Uniform Statistical Convergence on Time Scales

Abstract

We will introduce the concept of m - and ( λ , m ) -uniform density of a set and m - and ( λ , m ) -uniform statistical convergence on an arbitrary time scale. However, we will define m -uniform Cauchy function on a time scale. Furthermore, some relations about these new notions are also obtained.

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Academic Editor:Alberto Cabada

Department of Mathematics, Firat University, 23119 Elazig, Turkey

Received 22 March 2014; Accepted 30 May 2014; 18 June 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The idea of statistical convergence was known to Zygmund [1] as early as 1935 and in particular after 1951 when Fast [2] and Steinhaus [3] reintroduced statistical convergence for sequences of real numbers. Later, Schoenberg [4] independently gave some basic properties of statistical convergence. Several generalizations and applications of this notion have been discussed in the theory of Fourier analysis, ergodic theory, number theory, measure theory, trigonometric series, turnpike theory, and Banach spaces under different names (see [5-11]).

Statistical convergence depends on the density of subsets of the set N . Recall that a subset A of N is said to have "asymptotic density" δ ( A ) if [figure omitted; refer to PDF] where the vertical bars denote the cardinality of the enclosed set. It is clear that any finite subset of N has zero asymptotic density and δ ( ^{A c} ) = 1 - δ ( A ) (see [12]).

A sequence _{{ x k } k ∈ N} is said to be statistically convergent to a real number L if [figure omitted; refer to PDF] for each [varepsilon] > 0 . And we write _{x k} st [arrow right] L or S - lim ... _{x k} = L . The set of all statistically convergent sequences is denoted by S (see [2, 3, 5, 6, 9, 13]).

The generalized de la Vallée-Poussin mean is defined by [figure omitted; refer to PDF] where λ = ( _{λ n} ) is a nondecreasing sequence of positive numbers such that _{λ n + 1} ...4; _{λ n} + 1 , _{λ 1} = 1 , _{λ n} [arrow right] ∞ as n [arrow right] ∞ and _{I n} = [ n - _{λ n} + 1 , n ] . The set of all such sequences will be denoted by Λ (see [14]).

A sequence x = ( _{x k} ) is said to be ( V , λ ) -summable to a number L if _{t n} ( x ) [arrow right] L as n [arrow right] ∞ . ( V , λ ) -summability reduces to ( C , 1 ) summability when λ = ( _{λ n} ) = ( n ) (see [14]).

We write [figure omitted; refer to PDF] for the sets of sequences x = ( _{x k} ) which are strongly Cesàro summable and strongly ( V , λ ) -summable, respectively. Strong ( V , λ ) -summability reduces to strong ( C , 1 ) summability when _{λ n} = n .

The notion of λ -statistical convergence was introduced by Mursaleen [15] as follows.

Let K ⊂ N and define the λ -density of K by [figure omitted; refer to PDF] _{δ λ} ( K ) reduces to the asymptotic density δ ( K ) in case of _{λ n} = n for all n ∈ N (see [15]).

A sequence x = ( _{x k} ) is said to be λ -statistically convergent to L if for every [varepsilon] > 0 (see [15]) [figure omitted; refer to PDF]

After the concept of almost λ -statistical convergence was studied by Savas [16], many authors have studied statistical convergence (see [6, 9, 10]).

Statistical convergence is applied to time scales for different purposes by various authors (see [17, 18]).

We here recall some basic concepts and notations from the theory of time scales. A time scale is an arbitrary nonempty closed set of real numbers. We use the symbol T to denote a time scale. A time scale has the topology that it inherits from the real numbers with the standard topology. The theory of time scale was introduced by Hilger in his Ph.D. thesis supervised by Auldbach in 1988 (see [19]). It allows unifying the usual differential and integral calculus for one variable. One can replace the range of definition ( R ) of the functions under consideration by an arbitrary time scale T . Now, time scale theory has been applied to different areas by many authors (see [20-24]).

The forward jump operator σ : T [arrow right] T can be defined by [figure omitted; refer to PDF] for t ∈ T . And the graininess function μ : T [arrow right] [ 0 , ∞ ) is defined by μ ( t ) = σ ( t ) - t . In this definition, we put inf ... [varphi] = sup ... T , where [varphi] is an empty set. A half open interval on an arbitrary time scale T is given by [figure omitted; refer to PDF] Open intervals or closed intervals can be defined similarly (see [20, 21]).

Now, let A denote the family of all left closed and right open intervals of T of the form [ a , b _{) T} . Let s : A [arrow right] [ 0 , ∞ ) be the set function on A such that [figure omitted; refer to PDF]

Then, it is known that s is a countably additive measure on A . Now, the Caratheodory extension of the set function s associated with family A is said to be the Lebesgue Δ -measure on T and is denoted by _{μ Δ} . In this case, it is known that if a ∈ T - { max ... T } , then the single point set { a } is Δ -measurable and _{μ Δ} ( a ) = σ ( a ) - a . If a , b ∈ T and a ...4; b , then _{μ Δ} ( ( a , b _{) T} ) = b - σ ( a ) . If a , b ∈ T - { max ... T } , a ...4; b ; _{μ Δ} ( ( a , b _{] T} ) = σ ( b ) - σ ( a ) and _{μ Δ} ( [ a , b ] _{) T} ) = σ ( b ) - a (see [18]).

In this study, we will give some notations for m -uniform and ( λ , m ) -uniform density of a set and m -uniform and ( λ , m ) -uniform statistical convergence and some properties of m -uniform and ( λ , m ) -uniform statistical convergence on time scales.

Definition 1 (see [25]).

A subset E of N is said to be uniformly dense if [figure omitted; refer to PDF] uniformly in m or, equivalently, [figure omitted; refer to PDF] uniformly in m , where m = 0,1 , 2,3 , ... and _{χ E} is characteristic function. Subsequently, uniform density was studied by Baláz and Salát [26], Brown and Freedman [27], and Maddox [28].

The notion of m -uniform statistical convergence is first introduced by Nuray [29] as follows.

Definition 2 (see [29]).

Let x = ( _{x k} ) be a real or complex valued sequence. If [figure omitted; refer to PDF] uniformly in m , x = ( _{x k} ) is said to be m -uniform statistically convergence to L for [varepsilon] > 0 .

Based on this notion, we give the following definitions to generalize m -uniform statistical convergence.

Definition 3.

Let K ⊂ N and define the ( λ , m ) -uniform density of K by [figure omitted; refer to PDF] ^{δ λ m} ( K ) reduces to the ^{δ m} ( K ) in case of _{λ n} = n for all n ∈ N .

Definition 4.

A sequence x = ( _{x k} ) is said to be ( λ , m ) -uniform statistically convergent to L if [figure omitted; refer to PDF] for every [varepsilon] > 0 uniformly in m .

In [30], Borwein introduced and studied strongly summable functions. His definition is as follows.

Definition 5 (see [30]).

A real-valued function x ( t ) , measurable (in Lebesgue sense) on the interval ( 1 , ∞ ) , is said to be strongly summable to L = _{L x} if [figure omitted; refer to PDF] [ _{W p} ] will denote the space of real-valued function x , measurable (in the Lebesgue sense) on the interval ( 1 , ∞ ) .

Furthermore, Nuray [31] studied λ -strong summable and λ -statistically convergent functions as in the following.

Definition 6 (see [31]).

Let λ ∈ Λ , let p be a real number, and let x ( t ) be a real-valued function which is measurable (in Lebesgue sense) on the interval ( 1 , ∞ ) , if [figure omitted; refer to PDF] then, one says that x ( t ) is _{λ p} strongly summable to L . Strongly summable number sequences and statistically convergent number sequences were studied by Maddox [32], Nuray and Aydin [33], and Et et al. [34].

There are some studies about statistical convergence on time scales in the literature. For instance, Seyyidoglu and Tan [17] gave some new notations such as Δ -convergence and Δ -Cauchy by using Δ -density and investigated their relations. Turan and Duman [18] introduced the concept of density and statistical convergence of delta measurable real-valued functions defined on time scales as follows.

Definition 7 (see [18]).

Suppose that Ω is a Δ -measurable subset of T . Then, for t ∈ T , one defines the set Ω ( t ) by [figure omitted; refer to PDF]

In this case, one defines the density of Ω on T , denoted by _{δ T} ( Ω ) , [figure omitted; refer to PDF] provided that the above limit exists. Furthermore, f is statistically convergent to a real number L on T if, for every [varepsilon] > 0 , [figure omitted; refer to PDF] where f : T [arrow right] R is a Δ -measurable function (see [17, 18]). Lebesgue Δ -measure _{μ Δ} is introduced by Guseinov [20].

Definition 8 (see [18]).

Let f : T [arrow right] R be a Δ -measurable function. f is statistical Cauchy on T if, for each [varepsilon] > 0 , there exists a number _{t 1} > _{t 0} ∈ T such that [figure omitted; refer to PDF]

2. Main Results and Preliminaries

It is well known that the notion of statistical convergence is closely related to the density of the subset of N . So, in this section, we will first define m -uniform and ( λ , m ) -uniform density of the subset of the time scale. By using these definitions, we will focus on constructing a concept of m -uniform (or ( λ , m ) -uniform) statistical convergence and m -uniform statistical Cauchy function on time scales. In following definitions, notations _{Δ m} and _{Δ ( λ , m )} shows that Δ depends on m and ( λ , m ) , respectively.

Definition 9.

Let Ω be a _{Δ m} -measurable subset of T . Then, one defines the set Ω ( t , m ) by [figure omitted; refer to PDF] for t ∈ T . In this case, one defines the m -uniform density of Ω on T , denoted by ^{δ T m} ( Ω ) , as follows: [figure omitted; refer to PDF] provided that the above limit exists.

Definition 10.

Let f : T [arrow right] R be a _{Δ m} -measurable function. Then, one says that f is m -uniform statistically convergent to a real number L on T if [figure omitted; refer to PDF] uniformly in m for every [varepsilon] > 0 . In this case, one writes ^{s ^ T m} - _{lim ... t [arrow right] ∞} ( f ( t ) ) = L . The set of all m -uniform statistically convergent functions on T will be denoted by ^{s ^ T m} .

In case of T = N , [ m + _{t 0} - 1 , t + m ) is [ m , n + m ) for t = n and _{t 0} = 1 . In this instance, m -uniform statistical convergence on time scales is reduced to classical m -uniform statistical convergence which is given by Definition 2. This shows that our results are generalizations of classical results.

Similarly, we can define m -uniform statistical Cauchy functions on a time scale based on Definition 8.

Definition 11.

Let f : T [arrow right] R be a _{Δ m} -measurable function. f is an m -uniform statistical Cauchy function on T if there exists a number _{t 1} > _{t 0} ∈ T such that [figure omitted; refer to PDF] for each [varepsilon] > 0 uniformly in m . One can easily see that this definition is a generalization of Definition 8.

Definition 12.

Let Ω ( t , m , λ ) be a _{Δ ( λ , m )} -measurable subset of T . Then, one defines the set Ω ( t , m , λ ) by [figure omitted; refer to PDF] for t ∈ T . In this case, one defines the ( λ , m ) -uniform density of Ω on T denoted by ^{δ T ( λ , m )} ( Ω ) , as follows: [figure omitted; refer to PDF] provided that the above limit exists.

Definition 13.

Let f : T [arrow right] R be a _{Δ ( λ , m )} -measurable function. One says that f is ( λ , m ) -uniform statistically convergent to a real number L on T if [figure omitted; refer to PDF] uniformly in m for every [varepsilon] > 0 . In this case, one writes ^{s ^ T ( λ , m )} - _{lim ... t [arrow right] ∞} ... ( f ( t ) ) = L . The set of all ( λ , m ) -uniform statistically convergent functions on T will be denoted by ^{s ^ T ( λ , m )} .

Hence, we have generalized Definition 3 to an arbitrary time scale. We can easily get classical ( λ , m ) -uniform statistical convergence by taking _{t 0} = 1 in Definition 13.

Proposition 14.

If f , g : T [arrow right] R with ^{s ^ T ( λ , m )} - _{lim ... t [arrow right] ∞} f ( t ) = _{L 1} and ^{s ^ T ( λ , m )} - _{lim ... t [arrow right] ∞} g ( t ) = _{L 2} , then the following statements hold:

(i) ^{s ^ T ( λ , m )} - _{lim ... t [arrow right] ∞} ( f ( t ) + g ( t ) ) = _{L 1} + _{L 2} ,

(ii) ^{s ^ T ( λ , m )} - _{lim ... t [arrow right] ∞} ( c f ( t ) ) = c _{L 1} ( c ∈ R ) .

Theorem 15.

For f : T [arrow right] R to be any _{Δ ( λ , m )} -measurable function, f is ( λ , m ) -uniform statistically convergent on T if and only if f is a ( λ , m ) -uniform statistical Cauchy function on T .

Proof.

We can prove this by using techniques similar to Theorem 3 of [29].

Theorem 16.

Consider ^{s ^ T m} ⊂ ^{s ^ T ( λ , m )} if and only if [figure omitted; refer to PDF]

Proof.

For given [varepsilon] > 0 , we have [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] Hence by using (28) and taking the limit as t [arrow right] ∞ , we get f ( s ) [arrow right] L ( ^{s ^ T m} ) which implies f ( s ) [arrow right] L ( ^{s ^ T ( λ , m )} ) .

The definition of p -Cesàro summability on time scales was given by Turan and Duman [18] as follows.

Definition 17 (see [18]).

Let f : T [arrow right] R be a Δ -measurable function and 0 < p < ∞ . Then, f is strongly p -Cesàro summable on T if there exists some L ∈ R such that [figure omitted; refer to PDF] The set of all p -Cesàro summable functions on T will be denoted by _{[ W p ] T} .

Measure theory on time scales was first constructed by Guseinov [20] and Lebesgue Δ -integral on time scales introduced by Cabada and Vivero [35].

Definition 18.

Let f : T [arrow right] R be a _{Δ ( λ , m )} -measurable function and 0 < p < ∞ . One says that f is ( λ , m ) uniformly strongly p -summable on T if there exists some L ∈ R such that [figure omitted; refer to PDF]

In this case, one writes _{[ W ^ m p ] T} - lim ... f ( s ) = L . The set of all ( λ , m ) uniformly strongly p -summable functions on T will be denoted by _{[ W ^ m p ] T} .

Lemma 19.

Let f : T [arrow right] R be a _{Δ ( λ , m )} -measurable function and [figure omitted; refer to PDF] for [varepsilon] > 0 . In this case, we have [figure omitted; refer to PDF]

Proof.

This can be proved by using a method similar to the approach in [18].

Theorem 20.

Let f : T [arrow right] R be a _{Δ ( λ , m )} -measurable function, L ∈ R , and 0 < p < ∞ . Then, one gets the following.

(i) _{[ W ^ m p ] T} ⊂ ^{s ^ T ( λ , m )} .

(ii) If f is ( λ , m ) uniformly strongly p -summable to L , then ^{s ^ T ( λ , m )} - _{lim ... t [arrow right] ∞} ( f ( t ) ) = L .

(iii): If ^{s ^ T ( λ , m )} - _{lim ... t [arrow right] ∞} ( f ( t ) ) = L and f is a bounded function, then f is uniformly strongly p -summable to L .

Proof.

(i) Let [varepsilon] > 0 and _{[ W ^ m p ] T} - lim ... f ( s ) = L . We can write [figure omitted; refer to PDF] Therefore, _{[ W ^ m p ] T} - lim ... f ( s ) = L implies ^{s ^ T ( λ , m )} - lim ... f ( s ) = L .

(ii) Let f be ( λ , m ) uniformly strongly p -summable to L . For given [varepsilon] > 0 , let [figure omitted; refer to PDF] on time scale T . Then, it follows from Lemma 19 that [figure omitted; refer to PDF] Dividing both sides of the last inequality by _{μ Δ ( λ , m )} ( [ t + m - _{λ t} + _{t 0} - 1 , t + m _{) T} ) and taking limit as t [arrow right] ∞ , we obtain [figure omitted; refer to PDF] which yields ^{s ^ T ( λ , m )} - _{lim ... t [arrow right] ∞} ( f ( t ) ) = L .

(iii) Let f be bounded and statistically convergent to L on T . Then, there exists a positive number M such that | f ( s ) | ...4; M for all s ∈ T , and also [figure omitted; refer to PDF] where Ω ( t , m , λ ) is as before. Since [figure omitted; refer to PDF] we obtain [figure omitted; refer to PDF] Since [varepsilon] is arbitrary, the proof follows from (39) and (41).

Theorem 21.

Let f be a _{Δ m} measurable function. Then, ^{s ^ T ( λ , m )} - lim ... f ( s ) = L if and only if there exists a _{Δ m} measurable set Ω ⊂ T such that ^{δ T m} ( Ω ) = 1 and _{lim ... t} f ( t ) = L , t ∈ Ω ( t , m , λ ) .

Proof.

It can be easily proved by using similar way in the study of Turan and Duman (see [18, Theorem 3.9]).

3. Conclusions

In this study, we introduced the historical development of the notion of statistical convergence. Then we presented some fundamental notions based on statistical convergence. The concepts of m - and ( λ , m ) -uniform density and uniform statistical convergence were defined on an arbitrary time scale. However, we defined m -uniform Cauchy functions on a time scale in general. Furthermore, we obtained some relations between these new notions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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