Convergence Of Double Sequences Research Articles

Ideal Relative Uniform Convergence of Double Sequence of Positive Linear Functions

In this article, we look into the concept of ideal relative uniform convergence of a double sequence of functions. In addition, we define ideal relative uniform Cauchy and ideal regular relative uniform convergence of a double sequence of positive linear functions defined on a compact domain D with respect to the scale function σ(x) defined on D. We also introduced several classes of ideal relative uniform convergent double sequences of functions and investigated their algebraic and topological properties.

On ideal convergence of double sequences in 2—fuzzy n—normed linear space

On ideal convergence of double sequences in 2—fuzzy n—normed linear space

On strongly almost convergence of double sequences via complex uncertain variable

On strongly almost convergence of double sequences via complex uncertain variable

Tauberian theorems for weighted means of double sequences in intuitionistic fuzzy normed spaces

We define weighted mean summability method of double sequences in intuitionistic fuzzy normed spaces(IFNS), and obtain necessary and sufficient Tauberian conditions under which convergence of double sequences in IFNS follows from their weighted mean summability. This study reveals also Tauberian results for some known summation methods in the special cases.

Some New Observations on Wijsman ℐ 2 -Lacunary Statistical Convergence of Double Set Sequences in Intuitionistic Fuzzy Metric Spaces

In this study, we investigate the notions of the Wijsman ℐ 2 -statistical convergence, Wijsman ℐ 2 -lacunary statistical convergence, Wijsman strongly ℐ 2 -lacunary convergence, and Wijsman strongly ℐ 2 -Cesàro convergence of double sequence of sets in the intuitionistic fuzzy metric spaces (briefly, IFMS). Also, we give the notions of Wijsman strongly ℐ 2 ∗ -lacunary convergence, Wijsman strongly ℐ 2 -lacunary Cauchy, and Wijsman strongly ℐ 2 ∗ -lacunary Cauchy set sequence in IFMS and establish noteworthy results.

Regularly ideal invariant convergence of double sequences

In this paper, we introduce the notions of regularly invariant convergence, regularly strongly invariant convergence, regularly p-strongly invariant convergence, regularly (mathcal{I}_{sigma },mathcal{I}^{sigma }_{2})-convergence, regularly (mathcal{I}_{sigma }^{*},mathcal{I}^{sigma *}_{2})-convergence, regularly (mathcal{I}_{sigma },mathcal{I}^{sigma }_{2} )-Cauchy double sequence, regularly (mathcal{I}_{sigma }^{*},mathcal{I}^{sigma *}_{2})-Cauchy double sequence and investigate the relationship among them.

Multidimensional Strongly Deferred Invariant Convergence

The primary purpose of this article is to introduce double deferred invariant statistical convergence and double strongly invariant convergence for double sequences. Additionally, some implications are examined.

On the statistical convergence of (λ, μ) type in paranormed spaces

The idea of (λ, μ) statistical convergence was given by Mursaleen. The fundament aim of the article is to define and analyze the notion of (λ, μ) -statistical convergence of double sequences and present some relations between the sets of S(λ, μ) and S(θ, γ); [V, λ, μ] and [V, θ, γ].

The spaces of B(r,s,t,u) strongly almost convergent double sequences and matrix transformations

Most recently, some new double sequence spaces B(Mu), B(Cϑ) where ϑ={b,bp,r,f,f0} and B(Lq) for 0<q<∞ have been introduced as the domain of four-dimensional generalized difference matrix B(r,s,t,u) in the double sequence spaces Mu, Cϑ where ϑ={b,bp,r,f,f0} and Lq for 0<q<∞, and some topological properties, dual spaces, some new four-dimensional matrix classes and matrix transformations related to these spaces have also been studied by Tuğ and Başar and Tuğ (see [1–4]). In this paper, we introduce new strongly almost convergent double sequence spaces B[Cf] and B[Cf0] whose B(r,s,t,u)-transforms are in the spaces [Cf] and [Cf0], respectively. The main body of this paper is designed by the investigation of the following hypothesis. Firstly, we examine some topological properties and inclusion relations including the new double sequence spaces B[Cf] and B[Cf0]. Also, we determine the α-dual, β(bp)-dual and γ-dual of the space B[Cf]. Finally, we give the necessary and sufficient conditions on an infinite matrix transforming from [Cf] over Cf, and we also characterize the classes ([Cf];Mu), (B[Cf];Cf) and (B[Cf];Mu).

(e)-Convergence for double sequences

We define the notion of (e)-summability of double sequences and series of complex numbers. We also obtain a criteria for this summability method with regards to Berezin symbols of an diagonal operator, and show regularity of (e)-summability method for double sequences.

Applications of Tauberian theorems for double sequences of fuzzy numbers via De La Vallée Poussin mean

Tauberian theorem serves the purpose to recuperate Pringsheim’s convergence of a double sequence from its (C, 1, 1) summability under some additional conditions known as Tauberian conditions. In this article, we intend to introduce some Tauberian theorems for fuzzy number sequences by using the de la Vallée Poussin mean and double difference operator of order r . We prove that a bounded double sequence of fuzzy number which is Δ u r - convergent is ( C , 1 , 1 ) Δ u r - summable to the same fuzzy number L . We make an effort to develop some new slowly oscillating and Hardy-type Tauberian conditions in certain senses employing de la Vallée Poussin mean. We establish a connection between the Δ u r - Hardy type and Δ u r - slowly oscillating Tauberian condition. Finally by using these new slowly oscillating and Hardy-type Tauberian conditions, we explore some relations between ( C , 1 , 1 ) Δ u r - summable and Δ u r - convergent double fuzzy number sequences.

On M λ m , n -Statistical Convergence

In this paper, we introduce a new type of statistical convergence method for double sequences by using the M , λ m , n -method of summability which was defined by Natarajan. We also obtain some inclusion relations between statistical convergence and M λ m , n -statistical convergence for double sequences.

Generalized Weighted Statistical Convergence for Double Sequences of Fuzzy Numbers and Associated Korovkin-Type Approximation Theorem

We define the notions of weighted λ,μ-statistical convergence of order γ1,γ2 and strongly weighted λ,μ-summability of γ1,γ2 for fuzzy double sequences, where 0&lt;γ1,γ2≤1. We establish an inclusion result and a theorem presenting a connection between these concepts. Moreover, we apply our new concept of weighted λ,μ-statistical convergence of order γ1,γ2 to prove Korovkin-type approximation theorem for functions of two variables in a fuzzy sense. Finally, an illustrative example is provided with the help of q-analogue of fuzzy Bernstein operators for bivariate functions which shows the significance of our approximation theorem.

On mathcal{I}_{{theta}_{2}}-convergence in fuzzy normed spaces

In this study, first, lacunary convergence of double sequences is introduced in fuzzy normed spaces, and basic definitions and theorems about lacunary convergence for double sequences are given in fuzzy normed spaces. Then, we introduce the concept of lacunary ideal convergence of double sequences in fuzzy normed spaces, and the relation between lacunary convergence and lacunary ideal convergence is investigated for double sequences in fuzzy normed spaces. Finally, in fuzzy normed spaces, we give the concept of limit point and cluster point for double sequences, and the theorems related to these concepts are given.

On Borel convergence of double sequences

In this paper, we introduce the concept of (Ber)-convergence of bounded double sequences in the Fock space F(C²). We show that every (Ber)-convergent double sequence is Borel convergent. Namely, we prove the following theorem by using the Berezin symbol method: If the {x_{ij}}_{i,j=0}^{∞} is regularly convergent to x, then lim_{k,l→∞}e^{-k-l}∑_{i,j=0}^{∞}x_{ij}((k^{i}t^{j})/(i!j!))=x.

Generalized almost convergence of double sequences in modular function spaces

‎‎‎‎This article deals with almost convergence of double sequences using a new generalization of fractional-order difference operator in modular spaces and application to the Korovkin-type approximation in the context of modular spaces for positive linear operators‎. ‎We then obtain several inclusion relations and present some examples‎, ‎include proper non-trivial extensions of the corresponding classical ones‎. ‎Further‎, ‎we extend our study to new modular forms of Korovkin-type approximation theorems‎. ‎Finally‎, ‎we give an example using bivariate Chlodowsky-Szasz-Kantorovich-Charlier-type operators and outline possible further extensions and improvements‎, ‎in order to illustrate the effectiveness of the proposed methods‎.

Generalized Statistical Convergence of Double Sequences in Paranormed Spaces

We introduce the notion of (λ, µ)-statistical convergence of double sequences in a setting of paranormed space and prove that every convergent sequence is (λ, µ)-statistically convergent but not conversely by supporting an illustrative example. We also define the notions of (λ, µ)-statistical Cauchy and strongly (λ, µ)p-summable double sequences in a paranormed space and obtain their relationship with (λ, µ)- statistical convergence.

The λ ̅-statistically convergent double sequences in fuzzy normed spaces

We introduce double ? ?-statistically convergent sequences and double ? ?- statistically Cauchy sequences in the fuzzy normed spaces. We study [V,? ?] and [C,1]-summabilities for double sequences. In addition, we obtain the relation between these concepts and ? ?-statistically convergence.

Relative weighted almost convergence based on fractional-order difference operator in multivariate modular function spaces

In the present paper, we introduce the concept of relative weighted almost convergence and its weighted statistical extensions in multivariate modular function spaces based on a new type fractional-order double difference operator $$\nabla ^{a,b,c}_h$$ . We first define the concepts of weighted almost $$\nabla $$ -statistical convergence and statistical weighted almost $$\nabla $$ -convergence of double sequences. By using the notion of relative uniform convergence involving a scale function $$\sigma $$ , we introduce modular relative weighted almost statistical convergence and modular relative statistical weighted almost convergence of double sequences. We then obtain some inclusion relations between these proposed methods and provide some counterexamples that show that these are non-trivial and proper extensions of the existing literature on this topic. Moreover, we apply the relative statistical weighted almost convergence of a double sequence of positive linear operators to prove some Korovkin-type theorems in multivariate modular spaces by considering several kinds of test functions. Finally, we present a non-trivial application to generalized Boolean sum (GBS) operators of bivariate generalized Bernstein–Durrmeyer operators.

On the Lacunary (A, φ)-Statistical Convergence of Double Sequences

We extend some results known from the literature for ordinary (single) sequences to multiple sequences of real numbers. Further, we introduce a concept of double lacunary strong (A, φ) -convergence with respect to a modulus function. In addition, we also study some relationships between double lacunary strong (A, φ) -convergence with respect to a modulus and double lacunary statistical convergence.