Generalized Statistical Convergence in Intuitionistic Fuzzy 2 Normed Space

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1 Appl Mah If Sci 9, No L, (205) 59 Applied Mahemaics & Iformaio Scieces A Ieraioal Joural hp://dxdoiorg/02785/amis/09l07 Geeralized Saisical Covergece i Iuiioisic Fuzzy 2 Normed Space Ekrem Savas Deparme of Mahemaics, Isabul Ticare Uiversiy, Uskudar-Isabul, Turkey Received: 8 Nov 203, Revised: 9 Mar 204, Acceped: 0 Mar 204 Published olie: Feb 205 Absrac: I his paper, we shall iroduce he ew oio amely, I - saisical covergece by usig ideal wih respec o he iuiioisic fuzzy orm (µ,v) 2 We also sudy he relaio bewee I - saisical covergece ad I -saisical covergece subjclass[2000] 40A05; 40B50; 46A9; 46A45 Keywords: Saisical covergece; ideal covergece; -saisical covergece; -orm; 2-orm; iuiioisic fuzzy 2-ormed space Iroducio The coceps of fuzzy se ad fuzzy se operaios were firs iroduced by Zadeh [28] Subsequely several auhors have discussed various aspecs of he heory ad applicaios of fuzzy ses, such as fuzzy opological spaces, similariy relaios ad fuzzy orderig, fuzzy measures of fuzzy eves ad fuzzy mahemaical programmig, populaio dyamics, chaos corol, compuer programmig, oliear dyamical sysems, oliear operaor, ec Recely, he fuzzy opology proves o be a very useful ool o deal wih such siuaios where he use of classical heories breaks dow I [0], Park iroduced he cocep of iuiioisic fuzzy meric space ad laer o Saadai ad Park [8] iroduced he cocep of iuiioisic fuzzy ormed space Recely Mursalee ad Lohai [2] defied he cocep of iuiioisic fuzzy 2-ormed space which is geeralizaio of he oio of iuiioisic fuzzy The oio of saisical covergece was iroduced by Fas [2] ad Schoeberg [27] idepedely A lo of developmes have bee made i his areas afer he works of Salá [9], ad Fridy [3] Over he years ad uder differe ames saisical covergece has bee discussed i he heory of Fourier aalysis, ergodic heory ad umber heory Recely, Mursalee ad Mohiuddie [2] sudied he lacuary saisical covergece wih respec o he iuiioisic fuzzy ormed space I [], Mohiuddie ad Lohai iroduced he cocep of saisical covergece wih respec o he iuiioisic fuzzy ormed space Recely, Mursalee [3] sudied he cocep of saisical covergece of sequeces i radom 2-ormed space Quie recely, Savas [23] iroduced saisical covergece i radom 2-ormed space More ivesigaios i his direcio ad more applicaios ca be foud i ([6], [2], [22] ad [24]) where may impora refereces ca be foud I his paper, we shall sudy I [V,]- summable ad I saisical covergece o he iuiioisic fuzzy 2 ormed space (µ,v) 2 We maily examie he relaio bewee hese wo ew mehods i iuiioisic fuzzy ormed space(µ,v) 2 Firs, we recall some oaios ad basic defiiios which we will used hroughou he paper Defiiio [25] A biary operaio : [0,] [0,] [0,] is said o be a coiuous -orm if i saisfies he followig codiios: (a) is associaive ad commuaive, (b) is coiuous, (c) a =a for all a [0,] (d) a b c d wheever a c ad b d for each a,b,c,d [0,] Defiiio 2 [25] A biary operaio : [0,] [0,] [0,] is said o be a coiuous -coorm if i saisfies he followig codiios: Correspodig auhor ekremsavas@yahoocom, esavas@iicuedur c 205 NSP

2 60 E Savas: Geeralized Saisical Covergece i (a) is associaive ad commuaive, (b) is coiuous, (c) a 0=a for all a [0,] (d) a b c d wheever a c ad b d for each a,b,c,d [0,] Usig he aios of coiuous -orm ad -coorm, Saadai ad Park [8] have recely iroduced he cocep of iuiioisic fuzzy orm space as follows: Defiiio 3 The five-uple (X, µ,v,, ) is said o be a iuiioisic fuzzy orm space (for shor, IFNS) if X is a vecor space, is coiuous -orm, is coiuous -coorm, ad µ,v are fuzzy ses o X (0, ) saisfyig he followig codiios For every x,y X ad s, > 0 (a) µ(x,)+v(x,), (b) µ(x,)>0, (c) µ(x,)= if ad oly if x=0, (e) µ(x,) µ(y,s) µ(x+y,+ s), (f) µ(x,) :(0, ) [0,] is coiuous, (g) lim µ(x,)= ad lim 0 µ(x,)=0, (h) v(x,)<, (i) v(x,)= if ad oly if x=0, (d) µ(αx,)=µ(x, (k) µ(x,) µ(y,s) v(x+y,+ s), (l) v(x,) :(0, ) [0,] is coiuous, (m) lim v(x,)=0 ad lim 0 v(x,)= (j) v(αx,)=v(x, I his case(µ,v) is called a iuiioisic fuzzy orm I [4],Gähler iroduced he followig cocep of 2-ormed space Defiiio 4 Le X be a real vecor space of dimesio d, where 2 d < A 2-orm o X is a fucio : X X R which saisfies, (a) x, y = 0 if ad oly if x ad y are liearly depede; (b) x,y = y,x ; (c) αx, y = x, y ; (d) x,y+z x,y + x,z The pair(x,, ) is he called a 2-ormed space A rivial example of a 2-ormed space is X = R 2, equipped wih he Euclidea 2-orm x,x 2 E = he volume of he parallellogram spaed by he vecors x,x 2 which may be give expicily by he formula x,x 2 E = de(x i j ) =abs(de(< x i,x j >)) where x i =(x i,x i2 ) R 2 for each i=,2 Mursalee ad Lohai [2] used he idea of 2-ormed space o defie he iuiioisic fuzzy 2-ormed space Defiiio 5 The five-uple (X, µ,v,, ) is said o be a iuiioisic fuzzy 2-orm space (for shor, IF2NS) if X is a vecor space, is coiuous -orm, is coiuous -coorm, ad µ, v are fuzzy ses o X X (0, ) saisfyig he followig codiios for every x,y X ad s, > 0 (a) µ(x,y;)+v(x,y;), (b) µ(x,y;)>0, (c) µ(x,y;) = if ad oly if x ad y are liearly depede, (d) µ(αx,y;)=µ(x,y; (e) µ(x,y;) µ(x,z;s) µ(x,y+z;+ s), (f) µ(x,y;) :(0, ) [0,] is coiuous, (g) lim µ(x,y;)= ad lim 0 µ(x,y : )=0, (h) µ(x,y;)=µ(y,x;) (i) v(x,y;)<,, (j) v(x,y;) = 0 if ad oly if x ad y are liearly depede, (k) v(αx,y;)=v(x,y; (l) µ(x,y;) µ(x,z;s) v(x,y+z;+ s), (m) v(x,y;) :(0, ) [0,] is coiuous, () lim v(x,y;)=0 ad lim 0 v(x,y;)= (o) v(x,y;)=v(y,x;) I his case (µ,v) is called a iuiioisic fuzzy 2- orm o X, ad we deoe i by(µ,v) 2 Example Le (X, ) be a 2-ormed space, ad le a b= ab ad a b=mia+b, for all a,b [0,] For all x X ad every > 0, cosider µ(x,z;) := + x,z ad v(x,z;) := x,z + x,z The (X, µ, v,, ) is a iuiioisic fuzzy 2-ormed space Defiiio 6 Le (X, µ,v,, ) be a iuiioisic fuzzy 2 ormed space A sequece x=(x k ) is said o be coverge o L X wih respec o (µ,v) 2 if, for every ε > 0 ad > 0, here exiss k 0 N such ha µ(x k L,z;)> ε ad v(x k L,z;)<ε for all k k 0 ad for all z X I his case we wrie (µ,v) 2 limx=l or x k (µ,v) 2 L as k The family I 2 Y of subses a oempy se Y is said o be a ideal i Y if (i) /0 / I ; (ii) A,B I imply A B I ; (iii) A I, B A imply B I, while a admissible ideal I of Y furher saisfiesx I for each x Y If I is a ideal i Y he he collecio F(I) = M Y : M c I forms a filer i Y which is called he filer associaed wih I Defiiio ([9]) Le I 2 N be a orivial ideal i N The sequece x N i X is said o be I -coverge o x X, if for each ε > 0 he se A(ε)= N: x x ε belogs o I Defiiio 2Le(X, µ,v,, ) be a iuiioisic fuzzy 2 ormed space The, a sequece x =(x k ) is said o be I -saisically coverge o L X wih respec o (µ,v) 2, if, for every ε > 0 ad > 0, ad for o zero z X such ha N: k :µ(x k L,z;) ε or v(x k L,z;) ε δ I c 205 NSP

3 Appl Mah If Sci 9, No L, (205) / wwwauralspublishigcom/jouralsasp 6 ) (µ,v) I his case we wrie x k L ( (I) 2 Saisical covergece i IF2NS I his secio we sudy he cocep of I - - saisical covergece i he iuiioisic fuzzy 2 ormed space (µ,v) 2 Before proceedig furher, we recall he defiiio of desiy ad relaed coceps which form he backgroud of he prese work Defiiio 2 Le K be subse ofn, he se of aural umbers The he asympoic desiy of K deoed by δ(k), is defied as δ(k)=lim k :k K, where he verical bars deoe he cardialiy of he eclosed se A sequece x = (x k ) is said o be saisically coverge o he umber L if for each ε > 0, he se K(ε) = k : x k L > ε has asympoic desiy zero, ie lim k : x k L ε =0 I his case we wrie s limx=l (see [2], [3]) Noe ha every coverge sequece is saisically coverge o he same limi, bu coverse eed o be rue Le =( ) be a o-decreasig sequece of posiive umbers edig o such ha + +, = The collecio of such a sequece will be deoed by The geeralized de Valée-Pousi mea is defied by (x)= x k, where I =[ +,] Le (X, µ, v,, ) be a iuiioisic fuzzy 2 ormed space A sequece x = (x k ) is said o be I -[V,]-summable o L X wih respec o (µ,v) 2 if for ay δ > 0, > 0, ad for o zero z X such ha N:µ( (x) L,z;) δ or v( (x) L,z;) δ I Now we defie he I - - saisical covergece wih respec o iuiioisic fuzzy 2 ormed space Defiiio 3Le(X, µ,v,, ) be a iuiioisic fuzzy 2 ormed space A sequece x=(x k ) is said o be I - saisically coverge or I S coverge o L wih respec o(µ,v) 2,, if for every ε > 0 ad δ > 0, ad > 0, ad for o zero z X such ha N: : µ(x k L,z;) ε or v(x k L,z;) ε δ I I his case we wrie I limx = L or x k L I We shall deoe by S (µ,v) (µ,v) 2 (I), S 2 (I) ad [V,] (µ,v) 2 (I) he collecios of all I -saisically coverge, I coverge ad I -[V,] (µ,v) 2 -coverge sequeces respecively Theorem Le (X, µ, v,, ) be a iuiioisic fuzzy 2 ormed space If a sequece x = (x k ) i X is I saisically coverge sequeces wih respec o (µ,v) 2, he limi is uique ProofThis ca be proved by usig he echiques similar o hose used i Theorem of Savas[23] Theorem 2Le (X, µ, v,, ) be a iuiioisic fuzzy 2 ormed space Le =( ) The (i)x L[V,] (µ,v) 2 (I) x k L (I) ad he iclusio [V,] (µ,v) 2 (I) (I) is proper for every ideal I (ii)if x m(x), ( he space of ) all bouded sequeces of X ad x k L (I) he x k L[V,] (µ,v) 2 (I) (iii)(i) m(x)=[v,] (µ,v) 2(I) m(x) Proof(i) Le ε > 0 ad x k L[V,] (µ,v) 2 (I) We have & µ(x k L,z;)< ε or v(x k L,z;)>ε ε k I r : µ(x k L,z;) ε or v(x k L,z;) ε So for a give δ > 0, : µ(x k L,z;) ε or v(x k L,z;) ε δ µ(x k L,z;) ( ε)δ or v(x k L,z;) εδ ie N: : µ(x k L,z;) ε or v(x k L,z;) ε δ N: k I µ(x k L,z;) ε or v(x k L,z;) ε k I εδ Sice x k L[V,] (µ,v) 2(I), so he se o he righ-had side belogs o I ad so i follows ha x k L (I) To show ha (I) [V,] (µ,v) 2(I), ake a fixed A I Le (R, ) deoe he space of all real umbers wih he usual orm, ad le a b=ab ad ab=mia+ c 205 NSP

4 62 E Savas: Geeralized Saisical Covergece i b, for all a,b [0,] For all x R ad every > 0, cosider µ(x,z;) := x,z + x,z ad v(x,z;) := + x,z The (R, µ, v,, ) is a iuiioisic fuzzy 2 ormed space Now we defie a sequece x=(x k ) by x k = (k,0) for [ ] + k, / A (k,0) + k, A θ, oherwise The x / m(x) ad for every ε > 0(0<ε < ) sice : µ(x k 0,z;) ε or v(x k 0,z;) ε = [ ] 0 as ad / A, so for every δ > 0, N: k I : µ(x k 0,z;) ε or v(x k 0,z;) ε δ A,2,,m for some ( m N Sice ) I is admissible, i follows ha x k θ (I) Obviously S (µ,v) (µ(x k θ,z;) or v(x k θ,z;)) ie x k θ[v,] (µ,v) 2 (I) Noe ha if A I is fiie he x k θ This example also shows ha I -covergece is more geeral ha -covergece (ii) Suppose ha x k L (I) ad x m(x) Le µ(x k L,z;) M or v(x k L,z;) M k Le ε > 0 be give Now = & µ(x k L,z;) ε v(x k L,z;) ε + & µ(x k L,z;)> ε v(x k L,z;)<ε + M : µ(x k L,z;) ε or v(x k L,z;) ε +ε Noe ha N: : µ(x k L,z;) ε or v(x k L,z;) ε ε M A(ε) IIf (A(ε)) c he µ(x k L,z;)> 2ε or v(x k L,z;)<2ε Hece N: µ(x k L,z;) 2ε or v(x k L,z;) 2ε A(ε) ad so belogs o I This shows ha x k L[V,] (µ,v) 2(I) (iii) This readily follows from(i) ad(ii) = Theorem 3Le (X, µ, v,, ) be a iuiioisic fuzzy 2 ormed space The S (µ,v) (I) S (µ,v) (I) if lim if > 0 Proof(i) For give ε > 0, k :µ(x k L,z;) ε or v(x k L,z;) ε : µ(x k L,z;) ε or v(x k L,z;) ε If lim if : µ(x k L,z;) ε or v(x k L,z;) ε = α he from defiiio N: < 2 α is fiie For δ > 0, N: : µ(x k L,z;) ε or v(x k L,z;) ε δ N: : µ(x k L,z;) ε or v(x k L,z;) ε α2 δ N: < α 2 Sice I is admissible, he se o he righ-had side belogs o I ad his compleed he proof Refereces [] P Das, E Savaş, SKr Ghosal, O geeralizaios of cerai summabiliy mehods usig ideals, Appl Mah Le 24, (20) [2] H Fas, Sur la covergece saisique, Colloq Mah, 2, (95) [3] J A Fridy, O saisical covergece, Aalysis, 5, (985) [4] S Gähler, 2-merische Raume ad ihre opologische Srukur, Mah Nachr, 26, 5-48 (963) [5] S Gähler, Liear 2-ormiere Raume, Mah Nachr 28 (965) -43 J Mah, 27, (200) [6] I Goleţ, O probabilisic 2-ormed spaces, Novi Sad J Mah, 35, 9502 (2006) [7] M Gürdal ad S Pehliva, Saisical covergece i 2- ormed spaces, Souh Asia Bull Mah, 33, (2009) [8] M Gürdal, S Pehliva, The saisical covergece i 2- Baach spaces, Thai J Mah, 2, 073(2004) [9] P Kosyrko, T Šalá, W Wilczyki, I -covergece, Real Aal Exchage, 26, ( ) [0] J H Park, Iuiioisic fuzzy meric spaces, Chaos Solios Fracals, 22, (2004) [] S A Mohiuddie ad Q M Daish Lohai, O geeralized saisical covergece i iuiioisic fuzzy ormed space, Chaos, Solios ad Fracals, 42, (2009) [2] M Mursalee ad Q M Daish Lohai, Iuiioisic fuzzy 2- ormed space ad some relaes coceps Chaos, Solios ad Fracals, 42, (2009) [3] M Mursalee, Saisical covergece i radom 2-ormed spaces, Aca Sci Mah(Szeged), 76, 0-09 (200) [4] M Mursalee, Saisical covergece, Mah Slovaca, 50, -5 (2000) c 205 NSP

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