ABSTRACT The article explains and analyzes the conception of rough lacunary ideal statistical convergence on random ‐normed spaces for double sequences. The geometrical and topological results of rough lacunary ideal …
ABSTRACT The article explains and analyzes the conception of rough lacunary ideal statistical convergence on random ‐normed spaces for double sequences. The geometrical and topological results of rough lacunary ideal statistical limit points, rough lacunary ideal cluster points, and rough lacunary ideal boundedness are considered on these spaces. Through an exploration of properties associated with rough lacunary ideal convergence, equivalent conditions establish for set of lacunary ideal statistical limit points in the context of rough lacunary ideal statistically convergent double sequences.
ABSTRACT The article explains and analyzes the conception of rough lacunary ideal statistical convergence on random ‐normed spaces for double sequences. The geometrical and topological results of rough lacunary ideal …
ABSTRACT The article explains and analyzes the conception of rough lacunary ideal statistical convergence on random ‐normed spaces for double sequences. The geometrical and topological results of rough lacunary ideal statistical limit points, rough lacunary ideal cluster points, and rough lacunary ideal boundedness are considered on these spaces. Through an exploration of properties associated with rough lacunary ideal convergence, equivalent conditions establish for set of lacunary ideal statistical limit points in the context of rough lacunary ideal statistically convergent double sequences.
In this article we introduce the notion of I-convergent and I-Cauchy double sequences in a fuzzy normed linear space and establish some basic results related to these notions. Further, we …
In this article we introduce the notion of I-convergent and I-Cauchy double sequences in a fuzzy normed linear space and establish some basic results related to these notions. Further, we define I-limit points and I-cluster points of a double sequenc
We introduce the notions of lacunary -convergence and lacunary -Cauchy in the topology induced by random -normed spaces and prove some important results.
We introduce the notions of lacunary -convergence and lacunary -Cauchy in the topology induced by random -normed spaces and prove some important results.
x * ∈ X is said to be an r-limit point of a sequence (xi ) in some normed linear space (X,∥ · ∥) if (r ≥ 0). The set …
x * ∈ X is said to be an r-limit point of a sequence (xi ) in some normed linear space (X,∥ · ∥) if (r ≥ 0). The set of all r-limit points of (xi , denoted by LIM r x i , is bounded closed and convex. Further properties, in particular the relation between this rough convergence and other convergence notions, and the dependence of LIM r x i on the roughness degree r, are investigated. For instance, the set-valued mapping r ↦ LIM r x i is strictly increasing and continuous on (), where . For a so-called ρ-Cauchy sequence (xi ) satisfying it is shown in case X = R n that r = (n/(n + 1))ρ (or for Euclidean space) is the best convergence degree such that LIM r x i ≠ Ø.
Abstract In this work, using the concept of natural density, we introduce the notion of rough statistical convergence. We define the set of rough statistical limit points of a sequence …
Abstract In this work, using the concept of natural density, we introduce the notion of rough statistical convergence. We define the set of rough statistical limit points of a sequence and obtain two statistical convergence criteria associated with this set. Later, we prove that this set is closed and convex. Finally, we examine the relations between the set of statistical cluster points and the set of rough statistical limit points of a sequence. Keywords: Natural densityRough convergenceStatistical convergenceAMS Subject Classification: 40A05 ACKNOWLEDGMENTS The author is grateful to the referees and the editor in chief for their corrections and suggestions, which have greatly improved the readability of the paper.
Abstract For given r, ρ ≥ 0, a sequence (xi ) in some normed linear space X is said to be r-convergent if the r-limit set defined by is nonempty, …
Abstract For given r, ρ ≥ 0, a sequence (xi ) in some normed linear space X is said to be r-convergent if the r-limit set defined by is nonempty, and it is called a ρ-Cauchy sequence if This article investigates different aspects of this rough convergence, especially in infinite dimensional spaces, such as analytical properties of r-limit set, relation to other convergence notions, and the dependence of the r-limit set on the roughness degree r. Moreover, by using the Jung constant we find the minimal value of r such that an arbitrary ρ-Cauchy sequence in X is certainly r-convergent.
Given an n ‐normed space with n ≥ 2, we offer a simple way to derive an ( n − 1)‐norm from the n ‐norm and realize that any n …
Given an n ‐normed space with n ≥ 2, we offer a simple way to derive an ( n − 1)‐norm from the n ‐norm and realize that any n ‐normed space is an ( n − 1)‐normed space. We also show that, in certain cases, the ( n − 1)‐norm can be derived from the n ‐norm in such a way that the convergence and completeness in the n ‐norm is equivalent to those in the derived ( n − 1)‐norm. Using this fact, we prove a fixed point theorem for some n ‐Banach spaces.
In this paper we extend the notion of rough convergence using theconcept of ideals which automatically extends the earlier notions ofrough convergence and rough statistical convergence. We define the setof …
In this paper we extend the notion of rough convergence using theconcept of ideals which automatically extends the earlier notions ofrough convergence and rough statistical convergence. We define the setof rough ideal limit points and prove several results associated with thisset.
In (16) K. Menger proposed the probabilistic concept of distance by replacing the number d(p,q), as the distance between points p,q, by a distribution function Fp,q. This idea led to …
In (16) K. Menger proposed the probabilistic concept of distance by replacing the number d(p,q), as the distance between points p,q, by a distribution function Fp,q. This idea led to development of probabilistic analysis (3), (11) (18). In this paper, generalized probabilistic 2-normed spaces are studied and topological properties of these spaces are given. As an example, a space of random variables is considered, connections with the generalized deterministic 2-normed spaces studied in (14) being also given.
In this paper, we introduce and study the notion of rough I2 -lacunary statistical convergence of double sequences in normed linear spaces. We also introduce the notion of rough I2 …
In this paper, we introduce and study the notion of rough I2 -lacunary statistical convergence of double sequences in normed linear spaces. We also introduce the notion of rough I2 -lacunary statistical limit set of a double sequence and discuss some properties of this set.
The main purpose of this work is to define rough statistical convergence in probabilistic normed spaces. We have proved some basic properties as well as some examples which shows this …
The main purpose of this work is to define rough statistical convergence in probabilistic normed spaces. We have proved some basic properties as well as some examples which shows this idea of convergence in probabilistic normed spaces is more generalized as compared to the rough statistical convergence in normed linear spaces. Further, we have shown the results on sets of statistical limit points and sets of cluster points of rough statistically convergent sequences in these spaces.
In this paper we introduce and study the concept of ${\cal I}$-convergence of sequences in metric spaces, where ${\cal I}$ is an ideal of subsets of the set $\N$ of …
In this paper we introduce and study the concept of ${\cal I}$-convergence of sequences in metric spaces, where ${\cal I}$ is an ideal of subsets of the set $\N$ of positive integers. We extend this concept to ${\cal I}$-convergence of sequence of real functions defined on a metric space and prove some basic properties of these concepts.