scholarly journals Statistical Convergence of Double Sequences on Probabilistic Normed Spaces

2007 ◽  
Vol 2007 ◽  
pp. 1-11 ◽  
Author(s):  
S. Karakus ◽  
K. Demırcı
Keyword(s):  
Statistical Convergence ◽  
Double Sequences ◽  
Normed Spaces ◽  
Probabilistic Normed Spaces ◽  
Single Sequence

The concept of statistical convergence was presented by Steinhaus in 1951. This concept was extended to the double sequences by Mursaleen and Edely in 2003. Karakus has recently introduced the concept of statistical convergence of ordinary (single) sequence on probabilistic normed spaces. In this paper, we define statistical analogues of convergence and Cauchy for double sequences on probabilistic normed spaces. Then we display an exampl e such that our method of convergence is stronger than usual convergence on probabilistic normed spaces. Also we give a useful characterization for statistically convergent double sequences.

scholarly journals Statistical convergence of double sequences on probabilistic normed spaces defined by $[ V, \lambda, \mu ]$-summability

2014 ◽  
Vol 33 (2) ◽  
pp. 59-67
Author(s):  
Pankaj Kumar ◽  
S. S. Bhatia ◽  
Vijay Kumar
Keyword(s):  
Statistical Convergence ◽  
Convergence Criteria ◽  
Double Sequences ◽  
Normed Spaces ◽  
Real Numbers ◽  
Positive Real ◽  
Probabilistic Normed Spaces

In this paper, we aim to generalize the notion of statistical convergence for double sequences on probabilistic normed spaces with the help of two nondecreasing sequences of positive real numbers $\lambda=(\lambda_{n})$ and $\mu = (\mu_{n})$  such that each tending to zero, also $\lambda_{n+1}\leq \lambda_{n}+1, \lambda_{1}=1,$ and $\mu_{n+1}\leq \mu_{n}+1, \mu_{1}=1.$ We also define generalized statistically Cauchy double sequences on PN space and establish the Cauchy convergence criteria in these spaces.

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

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Ayhan Esi
Keyword(s):  
Statistical Convergence ◽  
The Other ◽  
Normed Spaces ◽  
Summability Methods ◽  
Probabilistic Normed Spaces ◽  
New Type ◽  
De La Vallée Poussin

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.

On certain spaces of double sequences by de la Vallée-Poussin mean

2015 ◽  
Vol 08 (04) ◽  
pp. 1550079 ◽  
Author(s):  
Kuldip Raj ◽  
Suruchi Pandoh
Keyword(s):  
Statistical Convergence ◽  
Orlicz Function ◽  
Double Difference ◽  
Double Sequences ◽  
Normed Spaces ◽  
Interval Numbers ◽  
Convergence Spaces ◽  
Difference Sequences ◽  
Inclusion Relations ◽  
De La Vallée Poussin

In this paper, we introduce some [Formula: see text]-convergence spaces of double difference sequences of interval numbers with Musielak–Orlicz function [Formula: see text] over [Formula: see text]-normed spaces. We also make an effort to study some topological properties and inclusion relations between these spaces. Furthermore, we study [Formula: see text]-statistical convergence of double difference sequences of interval numbers.

scholarly journals On Cluster Points, Continuity, and Boundedness Associated with the Generalized Statistical Convergence in Probabilistic Normed Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Pratulananda Das ◽  
Kaustubh Dutta ◽  
Vatan Karakaya
Keyword(s):  
Statistical Convergence ◽  
Normed Spaces ◽  
Statistical Cluster ◽  
Probabilistic Normed Spaces ◽  
Extremal Limit ◽  
Limit Points ◽  
Appl Math ◽  
Cluster Points

We consider the recently introduced notion ofℐ-statistical convergence (Das, Savas and Ghosal, Appl. Math. Lett., 24(9) (2011), 1509–1514, Savas and Das, Appl. Math. Lett. 24(6) (2011), 826–830) in probabilistic normed spaces and in the following (Şençimen and Pehlivan (2008 vol. 26, 2008 vol. 87, 2009)) we introduce the notions like strongℐ-statistical cluster points and extremal limit points, and strongℐ-statistical continuity and strongℐ-statisticalD-boundedness in probabilistic normed spaces and study some of their important properties.

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

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
S. A. Mohiuddine ◽  
Abdullah Alotaibi
Keyword(s):  
Double Sequences ◽  
Normed Spaces ◽  
Summability Methods ◽  
Probabilistic Normed Spaces ◽  
De La Vallée Poussin

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.

On ideal convergence in probabilistic normed spaces

2012 ◽  
Vol 62 (1) ◽  
Author(s):  
M. Mursaleen ◽  
S. Mohiuddine
Keyword(s):  
Normed Space ◽  
Statistical Convergence ◽  
Normed Spaces ◽  
Probabilistic Normed Space ◽  
Ideal Convergence ◽  
Probabilistic Normed Spaces ◽  
Real Anal ◽  
The Relationship ◽  
Cluster Points ◽  
Interesting Generalization

AbstractAn interesting generalization of statistical convergence is I-convergence which was introduced by P.Kostyrko et al [KOSTYRKO,P.—ŠALÁT,T.—WILCZYŃSKI,W.: I-Convergence, Real Anal. Exchange 26 (2000–2001), 669–686]. In this paper, we define and study the concept of I-convergence, I*-convergence, I-limit points and I-cluster points in probabilistic normed space. We discuss the relationship between I-convergence and I*-convergence, i.e. we show that I*-convergence implies the I-convergence in probabilistic normed space. Furthermore, we have also demonstrated through an example that, in general, I-convergence does not imply I*-convergence in probabilistic normed space.

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