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Search Results: 1 - 10 of 467715 matches for " Vakeel A. Khan "
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On Some I-Convergent Double Sequence Spaces Defined by a Modulus Function  [PDF]
Vakeel. A. Khan, Nazneen Khan
Engineering (ENG) , 2013, DOI: 10.4236/eng.2013.55A006
Abstract: In 2000, Kostyrko, Salat, and Wilczynski introduced and studied the concept of I-convergence of sequences in metric spaces where I is an ideal. The concept of I-convergence has a wide application in the field of Number Theory, trigonometric series, summability theory, probability theory, optimization and approximation theory. In this article we introduce the double sequence spaces \"\" and \"\",for a modulus function f and study some of the properties of these spaces.

Statistically Convergent Double Sequence Spaces in 2-Normed Spaces Defined by Orlicz Function  [PDF]
Vakeel A. Khan, Sabiha Tabassum
Applied Mathematics (AM) , 2011, DOI: 10.4236/am.2011.24048
Abstract: The concept of statistical convergence was introduced by Stinhauss [1] in 1951. In this paper, we study con- vergence of double sequence spaces in 2-normed spaces and obtained a criteria for double sequences in 2-normed spaces to be statistically Cauchy sequence in 2-normed spaces.
I-Pre-Cauchy Double Sequences and Orlicz Functions  [PDF]
Vakeel A. Khan, Nazneen Khan, Ayhan Esi, Sabiha Tabassum
Engineering (ENG) , 2013, DOI: 10.4236/eng.2013.55A008
Abstract: Let \"\" be a double sequence and let M be a bounded Orlicz function. We prove that x is I-pre-Cauchy if and only if \"\"This implies a theorem due to Connor, Fridy and Klin [1], and Vakeel A. Khan and Q. M. Danish Lohani [2]
On λ-Zweier Convergent Sequence Spaces
Khan Vakeel A.
Theory and Applications of Mathematics & Computer Science , 2012,
Abstract: In this paper we introduce a new concept of λ-Zweier convergence and λ-statistical Zweier convergence and give some relations between these two kinds of convergence.
An Extension of Kuttner’s Theorem
Vakeel A. Khan
Theory and Applications of Mathematics & Computer Science , 2011,
Abstract: If 0 < p < 1 and X is a locally convex FK - space, then X l∞ whenever X w0(p) (Kuttner's theorem see (B.Thorpe,1981). The aim of this paper is to give some extensions of this theorem by replacing w0(p) by [cA , M ]0.
On a New I-Convergent Double-Sequence Space
Vakeel A. Khan,Nazneen Khan
International Journal of Analysis , 2013, DOI: 10.1155/2013/126163
Abstract: The sequence space was introduced and studied by Mursaleen (1983). In this article we introduce the sequence space 2 and study some of its properties and inclusion relations. 1. Introduction and Preliminaries Let , , and be the sets of all natural, real, and complex numbers, respectively. We write showing the space of all real or complex sequences. Definition 1. A double sequence of complex numbers is defined as a function . We denote a double sequence as where the two subscripts run through the sequence of natural numbers independent of each other [1]. A number is called a double limit of a double sequence if for every there exists some such that (see??[2]). Let and denote the Banach spaces of bounded and convergent sequences, respectively, with norm . Let denote the space of sequences of bounded variation; that is, where is a Banach space normed by (see??[3]). Definition 2. Let be a mapping of the set of the positive integers into itself having no finite orbits. A continuous linear functional on is said to be an invariant mean or -mean if and only if(i) when the sequence has for all ;(ii) , where ;(iii) for all . In case is the translation mapping , a -mean is often called a Banach limit (see [4]), and , the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequences (see [5]). If , then . Then it can be shown that where , . Consider where denote the th iterate of at . The special case of (5) in which was given by Lorentz [5, Theorem 1], and that the general result can be proved in a similar way. It is familiar that a Banach limit extends the limit functional on . Theorem 3. A -mean extends the limit functional on in the sense that for all if and only if has no finite orbits; that is to say, if and only if, for all , , (see [3]) Put assuming that . A straight forward calculation shows (see [6]) that For any sequence , , and scalar , we have Definition 4. A sequence is of -bounded variation if and only if (i) converges uniformly in ;(ii) , which must exist, should take the same value for all . We denote by , the space of all sequences of -bounded variation (see [7]): Theorem 5. is a Banach space normed by (see [8]). Subsequently, invariant means have been studied by Ahmad and Mursaleen [9], Mursaleen et al. [3, 6, 8, 10–14], Raimi [15], Schaefer [16], Savas and Rhoades [17], Vakeel et al. [18–20], and many others [21–23]. For the first time, I-convergence was studied by Kostyrko et al. [24]. Later on, it was studied by ?alát et al. [25, 26], Tripathy and Hazarika [27], Ebadullah et al. [18–20, 28], and
On Some I-Convergent Sequence Spaces Defined by a Modulus Function
Vakeel A. Khan,Khalid Ebadullah
Theory and Applications of Mathematics & Computer Science , 2011,
Abstract: In this article we introduce the sequence spaces cI0(f), cI(f) and lI∞(f) for a modulus function f and study some of the properties of these spaces.
On Casaro Sequence Space of Fuzzy Numbers Defined by a Modulus Function  [PDF]
Ayhan Esi, Vakeel A. Khan
Open Access Library Journal (OALib Journal) , 2014, DOI: 10.4236/oalib.1100920
Abstract: The main purpose of this paper is to introduce the sequence space cesF(f, p) of sequence of fuzzy numbers defined by a modulus function. Furthermore, some inclusion theorems have been discussed.
Aσ-Double Sequence Spaces and Double Statistical Convergence in 2-Normed Spaces Defined by Orlicz Functions
Khan Vakeel A.,Tabassum Sabiha,Esi Ayhan
Theory and Applications of Mathematics & Computer Science , 2012,
Abstract: The main aim of this paper is to introduce a new class of sequence spaces which arise from the notion of invariant means, de la Valee-Pousin means and double lacunary sequence with respect to an Orlicz function in 2-normed space. Some properties of the resulting sequence space were also examined. Further we study the concept of uniformly (λ-, σ)-statistical convergence and establish natural characterization for the underline sequence spaces.
On Paranorm Zweier -Convergent Sequence Spaces
Vakeel A. Khan,Khalid Ebadullah,Ayhan Esi,Nazneen Khan,Mohd Shafiq
Journal of Mathematics , 2013, DOI: 10.1155/2013/613501
Abstract: In this paper, we introduce the paranorm Zweier -convergent sequence spaces , , and , a sequence of positive real numbers. We study some topological properties, prove the decomposition theorem, and study some inclusion relations on these spaces. 1. Introduction Let , and be the sets of all natural, real, and complex numbers, respectively. We write the space of all real or complex sequences. Let ,? , and denote the Banach spaces of bounded, convergent, and null sequences, respectively, normed by . The following subspaces of were first introduced and discussed by Maddox [1]:? , ? , ? , ? , where is a sequence of strictly positive real numbers. After that Lascarides [2, 3] defined the following sequence spaces: ?where , for all . Each linear subspace of , for example, , is called a sequence space. A sequence space with linear topology is called a -space provided each map defined by is continuous for all . A -space is called an -space provided is a complete linear metric space. An FK-space whose topology is normable is called a BK-space. Let and be two sequence spaces and an infinite matrix of real or complex numbers , where . Then we say that defines a matrix mapping from to , and we denote it by writing . If for every sequence the sequence , the transform of is in , where By , we denote the class of matrices such that . Thus, if and only if series on the right side of (3) converges for each and every . The approach of constructing the new sequence spaces by means of the matrix domain of a particular limitation method has been recently employed by Altay et al. [4], Ba?ar and Altay [5], Malkowsky [6], Ng and Lee [7], and Wang [8]. ?eng?nül [9] defined the sequence which is frequently used as the transform of the sequence , that is, where , and denotes the matrix defined by Following Ba?ar and Altay [5], ?eng?nül [9] introduced the Zweier sequence spaces and as follows: Here we quote below some of the results due to ?eng?nül [9] which we will need in order to establish the results of this paper. Theorem 1 (see [9, Theorem??2.1]). The sets and are the linear spaces with the coordinate wise addition and scalar multiplication which are the BK-spaces with the norm Theorem 2 (see [9, Theorem??2.2]). The sequence spaces and are linearly isomorphic to the spaces and , respectively, that is, and . Theorem 3 (see [9, Theorem??2.3]). The inclusions strictly hold for . Theorem 4 (see [9, Theorem??2.6]). is solid. Theorem 5 (see [9, Theorem??3.6]). is not a solid sequence space. The concept of statistical convergence was first introduced by Fast [10] and also
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