiri, A., Hazarika, B.: Weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems. Int. J. Gen. Syst. 48(5), 492–506 (2019)
Mohiuddine, S.A., Alamri, B.A.S.: Generalization of equi-statistical convergence via weighted lacunary sequence with associated Korovkin and Voronovskaya type approximation theorems. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 1955–1973 (2019)
Fridy, J.A., Orhan, C.: Statistical limit superior and limit inferior. Proc. Am. Math. Soc. 125(12), 3625–3631 (1997)
Tripathy, B.C.: On statistically convergent and statistically bounded sequences. Bull. Malays. Math. Soc. 20(1), 31–33 (1997)
Bhardwaj, V.K., Gupta, S.: On some generalizations of statistical boundedness. J. Inequal. Appl. 2014(12), 1 (2014)
Bhardwaj, V.K., Gupta, S., Mohiuddine, S.A., Kılıçman, A.: On lacunary statistical boundedness. J. Inequal. Appl. 2014(311), 1 (2014)
Et, M., Mohiuddine, S.A., Şengül, H.: On lacunary statistical boundedness of order α. Facta Univ., Ser. Math. Inform. 31(3), 707–716 (2016)
Savaş, R.: Lacunary statistical boundedness of measurable functions. Math. Nat. Sci. 6, 15–19 (2020)
Hilger, S.: Analysis on measure chains-a unified approach to continuous and discrete calculus. Results Math. 18(1–2), 18–56 (1990)
Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)
Guseinov, G.S.: Integration on time scales. J. Math. Anal. Appl. 285(1), 107–127 (2003)
Cabada, A., Vivero, D.R.: Expression of the Lebesgue Δ-integral on time scales as a usual Lebesgue integral: application to the calculus of Δ-antiderivatives. Math. Comput. Model. 43(1–2), 194–207 (2006)
Seyyidoglu, M.S., Tan, N.O.: A note on statistical convergence on time scale. J. Inequal. Appl. 2012(219), 1 (2012)
Turan, C., Duman, O.: Statistical convergence on time scales and its characterizations. Springer Proc. Math. Stat. 41, 57–71 (2013)
Seyyidoglu, M.S., Tan, N.O.: On a generalization of statistical cluster and limit points. Adv. Differ. Equ. 2015(55), 1 (2015)
Turan, C., Duman, O.: Convergence methods on time scales. AIP Conf. Proc. 1558(1), 1120–1123 (2013)
Turan, C., Duman, O.: Fundamental properties of statistical convergence and lacunary statistical convergence on time scales. Filomat 31(14), 4455–4467 (2017)
Altin, Y., Koyunbakan, H., Yilmaz, E.: Uniform statistical convergence on time scales. J. Appl. Math. 2014(471437), 1–6 (2014)
Yilmaz, E., Altin, Y., Koyunbakan, H.: λ-statistical convergence on time scales. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 23(1), 69–78 (2016)
Altin, Y., Er, B.N., Yilmaz, E.: \({\Delta _{\lambda}}\)-statistical boundedness on time scales. Commun. Stat., Theory Methods 50(3), 738–746 (2021)
Sozbir, B., Altundag, S.: Weighted statistical convergence on time scale. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 26(2), 137–143 (2019)
Sozbir, B., Altundag, S.: \(\alpha \beta -\)statistical convergence on time scales. Facta Univ., Ser. Math. Inform. 35(1), 141–150 (2020)
Sözbir, B., Altundağ, S.: On asymptotically statistical equivalent functions on time scales. Math. Methods Appl. Sci. (2021). https://doi.org/10.1002/mma.7587
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Sözbir, B., Altundağ, S. Lacunary statistical boundedness on time scales. Adv Differ Equ 2021, 513 (2021). https://doi.org/10.1186/s13662-021-03656-7
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DOI: https://doi.org/10.1186/s13662-021-03656-7
Keywords
- Lacunary statistical convergence
- Statistical boundedness
- Lacunary statistical boundedness
- Time scales