Statistical Summability through de la Vallée-Poussin Mean in Probabilistic Normed Spaces

Abstract

Two concepts—one of statistical convergence and the other of de la Vallée-Poussin mean—play an important role in recent research on summability theory. In this work we define a new type of summability methods and statistical completeness involving the ideas of de la Vallée-Poussin mean and statistical convergence in the framework of probabilistic normed spaces.

1. Introduction, Definitions, and Preliminaries

Fast [1] presented the following definition of statistical convergence for sequences of real numbers. Let , the set of natural numbers, and . The natural density of is defined by if the limit exists, where denotes the cardinality of .

The sequence is said to be statistically convergent to the number if for every the set has natural density zero; that is, for each , Note that every convergent sequence is statistically convergent to the same limit, but its converse need not be true.

In 1985, Fridy [2] has defined the notion of statistically Cauchy sequence and proved that it is equivalent to statistical convergence and since then a large amount of work has appeared. Various extensions, generalizations, variants, and applications have been given by several authors so far, for example, [3–8] and references therein. In the recent past, Mursaleen [9] presented a generalization of statistical convergence by using de la Vallée-Poussin mean which is known -statistical convergence and further studied by Çolak and Bektas [10, 11]. For more details related to this concept we refer to [12–18].

Let be a nondecreasing sequence of positive numbers tending to such that The generalized de la Vallée-Poussin mean is defined by where .

A sequence is said to be -summable to a number if In this case is called -limit of .

Let be a set of positive integers; then is said to be -density of .

In case , -density reduces to the natural density. Also, since , for every .

The number sequence is said to be -statistically convergent to the number if, for each , , where ; that is, In this case we write and we denote the set of all statistically convergent sequences by .

A distribution function is an element of , where is left-continuous, nondecreasing, and and the subset is the set . Here denotes the left limit of the function at the point . The space is partially ordered by the usual pointwise ordering of functions; that is, if and only if for all .

A triangle function is a binary operation on , namely, a function that is associative, commutative nondecreasing and which has as unit; that is, for all , we have(i),(ii),(iii) whenever ,(iv).

Here is the d.f. defined by We remark that the set as well as its subsets can be partially ordered by the usual pointwise order: in this order, is the maximal element in .

There are two definitions of probabilistic normed space, the original one by Šerstnev [19] who used the idea of Menger [20] to define such space and the other one by Alsina et al. [21].

According to Šerstnev, a probabilistic normed space (for short, PN-space) is a triple , where is a real linear space, is a triangle function, and is the probabilistic norm; that is, is a map from into that satisfies the following conditions:(i) if and only if , where is the null vector of ,(ii) for all , with , and ,(iii) whenever .

Here denotes the value of at .

In this paper, using the notions of statistical convergence and de la Vallée-Poussin mean, we define and study a new type of summability methods in the setting of probabilistic normed spaces. We also introduce a new type of statistical completeness through de la Vallée-Poussin mean in this framework.

2. Statistical Summability through de la Vallée-Poussin Mean

Here we introduce the notions of -summable and statistically -summable in PN-space and give some of its properties. We will assume throughout this paper that is a probabilistic normed space.

Definition 1. A sequence is said to be -summable in PN-space or simply -summable to if for each , there exists a positive integer such that for all . In this case one writes and is called the -limit of the sequence .

Definition 2. A sequence is said to be statistically -summable in or simply -summable to if for each , the set has natural density zero (briefly, ; that is, In this case one writes , and is called the -limit of . One may write (8) in the alternative form as

Theorem 3. If a sequence is statistically -summable in PN-space, that is, exists, then it is unique.

Proof. Suppose that there exist two elements with such that and . Let be given. Choose such that Then, for any , we define Since implies and, similarly, we have . Now, let . It follows that and hence the complement is nonempty set and . Now, if , then Since was arbitrary, we get for all . Hence . This means that -limit is unique.

The following theorem gives the algebraic properties of statistically -summable sequences in PN-spaces.

Theorem 4. Let and be two sequences. If and , then(i),(ii), .

Proof of the theorem is straightforward and so omitted.

Theorem 5. If a sequence is -summable to in PN-space; then it is statistically -summable to the same limit.

Proof. Let . Then for every and , there is a positive integer such that for all . Since the set is contained in . As we know, every finite subset of has natural density zero; that is, . Hence, a sequence is -summable to .

Remark 6. The converse of the above theorem is not true in general, which is verified by the following example.

Example 7. Let denote the space of all real numbers with the usual norm and for all . Let for all and . In this case, we observe that is a PN-space. Define a sequence by For and , write It is easy to see that hence Therefore, the sequence is not -summable. But the set has natural density zero since . From here, we conclude that the converse of Theorem 5 need not be true.

Theorem 8. A sequence is statistically -summable in PN-space to if and only if there exists a subset such that and .

Proof. Suppose that there exists a subset such that and . Then there exists a positive integer such that for Put and . Then and which implies that . Hence is statistically -summable to in PN-space.
Conversely, let sequence is statistically -summable to . For and , write Then and
Now we have to show that for , is -summable to . Suppose that is not -summable to . Therefore there is such that for infinitely many terms. Let and with . Then and, by (21), . Hence , which contradicts (22) and therefore is -summable to .

Similarly we can prove the following dual statement.

Theorem 9. A sequence is -statistically summable in PN-space to if and only if there exists a subset such that and .

3. Statistically Complete through de la Vallée-Poussin Mean

In this section, we define the notions of statistically -Cauchy and statistically -complete with respect to probabilistic normed space and prove related results.

Definition 10. A sequence is said to be statistically -Cauchy in or simply -Cauchy if, for every and , there exists a number such that, for all , the set has natural density zero (briefly, ; that is,

Theorem 11. A sequence is statistically -summable in PN-space; then it is statistically Cauchy.

Proof. Suppose that . Let be a given number and choose such that Then, for , we have , where which implies that Let . Then .
Now, let We need to show that . Let . Then and hence ; that is, . Otherwise, if , then which is not possible. Therefore and hence a sequence is statistically -Cauchy in PN-space.

Definition 12. Let be a PN-space. Then,(i)PN-space is said to be complete if every Cauchy sequence is convergent in ;(ii)PN-space is said to be statistically -complete or simply -complete if every statistically -Cauchy sequence in is statistically -summable.

Theorem 13. Every probabilistic normed space is statistically -complete but not complete in general.

Proof. Suppose that is statistically -Cauchy in PN-space but not statistically -summable. Then there exists such that This implies that . Since if , then ; that is, , which leads to a contradiction, since was statistically Cauchy. Hence must be statistically summable in PN-space.
A probabilistic normed space is not complete in general; we verify this by the following example.
Example 14. Let and for . Then is a probabilistic normed space but not complete, since the sequence is Cauchy with respect to but not convergent with respect to the present PN-space.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author thanks the referees for their comments.