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On strong almost convergence
Abstract
The concept of strong almost convergence was introduced in (2), where the matrices summing every strongly almost convergent sequence, leaving the limit invariant, were characterized.(Received June 12 1978)
... In this section, by using some results of Maddox [23,24], we prove some core inclusion results for the uniform statistical convergence which are the analogs of those given in [15] and [29]. ...
... In order to state our first result, we need the following Lemma of Maddox [23,24]. ...
... we immediately get the following result from Maddox [24,Theorem 5]. ...
This paper is a continuation of the so far performed studies on the concept of uniform statistical convergence. We first characterize two inequalities concerning the uniform statistical limit superior that lead to two core inclusion results for bounded real sequences. Using the inverse Fourier transformation, we also give a criterion on uniform statistical convergence, and study some factorization results of the space of uniformly statistically convergent sequences. These results are used to give a Korovkin-type approximation theorem.
... Several authors including Duran (see, [5]), Ganie et al (see, [1,2,3,4,22]), King (see, [10]), Lorentz (see, [12]) and many others have studied almost convergent sequences. Maddox (see, [15,14]) has defined x to be strongly almost convergent to a number if . m in uniformly 0, ...
... We have (see, [13,14]) that ...
The sequence space introduced by M. Et and have studied its various properties. The aim of the present paper is to introduce the new pranormed generalized difference sequence space. and , We give some topological properties and inclusion relations on these spaces. 2010 AMS Mathematical Subject Classification: 46A45; 40C05.
... < ε + ε + ε + ε = 4ε, By equation (3), (4), (7) and (8). ...
In this article we have introduced the notion of almost convergence, strongly almost convergence, almost null and strongly almost null of double sequences in Pringsheim sense of bi-complex numbers. We have proved that these are linear spaces and with the help of the Euclidean norm defined on bi-complex numbers, we have proved their different algebraic and topological properties. Suitable examples have been discussed.
... Quite recently Subramanian and Misra [29] have studied the space χ 2 M (p, q, u) of double sequences and gave some inclusion relations. Spaces are strongly summable sequences were discussed by Kuttner [31], Maddox [32], and others. The class of sequences which are strongly Cesàro summable with respect to a modulus was introduced by Maddox [8] as an extension of the definition of strongly Cesàro summable sequences. ...
In this paper we introduce the modular sequence space of χ 2 and examine some topological properties of these space also establish some duals results among them. Lindenstrauss and Tzafriri [7] used the idea of Orlicz function to define the sequence space ℓ M which is called an Orlicz sequence space. Another generalization of Orlicz sequence spaces is due to Woo [9]. We define the sequence spaces χ 2 f λ and χ 2λ g , where f = (fmn) and g = (gmn) are sequences of modulus functions such that fmn and gmn be mutually complementary for each m, n.
... Quite recently Subramanian and Misra [29] have studied the space χ 2 M (p, q, u) of double sequences and gave some inclusion relations. Spaces are strongly summable sequences were discussed by Kuttner [31], Maddox [32], and others. The class of sequences which are strongly Cesàro summable with respect to a modulus was introduced by Maddox [8] as an extension of the definition of strongly Cesàro summable sequences. ...
In this paper we introduce the sequence spaces χ 2I F and Λ 2I F for the sequence of moduli F = (f mn) and study some of the general properties of these spaces.
... Spaces are strongly summable sequences were discussed by Kuttner [31], Maddox [32], and others. The class of sequences which are strongly Cesàro summable with respect to a modulus was introduced by Maddox [8] as an extension of the definition of strongly Cesàro summable sequences. ...
In this paper we introduce The Cesàro mean of order one χ 2 spaces defined by modulus is associated with a fixed multiplier sequence of non-zero scalars, study their general properties of these spaces and also establish some duals results among them.
... Spaces are strongly summable sequences were discussed by Kuttner [31], Maddox [32], and others. The class of sequences which are strongly Cesàro summable with respect to a modulus was introduced by Maddox [8] as an extension of the definition of strongly Cesàro summable sequences. ...
In this paper, we introduce the modular sequence space of Γ 2 of fuzzy numbers using modulus function and examine some topological properties of these space also establish some duals results among them. Lindenstrauss and Tzafriri [7] used the idea of Orlicz function to define the sequence space M which is called an Orlicz sequence space. Another generalization of Orlicz sequence spaces is due to Woo [9]. We define the sequence spaces Γ 2 fλ and Γ 2λ g , where f = (f mn) and g = (g mn) are sequences of modulus functions such that f mn and g mn be mutually complementary for each m, n. This paper deals with the modular of Γ 2 defined by asymptotically λ− and σ invariant modulus of fuzzy numbers. If f mn and g mn are mutually complementary function then the 1320 N. Subramanian, P. Anbalagan and P. Thirunavukarasu two sequences X and Y of fuzzy numbers are said to be asymptotically λ− invariant statistical equivalent of multiple 0 provided that for every > 0, X S Γ 2F f,σ,μ,λ ≈ Y = 1 μrs m, n ∈ I rs : f d 1 λmn X σ mn (η) Y σ mn (η) 1/m+n , 0 ≥ → 0 as r, s → ∞, uniformly in η X S Γ 2F λ g,σ,μ ≈ Y = 1 μrs m, n ∈ I rs : 1 λmn g d X σ mn (η) Y σ mn (η) 1/m+n , 0 ≥ → 0 as r, s → ∞, uniformly in η. Mathematics Subject Classification: 40A05, 46S40, 40D05
... The notion of strong almost convergence was considered by Maddox [39]. In [40], Maddox defined a generalization of strong almost convergence. Related articles with the topic almost convergence and strong almost convergence can be seen in [3,[8][9][10][11][12][13][14][15]53]. ...
In the present paper we introduce and study Orlicz lacunary convergent triple sequences over n-normed spaces. We make an effort to present the notion of -ideal convergence in triple sequence spaces. We examine some topological and algebraic features of new formed sequence spaces. Some inclusion relations are obtained in this paper. Finally, we investigate ideal convergence in these spaces.
... Convergence of sequences plays a significant role in the fundamental theory of mathematical analysis. The notion of strongly almost convergence of real sequences was introduced by Maddox [14] in 1979. It is also reported by Lorentz [13] and later it was studied from the sequence-space point of view via summability by Mursaleen [17], King [8], Savas [21], and many others. ...
The aim of this article is to introduce the characterization of strongly almost convergence in complex uncertain double sequences. We investigate the concept of strongly almost convergence almost surely, convergence in measure, convergence in mean, convergence in distribution, and convergence uniformly almost surely of complex uncertain double sequences. Finally, we present the interrelationships among those newly introduced notions.
... f(Ψ kn − Ψ k−1, n ) = (1.5) olduğundan (Maddox, 1979). ...
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.
... f(Ψ kn − Ψ k−1, n ) = (1.5) olduğundan (Maddox, 1979). ...
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.
... elde edilir (Maddox, 1979). ...
In this study, invariant convergent sequence spaces defined with the help of the Modulus function were defined and some scope relations were established beyween them. Spaces of [ωσ(f)],ω̅σ (f) and ω̿σ (f) is extended to [ωσ(f)(p)],ω̅σ (f)(p) and ω̿σ (f)(p)spaces. Topological properties of generalized sequence spaces are studied.
... elde edilir (Maddox, 1979). ...
Bu çalışmada Modülüs fonksiyon yardımı ile tanımlanan invaryant yakınsak dizi uzayları tanımlanarak aralarında bazı kapsam bağıntıları kuruldu. [ω_σ (f)],ω ̅_(σ ) (f) ve ω ̿_(σ ) (f) uzayları [ω_σ (f)(p)],ω ̅_(σ ) (f)(p) ve ω ̿_(σ ) (f)(p) uzaylarına genişletildi. Genelleştirilen bu dizi uzaylarının topolojik özellikleri incelendi.
... Chen, Ning, and Wang (2016) explored the idea of convergence in the complex uncertain sequences using complex uncertain variable and it has been applied further by many researchers like Tripathy and Nath (2017), Tripathy and Dowari (2018), Tripathy (2019, 2020), Das et al. (2020), Roy, Saha, and Tripathy (2020). Maddox (1979) introduced the concept of strongly almost convergences on real sequences. King (1966) studied the almost summable sequence of complex numbers and Mursaleen (1984) established the absolute almost convergent sequence. ...
The aim of this treatise is to introduce the concept of strongly almost convergence in complex uncertain sequences. Also, we investigate the strongly almost convergence in almost surely (in shortly a.s.), convergence in measure, convergence in mean, convergence in distribution and convergence in uniformly almost surely of complex uncertain sequences and study the relationships among them.
... f(Ψ kn − Ψ k−1, n ) = (1.5) olduğundan (Maddox, 1979). ...
... Kuttner [3] examined spaces of strongly summable sequences and later on it was discussed by Maddox [5], and others. Also Maddox [4] studied the set of sequences which are strongly Cesàro summable with respect to a modulus as a generalization of the notion of strongly Cesàro summable sequences. ...
... Spaces of strongly summable sequences were examined by Kuttner [4] and later on Maddox [6] continue to discuss it. Also Maddox [5] presented the class of sequences which are strongly Cesàro summable with respect to a modulus as a generalization of the notion of strongly Cesàro summable sequences. ...
The concept of P-convergent double sequence spaces was presented in 1988 by Moricz and Rhoades, and also in 2005, the notion of almost lacunary strongly P-convergent double sequences was introduced by Savas and Patterson, using Orlicz functions. Following these two concepts, we present the double almost lacunary strongly P-convergent of weight g via Orlicz functions in 2-Normed spaces. Additionally, inclusion theorems and extension of existing results in the literature will be established.
... Existing work on statistical convergence appears to have been restricted to real or complex sequences, but several authors extended the idea to apply it to sequences of fuzzy numbers and also introduced and discussed the concept of statistical sequences of fuzzy numbers. For some very interesting investigations concerning statistical convergence, one may consult the papers of Cakalli [3], Miller [25] , Maddox [23] and many others [1,2,22], where more references on this important summability method can be found. ...
... Recall that is strongly almost convergent [20] to some number if ...
We first define the notion of lacunary statistical convergence of order (α,β) , and taking this notion into consideration, we introduce some seminormed difference sequence spaces over n -normed spaces with the help of Musielak-Orlicz function M=(Mk) of order (α,β) . We also examine some topological properties and prove inclusion relations between the resulting sequence spaces.
... Let c denote the space of all almost convergent sequences. Lorentz [12] proved that c = {x G loo lim t mn (x) exists, uniformly in n}, m-too where t mn (x) = (m + x k+n-The space [c] of strongly almost convergent sequences was introduced by Maddox [15] and also independently by Freedman et al. [7] as follows. ...
... Existing work on statistical convergence appears to have been restricted to real or complex sequence, but several authors extended the idea to apply to sequences of fuzzy numbers and also introduced and discussed the concept of statistically convergent sequences of fuzzy numbers. For some very interesting investigations concerning statistical convergence, one may consult the papers of Cakalli [11], Miller [13], Maddox [14] and many others, where more references on this important summability method can be found. ...
In this paper we have introduced the concept of asymptotically double lacunary statistically equivalent of interval numbers and strong asymptotically double lacunary statistically equivalent of interval numbers. We have investigated the relations related to these spaces.
... Later on, statistical convergence turned out to be one of the most active areas of research in summability theory after the works of Fridy [20] andSalát [32]. For some very interesting investigations concerning statistical convergence, one may consult the papers of Cakalli [14], Miller [16], Maddox [18] and many others, where more references on this important summability method can be found. ...
The main purpose of this paper is to introduce the necessary and sufficient conditions for the matrix of interval numbers Ā = (ānk) such that Ā-transform of x = (x k) belongs to the sets c0S(i) ∩ ℓi∞, cS(i) ∩ ℓi∞, where in particular x ∈ c0S(i) ∩ ℓi∞ and x ∈ cS(i) ∩ ℓi∞ respectively.
... Quite recently Subramanian and Misra [29] have studied the space χ 2 M (p, q, u) of double sequences and gave some inclusion relations. Spaces of strongly summable sequences were discussed by Kuttner [31] , Mad- dox [32], and others. The class of sequences which are strongly Cesàro summable with respect to a modulus was introduced by Maddox [8] as an extension of the definition of strongly Cesàro summable sequences. ...
In this paper we introduce the χ2 fuzzy numbers defined by a modulus, study some of their properties and inclusion results.
... Spaces are strongly summable sequences were discussed by Kuttner [31], Maddox [32], and others. The class of sequences which are strongly Cesàro summable with respect to a modulus was introduced by Maddox [8] as an extension of the definition of strongly Cesàro summable sequences. ...
In this paper we introduce The Ces̀aro mean of order one ?2 spaces defined by modulus is associated with a fixed multiplier sequence of non-zero scalars, study their general properties of these spaces and also establish some duals results among them.
... Quite recently Subramanian and Misra [29] have studied the space χ 2 M (p, q, u) of double sequences and gave some inclusion relations. Spaces are strongly summable sequences were discussed by Kuttner [31], Maddox [32], and others. The class of sequences which are strongly Cesàro summable with respect to a modulus was introduced by Maddox [8] as an extension of the definition of strongly Cesàro summable sequences. ...
In this paper we introduce a new concept for almost lacunary χ2 sequence spaces strong P- convergent to zero with respect to an modulus function and examine some properties of the resulting sequence spaces. We also introduce and study statistical convergence of almost lacunary χ2 sequence spaces and also some inclusion theorems are discussed.
... Quite recently Subramanian and Misra [29] have studied the space χ 2 M (p, q, u) of double sequences and gave some inclusion relations. Spaces are strongly summable sequences were discussed by Kuttner [31], Maddox [32], and others. The class of sequences which are strongly Cesàro summable with respect to a modulus was introduced by Maddox [8] as an extension of the definition of strongly Cesàro summable sequences. ...
This is review article . So the paper start with introduction onwards.
... The purpose of this paper is to introduce and study a concept of lacunary almost generalized ∆ m −convergence function and to examine these new sequence spaces which also generalize the well known Orlicz sequence space l M and strongly summable sequence [C, 1, p], [C, 1, P ] 0 and [C, 1, p] ∞ (see [18]). ...
In this paper we introduce a new concept for almost double lacunary �m -
sequence spaces de�ned by Orlicz function and give inclusion relations.The re-
sults here in proved are analogous to those by Ayhan Esi [General Mathematics
(2009),2(17) 53-66].
... Quite recently Subramanian and Misra [29] have studied the space χ 2 M (p, q, u) of double sequences and gave some inclusion relations. Spaces are strongly summable sequences were discussed by Kuttner [31], Maddox [32], and others. The class of sequences which are strongly Cesàro summable with respect to a modulus was introduced by Maddox [8] as an extension of the definition of strongly Cesàro summable sequences. ...
The main aim of this treatise is to study the concept of strongly almost convergence of complex uncertain triple sequences and to investigate the interrelationship among different aspect of this concept on the complex uncertain triple sequences.
This book collects select papers presented at the “International Conference on Mathematical Analysis and Application in Modeling,” held at Jadavpur University, Kolkata, India, on 9–12 January 2018. It discusses new results in cutting-edge areas of several branches of mathematics and applications, including analysis, topology, dynamical systems (nonlinear, topological), mathematical modeling, optimization and mathematical biology. The conference has emerged as a powerful forum, bringing together leading academics, industry experts and researchers, and offering them a venue to discuss, interact and collaborate in order to stimulate the advancement of mathematics and its industrial applications.
In this paper, by using a nonnegative real-valued Lebesque measurable function in the interval we introduce the concepts of strong -summability and -statistical convergence of weight where for any sequence in with . We also examine some relations between - statistical convergence of weight g and strong -summability of weight g.
The main goal of this paper is to define a contractible topological
vector space in quasi-reflexive space modeled in a non-reflexive
Banach space and to study the convergence of generalized vector
variational inequalities and generalized vector complementarity
problems using weakly absolute summability theorem in it. We
obtained an unconditional Schauder basis from a rho-idempotent
infinite dimensional real or complex matrix developed by Das and
Behera [6] and from this basis set, we obtained
the solution of the problem GVVIP. The existence of solution
of the dual problem GDVVIP, and generalized vector
complementarity problem GVCP are established using weakly
absolute almost summability theorems and vector univalent function.
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.
In this chapter, we recall the notion of almost convergence and statistical convergence for single sequences x=(x
k
). We present here a brief survey on developments of almost convergence, statistical convergence, and some related methods, e.g., absolute almost convergence and strong almost convergence for single sequences.
In this paper, we discuss almost convergent sequences. The concept of almost convergence was first introduced by G. G. Lorentz in 1948. Since then, various types of generalizations for almost convergence have been constructed and studied. Our concern is also to generalize the almost convergence. We construct a generalized almost convergence (GAC) in a natural way through the use of Lorentz-type definition, and give an analogue of Lorentz’s result. It should be noted that the definition of our GAC does not require normed space nor boundedness of sequences, although Lorentz-type definitions usually require it. On the other hand, we also show that our GAC eventually has a certain type of boundedness in spite of its definition.
In this paper, we define and study the notion of lacunary statistical convergence and lacunary of statistical Cauchy sequences in random on G3 over p-metric spaces defined by Musielak-Orlicz functions.
In this paper, we introduce the modular sequence space of Γ2 of fuzzy numbers using modulus function and examine some topological properties of these space also establish some duals results among them. Lindenstrauss and Tzafriri [7] used the idea of Orlicz function to define the sequence space lM which is called an Orlicz sequence space. Another generalization of Orlicz sequence spaces is due to Woo [9]. We define the sequence spaces Γ2fλ and Γ2λg, where f = (fmn) and g = (gmn) are sequences of modulus functions such that fmn and gmn be mutually complementary for each m, n. This paper deals with the modular of Γ2 defined by asymptotically λ- and σ invariant modulus of fuzzy numbers. If fmn and gmn are mutually complementary function then the two sequences X and Y of fuzzy numbers are said to be asymptotically λ- invariant statistical equivalent of multiple 0̄ provided that for every ε > 0,∞, uniformly in η.
The purpose of this paper is to study lacunary double statistical convergence of order α. Furthermore some inclusion relations have been examined. We also introduce a new sequence space by combining Orlicz function and lacunary double statistical convergence.
In this paper we introduce the strongly generalized difference sequence spaces of modulus function and A(i) = a(i(k,l))(i(m,n)) is a non-negative four dimensional matrix of complex numbers and (p(i(mn))) is a sequence of positive real numbers. We also give natural relationship between strongly generalized difference summable sequences with respect of modulus. We examine some topological properties of the above spaces and investigate some inclusion relations between these spaces.
In this paper, we define and study lacunary double almost statistical convergence of order α. Further, some inclusion relations have been examined. We also introduce a new sequence space by combining lacunary double almost statistical convergence and Orlicz function.
In this paper, we define the Cesàro χ2 sequence space Cesq (χ2f) defined by a Musielak and establish some general properties of the space.
In this paper we introduce a new class of ideal convergent (briefly I-convergent) sequence spaces defined by an infinite matrix, a sequence of modulus functions over n-normed spaces. We study some linear topological structures and algebraic properties of these spaces. We also give some inclusion relations between these new sequence spaces.
We establish a theorem of Korovkin type for approximation processes of positive linear operators and give a quantitative version of this result. The most typical examples of these processes in question are summation processes of positive linear operators which can be obtained by very general summability methods including convergence, almost convergence and so on. Also, an example of applications is given by the Bernstein-Lototsky-Schnabl functions on compact convex subsets of a real locally convex Hausdorff vector space.
We introduce the χ 2 fuzzy numbers defined by a modulus, study some of their properties and inclusion results.
In this paper we have studied simüar problems to those established for real bounded sequences related to the characterization of infinite matrix that transform a subspace Ei of l°° (N) into another E2 C z°° (IN), underlying the case in which Ei is the space of almost convergent sequences andi5'2 is the same. In fact, we have studied the above problems in the context of real bounded continuous func-tions defined on a separated completely regular and pay special attention to the notion of almost con-vergent functions. We characterize one type of kernel of summability that allows us to transform al-most convergent functions into other types of almost convergent functions. This theorem generalizes some result from the theory of sequences, such as Kojima-Suchur, Silverman-Toeplitz, Lorentz and Maddox and from the bounded functions on amenable semigroups. This paper continues the theory that we began in (4).
In his important paper (1), Lorentz defined the space f of almost convergent sequences, using the idea of Banach limits. If x ∈ l ∞ ( R ), the space of bounded real sequences, and
where the inf is taken over all sets n (1), n (2), …, n ( r ) of natural numbers, then a Banach limit L may be defined as a linear functional on l ∞ ( R ) which satisfies
Stuttgart, Univ., Fachbereich Mathematik, Habil.-Schr. 1971.