On Zweier paranorm I-convergent double sequence spaces

PURE ATHEATICS RESEARCH ARTICLE On Zweier paranorm I-convergent double sequence spaces Vaeel A. Khan 1 *, Nazneen Khan 1 and Yasmeen Khan 1 Received: 24 August 2015 Accepted: 31 October 2015 Published: 25 January 2016 *Corresponding author: Vaeel A. Khan Department of athematics, Aligarh uslim University, Aligarh 202002, India E-mail: vahanmaths@gmail.com Reviewing editor: Lishan Liu, Qufu Normal University, China Additional information is available at the end of the article Abstract: In this article, we introduce the Zweier Paranorm I-convergent double sequence spaces 2 I (q), 2 I (q) and 0 2 I (q) for q =(q ), a sequence of positive real numbers. We study some algebraic and topological properties on these spaces. Subjects: Engineering Technology; athematics Statistics; Science; Technology Keywords: ideal; filter; I-convergence; I-nullity; paranorm 1. Introduction Let IN, IR and IC be the sets of all natural, real and complex numbers, respectively. We write ω ={x =(x ) : x IR o r IC }, the space of all real or complex sequences. Let, c and c 0 denote the Banach spaces of bounded, convergent and null sequences, respectively, normed by x = sup x. The following subspaces of ω were first introduced and discussed by addox (1969). ABOUT THE AUTHORS Vaeel A. Khan received the Phil and PhD degrees in athematics from Aligarh uslim University, Aligarh, India. Currently he is a senior assistant professor at Aligarh uslim University, Aligarh, India. A vigorous researcher in the area of Sequence Spaces, he has published a number of research papers in reputed national and international journals, including Numerical Functional Analysis and Optimization (Taylor s and Francis), Information Sciences (Elsevier), Applied athematics Letters Applied athematics (Elsevier), A Journal of Chinese Universities (Springer- Verlag, China). Nazneen Khan received the Phil and PhD degrees in athematics from Aligarh uslim University, Aligarh, India. Currently she is an assistant professor at Taibah University, Kingdom of Saudi Arabia, adina. Her research interests are Functional Analysis, sequence spaces and double sequences. Yasmeen Khan received Sc and Phil from Aligarh uslim University, and is currently a PhD scholar at Aligarh uslim University. Her research interests are Functional Analysis, sequence spaces and double sequences PUBLIC INTEREST STATEENT The term sequence has a great role in analysis. Sequence spaces play an important role in various fields of real analysis, complex analysis, functional analysis and Topology. They are very useful tools in demonstrating abstract concepts through constructing examples and counter examples. Convergence of sequences has always remained a subject of interest to the researchers. Later on, the idea of statistical convergence came into existence which is the generalization of usual convergence. Statistical convergence has several applications in different fields of athematics lie Number Theory, Trigonometric Series, Summability Theory, Probability Theory, easure Theory, Optimization and Approximation Theory. The notion of Ideal convergence (I-convergence) is a generalization of the statistical convergence and equally considered by the researchers for their research purposes since its inception. 2016 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license. Page 1 of 9

l(p) := {x ω : x p < }, (p) := {x ω : sup x p < }, c(p) := {x ω : lim x l p = 0, for some l IC }, c 0 (p) := {x ω : lim x p = 0, }, where p =(p ) is a sequence of strictly positive real numbers. After then Lascarides (1971, 1983) defined the following sequence spaces {p} ={x ω : there exists r < 0 such that sup x r p t < }, c 0 {p} ={x ω : there exists r < 0 such that lim x r p t = 0, }, l{p} ={x ω : there exists r < 0 such that x r p t < }, where t = p 1, for all IN. A double sequence of complex numbers is defined as a function x : N N C. We denote a double sequence as (x ) 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 (x ) if for every ε > 0 there exists some N = N(ε) N such that (Khan & Sabiha, 2011) x a <ε, i, j N =1 Therefore we have, ω ={x =(x 2 ) IR or IC }, the space of all real or complex double sequences. Each linear subspace of ω, for example, λ, μ ω is called a sequence space. The notion of I-convergence is a generalization of the statistical convergence. At the initial stage it was studied by Kostyro, Šalát, and Wilczynsi (2000). Later on it was studied by Šalát, Tripathy, and Ziman (2004), Tripathy and Hazaria (2009) and Demirci (2001). 2. Preliminaries and definitions Here, we give some preliminaries about the notion of I-convergence and Zweier sequence spaces. For more details one refer to Das, Kostyro, ali, and Wilczyńsi (2008), Gurdal and Ahmet (2008), Khan and Khan (2014a, 2014b), ursaleen and ohiuddine (2010, 2012), Esi and Sapsizoğlu (2012), Fadile Karababa and Esi (2012), Khan et al. (2013b). Definition 2.1 If (X, ρ) is a metric space, a set A X is said to be nowhere dense if its closure Ā contains no sphere, or equivalently if Ā has no interior points. Definition 2.2 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) I (ii) I is finitely additive i.e. A, B I A B I. (iii) I is hereditary i.e. A I, B A B I. Page 2 of 9

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. Definition 2.3 A double sequence (x ) is said to be (i) I-convergent to a number L if for every ε >0, {(i, j) N N : x L ε} I. In this case we write I lim x = L. (ii) A double sequence (x ) is said to be I-null if L = 0. In this case, we write I lim x = 0. (iii) A double sequence (x ) is said to be I-cauchy if for every ε >0 there exist numbers m = m(ε), n= n(ε) such that {(i, j) N N : x x mn ε} I. (iv) A double sequence (x ) is said to be I-bounded if there exists > 0 such that {(i, j) N N : x < }. Definition 2.4 A double sequence space E is said to be solid or normal if (x ) E implies (α x ) E for all sequence of scalars (α ) with α < 1 for all () N N. Definition 2.5 Let X be a linear space. A function g : X R is called a paranorm, if for all x, y, z X, (i) g(x) =0 if x = θ, (ii) g( x) =g(x), (iii) g(x + y) g(x)+g(y), (iv) If (λ n ) is a sequence of scalars with λ n λ (n ) and x n, a X with x n a (n ), in the sense that g(x n a) 0 (n ), in the sense that g(λ n x n λa) 0 (n ).The concept of paranorm is closely related to linear metric spaces. It is a generalization of that of absolute value (see Lascarides, 1971; Tripathy & Hazaria, 2009). A sequence space λ with linear topology is called a K-space provided each of maps p i IC defined by p i (x) =x i is continuous for all i IN. 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 =(a n ) is an infinite matrix of real or complex numbers a n, where n, IN. 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 =(x ) λ the sequence Ax = {(Ax) n }, the A transform of x is in μ, where (Ax) n = a n x, (n IN) (1) Page 3 of 9

By (λ : μ), we denote the class of matrices A such that A : λ μ. Thus, A (λ:μ) if and only if series on the right side of (1) converges for each n IN and every x λ. The approach of constructing new sequence spaces by means of the matrix domain of a particular limitation method have been recently employed by Altay, Başar, and ursaleen (2006), Başar and Altay (2003), alowsy(1997), Ng and Lee (1978) and Wang (1978). Şengönül (2007) defined the sequence y =(y i ) which is frequently used as the Z p transform of the sequence x =(x i ) i.e. y i = px i +(1 p)x i 1 where x 1 = 0, 1 < p < and Z p denotes the matrix Z p =(z i ) defined by { z i = p, (i = ), 1 p, (i 1 = ); (i, IN), 0, otherwise. Following Basar and Altay (2003), Şengönül (2007), introduced the Zweier sequence spaces and 0 as follows ={x =(x ) ω : Z p x c} 0 ={x =(x ) ω : Z p x c 0 } Here, we quote below some of the results due to Şengönül (2007) which we will need in order to establish the results of this article. Theorem 2.1 The sets and 0 are the linear spaces with the co-ordinate wise addition and scalar multiplication which are the BK-spaces with the norm x = x 0 = Z p x c. Theorem 2.2 The sequence spaces and 0 are linearly isomorphic to the spaces c and c 0, respectively, i.e. c and 0 c 0. Theorem 2.3 The inclusions 0 strictly hold for p 1. The following Lemma and the inequality has been used for establishing some results of this article. Lemma 2.4 If I 2 N and N. If I then N I (Şengönül, 2007). Let p =(p ) be the bounded sequence of positive reals numbers. For any complex λ, whenever H = sup p <, we have λ p max(1, λ H ). Also, whenever H = sup p we have a + b p C( a p + b p ) where C = max(1;2 H 1 ). (addox, 1969) cf. (Khan Ebadullah, Ayhan Esi, Khan, & Shafiq, 2013a; Khan & Khan, 2014b; Khan & Sabiha, 2011; alowsy, 1997; Ng & Lee, 1978). Recently Khan Ebadullah, Ayhan Esi, Khan, and Shafiq (2013a) introduced various Zweier sequence spaces the following sequence spaces. I ={x =(x ) ω : { IN : I lim Z p x = L, for some L} I}, I 0 ={x =(x ) ω : { IN : I lim Zp x = 0} I}, I ={x =(x ) ω : { IN : sup Z p x < } I}. We also denote by m I = I I and m I 0 = I I 0. Page 4 of 9

In this article, we introduce the following sequence spaces. For any ε >0, we have 2 I (q) ={x =(x ) 2 ω : {(i, j) IN IN : Z p x L q ε} I, for some L IC }; 2 I 0 (q) ={x =(x ) 2 ω : {(i, j) IN IN : Zp x q ε} I}; 2 (q) ={x =(x ) 2 We also denote by 2 mi (q) = 2 (q) 2 I (q) and 2 mi 0 (q) = 2 (q) 2 I 0 (q) ω : sup Z p x q < }. where q =(q ) is a double sequence of positive real numbers. Throughout the article, for the sae of convenience now we will denote by Z p x = x for all x 2 ω. 3. ain results Theorem 3.1 The sequence spaces 2 I (q), 0 2 I (q), 2 I (q) are linear spaces. Proof We shall prove the result for the space 2 I (q). The proof for the other spaces will follow similarly. Let (x ), (y ) 2 I (q) and let α, β be scalars. Then for a given ε >0. we have {(i, j) IN IN : x L 1 q ε 2 1, for some L 1 IC } I {(i, j) IN IN : y L 2 q ε 2 2, for some L 2 IC } I where 1 = D.max{1, sup α q } 2 = D.max{1, sup β q } and D = max{1, 2 H 1 } where H = sup q 0. Let A 1 = {(i, j) IN IN : x L 1 q < ε 2 1, for some L 1 IC } A 2 = {(i, j) IN IN : y L 2 q < ε 2 2, for some L 2 IC } be such that A c, 1 Ac 2 I. Then A 3 = {(i, j) IN IN : (αx + ) (αl + βl βy 1 2 ) q ) <ε} {(i, j) IN IN : α q x L 1 q < ε α q.d} 2 1 {(i, j) IN IN : β q y L 2 q < ε β q.d} 2 2 Page 5 of 9

Thus A c 3 Ac 1 Ac 2 I. Hence (αx + βy ) 2 I (q). Therefore 2 I (q) is a linear space. Proof of 2 Z I 0 (q) follows since it is a special case of 2 Z I (q). Remar The sequence spaces 2 m I Z (q), 2 mi Z 0 (q), are linear spaces since each is an intersection of two of the linear spaces in Theorem 3.1. Theorem 3.2 Let (q ) 2. Then 2 m I (q) and 2 mi (q) are paranormed spaces, paranormed by 0 g(x )=sup x q where = max{1, sup q }. Proof Let x =(x ), y =(y ) 2 mi (q). (1) Clearly, g(x )=0 if and only if x = 0. (2) g(x )=g( x ) is obvious. q (3) Since 1 and > 1, using inowsi s inequality, we have g(x + y )=g(x + y )=sup x + y q sup( x q = g(x )+g(y )=g(x )+g(y ). Therefore, g(x + y ) g(x )+g(y ), for all x, y 2 m I (q). z + y q ) sup x q + sup y q (4) Let (λ ) be a double sequence of scalars with (λ ) λ, (i, j ) and x =(x ), x 0 =(x i 0 j ) 0 2 m I (q) z with g(x ) g(x 0 ), (i, j ). Note that g(λx ) max{1, λ }g(x ). Then since the inequality g(x ) g(x x 0 )+g(x 0 ) holds by subadditivity of g, the sequence {g(x )} is bounded. Therefore, g(λ x q q ) g(λx ) = g(λ 0 x ) g(λx )+g(λx ) g(λx ) λ λ 0 g(x ) + λ g(x ) g(x ) 0 0 as (i, j ). That is to say that the scalar multiplication is continuous. Hence 2 m I z (q) is a paranormed space. Theorem 3.3 2 mi (q) is a closed subspace of 2 (q). Proof Let (x (mn) ) be a Cauchy sequence in 2 m I (q) such that x (mn) x. We show that x 2 m I (q). Since (x (mn) ) 2 m I (q), then there exists (a mn ) such that {(i, j) IN IN : x (mn) a mn ε} I We need to show that (1) (a mn ) converges to a. (2) If U = {(i, j) IN IN : x a <ε} then Uc I. Since (x (mn) ) is a Cauchy sequence in 2 m I (q) then for a given ε >0, there exists (i, j 0 0 ) IN IN such that sup x (mn) x (pq) < ε 3, for all (m,n),(p,q) (i, j ) 0 0 For a given ε >0, we have B mn,pq = {(i, j) IN IN : x (mn) x (pq) < ε 3 } B pq = {(i, j) IN IN : x (pq) a pq < ε 3 } B mn = {(i, j) IN IN : x (mn) a mn < ε 3 } Page 6 of 9

Then B c mn,pq, Bc pq, Bc mn I. Let B c = B c mn,pq Bc pq Bc mn, where B = {(i, j) IN IN : a pq a mn <ε}. Then B c I. We choose (i 0, j 0 ) B c, then for each (m, n), (p, q) (i 0, j 0 ), we have {(i, j) IN IN : a pq a mn <ε} {(i, j) IN IN : x (pq) a pq < ε 3 } {(i, j) IN IN : x (mn) x (pq) < ε 3 } {(i, j) IN IN : x (mn) a mn < ε 3 } Then (a mn ) is a Cauchy sequence of scalars in IC, so there exists a scalar a IC such that a mn a, as (m, n). For the next part let 0 <δ<1 be given. Then we show that if U = {(i, j) IN IN : x a q <δ}, then U c I. Since x (mn) x, then there exists (p 0, q 0 ) IN IN such that P = {(i, j) IN IN : x (p 0,q 0 ) x 3D ) } which implies that P c I. The number (p 0, q 0 ) can be so chosen that together with (1), we have Q = {(i, j) IN IN : a p0 q 0 a q such that Q c I Since {(i, j) IN IN : x (p 0 q 0 ) a p0 q q δ} I. Then we have a subset S of IN IN such that IN IN, 0 where S = {(i, j) IN IN : x (p 0 q 0 ) a p0 q 0 q 3D ) } Let U c = P c Q c S c, where U = {(i, j) IN IN : x a q <δ}. Therefore for each (i, j) U c, we have 3D ) }. {(i, j) IN IN : x a q <δ} {(i, j) IN IN : x (p 0 q 0 ) x q 3D ) } {(i, j) IN IN : x (p 0 q 0 ) a p0 q q 0 3D ) } {(i, j) IN IN : a p0 q a q 0 3D ) }. Then the result follows. Theorem 3.4 The spaces 2 m I (q) and 2 mi 0 (q) are nowhere dense subsets of 2 (q). Proof Since the inclusions 2 m I (q) 2 (q) and 2 mi 0 (q) 2 (q) are strict so in view of Theorem 3.3 we have the following result. Theorem 3.5 The spaces 2 m I (q) and 2 mi 0 (q) are not separable. Page 7 of 9

Proof We shall prove the result for the space 2 m I (q). The proof for the other spaces will follow similarly. Let be an infinite subset of IN IN of such that I. Let q = { 1, if (i, j), 2, otherwise. Let P 0 ={x =(x ) : x = 0 or 1, for (i, j) and x = 0, otherwise}. Clearly P 0 is uncountable. Consider the class of open balls B 1 ={B(x, 1 2 ) : x P 0 }. Let C 1 be an open cover of 2 m I (q) containing B 1. Since B 1 is uncountable, so C 1 cannot be reduced to a countable subcover for 2 m I (q). Thus 2 m I (q) is not separable. Theorem 3.6 Let h = inf (a) H < and h > 0. (b) 2 I 0 (q) = 2 I 0. q and H = sup q. Then the following results are equivalent. Proof Suppose that H < and h > 0, then the inequalities min{1, s h } s q max{1, s H } hold for any s > 0 and for all (i, j) IN IN. Therefore, the equivalence of (a) and (b) is obvious. Theorem 3.7 Let (q ) and (r ) be two sequences of positive real numbers. Then 2 m I Z (r) = 0 2 m I Z (q) if 0 and only if lim inf q > 0, and lim inf r > 0, where K IN IN such that K c I. () K r () K q Proof Let lim inf q > 0 and (x ) () K r 2 mi (r). Then there exists β >0 such that q 0 >βr, for all sufficiently large (i, j) K. Since (x ) 2 m I (r) for a given ε >0, we have 0 B 0 = {(i, j) IN IN : x r ε} I Let G 0 = K c B 0. Then G 0 I. Then for all sufficiently large (i, j) G 0, {(i, j) IN IN : x q ε} {(i, j) IN IN : x βr ε} I. Therefore (x ) 2 mi 0 (q). The converse part of the result follows obviously. The other inclusion follows by symmetry of the two inequalities. 4. Conclusion The notion of Ideal convergence (I-convergence) is a generalization of the statical convergence and equally considered by the researchers for their research purposes since its inception. Along with this the very new concept of double sequences has also found its place in the field of analysis. It is also being further discovered by mathematicians all over the world. In this article, we introduce paranorm ideal convergent double sequence spaces using Zweier transform. We study some topological and algebraic properties. Further we prove some inclusion relations related to these new spaces. Page 8 of 9

Acnowledgements The authors would lie to record their gratitude to the reviewer for his careful reading and maing some useful corrections which improved the presentation of the paper. Funding The authors received no direct funding for this research. Author details Vaeel A. Khan 1 E-mail: vahanmaths@gmail.com Nazneen Khan 1 E-mail: nazneen4maths@gmail.com Yasmeen Khan 1 E-mail: yasmeen9828@gmail.com 1 Department of athematics, Aligarh uslim University, Aligarh, 202002, India. Citation information Cite this article as: On Zweier paranorm I-convergent double sequence spaces, Vaeel A. Khan, Nazneen Khan & Yasmeen Khan, Cogent athematics (2016), 3: 1122257. References Altay, B., Başar, F., & ursaleen,. (2006). On the Euler sequence space which include the spaces lp and l.i. Information Sciences, 176, 1450 1462. Başar, F., & Altay, B. (2003). On the spaces of sequences of p-bounded variation and related matrix mappings. Urainian athematical Journal, 55, 136 147 Das, P., Kostyro, P., ali, P., & Wilczyńsi, W. (2008). I and I*-convergence of double sequences. athematica Slovaca, 58, 605 620. Demirci, K. (2001). I-limit superior and limit inferior. athematical Communications, 6, 165 172. Esi, A., & Sapsizoğlu, A. (2012). On some lacunary σ -strong Zweier convergent sequence spaces. ROAI Journal, 8, 61 70. Fadile Karababa, Y., & Esi, A. (2012). On some strong Zweier convergent sequence spaces. Acta University Apulennsis, 29, 9 15. Gurdal,., & Ahmet, S. (2008). Extremal I-limit points of double sequences. Applied athematics E-Notes, 8, 131 137. Khan, V. A., Ebadullah, K., Esi, A., Khan, N., & Shafiq,. (2013a). On Paranorm Zweier I-convergent sequence spaces. Journal of athematics, 2013, 613501, 1 6. Khan, V. A., Ebadullah, K., Esi, A., & Shafiq,. (2013b). On some Zweier I-convergent sequence spaces defined by a modulus function. Afria atematia. doi:10.1007/s13370-013-0186-y Khan, V. A., & Khan, N. (2014a). On some Zweier I-convergent double sequence spaces defined by a modulus function. Analysis. doi:10.1515/anly-2014-1242 Khan, V. A., & Khan, N. (2014b). On some Zweier I-convergent double sequence spaces defined by Orlicz function. Journal of Applied athematics & Informatics, 32, 687 695. Khan, V. A., & Sabiha, T. (2011). On some new double sequence spaces of invariant means defined by Orlicz function. Communications Faculty of Sciences, 60, 11 21. Kostyro, P., Šalát, T., & Wilczynsi, W. (2000). I-convergence. Real Analysis Exchange, 26, 669 686. Lascarides, C. G. (1971). A study of certain sequence spaces of addox and generalization of a theorem of Iyer. Pacific Journal of athematics, 38, 487 500. Lascarides, C. G. (1983). On the equivalence of certain sets of sequences. Indian Journal of athematics, 25, 41 52. addox, I. J. (1969). Some properties of paranormed sequence spaces. Journal London athematical Society, 1, 316 322. alowsy, E. (1997). Recent results in the theory of matrix transformation in sequence spaces. atematici Vesni, 49, 187 196. ursaleen,., & ohiuddine, S. A. (2010). On ideal convergence of double sequences in probabilistic normed spaces. athematical Reports, 12, 359 371. ursaleen,., & ohiuddine, S. A. (2012). On ideal convergence in probabilistic normed spaces. athematica Slovaca, 62, 49 62. Ng, P. N., & Lee, P. Y. (1978). Cesaro sequence spaces of nonabsolute type. Commentationes athematicae (Prace atematyczne), 20, 429 433. Šalát, T., Tripathy, B. C., & Ziman,. (2004). On some properties of I-convergence. Tatra ountains athematical Publications, 28, 279 286. Şengönül,. (2007). On the Zweier sequence space. Demonstratio athematica, XL, 181 196. Tripathy, B. C., & Hazaria, B. (2009). Paranorm I-convergent sequence spaces. athematica Slovaca, 59, 485 494. Wang, C. S. (1978). On Nörlund sequence spaces. Tamang Journal of athematics, 9, 269 274. 2016 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license. You are free to: Share copy and redistribute the material in any medium or format Adapt remix, transform, and build upon the material for any purpose, even commercially. The licensor cannot revoe these freedoms as long as you follow the license terms. Under the following terms: Attribution You must give appropriate credit, provide a lin to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits. Page 9 of 9