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Strongly σ-convergent sequences
... A sequence x = (x k ) is said to be strongly Cesaro summable to the number if A sequence x = (x k ) is said to be statistically convergent to the number ∈ R if for every > 0, The idea of statistical convergence was introduced by Steinhaus in [2] and Fast in [3] independently and since then has been studied by other authors including [4][5][6][7][8][9][10][11] and [12]. The following relationship between statistical convergence and strong Cesaro summability is known [4]: ...
... where ∞ denotes the set of all bounded sequences. Several authors including Raimi [18], Schaefer [19], Mursaleen [20,21], Savaş [11,22] and others have studied invariant convergent sequences. ...
... In the case σ(n) = n + 1, the space [V σ ] is the set of strongly almost convergent sequences [ĉ]. The concept of strong sinvariant convergent sequence was defined by Savaş in [11]: ...
... In section 3 we prove the main results of this paper. The results which we give in this paper are more general than those of Nuray and Gülcü [16], Bhardwaj and Singh [1] Savaş and Nuray [22] and Savaş [23]. ...
... The special case of (1) in which σ (n) = n + 1 was given by Lorentz [9]. Several authors including Schaefer [24], Mursaleen [14], Savaş [23] and many others have studied invariant convergent sequences. ...
... Giving particular values to M, θ, X and q we obtain some sequence spaces which are the special cases of the sequence spaces that we have defined, for example i) If X = C, M (x) = x, θ = (2 r ) and q (x) = |x| , then [V σ , θ, M, p, q] Z = [V σ ] Z (p k ) , ( see [23] ). ii) If X = C , q (x) = |x| and θ = (2 r ) , then [V σ , θ, M, p, q] Z = [V σ , M ] Z (p k ) , ( see [16] ). ...
The purpose of this paper is to introduce the space of sequences that are strongly (Vσ,λ,q)-summable with respect to an Orlicz function. We give some relations related to these sequence spaces. We also show that the spaces [Vσ,λ,M,p,q]1∩ℓ∞(q) may be represented as a space.
... Several authors studied invariant mean and invariant convergent sequence (for examples, see [24][25][26][27][28][29][30][31][32][33]). Savaş and Nuray [26] introduced the concepts of σ-statistical convergence and lacunary σ-statistical convergence and gave some inclusion relations. ...
In this paper, we introduce the concepts of -invariant arithmetic convergence, -invariant arithmetic convergence, strongly q-invariant arithmetic convergence for real sequences, and give some inclusion relations.
... Several authors have studied invariant convergent sequences (see, [4,24,25,30,35,[38][39][40][41]44]). Recently, the concepts of σ -uniform density of the set A ⊆ N, I σ -convergence and I * σ -convergence of sequences of real numbers were defined by Nuray et al. [35]. ...
In this paper, we introduce the notions of regularly invariant convergence, regularly strongly invariant convergence, regularly p -strongly invariant convergence, regularly ( I σ , I 2 σ ) -convergence, regularly ( I σ ∗ , I 2 σ ∗ ) -convergence, regularly ( I σ , I 2 σ ) -Cauchy double sequence, regularly ( I σ ∗ , I 2 σ ∗ ) -Cauchy double sequence and investigate the relationship among them.
... In [17], Savaş generalized the concept of strongly σ-convergence as below: ...
In this study, we defined concepts of Wijsman quasi-invariant convergence, Wijsman quasi-strongly invariant convergence and Wijsman quasi-strongly q-invariant convergence. Also, we give the concept of Wijsman quasi-invariant statistically convergence. Then, we study relationships among these concepts. Furthermore, we investigate relationship between these concepts and some convergence types given earlier for sequences of sets, too.
... Several authors have studied invariant convergent sequences [see, Mursaleen (1983), Nuray and Savaş (1994), Pancaroğlu and Nuray (2013a, 2013b, Raimi (1963), Savaş (1989aSavaş ( , 1989b, Savaş and Nuray (1993), Schaefer (1972) and Ulusu et al. (2018)]. Nuray et al. (2011) defined the notions of invariant uniform density of subsets of ℕ, ℐ -convergence and investigated relationships between ℐ -convergence and -convergence also ℐ -convergence and [ ] -convergence. ...
... The space , the set of bounded sequences whose invariant means are equal, can be shown that Several authors studied on the notions of invariant mean and invariant convergent sequence (for examples, see [1][2][3][4][5][6][7][8]). ...
In this study, firstly, we introduce the notion of lacunary invariant uniform density of any subset E of the set N (the set of all natural numbers). Then, as associated with this notion, we give the definition of lacunary I_σ-convergence for real sequences. Furthermore, we examine relations between this new type convergence notion and the notions of lacunary invariant summability, lacunary strongly q-invariant summability and lacunary σ-statistical convergence which are studied in this area before. Finally, introducing the notions of lacunary I_σ^*-convergence and I_σ-Cauchy sequence, we give the relations between these notions and the notion of lacunary I_σ-convergence.
... İnvaryant yakınsaklık kavramı başta Mursaleen (1979Mursaleen ( , 1983, Mursaleen ve Edely (2009), Nuray ve Savaş (1994), Nuray vd. (2011), Pancaroğlu ve Nuray (2013), Raimi (1963), Savaş (1989aSavaş ( , 1989b, Savaş ve Nuray (1993), Schaefer (1972) ve Ulusu ve Nuray olmak üzere pek çok yazar tarafından çalışılmıştır. Aşağıdaki şartları sağlayan ⊆ 2 ℕ kümesi bir ideal olarak adlandırılır; 1) ∅ ∈ , 2) Her bir , ∈ için ∪ ∈ , 3) Her bir ∈ ve her bir ⊆ E için ∈ . ...
... Reel sayılarda yakınsaklık kavramı Fast (1951), Schoenberg (1959) İnvaryant yakınsaklık ile ilgili tanım, teorem ve özellikler Raimi (1963), Mursaleen (1979Mursaleen ( , 1983, Mursaleen ve Edely (2009), Nuray ve Savaş (1994), Nuray vd. (2011), Pancaroğlu ve Nuray (2013a), Savaş (1989aSavaş ( , 1989b, Savaş ve Nuray (1993) ve Schaefer (1972) gibi birçok araştırmacı tarafından çalışılmıştır. Kuvvetli σ-yakınsaklık kavramı Mursaleen (1983) ...
... On ℐ-lacunary statistical convergence of set sequences was studied by Ulusu and Dündar [9]. Also, after these important studies, the notions of statistical convergence, ideal convergence and ℐ-statistical Several authors including Raimi [10], Schaefer [11], Mursaleen [12,13], Savaş [14,15], Mursaleen and Edely [16], Pancarog lu and Nuray [17,18] and some authors have studied invariant convergent sequences. The notion of strong -convergence was defined by Savaş [16]. ...
We investigate the notions of strongly asymptotically I_σ0-equivalence, f-asymptotically I_σ0-equivalence, strongly f-asymptotically I_σ0-equivalence and asymptotically I_σ0-statistical equivalence for sequences of sets. Also, we investigate some relationships among these concepts.
... An admissible ideal I 2 ⊂ 2 N×N satisfies the property (AP2) if for every countable family of mutually disjoint sets {E 1 , E 2 , ...} belonging to I 2 , there exists a countable family of sets {F 1 , F 2 , ...} such that E j ∆F j ∈ I 0 2 , i.e., E j ∆F j is included in the finite union of rows and columns in N × N for each j ∈ N and F = ∞ j=1 F j ∈ I 2 (hence F j ∈ I 2 for each j ∈ N). Several authors have studied convergence, invariant convergence and Cauchy sequences (see, [1,3,5,[8][9][10][12][13][14][15][16][17]). ...
In this paper, we study the concepts of invariant convergence, p-strongly invariant convergence ([V^2_σ]_p), I_2-invariant convergence (I^σ_2), I^*_2-invariant convergence (I^σ*_2) of double sequences and investigate the relationships among invariant convergence, [V^2_σ]_p, I^σ_2 and I^σ*_2. Also, we introduce the concepts of I^σ_2-Cauchy double sequence and I^σ*_2-Cauchy double sequence.
... On ℐ-lacunary statistical convergence of set sequences was studied by Ulusu and Dündar [9]. Also, after these important studies, the notions of statistical convergence, ideal convergence and ℐ-statistical Several authors including Raimi [10], Schaefer [11], Mursaleen [12,13], Savaş [14,15], Mursaleen and Edely [16], Pancarog lu and Nuray [17,18] and some authors have studied invariant convergent sequences. The notion of strong -convergence was defined by Savaş [16]. ...
... The concept of lacunary strongly σ-convergence was introduced by Savaş [17] as below: Pancaroǧlu and Nuray [13] defined the concept of lacunary invariant summability and the space [V σθ ] q as follows: ...
In this paper, the concept of lacunary invariant uniform density of any subset A of the set N×N is defined. Associate with this, the concept of lacunary I_2-invariant convergence for double sequences is given. Also, we examine relationships between this new type convergence concept and the concepts of lacunary invariant convergence and p-strongly lacunary invariant convergence of double sequences. Finally, introducing lacunary I^*_2-invariant convergence concept and lacunary I_2-invariant Cauchy concepts, we give the relationships among these concepts and relationships with lacunary I_2-invariant convergence concept.
... Mursaleen [4] defined the concept of stronglyconvergent sequence. Then, using a positive real number , Savaş [9] generalized the concept of strongly -convergent sequence. Following, Savaş and Nuray [10] defined the concept of lacunary -statistically convergent sequence. ...
In this study, we introduce the concepts of Wijsman asymptotically J-invariant equivalence (WLJσ) , Wijsman asymptotically strongly p-invariant equivalence ([WLVσ)]p) and Wijsman asymptotically J*-invariant equivalence (WLJ*σ). Also, we investigate the relationships among the concepts of Wijsman asymptotically invariant equivalence, Wijsman asymptotically invariant statistical equivalence, WLJσ, [WLVσ)]p and WLJ*σ.
... On ℐ-lacunary statistical convergence of set sequences was studied by Ulusu and Dündar [9]. Also, after these important studies, the notions of statistical convergence, ideal convergence and ℐ-statistical Several authors including Raimi [10], Schaefer [11], Mursaleen [12,13], Savaş [14,15], Mursaleen and Edely [16], Pancarog lu and Nuray [17,18] and some authors have studied invariant convergent sequences. The notion of strong -convergence was defined by Savaş [16]. ...
In this paper, we introduce the concepts of strongly asymptotically lacunary I-invariant equivalence, f-asymptotically lacunary I-invariant equivalence, strongly f-asymptotically lacunary I-invariant equivalence and asymptotically lacunary I-invariant statistical equivalence for sequences of sets. Also, we investigate some relationships among these concepts.
... Definition 1 (see [14]). A set of positive integers is said to have uniform invariant density of zero if ...
We introduce and study the concept of invariant convergence for sequences of sets with respect to modulus function
f
and give some inclusion relations.
... Recently strongly σ-convergent sequences have been discussed and this concept of strong σ-convergence has been generalized by Sava § [5] in the following way: Fatih NURAY = <*: lim m 1/m £ |x. ί(ll) -L|" = 0 uniformly in n ( ...
In the study done here regarding the theory of summability, we introduce some new concepts in fuzzy normed spaces. First, in the beginning of the original part of our study, we define the lacunary invariant statistical convergence. Then, we examine some characteristic features like uniqueness, linearity of this new notion and give its important relation with pre-given concepts.
In this manuscript, we present the ideas of asymptotically -equivalence, asymptotically -equivalence, asymptotically -equivalence and asymptotically -equivalence for real sequences. In addition to, investigate some connections among these new ideas and we give some inclusion theorems about them.
In this paper, we introduce the concepts of invariant convergence in probability, statistically invariant convergence in probability, invariant convergence almost surely, invariant convergence in distribution and invariant convergence in Lp-norm for sequences of random variables. Also, we investigate some inclusion relations.
In this study, some scopes are established by defining some sequence spaces with the help of invariant convergence. Just as the spaces lσ and lσσ are generalized to the spaceslσ(p) and lσσ(p), so do the spaces [ωσ], ω̅σ and ω̿σ it has been extended to the spaces [ωσ(p)], ω̅σ (p) ve ω̿σ (p) and modulus functions have been implemented.
Keywords: Invariant convergence, sequence spaces, module function.
ÖZET: Bu çalışmada invaryant yakınsaklık yardımı ile bazı dizi uzayları tanımlanarak bazı kapsamlar kuruldu. lσ ve lσσ uzaylarının lσ(p) ve lσσ(p) uzaylarına genelleştirildiği gibi [ωσ], ω̅σveω̿σuzaylarını da [ωσ(p)],ω̅σ(p)veω̿σ(p)uzaylarına genişletildi ve modülüs fonksiyonlar uygulandı.
Anahtar Kelimeler: İnvaryant yakınsaklık, dizi uzayları, modülüs fonsiyonu.
The Space of Modulus Functions Defined by Invariant Convergence
ABSTRACT: : In this study, some scopes are established by defining some sequence spaces with the help of invariant convergence. Just as the spaces lσ and lσσ are generalized to the spaceslσ(p) and lσσ(p), so do the spaces [ωσ], ω̅σ and ω̿σ it has been extended to the spaces [ωσ(p)], ω̅σ (p) ve ω̿σ (p) and modulus functions have been implemented.
Keywords: Invariant convergence, sequence spaces, module function.
In this paper, we study the concepts of Wijsman lacunary -invariant convergence Wijsman lacunary -invariant convergence Wijsman p-strongly lacunary invariant convergence of sequences of sets and investigate the relationships among Wijsman lacunary invariant convergence, , and . Also, we introduce the concepts of -Cauchy sequence and -Cauchy sequence of sets.
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Journal of Mathematical Sciences and Modelling (JMSM) (J. Math. Sci. Model.) is an international research journal of rapid publication.
JMSM appeal work at the interface between analysis with applications, applied mathematics, numerical computation, computer science with application, computational intelligence and soft computing, ordinary and partial differential equations, optimization theory, control theory, engineering sciences, physical, applications of biological, chemical, social and behavioral sciences, algorithms, computational research with analysis and numerical results.
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In this paper, the concepts of asymptotically \mathcal{I}_{\sigma}-equivalence, \sigma-asymptotically equivalence, st\-rong\-ly \sigma-asymptotically equivalence and strongly \sigma-asymptotically p-equivalence for real number sequences are de\-fi\-ned. Also, we give relationships among these new type equivalence concepts and the concept of \linebreak S_{\sigma}-asymptotically equivalence which is studied in [Savaş, E. and Patterson, R. F., \sigma-asymptotically lacunary statistical equivalent sequences, Cent. Eur. J. Math., 4 (2006), No. 4, 648–655].
In this paper, the concepts of σ-uniform density of subsets A of the set N of positive integers and corresponding I_σ-convergence were introduced. Furthermore, inclusion relations between I_σ-convergence and invariant convergence also I_σ-convergence and [V_σ ]_p-convergence were given.
In this paper, the concepts of σ-uniform density of subsets A of the set N of positive integers and corresponding I_{σ}-convergence of functions defined on discrete countable amenable semigroups were introduced. Furthermore, for any Folner sequence inclusion relations between I_{σ}-convergence and invariant convergence also I_{σ}-convergence and [V_{σ}]_{p}-convergence were given. We introduce the concept of I_{σ}-statistical convergence and I_{σ}-lacunary statistical convergence of functions defined on discrete countable amenable semigroups. In addition to these definitions, we give some inclusion theorems. Also, we make a new approach to the notions of [V,\lamda]-summability,σ-convergence and σ-statistical convergence of Folner sequences by using ideals and introduce new notions, namely, I_{σ}-[V,\lamda]-summability, I_{σ}-\lamda-statistical convergence of Folner sequences. We mainly examine the relation between these two methods as also the relation between I_{σ}-statistical convergence and I_{σ}-\lamda-statistical convergence of Folner sequences introduced by the author recently.
The main object of this paper is to introduce and study a new concept of lacunary strong σ-convergence with respect to an Orlicz function M. We examine some topological properties of the resulting sequence spaces and establish some elementary connections between lacunary strong σ-convergence and lacunary strong σ-convergence with respect to an Orlicz function which satisfies Δ 2 -condition. We also give the relation between strong σ-convergence with respect to an Orlicz function and lacunary strong σ-convergence with respect to an Orlicz function.
The concepts of σ-uniform density of subsets A of the set ℕ of positive integers and corresponding I σ -convergence are introduced. Furthermore, inclusion relations between I σ -convergence and invariant convergence also I σ -convergence and [V σ ] p -convergence are given.
We define and study two concepts which arise from the notion of lacunary strong convergence and invariant means, namely lacunary strong σ-convergence with respect to an Orlicz function and lacunary σ-statistical convergence, and we establish the relationship between these two concepts.
For a normal sequence space λ and a sequence of modulus functions ℱ=(f k ) let λ(ℱ)={(x k ):(f k (|x k |))∈λ}. We define an F-seminorm for λ(ℱ) and examine the completeness of λ(ℱ) according to this F-seminorm. As concrete examples of the spaces which of type λ(ℱ) we consider spaces which are strongly summable and strongly bounded with respect to ℱ sequences.
The object of this paper is to introduce some new strongly invariant A-summable sequence spaces defined by a sequence of modulus functions ℱ=(f k ) in a seminormed space, when A=(a n,k ) is a non-negative regular matrix. Various algebraic and topological properties of these spaces, and some inclusion relations between these spaces are discussed. Finally, we study some relations between A-invariant statistical convergence and strong invariant A-summability with respect to a sequence of modulus functions in a seminormed space.
In this paper we introduce a new concept of -strong conver- gence with respect to an Orlicz function and examine some properties of the resulting sequence spaces. It is also shown that if a sequence is -strongly convergent with respect to an Orlicz function then it is -statistically conver- gent.
The purpose of this paper is to introduce and study some sequence spaces which are defined bycombining the concepts ofaOrlicz function, invariant meanand lacunary convergence. We also examine some topological properties of these spaces and establish some elementary connections between lacunary (w)σ -convergence and lacunary (w)σ -convergence with respect to an Orlicz functions which satisfy a Δ2 -condition.
In this paper, the concepts of σ-uniform density of subsets A of the set N of positive integers and corresponding I σ-convergence were introduced. Furthermore, inclusion relations between Iσ-convergence and invariant convergence also Iσ-convergence and [V σ ] p-convergence were given.
We define the sequence spaces [wθ,M,p,u,Δ]σ, [wθ,M,p,u,Δ]σ0, and [wθ,M,p,u,Δ]σ∞ which are defined by combining the concepts of Orlicz functions, invariant means,
and lacunary convergence. We also study some inclusion relations and linearity properties of the above-mentioned spaces. These are generalizations of those defined and studied by
Savaş and Rhoades in 2002 and some others before.
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