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Statistical Summability of Double Sequences through de la Vallée-Poussin Mean in Probabilistic Normed Spaces
The purpose of this paper is to define some new types of summability methods for double sequences involving the ideas of de la Vallée-Poussin mean in the framework of probabilistic normed spaces and establish some interesting results.
1. Introduction and Preliminaries
Throughout the paper, the symbols and will denote the set of all natural and real numbers, respectively. The notion of convergence for double sequence was introduced by Pringsheim : we say that a double sequence of reals is convergent to in Pringsheim’s sense (briefly, () convergent) provided that given there exists a positive integer such that whenever .
The idea of statistical convergence is a generalization of convergence of real sequences which was first presented by Fast  and Steinhaus , independently. Some of its basic properties and interesting concepts, especially, the notion of statistically Cauchy sequence, were proved by Schoenberg , Šalát , and Fridy . See, for instance, [7–16] and references therein. Mursaleen and Edely  introduced the two-dimensional analogue of natural (or asymptotic) density as follows: let and , where . Then are called the upper and lower asymptotic densities of a two-dimensional set , respectively, where the vertical bars stand for cardinality of the enclosed set. If , then is called the double natural density of the set . In the same paper, using the notion of double natural density, they extended the idea of statistical convergence from single to double sequences (for recent work, see [18–23]).
The double sequence is statistically convergent to the number if, for each , the set has double natural density zero. We denote this by (or ).
Mursaleen initiated the notion of -statistical convergence (single sequences) with the help of de la Vallée-Poussin mean, in . For detail of -statistical convergence, one can be referred to [25–31] and many others. In , Mursaleen et al. presented the notion of -statistical convergence and -statistically bounded for double sequences and showed that -statistically bounded double sequences are -statistical convergence if and only if -statistical limit infimum of is equal to -statistical limit supremum of (also see ).
Suppose that and are two nondecreasing sequences of positive real numbers such that and each tends to infinity.
Recall that -density of the set is given by provided that the limit exists.
We remark, that, for and , the above density reduces to the double natural density.
The generalized double de la Vallée-Poussin mean is defined as where and .
We say that is -statistically convergent to the number if, for every , We denote this by .
The symbol will denote the set of all distribution functions (d.f.) which are nondecreasing, left continuous on , equal to zero on , and such that . The space is partially ordered by the usual pointwise ordering of functions.
A triangular norm (or a -norm)  is a binary operation which satisfies the following conditions. For all (i), (ii), (iii) whenever , (iv).
In the literature, we have two definitions of probabilistic normed space or, briefly, PN-space; the original one is given by Šerstnev  in 1962 who used the concept of Menger  to define such space and the other one by Alsina et al.  (for more details, see [38–40]).
According to Šerstnev , a probabilistic normed space is a triple , where is a real linear space, is the probabilistic norm, that is, is a function from into , for , the d.f. is denoted by , which is the value of at , and is a -norm that satisfies the following conditions: (i); (ii) for all if and only if ; (iii) for all , with and ; (iv) for all and .
2. Main Results
We define the notions of -summable, statistically summable, statistically -Cauchy, and statistically -complete for double sequences with respect to PN-space and establish some interesting results.
Definition 1. A double sequence is said to be -summable in (or, shortly, -summable) to if for each , there exists such that for all . In this case, one writes .
Definition 2. A double sequence is said to be statistically -summable in (or, shortly, -summable) to if , where ; that is, if, for each , , or equivalently In this case, we write , and is called the -limit of .
Definition 3. A double sequence is said to be statistically -Cauchy in (or, shortly, -Cauchy) if, for every and , there exist such that, for all , , the set has double natural density zero; that is,
Theorem 4. If a double sequence is statistically -summable in , that is, exists, then -limit of is unique.
Proof. Assume that and . We have to prove that . For 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 obtain for all . Hence . This means that -limit is unique.
Theorem 5. If a double sequence is -summable to , then it is -summable to the same limit.
Proof. Let us consider that . For every and , there exists a positive integer such that holds for all . Since is contained in , hence ; that is, is -summable to .
Example 6. This example proves that the converse of Theorem 5 need not be true. We denote by the set of all real numbers with the usual norm and for all . Assume that for all and all . Here, we observe that is a PN-space. The double sequence is defined by For and , write It is easy to see that and hence We see that the sequence is not -summable in . But the set has double natural density zero since . From here, we conclude that the converse of Theorem 5 need not be true.
Theorem 7. A double sequence is -summable to if and only if there exists a subset such that and .
Proof. Assume that there exists a subset = ; < < < < such that and . Then there exists such that holds for all . Put and = , . Then = and which implies that . Hence is statistically -summable to in PN-space.
Conversely, suppose that is -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 .
Theorem 8. If a double sequence is statistically -summable in PN-space, then it is statistically -Cauchy.
Proof. Suppose that . Let be a given number so that we choose such that Then, for , we have where which implies that Let . Then .
Now, let We need to show that . Let . Then , , and in particular . Then which is not possible. Hence . Therefore, by (26) . Hence, is statistically -Cauchy in PN-space.
Definition 9. Let be a PN-space. Then, (i)PN-space is said to be complete if every Cauchy double sequence is -convergent in ;(ii)PN-space is said to be statistically -complete (or, shortly, -complete) if every statistically -Cauchy sequence in PN-space is statistically -summable.
Theorem 10. Every probabilistic normed space is -complete but not complete in general.
Proof. Suppose that is -Cauchy but not -summable. Then there exist such that, for all , , the set has double natural density zero; that is, and This implies that , since if . Therefore ; that is, , which leads to a contradiction, since was -Cauchy. Hence must be -summable.
To see that a probabilistic normed space is not complete in general, we have the following example.
Example 11. Let and for . Then is a probabilistic normed space but not complete, since the double sequence is Cauchy with respect to but not -convergent with respect to the present PN-space.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (303/130/1433). The authors, therefore, acknowledge with thanks DSR technical and financial support.
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