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.

scholarly journals Some remarks on functions with values in probabilistic normed spaces

2007 ◽  
Vol 57 (3) ◽  
Author(s):  
Ioan Goleţ
Keyword(s):  
Normed Space ◽  
Topological Properties ◽  
Normed Spaces ◽  
Probabilistic Normed Space ◽  
Random Functions ◽  
New Class ◽  
Probabilistic Normed Spaces ◽  
Special Cases ◽  
Probabilistic Norm

AbstractIn this paper we consider an enlargement of the notion of the probabilistic normed space. For this new class of probabilistic normed spaces we give some topological properties. By using properties of the probabilistic norm we prove some differential and integral properties of functions with values into probabilistic normed spaces. As special cases, results for deterministic and random functions can be obtained.

scholarly journals On Lacunary Mean Ideal Convergence in Generalized Randomn-Normed Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Awad A. Bakery ◽  
Mustafa M. Mohammed
Keyword(s):  
Normed Space ◽  
Complex Numbers ◽  
Normed Spaces ◽  
Ideal Convergence ◽  
Cauchy Sequences ◽  
Limit Points ◽  
Positive Integers ◽  
Cluster Points

An idealIis a hereditary and additive family of subsets of positive integersℕ. In this paper, we will introduce the concept of generalized randomn-normed space as an extension of randomn-normed space. Also, we study the concept of lacunary mean (L)-ideal convergence andL-ideal Cauchy for sequences of complex numbers in the generalized randomn-norm. We introduceIL-limit points andIL-cluster points. Furthermore, Cauchy andIL-Cauchy sequences in this construction are given. Finally, we find relations among these concepts.

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 Compactness and D-Boundedness in Menger’s 2-probabilistic normed spaces

Filomat ◽  
2016 ◽  
Vol 30 (5) ◽  
pp. 1263-1272 ◽  
Author(s):  
P.K. Harikrishnan ◽  
Bernardo Guillén ◽  
K.T. Ravindran
Keyword(s):  
Normed Space ◽  
Convex Sets ◽  
Normed Spaces ◽  
Probabilistic Normed Space ◽  
Probabilistic Normed Spaces

The idea of convex sets and various related results in 2-Probabilistic normed spaces were established in [7]. In this paper, we obtain the concepts of convex series closedness, convex series compactness, boundedness and their interrelationships in Menger?s 2-probabilistic normed space. Finally, the idea of D-Boundedness in Menger?s 2-probabilistic normed spaces and Menger?s Generalized 2-Probabilistic Normed spaces are discussed.

scholarly journals On Asymptotically Double Lacunary Statistical Equivalent Sequences in Probabilistic Normed Space

2012 ◽  
Vol 20 (1) ◽  
pp. 89-100
Author(s):  
Ayhan Esi
Keyword(s):  
Normed Space ◽  
Normed Spaces ◽  
Probabilistic Normed Space ◽  
Basic Properties ◽  
Probabilistic Normed Spaces ◽  
Convergent Sequences

Abstract In this paper we study the concept of asymptotically double lacunary statistical convergent sequences in probabilistic normed spaces and prove some basic properties.

scholarly journals Completion of probabilistic normed spaces

1995 ◽  
Vol 18 (4) ◽  
pp. 649-652 ◽  
Author(s):  
Bernardo Lafuerza Guillén ◽  
José Antonio Rodríguez Lallena ◽  
Carlo Sempi
Keyword(s):  
Normed Space ◽  
Normed Spaces ◽  
Probabilistic Normed Space ◽  
Probabilistic Normed Spaces ◽  
Original Definition

We prove that every probabilistic normed space, either according to the original definition given by Šerstnev, or according to the recent one introduced by Alsina, Schweizer and Sklar, has a completion.

On I-convergence in random 2-normed spaces

2011 ◽  
Vol 61 (6) ◽  
Author(s):  
M. Mursaleen ◽  
Abdullah Alotaibi
Keyword(s):  
Banach Spaces ◽  
Normed Space ◽  
Statistical Convergence ◽  
Normed Spaces ◽  
Ideal Convergence

AbstractRecently the concepts of statistical convergence and ideal convergence have been studied in 2-normed and 2-Banach spaces by various authors. In this paper we define and study the notion of ideal convergence in random 2-normed space and construct some interesting examples.

$\bar \lambda $-statistically convergent double sequences in probabilistic normed spaces

2012 ◽  
Vol 62 (1) ◽  
Author(s):  
E. Savaş ◽  
S. Mohiuddine
Keyword(s):  
Normed Space ◽  
Double Sequences ◽  
Normed Spaces ◽  
Probabilistic Normed Space ◽  
Probabilistic Normed Spaces ◽  
Cauchy Sequences

AbstractThe purpose of this paper is to introduce and study the concepts of double $\bar \lambda $-statistically convergent and double $\bar \lambda $-statistically Cauchy sequences in probabilistic normed space.

scholarly journals Generalized Hyers-Ulam-Rassias Theorem in Menger Probabilistic Normed Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
M. Eshaghi Gordji ◽  
M. B. Ghaemi ◽  
H. Majani
Keyword(s):  
Normed Space ◽  
Additive Function ◽  
Additive Mapping ◽  
Additive Functions ◽  
Normed Spaces ◽  
Probabilistic Normed Space ◽  
Triangle Function ◽  
Probabilistic Normed Spaces

We introduce two reasonable versions of approximately additive functions in a Šerstnev probabilistic normed space endowed with triangle function. More precisely, we show under some suitable conditions that an approximately additive function can be approximated by an additive mapping in above mentioned spaces.

scholarly journals Ideal Convergence and Completeness of a Normed Space

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 897 ◽  
Author(s):  
Fernando León-Saavedra ◽  
Francisco Javier Pérez-Fernández ◽  
María del Pilar Romero de la Rosa ◽  
Antonio Sala
Keyword(s):  
Normed Space ◽  
Cauchy Sequence ◽  
Convergence Space ◽  
Normed Spaces ◽  
Ideal Convergence ◽  
Underlying Space ◽  
Phd Thesis

We aim to unify several results which characterize when a series is weakly unconditionally Cauchy (wuc) in terms of the completeness of a convergence space associated to the wuc series. If, additionally, the space is completed for each wuc series, then the underlying space is complete. In the process the existing proofs are simplified and some unanswered questions are solved. This research line was originated in the PhD thesis of the second author. Since then, it has been possible to characterize the completeness of a normed spaces through different convergence subspaces (which are be defined using different kinds of convergence) associated to an unconditionally Cauchy sequence.

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