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Abstract
The object of the paper is to introduce some new sequence spaces related with the concept of absolute and strong almost convergence.
... Das ve Sahoo (1992) (Sahoo, 1992) ...
... Das ve Sahoo (1992) (Sahoo, 1992) ...
... (1.3) olacak şekilde en az bir p tamsayısı vardır (Sahoo, 1992). Böylece m sabit olmak üzere her n için f(|Ψ mn − Ψ m−1, n |) < 1 (1.4) olur. ...
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.
... Das ve Sahoo (1992) (Sahoo, 1992) ...
... Das ve Sahoo (1992) (Sahoo, 1992) ...
... (1.3) olacak şekilde en az bir p tamsayısı vardır (Sahoo, 1992). Böylece m sabit olmak üzere her n için f(|Ψ mn − Ψ m−1, n |) < 1 (1.4) olur. ...
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.
... Das ve Sahoo (1992) (Sahoo, 1992) ...
... Das ve Sahoo (1992) (Sahoo, 1992) ...
... (1.3) olacak şekilde en az bir p tamsayısı vardır (Sahoo, 1992). Böylece m sabit olmak üzere her n için f(|Ψ mn − Ψ m−1, n |) < 1 (1.4) olur. ...
... Ayrıca ( ) = + 1 olduğunda, bu uzaylar [ω σ (f)], ω ̅ σ (f) ve ω ̿ σ (f) dizi uzaylarına indirgenir. (Sahoo, 1992). ...
... Bu da → ′ yakınsaması demektir (Mursaleen, 1983). [ω σ (f)(p)] uzayı Banach uzayı 0 < < 1 ise p-normlu uzaya indirgenir (Sahoo, 1992). Teoremin ispatı teorem 2'nin ispatının benzeri olduğundan tekrar vermeyeceğiz. ...
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.
... Ayrıca ( ) = + 1 olduğunda, bu uzaylar [ω σ (f)], ω ̅ σ (f) ve ω ̿ σ (f) dizi uzaylarına indirgenir. (Sahoo, 1992). ...
... Bu da → ′ yakınsaması demektir (Mursaleen, 1983). [ω σ (f)(p)] uzayı Banach uzayı 0 < < 1 ise p-normlu uzaya indirgenir (Sahoo, 1992). Teoremin ispatı teorem 2'nin ispatının benzeri olduğundan tekrar vermeyeceğiz. ...
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.
... By (X : Y ) , we denote the class of all such matrices. A sequence x is said to be A -summable to l if Ax converges to l which is called as the A -limit of x (see [2,7,8,15,17]). ...
... Following [2,7,8,13,16], we define the space V ∞ (θ) as follows: ...
... Let σ be one-to-one mapping of the set of positive integers into itself such that σ k (n) = σ(σ k−1 (n)), k = 1, 2, 3, . . . . A continuous linear functional ϕ on l ∞ is said to be an invariant mean or a σ-mean if and only if (1) ϕ ≥ 0 when the sequence has x n ≥ 0 for all n (2) ϕ(e) = 1 where e = (1, 1, . . . ) and (3) ϕ(x σ(n) ) = ϕ(x) or all x ∈ l ∞ . For a certain kinds of mapping σ every invariant mean ϕ extends the limit functional on space c, in the sense that ϕ(x) = lim x for all x ∈ c. ...
... In [11] Mursaleen, Gaur and Chishti introduced the following sequence space, which is generalized the sequence space [w] of Das and Sahoo [3], ...
This paper presents new definitions which are a natural combination of the definition for asymptotical equivalence and [w] σ,θ -statistical convergence. Using these definitions, we have proved the st-[w] σ,θ L -asymptotical equivalence analogues of J. A. Fridy and C. Orhan’s theorems in [Pac. J. Math. 160, No. 1, 43–51 (1993; Zbl 0794.60012)] and analogous results of G. Das and B. K. Patel [Indian J. Pure Appl. Math. 26, No. 1, 64–74 (1989; Zbl 0726.40002)].
... For more details of spaces for single and double sequences and related concepts, we refer to [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] and references therein. ...
... Substituting (18) and (19) in (17), we get − −1, , , − , −1, , + −1, −1, , ...
We introduce some double sequences spaces involving the notions of invariant mean (or σ -mean) and σ -convergence for double sequences while the idea of σ -convergence for double sequences was introduced by Çakan et al. 2006, by using the notion of invariant mean. We determine here some inclusion relations and topological results for these new double sequence spaces.
... Using the concept of invariant means, the following sequence spaces have been introduced and examined by Mursaleen et al. [14] as a generalization of Das and Sahoo [3]: ...
... Let x, y ∈ [w θ ,M, p,u,∆] 0 σ and α,β ∈ C, the set of complex numbers. In order to prove the result, we need to find some ρ 3 where H = max(1,sup p k ). ...
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.
... It is easy to see that c ⊂ [ĉ] ⊂ĉ ⊂ ℓ ∞ . In [3], Das Lindenstrauss and Tzafriri [11] used the idea of an Orlicz function to define the following sequence space ...
In this paper, we introduce some new sequence spaces in n-normed spaces defined by Museliak-Orlicz function. Also we investigate some topological properties and inclusion relations between these Spaces
... 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.
... We define, as r -• cx) uniformly m m}.If we put s = 0, then we obtain [to°(/)]. Note that, if we put f(x) = x then [w(f)} = [w] and [w 1 (/)j = [to 1 ] which were studied by Das and Sahoo[2] and E. Sava §[17] respectively. ...
The purpose of this paper is to introduce some sequence spaces and also give some inclusion relations between sequence spaces and s-convergence.
... f denote the set of all strongly almost convergent sequences. If x is strongly almost convergent to , Das and Sahoo (1992) The order of statistical convergence of a sequence of numbers was given by Gadjiev and Orhan (2002) and after then statistical convergence of order and strongly p Cesàro summability of order studied by Çolak (2010) and generalized by Çolak and Bektas (2011). ...
In this paper, we introduce the concept Ŝαλ – statistical convergence of order α. Also some relations between Ŝαλ – statistical convergence of order α and strong ŵαp(λ) – summability of order α are given. Furthermore some relations between the spaces ŵα(p)[λ, M] – and Ŝαλ – are examined. © 2014, Eduem - Editora da Universidade Estadual de Maringa. All rights reserved.
... In case í µí¼ is the translation mapping í µí± → í µí± + 1, í µí¼-mean is often called a Banach limit and í µí± í µí¼ , the set of bounded sequences all whose invariant means are equal, is the set of almost convergent sequences (see Lorentz [2]). Using the concept of invariant means Mursaleen et al. [3] introduced the following sequence spaces as a generalization of Das and Sahoo [4]: í µí±¡ í µí± í µí± (|í µí±¥ − ℓ|) í®í¸→ 0, uniformly in í µí±} (2) and investigated some of its properties. ...
The main objective of this paper is to introduce a new kind of sequence spaces by combining the concepts of modulus function, invariant means, difference sequences, and ideal convergence. We also examine some topological properties of the resulting sequence spaces. Further, we introduce a new concept of SθσΔm(I)-convergence and obtain a condition under which this convergence coincides with above-mentioned sequence spaces.
... It is easy to see that c ⊂ [ĉ] ⊂ĉ ⊂ ∞ . Das and Sahoo [4] defined the sequence space ...
The purpose of this paper is to introduce the spaces of sequences that are strongly almost (ω, λ, q)-summable with respect to a modulus function. We give some relations related to these sequence spaces. It is also shown that if a sequence is strongly (ω, λ, q)-summable with respect to a modulus function, then it is S(λ q)-statistically convergent.
... Quite recently, Mursaleen and Mohiuddine (see [12]) introduced the following double sequence spaces by using almost convergence, while such spaces for single sequences were studied by Das and Sahoo [4]. ...
In this paper, we introduce some new double sequence spaces with respect to an Orlicz function and define two new convergence methods related to the concepts of statistical convergence and lacunary statistical convergence for double sequences. We also present some inclusion theorems for our newly defined sequence spaces and statistical convergence methods
... Several authors including Lorentz [12], Duran [4] and King [8] have studied almost convergent sequences. Maddox [14] has defined x to be Manuscript [3] as follows: ...
The concept of strong almost convergence was introduced by Maddox in 1978 [Math. Proc. Camb. Philos. Soc., 83 (1978), 61-64] which has various applications. In this paper we introduce some new sequence spaces which arise from the notions of strong almost convergence and an Orlicz function in a seminormed space. A new concept of uniform statistical convergence in a seminormed space has also been introduced.
... In this section, we introduce the following sequence spaces, while such spaces for single sequences were studied by Das and Sahoo [2]. ...
In this paper, we define and study the Banach limit for double sequences and introduce some new spaces related to the concept of almost and strong almost convergence for double sequences. We characterize these spaces through some sublinear functionals and we also establish some inclusion relations.
... Maddox (1978) . This topic has been widely studied (Lorentz, 1948;Maddox, 1978;Maddox, 1979;Maddox, 1967;Maddox, 1986;Basarir, 1992;Das and Mishra, 1983;Das and Patel, 1989;Das and Sahoo, 1992). ...
In the present paper, we introduce some new sequence spaces derived by Riesz mean and the notions of almost and strongly almost convergence in a real 2-normed space. Some topological properties of these spaces are investigated. Further, new concepts of statistical convergence which will be called weighted almost statistical convergence, almost statistical convergence and R, pn statistical convergence in a real 2-normed space, are defined. Also, some relations between these concepts are investigated.
... A sequence x ∈ l ∞ , the space of bounded sequences x = (x k ), is said to be almost convergent [5] Recently, Das and Sahoo [2] introduced the following sequence spaces using the concept of almost convergence. By a lacunary sequence θ = (k r ), r = 0, 1, 2, · · · , where k 0 = 0, we shall mean an increasing sequence of non-negative integers h r = (k r − k r−1 ) → ∞, (r → ∞). ...
We introduce the sequence spaces (^ w(M)) and (^ w(M))µ of strong almost convergence and lacunary strong almost convergence re- spectively deflned by the Orlicz function M. We establish certain inclu- sion relations and show that the above spaces are same for any bounded sequences.
... Nanda [7] ...
In this paper we characterize the matrix classes (l(p, u),??) and (l(p, u), ?) which generalize the matrix classes given by Mursaleen [6]. .
... k (|t kn (x − l)|) = 0, uniformly in n, for some l Some sequence spaces are obtained by specializing F , θ, σ. For example, if θ = (2 r ) , σ (n) = n + 1and f k (x) = x for all k, then [w σ , F ] θ =ŵ (see, Das and Sahoo[3]). If σ (n) = n + 1 and f k (x) = f for all k, then [w σ , F ] θ = [ŵ (f )] θ and [w σ , F ] = [ŵ (f )] (see, Mursaleen and Chishti[12]). ...
The purpose of the paper is to introduce and study some sequence spaces which are defined by combining the concepts of lacunary convergence, invariant mean and the sequence of modulus functions. We also examine some topological properties of these spaces.
... If x = (x k ) is statistically convergent to l, then there is a convergent sequence y = (y k ) such that y is convergent to l and the set {k : x k = y k } has natural density 0, i.e., x k = y k " for almost all " k [10]. Statistical convergence of a sequence x has been studied by many authors including Fast [7], Fridy [10], Fridy and Miller [11], Connor [2], Connor and Kline [3], Salat [22], Pehlivan and Fisher [19], Maddox [16] and Kolk [14]. Fridy [12] introduced the concept of statistical limit points and statistical cluster points of real number sequences and studied some properties of the sets of statistical limit and cluster points. ...
In this note we introduce the concept of a lacunary statistical cluster (l.s.c.) point and prove some of its properties in finite dimensional Banach spaces. We develop the method suggested by S. Pehlivan and M.A. Mamedov [20] where it was proved that under some conditions optimal paths have the same unique stationary limit point and stationary cluster point. We also show that the set of l.s.c. points is nonempty and compact.
Generalised CR submanifolds of nearly tran-sasakian manifolds
The present paper is concerned with the concept of sequentially topologies in non-archimedean analysis. We give characterizations of such topologies.
The notion of almost convergence for ordinary (single) sequences was given by Lorentz (Acta Math. 80:167–190, 1948), and for double sequences by Moricz and Rhoades (Math. Proc. Camb. Philos. Soc. 104:283–294, 1988). In this chapter, we discuss the notion of almost convergence and almost Cauchy for double sequences. Some more related spaces for double sequences, associated sublinear functionals, and various inclusion relations are also studied.
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 chapter, we study the concepts of linear metric spaces, paranormed spaces, FK spaces and BK spaces which play an important role in our studies on sequence spaces. One of the main advantage of FK space theory is that it provides easy and short proofs of numerous classical results of summability theory and is the most powerful and widely used tool in the characterization of matrix mappings between sequence spaces. Moreover, it enables us to obtain the most important result on the continuity of matrix mappings between FK spaces. The results of the theory of FK and BK spaces are also applied to characterize the matrix classes.
In this paper, we introduce the concept Ŝαλ - statistical convergence of order α. Also some relations between Ŝαλ -statistical convergence of order α and strong ŵβp(μ) -summability of order β are given. Furthermore some relations between the spaces ŵα(p) [λ,f] and Ŝαλ are examined.
This book exclusively deals with the study of almost convergence and statistical convergence of double sequences. The notion of "almost convergence" is perhaps the most useful notion in order to obtain a weak limit of a bounded non-convergent sequence. There is another notion of convergence known as the "statistical convergence", introduced by H. Fast, which is an extension of the usual concept of sequential limits. This concept arises as an example of "convergence in density" which is also studied as a summability method. Even unbounded sequences can be dealt with by using this method. The book also discusses the applications of these non-matrix methods in approximation theory. Written in a self-contained style, the book discusses in detail the methods of almost convergence and statistical convergence for double sequences along with applications and suitable examples. The last chapter is devoted to the study convergence of double series and describes various convergence tests analogous to those of single sequences. In addition to applications in approximation theory, the results are expected to find application in many other areas of pure and applied mathematics such as mathematical analysis, probability, fixed point theory and statistics.
The purpose of this paper is to introduce and study some sequence spaces which are defined by combining the concepts of a sequence of Orlicz functions, invariant mean and lacunary convergence. We also examine some topological properties of these sequence spaces and establish some elementary connections between them.
The object of this paper is to introduce some new sequence spaces
related with the concept of lacunary strong almost convergence for double sequences and also to characterize these spaces through sublinear functionals that
both dominate and generate Banach limits and to establish some inclusion
relations.
Purpose
For a non-trivial ideal
, a lacunary sequence θ = (k
r
), and a modulus function f, the purpose of present paper is to introduce certain new notions:
-asymptotically equivalent,
-asymptotically equivalent, and
-asymptotically equivalent sequences of numbers.
Methods
We use an analytic method to obtain our results.
Results
Certain theorems on generalized equivalent sequences by the use of ideals, lacunary sequences, and a modulus function are obtained.
Conclusions
We observe that if the modulus function f satisfies
, then the notions
and
respectively coincide with the notions
and
. However, if f is bounded, the notions
and
coincides respectively with the notions
and
.
In this paper we define three classes of new sequence spaces with respect to an Orlicz function. We give some relations related to these sequence spaces. We also introduce the concept of
S*
q
‐statistically convergence and obtain some inclusion relations related to these new sequence spaces.
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.
The idea of dierence sequence spaces was introduced by Kizmaz (8) and this concept was generalized by Et and Colak (6). In this paper we define the space '( m,f,p,q,s) on a seminormed complex linear space by using modulus function and we give various properties and some inclusion relations on this space. Furthermore we study some of its properties, solidity, symmetricity etc.
In this paper, for an -strict pseudocontraction T, we prove strong
convergence of the modified Mann's iteration defined by
where , and in (0,1) satisfy:
(i) with
; (ii)
and
; (iii)
.Our results unify and improve some
existing results.
The idea of A-invariant mean and A-almost convergence is due to J. P. Duran [8], which is a generalization of the usual notion of Banach limit and almost convergence.
In this paper, we discuss some important properties of this method and prove that the space F(A) of A-almost convergent sequences is a BK space with ‖ · ‖∞, and also show that it is a nonseparable closed subspace of the space l
∞ of bounded sequences.
Идея A-инвариантных средних и A-почти сходимости принадлежит Дж. П. Дурану [8] и является обобщением принятых понятий Банахова предела и почти сходимости.
В настоящей работе обсуждаются некоторые важные свойства этого метода и устанавливается, что пространство F(A)
A-почти сходящихся после-довательностей является BK пространством с нормой ‖ · ‖∞, а также что оно есть несепарабельное эамкнутое подпространство пространства ℓ
∞ ограниченных последовательностей.
We introduce the vector-valued sequence spaces w ∞ (Δ m ,F,Q,p,u), w 1 (Δ m ,F,Q,p,u), and w 0 (Δ m ,F,Q,p,u), S u q and S 0u q , using a sequence of modulus functions and the multiplier sequence u=(u k ) of nonzero complex numbers. We give some relations related to these sequence spaces. It is also shown that, if a sequence is strongly Δ m u q -Cesàro summable with respect to the modulus function f, then it is Δ m u q -statistically convergent.
The object of this paper is to introduce a new concept of absolute almost convergence which emerges naturally as an absolute analogue of almost convergence, in the same way as convergence leads to absolute convergence.
The object of this paper is to introduce a new concept of absolute almost convergence which emerges naturally as an absolute analogue of almost convergence, in the same way as convergence leads to absolute convergence.
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
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)
Let m be the set of all real sequences x = ( x n ) with norm . A linear functional L on m is said to be a Banach limit (see Banach(1), p. 32) if it has the following properties:
(i) L ( x ) ≥ 0, if x ≥ 0 (i.e. x n ≥ 0, for all n ∈ N )
(ii) L { e ) = 1, where e = (1, 1, 1,…),
(iii) L ( Sx ) = L ( x ), where ( Sx ) n = x n +1 .
The sequence spaces I
- S Simons
S. SIMONS. The sequence spaces I(p,,) and m(p,, ), Proc. London Math. Sot. (3) 15 (1965), 422436.