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
• Received : March 29, 2014
• Accepted : June 09, 2014
• Published : September 30, 2014
PDF
e-PUB
PPT
Export by style
Share
Article
Author
Metrics
Cited by
TagCloud
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
PPT Slide
Lager Image
be the sets of all natural, real and complex numbers respectively. We write
PPT Slide
Lager Image
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
PPT Slide
Lager Image
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
PPT Slide
Lager Image
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
PPT Slide
Lager Image
Definition 1.3 ( [12] ). A double sequence ( xij ) ∈ ω is said to be I-convergent to a number L if for every ϵ > 0,
PPT Slide
Lager Image
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
PPT Slide
Lager Image
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
PPT Slide
Lager Image
Definition 1.6 ( [12] ). A double sequence ( xij ) ∈ ω is said to be I-bounded if there exists M > 0 such that
PPT Slide
Lager Image
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
PPT Slide
Lager Image
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
PPT Slide
Lager Image
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
PPT Slide
Lager Image
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
PPT Slide
Lager Image
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
PPT Slide
Lager Image
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,
PPT Slide
Lager Image
where x −1 = 0, p ≠ 1, 1 < p < ∞ and Zp denotes the matrix Zp = ( zik ) defined by
PPT Slide
Lager Image
Following Basar and Altay [15] , Şengönül [20] introduced the Zweier sequence spaces Z and Z 0 as follows
PPT Slide
Lager Image
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
PPT Slide
Lager Image
The space M is a Banach space with the norm
PPT Slide
Lager Image
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.
PPT Slide
Lager Image
We also denote by
PPT Slide
Lager Image
and
PPT Slide
Lager Image
2. Main results
In this article we introduce the following classes of zweier I-Convergent double sequence spaces defined by the Orlicz function.
PPT Slide
Lager Image
Also we denote by
PPT Slide
Lager Image
and
PPT Slide
Lager Image
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 ),
PPT Slide
Lager Image
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
PPT Slide
Lager Image
That is for a given ϵ > 0, we have
PPT Slide
Lager Image
PPT Slide
Lager Image
Let ρ 3 = max{2| α | ρ 1 , 2| β | ρ 2 }. Since M is non-decreasing and convex function, we have
PPT Slide
Lager Image
Now, by (1) and (2),
PPT Slide
Lager Image
Therefore ( αxij + βyij ) ∈ 2 ZI ( M ). Hence 2 ZI ( M ) is a linear space.
Theorem 2.2. The spaces
PPT Slide
Lager Image
are Banach spaces normed by
PPT Slide
Lager Image
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
PPT Slide
Lager Image
Proof . (a) Let
PPT Slide
Lager Image
Then there exists ρ > 0 such that
PPT Slide
Lager Image
Let ϵ > 0 and choose δ with 0 < δ < 1 such that M 1 ( t ) < ϵ for 0 ≤ t δ .
Write
PPT Slide
Lager Image
and consider for all
PPT Slide
Lager Image
we have
PPT Slide
Lager Image
We have
PPT Slide
Lager Image
For ( yij ) > δ , we have
PPT Slide
Lager Image
Since M 1 is non-decreasing and convex, it follows that
PPT Slide
Lager Image
Since M 1 satisfies the △ 2 -condition, we have
PPT Slide
Lager Image
Hence
PPT Slide
Lager Image
From (3), (4) and (5), we have
PPT Slide
Lager Image
Thus
PPT Slide
Lager Image
The other cases can be proved similarly.
(b) Let
PPT Slide
Lager Image
Then there exists ρ > 0 such that
PPT Slide
Lager Image
The rest of the proof follows from the following equality
PPT Slide
Lager Image
Theorem 2.4. The spaces
PPT Slide
Lager Image
are solid and monotone.
Proof . We shall prove the result for
PPT Slide
Lager Image
the result can be proved similarly. Let
PPT Slide
Lager Image
Then there exists ρ > 0 such that
PPT Slide
Lager Image
Let ( αij ) be a sequence of scalars with | αij | ≤ 1 for all
PPT Slide
Lager Image
Then the result follows from (6) and the following inequality for all
PPT Slide
Lager Image
By Lemma 1.12, a sequence space E is solid implies that E is monotone.
We have the space
PPT Slide
Lager Image
is monotone.
Theorem 2.5. The spaces
PPT Slide
Lager Image
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
PPT Slide
Lager Image
Consider the sequence ( xij ) defined by xij = 1 for all
PPT Slide
Lager Image
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
PPT Slide
Lager Image
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
PPT Slide
Lager Image
Then ( xij ) ∈ 2 ZI ( M ) and
PPT Slide
Lager Image
but ( yij ) ∉ 2 ZI ( M ) and
PPT Slide
Lager Image
Hence the spaces 2 ZI ( M ) and
PPT Slide
Lager Image
are not convergence free.
Theorem 2.7. The spaces
PPT Slide
Lager Image
and 2 ZI ( M ) are sequence algebras.
Proof . We prove that
PPT Slide
Lager Image
is a sequence algebra. For the space 2 ZI ( M ), the result can be proved similarly. Let
PPT Slide
Lager Image
Then
PPT Slide
Lager Image
Let ρ = ρ 1 . ρ 2 > 0. Then we can show that
PPT Slide
Lager Image
Thus
PPT Slide
Lager Image
Hence
PPT Slide
Lager Image
is a sequence algebra.
Theorem 2.8. If I is not maximal and I If , then the spaces 2 ZI ( M ) and
PPT Slide
Lager Image
are not symmetric.
Proof . Let A I be infinite and M ( x ) = x for all x = ( xij ). If
PPT Slide
Lager Image
Then
PPT Slide
Lager Image
by lemma 1.14. Let
PPT Slide
Lager Image
be such that K I and
PPT Slide
Lager Image
Let ϕ : K A and
PPT Slide
Lager Image
be bijections, then the map
PPT Slide
Lager Image
defined by
PPT Slide
Lager Image
is a permutation on
PPT Slide
Lager Image
but ( x π(i)π(j) ) ∉ 2 ZI ( M ) and
PPT Slide
Lager Image
Hence
PPT Slide
Lager Image
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
References
Kostyrko P. , Šalát T. , Wilczyński W. (2000) I-convergence. Real Anal. Exch. 26 (2) 669 - 686
Šalát T. , Tripathy B.C. , Ziman M. (2004) On some properties of I-convergence. Tatra Mt. Math. Publ. (28) 279 - 286
Demirci K. (2001) I-limit superior and limit inferior Math. Commun. 6 165 - 172
Das P. , Kostyrko P. , Malik P. , Wilczyński W. (2008) I and I*-Convergence of Double Sequences. Math. Slovaca 58 605 - 620    DOI : 10.2478/s12175-008-0096-x
Güurdal M. , Huban M.B. 2013 On I-Convergence of Double Sequences in the Topology induced by Random 2-Norms. Matematicki Vesnik 65 (3) 113 -
Esi Ayhan , Hazarika Bipan (2012) Lacunary summable sequence spaces of fuzzy numbers defined by ideal convergence and an Orlicz function Afrika Matematika November 1 - 7
Esi Ayhan , Kemal zdemir M. (2013) 0-strongly summable sequence spaces in n-normed spaces defined by ideal convergence and an Orlicz function Mathematica Slovaca 63 (4) 829 - 838    DOI : 10.2478/s12175-013-0137-y
Esi Ayhan , Hazarika Bipan (2013) λ-ideal convergence in intuitionistic fuzzy 2-normed linear space Journal of Intelligent Fuzzy Systems: Applications in Engineering and Technology 24 (4) 725 - 732
Esi Ayhan , Sharma S.K (2013) Some I-convergent sequence spaces defined by using sequence of moduli and n-normed space Journal of the Egyptian Mathematical Society 21 (2) 103 - 107    DOI : 10.1016/j.joems.2013.01.005
Esi Ayhan , Dutta Hemen , Khalaf Alias B 2013 Some Orlicz extended I-convergent Asummable classes of sequences of fuzzy numbers Journal of Inequalities and Applications
Gürdal M. , Ahmet S. (2008) Extremal I-Limit Points of Double sequences. Applied Mathematics E-Notes 8 131 - 137
Khan V.A. , Khan N. (2013) On a new I-convergent double sequence spaces. Int J of Anal Hindawi Publishing Corporation Article ID-126163 2013 1 - 7
Khan V.A. , Sabiha T. (2011) On Some New double sequence spaces of Invariant Means defined by Orlicz function. Commun. Fac. Sci. 60 11 - 21
Khan V.A. , Ebadullah K. On Zweier I-convergent sequence spaces defined by Orlicz functions. Submitted.
Başar F. , Altay B. (2003) On the spaces of sequences of p-bounded variation and related matrix mappings. Ukr. Math.J. 55
Malkowsky E. (1997) Recent results in the theory of matrix transformation in sequence spaces. Math.Vesnik. (49) 187 - 196
Ng P.N. , Lee P.Y. (1978) Cesaro sequence spaces of non-absolute type. Comment.Math. Prace.Mat. 20 (2) 429 - 433
Wang C.S. (1978) On Nörlund sequence spaces. Tamkang J.Math. (9) 269 - 274
Altay B. , Başar F. , Mursaleen M. (2006) On the Euler sequence space which include the spaces lpand l∞. I,Inform.Sci. 176 (10) 1450 - 1462    DOI : 10.1016/j.ins.2005.05.008
Şengönül M. (2007) On The Zweier Sequence Space. Demonstratio Math. XL (1) 181 - 196
Bhardwaj V.K. , Singh N. (2000) Some sequence spaces defined by Orlicz functions. Demons. Math. 33 (3) 571 - 582
Lindenstrauss J. , Tzafriri L. (1971) On Orlicz sequence spaces. Israel J. Math. 101 379 - 390    DOI : 10.1007/BF02771656
Kamthan P.K. , Gupta M. (1980) Sequence spaces and series. Marcel Dekker Inc New York
Tripathy B.C. , Hazarika B. (2009) Paranorm I-Convergent sequence spaces. Math Slovaca. 59 (4) 485 - 494    DOI : 10.2478/s12175-009-0141-4
Tripathy B.C. , Hazarika B. (2011) Some I-Convergent sequence spaces defined by Orlicz function. Acta Math. Appl. Sin.,Engl.Ser. 27 (1) 149 - 154    DOI : 10.1007/s10255-011-0048-z
Khan V.A. , Khan N. (2013) On some I-Convergent double sequence spaces defined by a sequence of modulii Ilirias J. Math 4 (2) 1 - 8
Khan V.A. , Khan N. (2013) On some I-Convergent double sequence spaces defined by a modulus function Eng. Sci. Res. 5 35 - 40
Khan V.A. , Khan N. (2013) I-Pre-Cauchy Double Sequences and Orlicz Functions Eng.Sci. Res. 5 52 - 56
Khan V.A. , Ebadullah K. , Esi Ayhan , Khan N. , Shafiq M. (2013) On Paranorm Zweier IConvergent Sequence Spaces Journal of Mathematics Hindawi Publishing Corporation Article ID 613501 2013 1 - 6