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ZWEIER I-CONVERGENT DOUBLE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTION†
ZWEIER I-CONVERGENT DOUBLE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTION†
Journal of Applied Mathematics & Informatics. 2014. Sep, 32(5_6): 687-695
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : March 29, 2014
  • Accepted : June 09, 2014
  • Published : September 30, 2014
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About the Authors
VAKEEL A. KHAN
NAZNEEN KHAN

Abstract
In this article we introduce the zweier double sequence spaces using the Orlicz function M . We study the algebraic properties and inclusion relations on these spaces. AMS Mathematics Subject Classification : 65H05, 65F10.
Keywords
1. Introduction
Let
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be the sets of all natural, real and complex numbers respectively. We write
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the space of all double sequences real or complex.
Let , c and c 0 denote the Banach spaces of bounded, convergent and null sequences respectively normed by
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At the initial stage the notion of I-convergence was introduced by Kostyrko,Šalát and Wilczyński [1] . Later on it was studied by Šalát, Tripathy and Ziman [2] , Demirci [3] and many others. I-convergence is a generalization of Statistical Convergence.
Now we have a list of some basic definitions used in the paper .
Definition 1.1 ( [4 , 5] ). Let X be a non empty set. Then a family of sets I⊆2 X (2 X denoting the power set of X) is said to be an ideal in X if
  • (i)
  • (ii) I is additive i.e A,B∈I ⇒ A ∪ B∈I.
  • (iii) I is hereditary i.e A∈I, B⊆A⇒B∈I.
For more details see [6 , 7 , 8 , 9 , 10] . An Ideal I⊆ 2 X is called non-trivial if I≠ 2 X . A non-trivial ideal I⊆ 2 X is called admissible if {{ x } : x X } ⊆I.
A non-trivial ideal I is maximal if there cannot exist any non-trivial ideal J≠I containing I as a subset.
For each ideal I, there is a filter £ (I) corresponding to I. i.e
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Definition 1.2. A double sequence of complex numbers is defined as a function x : ℕ × ℕ → ℂ. We denote a double sequence as ( xij ), where the two subscripts run through the sequence of natural numbers independent of each other. A number a ∈ ℂ is called a double limit of a double sequence ( xij ) if for every ϵ > 0 there exists some N = N ( ϵ ) ∈ N such that
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Definition 1.3 ( [12] ). A double sequence ( xij ) ∈ ω is said to be I-convergent to a number L if for every ϵ > 0,
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In this case we write I − lim xij = L .
Definition 1.4 ( [12] ). A double sequence ( xij ) ∈ ω is said to be I-null if L = 0. In this case we write
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Definition 1.5 ( [12] ). A double sequence ( xij ) ∈ ω is said to be I-Cauchy if for every ϵ > 0 there exist numbers m = m( ϵ ), n= n( ϵ ) such that
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Definition 1.6 ( [12] ). A double sequence ( xij ) ∈ ω is said to be I-bounded if there exists M > 0 such that
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Definition 1.7 ( [12] ). A double sequence space E is said to be solid or normal if ( xij ) ∈ E implies ( αijxij ) ∈ E for all sequence of scalars ( αij ) with | αij | < 1 for all i,j ∈ ℕ.
Definition 1.8 ( [12] ). A double sequence space E is said to be monotone if it contains the canonical preimages of its stepspaces.
Definition 1.9 ( [12] ). A double sequence space E is said to be convergence free if ( yij ) ∈ E whenever ( xij ) ∈ E and xij = 0 implies yij = 0.
Definition 1.10 ( [12] ). A double sequence space E is said to be a sequence algebra if ( xij . yij ) ∈ E whenever ( xij ), ( yij ) ∈ E .
Definition 1.11 ( [12] ). A double sequence space E is said to be symmetric if ( xij ) ∈ E implies ( x π(ij) ) ∈ E , where π is a permutation on
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Any linear subspace of ω , is called a sequence space.
A sequence space λ with linear topology is called a K-space provided each of maps pi ¢ defined by pi ( x ) = xi is continuous for all i
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A K-space λ is called an FK-space provided λ is a complete linear metric space.
An FK-space whose topology is normable is called a BK-space.
Let λ and μ be two sequence spaces and A = ( ank ) be an infinite matrix of real or complex numbers ank , where n, k
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Then we say that A defines a matrix mapping from λ to μ , and we denote it by writing A : λ μ .
If for every sequence x = ( xk ) ∈ λ the sequence Ax = {( Ax ) n }, the A transform of x is in μ , where
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By ( λ : μ ), we denote the class of matrices A such that A : λ μ .
Thus, A ∈ ( λ : μ ) if and only if the series on the right side of (1) converges for each n
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and every x λ . (see [14] ).
The approach of constructing the new sequence spaces by means of the matrix domain of a particular limitation method have been recently studied by Başar and Altay [15] , Malkowsky [16] , Ng and Lee [17] and Wang [18] , Başar, Altay and Mursaleen [19] .
Şengönül [20] defined the sequence y = ( yi ) which is frequently used as the Zp transform of the sequence x = ( xi ) i.e,
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where x −1 = 0, p ≠ 1, 1 < p < ∞ and Zp denotes the matrix Zp = ( zik ) defined by
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Following Basar and Altay [15] , Şengönül [20] introduced the Zweier sequence spaces Z and Z 0 as follows
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An Orlicz function is a function M : [0,∞) → ]undefined[0,∞), which is continuous, non-decreasing and convex with M (0) = 0, M ( x ) > 0 for x > 0 and M ( x ) → ∞ as x → ∞.(see]undefined [21 , 22] ).
Lindenstrauss and Tzafriri [22] used the idea of Orlicz functions to construct the sequence space
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The space M is a Banach space with the norm
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The space M is closely related to the space p which is an Orlicz sequence space with M ( x ) = xp for 1 ≤ p < ∞ (c.f [23] , [24] , [25] ).
The following Lemmas will be used for establishing some results of this article.
Lemma 1.12 ( [24] ). A sequence space E is solid implies that E is monotone.
Lemma 1.13 ( [26 , 27 , 28] ). Let K £ ( I ) and M N. If M I, thenM K ∉ I.
Lemma 1.14 ( [26 , 27 , 28] ). If I ⊂ 2 N and M N. If M I, then M K I.
Recently Vakeel.A.Khan et. al. [29] introduced and studied the following classes of sequence spaces.
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We also denote by
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and
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2. Main results
In this article we introduce the following classes of zweier I-Convergent double sequence spaces defined by the Orlicz function.
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Also we denote by
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and
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Throughout the article, for the sake of convenience, we will denote by Zp ( xk ) = x′ , Zp ( yk ) = y′ , Zp ( zk ) = z′ for x, y, z ω .
Theorem 2.1. For any Orlicz function M, the classes of sequences 2 ZI ( M ),
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are linear spaces.
Proof . We shall prove the result for the space 2 ZI ( M ). The proof for the other spaces will follow similarly. Let ( xij ), ( yij ) ∈ 2 ZI ( M ) and let α, β be scalars. Then there exists positive numbers ρ 1 and ρ 2 such that
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That is for a given ϵ > 0, we have
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Let ρ 3 = max{2| α | ρ 1 , 2| β | ρ 2 }. Since M is non-decreasing and convex function, we have
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Now, by (1) and (2),
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Therefore ( αxij + βyij ) ∈ 2 ZI ( M ). Hence 2 ZI ( M ) is a linear space.
Theorem 2.2. The spaces
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are Banach spaces normed by
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Proof . Proof of this result is easy in view of the existing techniques and therefore is omitted.
Theorem 2.3. Let M 1 and M 2 be Orlicz functions that satisfy the 2 - condition. Then
(a) X ( M 2 ) ⊆ X ( M 1 . M 2 );
(b) X ( M 1 ) ∩ X ( M 2 ) ⊆ X ( M 1 + M 2 ) For
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Proof . (a) Let
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Then there exists ρ > 0 such that
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Let ϵ > 0 and choose δ with 0 < δ < 1 such that M 1 ( t ) < ϵ for 0 ≤ t δ .
Write
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and consider for all
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we have
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We have
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For ( yij ) > δ , we have
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Since M 1 is non-decreasing and convex, it follows that
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Since M 1 satisfies the △ 2 -condition, we have
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Hence
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From (3), (4) and (5), we have
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Thus
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The other cases can be proved similarly.
(b) Let
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Then there exists ρ > 0 such that
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The rest of the proof follows from the following equality
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Theorem 2.4. The spaces
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are solid and monotone.
Proof . We shall prove the result for
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the result can be proved similarly. Let
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Then there exists ρ > 0 such that
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Let ( αij ) be a sequence of scalars with | αij | ≤ 1 for all
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Then the result follows from (6) and the following inequality for all
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By Lemma 1.12, a sequence space E is solid implies that E is monotone.
We have the space
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is monotone.
Theorem 2.5. The spaces
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are neither solid nor mono-tone in general.
Proof . Here we give a counter example. Let I = Iδ and M ( x ) = x 2 for all x ∈ [0,∞). Consider the K-step space XK ( M ) of X ( M ) defined as follows,
let ( xij ) ∈ X ( M ) and let ( yij ) ∈ XK ( M ) be such that
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Consider the sequence ( xij ) defined by xij = 1 for all
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Then ( xij ) ∈ 2 ZI ( M ) but its K-stepspace preimage does not belong to 2 ZI ( M ).
Thus 2 ZI ( M ) is not monotone. Hence 2 ZI ( M ) is not solid.
Theorem 2.6. The spaces
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and 2 ZI ( M ) are not convergence free in general.
Proof . Here we give a counter example. Let I = If and M ( x ) = x 3 for all x ∈ [0,∞). Consider the sequence ( xij ) and ( yij ) defined by
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Then ( xij ) ∈ 2 ZI ( M ) and
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but ( yij ) ∉ 2 ZI ( M ) and
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Hence the spaces 2 ZI ( M ) and
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are not convergence free.
Theorem 2.7. The spaces
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and 2 ZI ( M ) are sequence algebras.
Proof . We prove that
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is a sequence algebra. For the space 2 ZI ( M ), the result can be proved similarly. Let
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Then
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Let ρ = ρ 1 . ρ 2 > 0. Then we can show that
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Thus
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Hence
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is a sequence algebra.
Theorem 2.8. If I is not maximal and I If , then the spaces 2 ZI ( M ) and
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are not symmetric.
Proof . Let A I be infinite and M ( x ) = x for all x = ( xij ). If
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Then
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by lemma 1.14. Let
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be such that K I and
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Let ϕ : K A and
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be bijections, then the map
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defined by
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is a permutation on
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but ( x π(i)π(j) ) ∉ 2 ZI ( M ) and
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Hence
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and 2 ZI ( M ) are not symmetric.
Acknowledgements
The authors would like to record their gratitude to the reviewer for his careful reading and making some useful corrections which improved the presentation of the paper.
BIO
Vakeel A. Khan received M.Sc., M.Phil and Ph.D at Aligarh Muslim University. He is currently an Associate Professor at Aligarh Muslim University. He has published a number of research articles and some books to his name. His research interests include Functional Analysis, sequence spaces, I-convergence, invariant means, zweier sequences and so on.
Department of Mathematics, Aligarh Muslim University, Aligarh, 200-002, India.
e-mail:vakhanmaths@gmail.com
Nazneen Khan received M.Sc. and M.Phil. from Aligarh Muslim University, and is currently a Ph.D. scholar at Aligarh Muslim University. Her research interests are Functional Analysis, sequence spaces and double sequences.
Department of Mathematics, Aligarh Muslim University, Aligarh, 200-002, India.
e-mail:nazneen4maths@gmail.com
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