Mathematics Volume 2013, Article ID 613501, 6 pages http://dx.doi.org/10.1155/2013/613501 Research Article On Paranorm Zweier II-Convergent Sequence Spaces Vakeel A. Khan, 1 Khalid Ebadullah, 1 Ayhan Esi, 2 Nazneen Khan, 1 and ohd Sha q 1 1 Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India 2 Department of Mathematics, University of Adiyaman, Altinsehir, 02040 Adiyaman, Turkey Correspondence should be addressed to Vakeel A. Khan; vakhan@math.com Received 14 August 2012; Revised 6 November 2012; Accepted 21 November 2012 Academic Editor: Ali Jaballah Copyright 2013 Vakeel A. Khan et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we introduce the paranorm Zweier II-convergent sequence spaces ZZ II (qqq, ZZ 0 II(qqq, and ZZ II (qqqqq II (qqq for qq q qqq ), 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 N, R, and C be the sets of all natural, real, and complex numbers, respectively. We write ωωω xx x xx xx Ror C, (1) the space of all real or complex sequences. Let ll, cc, and cc 0 denote the Banach spaces of bounded, convergent, and null sequences, respectively, normed by xxxx = sup xx. e following subspaces of ωω were rst introduced and discussed by Maddox [1]: llllll ll lll l ll l l xx pp < }, ll (ppp pp ppp p pp p ppp xx pp < }, cccccc cc ccc c cc c ccc xx lll pp = 0, for some ll l ll, cc 0 (ppp pp ppp p pp p ppp xx pp = 0}, where pp p ppp ) is a sequence of strictly positive real numbers. A er that Lascarides [2, 3] de ned the following sequence spaces: ll pp = xx xxxx there exists rrrr such that sup xx rr pp tt <, cc 0 pp = xx xxxx there exists rrrr such that lim xx rr pp tt =0, ll pp = xx xxxx there exists rrrr such that xx rr pp tt <, where tt =pp 1, for all. Each linear subspace of ωω, for example, λλλλλλλλ, is called a sequence space. A sequence space λλ with linear topology is called a KKspace provided each map pp ii C de ned by pp ii (xxx x xx ii is continuous for all iiii. A KK-space λλ is called an FK-space provided λλ is a complete linear metric space. An FK-space whose topology is normable is called a BKspace. Let λλ and μμ be two sequence spaces and AA A AAA nnnn ) an in nite matrix of real or complex numbers aa nnnn, where nnnnnnn. (2)
2 Mathematics en we say that AA de nes a matrix mapping from λλ to μμ, and wedenoteitbywritingaaaaa A AA. If for every sequence xxxxxx ) λλthe sequence AAAA A {(AAAAA nn }, the AA transform of xx is in μμ, where (AAAA) nn = aa nnnn xx, (nn nn). By (λλ λ λλλ, we denote the class of matrices AA such that AAA λλλλλ. us, AA A AAA A AAA if and only if series on the right side of (3) converges for each nnnnand every xxxxx. e 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], Başar and Altay [5], Malkowsky [6], Ng and Lee[7], and Wang [8]. Şengönül [9] de ned the sequence yy y yyy ii ) which is frequently used as the ZZ pp transform of the sequence xx x xxx ii ), that is, (3) yy ii = pppp ii + 1 pp xx iiii, (4) where xx 1 = 0,pppp, 1<ppppand ZZ pp denotes the matrix ZZ pp = (zz iiii ) de ned by ppp (ii iii), ZZ iiii = 1 ppp (iiiiiii);(iiiiiii), 0, otherwise. Following Başar and Altay [5], Şengönül [9] introduced the Zweier sequence spaces ZZ and ZZ 0 as follows: ZZZ xx x xx ωωωωω pp xxxxx, ZZ 0 = xx x xx ωωωωω pp xxxxx 0. HerewequotebelowsomeoftheresultsduetoŞengönül[9] which we will need in order to establish the results of this paper. eorem 1 (see [9, eorem 2.1]). e sets ZZ and ZZ 0 are the linear spaces with the coordinate wise addition and scalar multiplicationwhicharethebk-spaceswiththenorm (5) (6) xx ZZ = xx ZZ0 = ZZ pp xx cc. (7) eorem 2 (see [9, eorem 2.2]). e sequence spaces ZZ and ZZ 0 are linearly isomorphic to the spaces cc and cc 0, respectively, that is, ZZZZZand ZZ 0 cc 0. eorem 3 (see [9, eorem 2.3]). e inclusions ZZ 0 ZZ strictly hold for pppp. eorem 4 (see [9, eorem 2.6]). ZZ 0 is solid. eorem 5 (see [9, eorem 3.6]). ZZ isnotasolidsequence space. e concept of statistical convergence was rst introduced by Fast [10] and also independently by Buck [11] and Schoenberg [12] for real and complex sequences. Further this concept was studied by Connor [13, 14], Connor et al. [15], and many others. Statistical convergence is a generalization of the usual notion of convergence that parallels the usual theory of convergence. A sequence xx x xxx ) is said to be statistically convergent to LL if for a given as lim 1 ii i xx ii LL = 0. (8) e notion of II-convergence is a generalization of the statistical convergence. At the initial stage, it was studied by Kostyrko et al. [16]. Later on, it was studied by Šalát et al. [17, 18], Demirci [19], Tripathy and Hazarika [20, 21], and Khanetal.[22 24]. Here we give some preliminaries about the notion of IIconvergence. Let X be a nonempty set. en a family of sets IIII XX (denoting the power set of XX) issaidtobeanidealifii is additive, that is, AAAAA A AA A AA A AA A AA, and hereditary, that is, AAAAA, BBBBBBBBBBB. A nonempty family of sets 2 XX issaidtobea lteron XX if and only if φφφ, for AAA AA A we have AAAAAA and for each AAA and AAAAAimplies BBB. An ideal IIII XX is called nontrivial if IIII XX. A non-trivial ideal IIII XX is called admissible if {{xxx x xxxxxxxxx. A non-trivial ideal II is maximal if there cannot exist any non-trivial ideal JJJJJ containing II as a subset. For each ideal II, there is a lter (II) corresponding to I. that is, (II)={KKKKKKKK cc III, where KK cc =NNNNN. e nition. A sequence (xx ) ωωissaidtobeii-convergent to a number LL if { k k k LLLLLLLLLLfor every. In this case we write II I III II =LL. e space cc II of all II-convergent sequences converging to LL isgivenby cc II = xx ωωω k xx LL III for some LLLL. e nition 7. A sequence (xx ) ωωissaidtobeii-null if LLLL. InthiscasewewriteII I III II =0. e nition 8. A sequence (xx ) ωωissaidtobeii-cauchy if for every there exists a number 0 Nsuch that { k NN N NNN xx mm for all 0. e nition. A sequence (xx ) ωωissaidtobeii-bounded if there exists MMMMsuch that {k >MMMMMM. e nition 10. Let (xx ), (yy ) be two sequences. We say that (xx ) = (yy ) for almost all k relative to II (a.a.k.r.i), if { k k k xx yy } II. e following lemma will be used for establishing some results of this paper. Lemma 11. If IIII NN and MMMMM. If MMMMM, then MMMMMMMM (see [20, 21]) cf. ([17, 18, 20 24]). (9)
Mathematics 3 RecentlyKhan andebadullah [25] introduced the following classes of sequence spaces: ZZ II = k xx x xx ωω ω ωω ω ωωω ωω pp xxxxx for some LL III ZZ II 0 = k xx x xx ωω ω ωω ω ωωω ωω pp xxxx III ZZ II = k xx x xx ωω ω ωωω ZZ pp xx < III (10) We also denote by mm II ZZ =ZZ II ZZ II, mm II ZZ 0 =ZZ II ZZ II 0. (11) In this paper we introduce the following classes of sequence spaces: ZZ II qq = xx x xx ωωω k ZZ pp xxxxx qq III for some LLLL ; ZZ II 0 qq = xx x xx ωωω k ZZ pp xx qq II ; ZZ II qq = xx x xx ωω ω ωωω ZZ pp xx qq <. We also denote by mm II ZZ qq =ZZ II qq ZZ II qq, mm II ZZ 0 qq =ZZ II qq ZZ II 0 qq, (12) (13) where qq q qqq ), is a sequence of positive real numbers. roughout the paper, for the sake of convenience now we will denote by ZZ pp xx x xx /,ZZ pp yy y yy /,ZZ pp zz z zz / for all xxx xxx xx x xx. 2. Main Results eorem 12. e classes of sequences ZZ II (qqqq qq II 0(qqqq qq II ZZ(qqq and mm II ZZ 0 (qqq are linear spaces. Proof. WeshallprovetheresultforthespaceZZ II (qqq. e proof for the other spaces will follow similarly. Let (xx ), (yy ) ZZ II (qqq, and let ααα αα be scalars. en for a given : we have k xx / LL 1 qq k yy / LL 2 qq 2MM 1, for some LL 1 C II 2MM 2, for some LL 2 C III (14) where Let MM 1 = DD D DDD 1, sup αα qq, MM 2 = DD D DDD 1, sup ββ qq, DD D DDD 1, 2 HHHH AA 1 = k xx / LL 1 qq < AA 2 = k yy / LL 2 qq < be such that AA cc 1,AA cc 2 II. en where HH H HHH qq 0. (15) 2MM 1, for some LL 1 C II 2MM 2, for some LL 2 C II AA 3 = k αααα / + ββββ/ αααα 1 + ββββ 2 qq < k αα qq xx / LL 1 qq < k ββ qq yy / LL 2 qq < 2MM 1 αα qq DD 2MM 2 ββ qq DD. (16) (17) us AA cc 3 =AA cc 1 AA cc 2 II. Hence (αααα +ββββ ) ZZ II (qqq. erefore ZZ II (qqq is a linear space. e rest of the result follows similarly. eorem 13. Let (qq ) ll. en mm II ZZ(qqq and mm II ZZ 0 (qqq are paranormed spaces, paranormed by gggggg g ggg xx qq /MM where MM M MMMMMM MMM qq }. Proof. Let xx x xxx ), yy y yyy ) mm II ZZ(qqq. (1) Clearly, gggggg g g ifandonlyifxxxx. (2) gggggg g ggggggg is obvious. (3) Since qq /MM M M and MM M M, using Minkowski s inequality, we have sup xx +yy qq /MM sup xx qq /MM + sup yy qq /MM (18) (4) Now for any complex λλ, we have (λλ ) such that λλ λλ, ( k. Let xx mm II ZZ(qqq such that xx LLL qq. erefore, ggggggggggg g ggg xx LLL qq /MM sup xx qq /MM + sup LLL qq /MM, where ee e eee ee ee ee. Hence gggggg nn xx λλλλλλ λ λλλλλλ nn xx )) + gggggggg g gg nn gggggg g λλλλλλλλ as ( k. Hence mm II ZZ(qqq is a paranormed space. e rest of the result follows similarly.
4 Mathematics eorem 14. mm II ZZ(qqq is a closed subspace of ll (qqq. Proof. Let (xx (nnn ) be a Cauchy sequence in mmii ZZ(qqq such that xx (nnn xx. Weshowthatxxxxx II ZZ(qqq. Since (xx (nnn ) mmii ZZ(qqq, then there exists aa nn such that Weneedtoshowthat k xx (nnn aa nn III (19) (1) (aa nn ) converges to aa, (2) if UUUUUUUUUUUU aaa a aaa, then UU cc II. (1) Since (xx (nnn ) is a Cauchy sequence in mmii ZZ(qqq thenfora given, there exists 0 Nsuchthat For a given,wehave sup xx (nnn xx (iii < 3, nnn nn n nn 0. (20) BB nnnn = k xx (nnn xx (iii < 3, BB ii = k xx (iii aa ii < 3, BB nn = k xx (nnn aa nn < 3. (21) en BB cc nnnn,bb cc ii,bb cc nn II. Let BB cc =BB cc nnnn BB cc ii BB cc nn, where BBBBBBBBBBBB ii aa nn < xxxxx. en BB cc II. We choose 0 BB cc, then for each nnn nnnnn 0,wehave k aa ii aa nn < k xx (iii k xx (nnn aa ii < 3 xx (iii < 3 k xx (nnn aa nn < 3. (22) en (aa nn ) is a Cauchy sequence of scalars in C, so there exists a scalar aaaasuch that aa nn aa, as nnnn. (2) Let 0<δδδδbe given. en we show that if UUUUUUU N xx aaa qq < δδδ, then UU cc II. Since xx (nnn xx, then there exists qq 0 Nsuch that PPP k xx (qq 0 ) xx (23) which implies that PP cc II. e number qq 0 can be so chosen that together with (23), we have such that QQ cc II. QQQ k aa qq0 aa qq (24) Since { k k k k (qq 0 ) aa qq0 qq δδδδδδ. enwehavea subset SS of N such that SS cc II, where SSS k xx (qq 0 ) aa qq0 qq. (25) Let UU cc =PP cc QQ cc SS cc, where UUUUUUUUUUUU aaa qq < δδδ. erefore for each k cc, we have k xx aa qq < δδ en the result follows. k xx (qq 0 ) xx qq k xx (qq 0 ) aa qq0 qq k aa qq0 aa qq. (26) Since the inclusions mm II ZZ(qqq q qq (qqq and mm II ZZ 0 (qqq q qq (qqq are strict, so in view of eorem 14 we have the following result. eorem 15. e spaces mm II ZZ(qqq and mm II ZZ 0 (qqq are nowhere dense subsets of ll (qqq. eorem 16. e spaces mm II ZZ(qqq and mm II ZZ 0 (qqq are not separable. Proof. Weshallprovetheresultforthespacemm II ZZ(qqq. e proof for the other spaces will follow similarly. Let MM be an in nite subset of N of increasing natural numbers such that MMMMM. Let qq = 1, if k k 2, otherwise. (27) Let PP 0 = {(xx ) xx = 0 oooo oo for k and xx = 0, otherwise}. Clearly PP 0 is uncountable. Consider the class of open balls BB 1 = {BBBBBB BBBB B BB B BB 0 }. Let CC 1 be an open cover of mm II ZZ(qqq containing BB 1. Since BB 1 is uncountable, so CC 1 cannot be reduced to a countable subcover for mm II ZZ(qqq. us mm II ZZ(qqq is not separable. eorem 17. Let GG G GGG qq < and II an admissible ideal. en the following is equivalent. (a) (xx ) ZZ II (qqq; (b) there exists (yy ) ZZZZZZ such that xx =yy, for a.a.k.r.i; (c) there exists (yy ) ZZZZZZ and (xx ) ZZ II 0(qqq such that xx =yy +zz for all and { k k k k LLL qq ) ε ε ; (d) there exists a subset KK K KKK 1 < 2 < } of N such that KKK (III and lim nnnn xx nn LLL qq nn =0.
Mathematics 5 Proof. (a) implies (b). Let (xx ) ZZ II (qqq. en there exists LLL L such that k xx / LL qq III (28) Let (mm tt ) be an increasing sequence with mm tt Nsuch that e ne a sequence (yy ) as For mm tt <k tttt,tttt. k tt xx / LL qq tt 1 III (29) yy =xx, k 1. (30) yy = xx, if xx / LL qq < tt 1, LLL otherwise. en (yy ) ZZZZZZ and form the following inclusion: (31) k tt xx yy k tt xx / LL qq III (32) we get xx =yy, for a.a.k.r.i. (b) implies (c). For (xx ) ZZ II (qqq. en there exists (yy ) ZZZZZZ such that xx =yy, for a.a.k.r.i. Let KKKKKKKKKKK yy }, then k. e ne a sequence (zz ) as Proof. Suppose that GGGGand h>0, then the inequalities min{1, ss h } ss qq max{1, ss GG } hold for any ssssandforall k k. erefore the equivalence of (a) and (b) is obvious. eorem 19. Let (qq ) and (rr ) be two sequences of positive real numbers. en mm II ZZ 0 (qqq q qq II ZZ 0 (rrr if and only if lim k inf (qq /rr )>0, where KK cc N such that KKKKK. Proof. Let lim k inf (qq /rr )>0and (xx ) mm II ZZ 0 (rrr. en there exists ββββsuch that qq > ββββ, for all sufficiently large k. Since (xx ) mm II ZZ 0 (rrr for a given,wehave BB 0 = k xx rr III (36) Let GG 0 =KK cc BB 0 en GG 0 II. en for all sufficiently large k 0, k xx qq k xx ββββ III (37) erefore (xx ) mm II ZZ 0 (qqq. e converse part of the result follows obviously. eorem 20. Let (qq ) and (rr ) be two sequences of positive real numbers. en mm II ZZ 0 (rrr r rr II ZZ 0 (qqq if and only if lim k inf (rr /qq )>0, where KK cc Nsuch that KKKKK. Proof. e proof follows similarly as the proof of eorem 19. zz = xx yy, if k k 0, otherwise. en zz ZZ II 0(qqq and yy ZZZZZZ. (c) implies (d). Suppose (c) holds. Let be given. Let PP 1 ={ / qq and (33) eorem 21. Let (qq ) and (rr ) be two sequences of positive real numbers. en mm II 0(rrr r rr II 0(qqq if and only if lim k inf (qq /rr ) > 0, and lim k inf (rr /qq ) > 0, where KKKKsuch that KK cc II. Proof. By combining eorems 19 and 20, we get the required result. KKKKK cc 1 = 1 < 2 < 3 < (II). (34) enwehavelim nnnn xx / nn LLL qq nn =0. (d) implies (a). Let KK K KKK 1 < 2 < 3 < } (III and lim nnnn xx / nn LLL qq nn =0. en for any, and Lemma 11,wehave k xx / LL qq KK cc xx / LL qq. (35) us (xx ) ZZ II (qqq. eorem 18. Let h = inf qq and GG G GGG qq. en the following results are equivalent. (a) GGGGand h>0. (b) ZZ II 0(qqq q qq II 0. Acknowledgment e authors would like to record their gratitude to the reviewer for his careful reading and making some useful corrections which improved the presentation of this paper. References [1] I. J. Maddox, Some properties of paranormed sequence spaces, the London Mathematical Society, vol. 1, pp. 316 322, 1969. [2] C. G. Lascarides, A study of certain sequence spaces of Maddox and a generalization of a theorem of Iyer, Paci c Mathematics, vol. 38, pp. 487 500, 1971. [3] C. G. Lascarides, On the equivalence of certain sets of sequences, Indian Mathematics, vol. 25, no. 1, pp. 41 52, 1983. [4] B. Altay, F. Başar, and M. Mursaleen, On the Euler sequence spaces which include the spaces ll pp and ll. I, Information Sciences, vol. 176, no. 10, pp. 1450 1462, 2006.
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