Abstract and Applied Analysis

Volume 2013, Article ID 215612, 5 pages

http://dx.doi.org/10.1155/2013/215612

## Statistical Summability of Double Sequences through de la Vallée-Poussin Mean in Probabilistic Normed Spaces

Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 21 September 2013; Accepted 26 October 2013

Academic Editor: Mohammad Mursaleen

Copyright © 2013 S. A. Mohiuddine and Abdullah Alotaibi. 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.

#### Abstract

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 [1]: 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 [2] and Steinhaus [3], independently. Some of its basic properties and interesting concepts, especially, the notion of statistically Cauchy sequence, were proved by Schoenberg [4], Šalát [5], and Fridy [6]. See, for instance, [7–16] and references therein. Mursaleen and Edely [17] 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 [24]. For detail of -statistical convergence, one can be referred to [25–31] and many others. In [32], 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 [33]).

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) [34] 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 [35] in 1962 who used the concept of Menger [36] to define such space and the other one by Alsina et al. [37] (for more details, see [38–40]).

According to Šerstnev [35], 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.

#### Acknowledgments

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.

#### References

- A. Pringsheim, “Zur theorie der zweifach unendlichen zahlenfolgen,”
*Mathematische Annalen*, vol. 53, no. 3, pp. 289–321, 1900. View at Publisher · View at Google Scholar · View at MathSciNet - H. Fast, “Sur la convergence statistique,”
*Colloquium Mathematicum*, vol. 2, pp. 241–244, 1951. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Steinhaus, “Sur la convergence ordinaire et la convergence asymptotique,”
*Colloquium Mathematicum*, vol. 2, pp. 73–74, 1951. View at Google Scholar - I. J. Schoenberg, “The integrability of certain function and related summability methods,”
*The American Mathematical Monthly*, vol. 66, pp. 361–375, 1959. View at Publisher · View at Google Scholar - T. Šalát, “On statistically convergent sequences of real numbers,”
*Mathematica Slovaca*, vol. 30, no. 2, pp. 139–150, 1980. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. A. Fridy, “On statistical convergence,”
*Analysis*, vol. 5, no. 4, pp. 301–313, 1985. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Çakalli and M. K. Khan, “Summability in topological spaces,”
*Applied Mathematics Letters*, vol. 24, no. 3, pp. 348–352, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Hazarika, “On generalized statistical convergence in random 2-normed spaces,”
*Scientia Magna*, vol. 8, no. 1, pp. 58–67, 2012. View at Google Scholar - S. A. Mohiuddine and M. A. Alghamdi, “Statistical summability through a lacunary sequence in locally solid Riesz spaces,”
*Journal of Inequalities and Applications*, vol. 2012, article 225, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - S. A. Mohiuddine and M. Aiyub, “Lacunary statistical convergence in random 2-normed spaces,”
*Applied Mathematics & Information Sciences*, vol. 6, no. 3, pp. 581–585, 2012. View at Google Scholar · View at MathSciNet - S. A. Mohiuddine, H. Şevli, and M. Cancan, “Statistical convergence in fuzzy 2-normed space,”
*Journal of Computational Analysis and Applications*, vol. 12, no. 4, pp. 787–798, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Mursaleen, “On statistical convergence in random 2-normed spaces,”
*Acta Scientiarum Mathematicarum*, vol. 76, no. 1-2, pp. 101–109, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Mursaleen and O. H. H. Edely, “Generalized statistical convergence,”
*Information Sciences*, vol. 162, no. 3-4, pp. 287–294, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Mursaleen, V. Karakaya, M. Ertürk, and F. Gürsoy, “Weighted statistical convergence and its application to Korovkin type approximation theorem,”
*Applied Mathematics and Computation*, vol. 218, no. 18, pp. 9132–9137, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Şahiner, M. Gürdal, S. Saltan, and H. Gunawan, “Ideal convergence in 2-normed spaces,”
*Taiwanese Journal of Mathematics*, vol. 11, no. 5, pp. 1477–1484, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - U. Yamancı and M. Gürdal, “On lacunary ideal convergence in random 2-normed space,”
*Journal of Mathematics*, vol. 2013, Article ID 868457, 8 pages, 2013. View at Publisher · View at Google Scholar - M. Mursaleen and O. H. H. Edely, “Statistical convergence of double sequences,”
*Journal of Mathematical Analysis and Applications*, vol. 288, no. 1, pp. 223–231, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Mursaleen and S. A. Mohiuddine, “Statistical convergence of double sequences in intuitionistic fuzzy normed spaces,”
*Chaos, Solitons & Fractals*, vol. 41, no. 5, pp. 2414–2421, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. A. Mohiuddine, H. Şevli, and M. Cancan, “Statistical convergence of double sequences in fuzzy normed spaces,”
*Filomat*, vol. 26, no. 4, pp. 673–681, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Çolak and Y. Altin, “Statistical convergence of double sequences of order $\stackrel{\u0303}{\alpha}$,”
*Journal of Function Spaces and Applications*, vol. 2013, Article ID 682823, 5 pages, 2013. View at Publisher · View at Google Scholar - S. A. Mohiuddine, A. Alotaibi, and M. Mursaleen, “Statistical convergence of double sequences in locally solid Riesz spaces,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 719729, 9 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. A. Mohiuddine, B. Hazarika, and A. Alotaibi, “Double lacunary density and some inclusion results in locally solid Riesz spaces,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 507962, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - H. Dutta, “A characterization of the class of statistically pre-Cauchy double sequences of fuzzy numbers,”
*Applied Mathematics & Information Sciences*, vol. 7, no. 4, pp. 1437–1440, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Mursaleen, “$\lambda $-statistical convergence,”
*Mathematica Slovaca*, vol. 50, no. 1, pp. 111–115, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Belen and S. A. Mohiuddine, “Generalized weighted statistical convergence and application,”
*Applied Mathematics and Computation*, vol. 219, no. 18, pp. 9821–9826, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - R. Çolak and C. A. Bektaş, “$\lambda $-statistical convergence of order $\alpha $,”
*Acta Mathematica Scientia B*, vol. 31, no. 3, pp. 953–959, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Dutta and T. Bilgin, “Strongly (${V}^{\lambda},A,{\mathrm{\Delta}}_{vm}^{n},p$)-summable sequence spaces defined by an Orlicz function,”
*Applied Mathematics Letters*, vol. 24, no. 7, pp. 1057–1062, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - O. H. H. Edely, S. A. Mohiuddine, and A. K. Noman, “Korovkin type approximation theorems obtained through generalized statistical convergence,”
*Applied Mathematics Letters*, vol. 23, no. 11, pp. 1382–1387, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Et, M. Çınar, and M. Karakaş, “On $\lambda $-statistical convergence of order $\alpha $ of sequences of function,”
*Journal of Inequalities and Applications*, vol. 2013, article 204, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - S. A. Mohiuddine and Q. M. D. Lohani, “On generalized statistical convergence in intuitionistic fuzzy normed space,”
*Chaos, Solitons & Fractals*, vol. 42, no. 3, pp. 1731–1737, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. A. Mohiuddine, A. Alotaibi, and M. Mursaleen, “A new variant of statistical convergence,”
*Journal of Inequalities and Applications*, vol. 2013, article 309, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - M. Mursaleen, C. Çakan, S. A. Mohiuddine, and E. Savaş, “Generalized statistical convergence and statistical core of double sequences,”
*Acta Mathematica Sinica*, vol. 26, no. 11, pp. 2131–2144, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Savaş and S. A. Mohiuddine, “$\stackrel{\u0305}{\lambda}$-statistically convergent double sequences in probabilistic normed spaces,”
*Mathematica Slovaca*, vol. 62, no. 1, pp. 99–108, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. P. Klement, R. Mesiar, and E. Pap,
*Triangular Norms*, vol. 8 of*Trends in Logic*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000. View at MathSciNet - A. N. Šerstnev, “Random normed spaces: problems of completeness,”
*Kazanskij Gosudarstvennyj Universitet Imeni V.I. Ul'janova-Lenina. Učenye Zapiski*, vol. 122, no. 4, pp. 3–20, 1962. View at Google Scholar · View at MathSciNet - K. Menger, “Statistical metrics,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 28, pp. 535–537, 1942. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Alsina, B. Schweizer, and A. Sklar, “On the definition of a probabilistic normed space,”
*Aequationes Mathematicae*, vol. 46, no. 1-2, pp. 91–98, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Schweizer and A. Sklar, “Statistical metric spaces,”
*Pacific Journal of Mathematics*, vol. 10, pp. 313–334, 1960. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Schweizer and A. Sklar,
*Probabilistic Metric Spaces*, Elsevier, New York, NY, USA, 1983. - B. Schweizer and A. Sklar,
*Probabilistic Metric Spaces*, Dover Publication, Mineola, NY, USA, 2nd edition, 2005.