New Convergence Definitions for Sequences of Sets

Kı̇şı̇, Ömer; Nuray, Fatıh

doi:10.1155/2013/852796

Cited by 24 publications

(23 citation statements)

References 14 publications

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“…Nuray and Rhodes [18] introduced Wijsman statistical convergence for set sequences by combining statistical convergence with this new concept. Similarly, Kisi and Nuray [19] defined Wijsman J −convergence for set sequences with an ideal J . Definition 7.…”

Section: Definition 6 ([17])mentioning

confidence: 99%

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A New Type of Generalization on W—Asymptotically J λ—Statistical Equivalence with the Number of α

Gümüş

Demir²

2018

Axioms

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In our paper, by using the concept of W-asymptotically J- statistical equivalence of order α which has been previously defined, we present the definitions of W-asymptotically Jλ-statistical equivalence of order α, W-strongly asymptotically Jλ-statistical equivalence of order α, and W-strongly Cesáro asymptotically J-statistical equivalence of order α where 0<α≤1. We also extend these notions with a sequence of positive real numbers, p=(pk), and we investigate how our results change if p is constant.

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Section: Definition 6 ([17])mentioning

confidence: 99%

“…Definition 8. ( [19]) Let (X, ρ) be a metric space and J ⊆ 2 N be a proper ideal in N. For any non-empty closed subsets, A, A k ⊂ X, we say that the sequence {A k } is Wijsman J −convergent to A, if for each ε > 0, and x ∈ X,, the set is…”

Section: Definition 6 ([17])mentioning

confidence: 99%

A New Type of Generalization on W—Asymptotically J λ—Statistical Equivalence with the Number of α

Gümüş

Demir²

2018

Axioms

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“…Definition 1.11. (Kişi and Nuray, [12]) Let (X, d) be a metric space and I ⊆ 2 N be a non-trivial ideal in N. For any non-empty closed subsets A, A k ⊆ X for all k ∈ N we say that the sequence {A k } is Wijsman I−convergent to A, if for each ε > 0 and for each x ∈ X the set, As an example, consider the following sequence. Let X = R 2 and {A k } be a sequence as follows: For any non-empty closed subsets A, A k ⊆ X for all k ∈ N we say that the sequence {A k } is Wijsman I−statistically convergent to A or S (I W )-convergent to A if for each ε > 0, for each x ∈ X and δ > 0 we have,…”

Section: Definition 13mentioning

confidence: 99%

On Wijsman ideal convergent set of sequences defined by an Orlicz function

Gümüş

2016

Filomat

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In this study, our main topics are Wijsman ideal convergence and Orlicz function. We define Wijsman ideal convergent set of sequences defined by an Orlicz function where I is an ideal of the subset of positive integers N. We also obtain some inclusion theorems. 1. Preliminaries and Notation Statistical convergence of sequences of points was introduced by Steinhaus [21] and Fast [7] and later Schoenberg reintroduced this concept and he established some basic properties of statistical convergence and also studied the concept as a summability method [20]. The last twenty years this concept has been applied in various areas. Let K be a subset of the set of all natural numbers N and K n = |k ≤ n : k ∈ K| where the vertical bars indicate the number of elements in the enclosed set. The natural density of K is defined by δ(K) := lim n→∞ n −1 |{k ≤ n : k ∈ K}. Now we recall some definitions and results on statistical convergence. Definition 1.1. (Fast, [7]) A number sequence x = (x k) is said to be statistically convergent to the number L if for every ε > 0, lim n→∞ 1 n |{k ≤ n : |x k − L| ≥ ε}| = 0. In this case we write st − lim x k = L. Statistical convergence is a natural generalization of ordinary convergence. If lim x k = L, then st − lim x k = L. The converse does not hold in general. I−convergence is an important notion in our area and that is based on the notion of an ideal of the subset of positive integers. Kostyrko et al. [14] introduced the notion of I−convergence in a metric space in 2000. Esi and Hazarika ([5] , [6]), Hazarika and Savas [9], Savas ([17] , [18] , [19]), Kişi et al. ([12] , [13]) and many others dealt with I−convergence and Orlicz function. Now we state the definitions of ideal and filter. Definition 1.2. A non-empty family of sets I ⊆ 2 N is called an ideal if and only if ∅ ∈ I, for each A, B ∈ I we have A ∪ B ∈ I and for each A ∈ I and each B ⊆ A we have B ∈ I.

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“…Statistical convergence is one of the main concepts of the summability theory that can be introduced in an arbitrary topological space without the requirement of a lineer structure or at least a group structure on that space, so it is natural to consider statistical convergence of sequences of sets in the realm of hyperspaces. Maddox [18] studied statistical convergence in locally convex spaces, Maio and Ko£inac [19] have considered it in topological spaces and there are some papers studying statistical convergence in hyperspaces [15], [22], [23], [25]. We add to all these the facts that are direct results of the topological view.…”

Section: Introductionmentioning

confidence: 99%