-statistical convergence in intuitionistic fuzzy normed linear space

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1 Soglaaai J. Sci. Techol. 40 (3), , Ma - Ju Oigial Aticle Geealized I -tatitical covegece i ituitioitic fuzz omed liea pace Nabaita Kowa ad Padip Debath* Depatmet of Mathematic, Noth Eate Regioal Ititute of Sciece ad Techolog, Nijuli, Auachal Padeh, Idia Received: 2 Novembe 2016; Revied: 4 Jaua 2017; Accepted: 8 Febua 2017 Abtact The otio of lacua ideal covegece i ituitioitic fuzz omed liea pace (IFNLS) wa itoduced i (Debath, 2012). A a cotiuatio of thi wo, i the peet pape, we itoduce ad tud the ew cocept of covegece i IFNLS. A aalogou poof of the ope poblem dicued above with epect to -tatitical -tatitical covegece i give. Alo, we ugget a ope poblem egadig the completee of the pace with epect to thi ew covegece, whoe poof could ope up a ew aea of eeach i oliea Fuctioal Aali i the ettig of IFNLS. Kewod: ituitioitic fuzz omed liea pace, -tatitical covegece, -covegece 1. Itoductio Zadeh (1965) itoduced the fuzz et theo i ode to model cetai ituatio whee data ae impecie o vague. Late, Ataaaov (1986) itoduced a o-tivial eteio of tadad fuzz et amel ituitioitic fuzz et which deal with both the degee of membehip (belogig-e) ad omembehip (o-belogige) fuctio of a elemet withi a et. Whe the ue of claical theoie bea dow i ome ituatio, fuzz topolog i coideed a oe of the mot impotat ad ueful tool fo dealig with impeciee. I *Coepodig autho adde: debath.padip@ahoo.com liea pace, if the iduced metic atifie the talatio ivaiace popet, a om ca be defied thee. B itoducig the om i uch pace we ca get a tuctue of the pace which i compatible with that metic o topolog ad thi eultig tuctue i called a omed liea pace. The idea of a fuzz om o a liea pace wa itoduced b Kataa (1984). Felbi (1992) itoduced a alteative idea of a fuzz om whoe aociated metic i of Kaleva ad Seiala (1984) tpe. Aothe otio of fuzz om o a liea pace wa give b Cheg-Modeo (1994) whoe aociated metic i that of Kamoil-Michale (1975) tpe. Agai, followig Cheg ad Modeo, Bag ad Samata (2003) itoduced aothe cocept of fuzz omed liea pace. I thi wa, thee ha bee a tematic developmet of fuzz omed liea pace

2 N. Kowa & P. Debath / Soglaaai J. Sci. Techol. 40 (3), , (FNLS) ad oe of the impotat developmet ove FNLS i the otio of ituitioitic fuzz omed liea pace (IFNLS). With the help of fuzz om, Pa (2004) gave the otio of a ituitioitic fuzz metic pace. Uig the cocept of Pa (2004), agai Saadati ad Pa (2006) itoduced the otio of IFNLS. The cocept of tatitical covegece wa itoduced b Steihau (1951) ad Fat (1951) ad late o Fid (1985) developed the topic futhe. To tud the covegece poblem though the cocept of deit, the otio of tatitical covegece i a ve fuctioal tool. Koto et al. (2000) itoduced a geealized otio of tatitical covegece i.e. I-covegece, which i baed o the tuctue of the ideal I of famil of ubet of atual umbe N. Kaau et al. (2008) tudied tatitical covegece o IFNLS. Futhe, Koto et al. (2005) tudied ome of baic popetie of I-covegece ad defied eteal I-limit poit. Fo ome impotat ecet wo o ummabilit method ad geealized covegece we efe to Nabiev et al. (2007), Savaş ad Güdal (2014), Savaş ad Güdal (2015a), Savaş ad Güdal (2015b), Yamacı ad Güdal (2013). Kizmaz (1981) itoduced the otio of diffeece equece pace, whee the pace l, c( ), c0( ) wee tudied. Futhe, i 1995, the otio of diffeece equece pace wee geealized b Et ad Cola (1995) i l ( ), c( ), ( c ). 0 Agai Tipath ad Ei (2006) itoduced aothe tpe of geealizatio of diffeece equece pace i.e. l ( ), c( ), c ( ) 0. Afte thi, Tipath et al. (2005) geealized the above b itoducig the otio of diffeece a follow: let, be o-egative itege, the fo Z a give equece pace we have, = ( ) = { ( ) w :( 1 1 ) = Z( ) Z}, whee ad 0 fo all N, (hee ad thoughout the pape b N, we deote the et of atual umbe) which i equivalet to the biomial epeetatio : = v0 v v 1 v. I thi biomial epeetatio, if we tae = 1, we get the pace l ( ), pace ( ) pace l, c( ), ( ) 0 l, c( ), c ( ) c c( ), ( ) ; if we tae = 1, we get the c ad if we tae = = 1, we get the 0. With the help of thi ew geealized diffeece otio i.e. the otio geealized b Tipath et al. (2005), Dutta et al. (2014), itoduced a ew geealized fom of tatitical covegece i.e. -tatitical covegece fo eal umbe equece a follow: a equece ( ) i aid to be L X, if fo eve ϵ > 0 ad eve δ > 0, { N : 1 { : -tatiticall coveget to - L ϵ} δ} I. I the cuet pape we ue the above geealized otio of covegece of equece i ode to itoduce a ew geealized tatitical covegece called the -tatitical covegece o IFNLS ad eted the wo to obtai ome impotat eult. 2. Pelimiaie Fit we ecall ome eitig defiitio ad eample which ae elated to the peet wo. Defiitio 2.1 (Saadati & Pa, 2006) The 5-tuple (X, µ, ν,, ) i aid to be a IFNLS if X i a liea pace, i a cotiuou t-om, i a cotiuou t- coom ad μ, ν fuzz et o X (0, ) atifig the followig coditio fo eve, X ad, t > 0: (a) μ(, t) + μ(, t) 1, (b) μ(, t) > 0, (c) μ(, t) = 1 if ad ol if = 0, (d) μ(α, t) = μ(, t ) fo each α 0, α (e) μ(, t) μ(, ) μ( +, t + ), (f) μ(, t) : (0, ) [0, 1] i cotiuou i t,

3 542 N. Kowa & P. Debath / Soglaaai J. Sci. Techol. 40 (3), , 2018 (g) lim t μ(, t)= 1 ad lim t 0 μ(, t)= 0, (h) ν(, t) < 1, (i) ν(, t) = 0 if ad ol if = 0, (j) ν(α, t) = ν(, t ) fo each α 0, α () ν(, t) ν(, ) ν( +, t + ), (l) ν(, t) : (0, ) [0, 1] i cotiuou i t, (m) lim t ν(, t) = 0 ad lim t ν(, t) = 1. I thi cae (μ, ν) i called a ituitioitic fuzz om. Whe o cofuio aie, a IFNLS will be deoted impl b X. Defiitio 2.2 (Saadati & Pa, 2006) Let (X, µ, ν,, ) be a IFNLS. A equece = {} i X i aid to be coveget to l ϵ X with epect to the ituitioitic fuzz om (μ, ν) if, fo eve α ϵ (0, 1) ad t > 0, thee eit 0 ϵ N, uch that μ( - l, t) > 1 α ad ν( - l, t) < α fo all 0. It i deoted b (μ, ν) - lim = l. Defiitio 2.3 (Saadati & Pa, 2006) Let (X, µ, ν,, ) be a IFNLS. A equece = {} i X i aid to be a Cauch equece with epect to the ituitioitic fuzz om (µ, ν) if, fo eve α (0, 1) ad t > 0, thee eit 0 N, uch that µ( m, t) > 1 α ad ν ( m, t) < α fo all, m 0. Defiitio 2.4 (Koto et al., 2000) If X i a o-empt et the a famil of et I P(X) i called a ideal i X if ad ol if (a) I, (b) A, B I implie A B I, (c) Fo each A I ad B A we have B I, whee P(X ) i the powe et of X. Defiitio 2.5 (Koto et al., 2000) If X i a o-empt et the a o-empt famil of et F P(X ) i called a filte o X if ad ol if (a) F, (b) A, B F implie A B F, (c) Fo each A F ad B A we have B F. A ideal I i called o-tivial if I ad X I. A otivial ideal I P(X) i called a admiible ideal i X if ad ol if it cotai all igleto, i.e., if it cotai {{}: X }. Defiitio 2.6 (Debath, 2012) Let (X, µ, ν,, ) be a IFNLS. Fo t > 0, we defie a ope ball B(,, t) with cete at X, adiu 0 < < 1 a B (,, t) = { X : µ(, t) > 1 ad ν(, t) < }. Defiitio 2.7 (Steihau, 1951) If K i a ubet of N, the et of atual umbe, the the atual deit of K, deoted b δ (K), i give b δ(k) = lim 1 { : K }, wheeve the limit eit, whee A deote the cadialit of the et A. Defiitio 2.8 (Kaau, 2008) Let (X, µ, ν,, ) be a IFNLS. A equece ( ) i X i aid to be tatiticall coveget to l X with epect to the ituitioitic fuzz om (μ, ν) if, fo eve ϵ >0 ad eve t >0, δ({ N, : µ( l, t) 1 ϵ o ν ( l, t) ϵ}) = Mai Reult I thi ectio we ae goig to dicu ou mai eult. Fit we defie ome impotat defiitio ad theoem o Defiitio 3.1 -tatiticall covegece. Let I 2 N ad let, be o-egative itege ad be the otio of diffeece equece. Let ( ) be a equece i a IFNLS (X, µ, ν,, ).The fo eve α (0, 1), ϵ > 0, δ > 0 ad t > 0, the equece ( ) i aid to be -tatiticall coveget to l X with epect to the ituitioitic fuzz om (µ, ν) if, we have

4 N. Kowa & P. Debath / Soglaaai J. Sci. Techol. 40 (3), , { N : 1 { : if {t > 0 : µ( ( l, t) α} ϵ} δ} I. Hee l i called the wite S(I, Eample 3.2 ) lim = l. - limit of the equece l, t) 1 α o ν ( ) ad we Coide the pace of all eal umbe with the uual om i.e. (R, ) ad let fo all a, b [0, 1], a b = ab ad a b = mi{a + b, 1}. Fo all R ad eve t > 0, let t t, ad t, t. The (R, µ, ν,, ) i a t IFNLS. We coide I = {A N: δ(a) = 0}, whee δ(a) deote atual deit of the et A, the I i a o-tivial admiible ideal. Defie a equece ( ) ={, if = i2, i N 0, ele. a follow: The fo eve α (0, 1), ϵ > 0, δ > 0 ad fo a t > 0, the et K (α, t) = { N: 1 { : if {t > 0 : µ( o ν( l, t) 1α l, t) α} ϵ} δ} I will be a fiite et. Sice α > 0 i fied, whe become ufficietl lage, the quatit µ( quatit ν(, t) become geate tha 1 α (ad imilal, the, t) become le tha α). Hece δ(k(α, t)) = 0, ad coequetl, K(α, t) I, i.e. S(I, ) lim = 0. (a) S( I, ) lim l, (b) { N : 1 { : if {t > 0 : µ( α} ϵ} δ} ad { N : 1 { : if {t > 0 : ν( α} ϵ} δ} I, (c) { N : 1 { : if {t > 0 : μ( l, t) 1 l, t) l, t) > 1- α ad ν( l, t) < α } < ϵ } < δ} F(I ), (d) { N : 1 { : if {t > 0 : μ( α} < ϵ} < δ} F(I ) ad { N : 1 { : if {t > 0 : ν( α} < ϵ } < δ} F(I ), (e) lim ( l, t) 1 ad lim ( l, t) 0. l, t) > 1- l, t) < Poof of the followig ca be obtaied uig imila techique a i (Debath, 2012). Theoem 3.4 Let ( ) be a equece i a IFNLS (X, µ, ν,, ). If the equece ( ) i -tatiticall coveget to l X with epect to the ituitioitic fuzz om (µ, ν), the S(I, ) lim i uique. Agai, ( 0, t) ={ t t, if = i 2, i N 1, ele. The lim (, t) doe ot eit ad coequetl the equece { } i ot coveget with epect to the ituitioitic fuzz om (, ). Lemma 3.3 Let ( ) be a equece i a IFNLS (X, µ, ν,, Theoem 3.5 Let =( ) be a equece i a IFNLS (X, µ, ν,, ). If (µ, ν )lim = l, the S (I, ) lim = l. Poof. Let u coide (µ, ν) lim = l. The fo α (0, 1), t > 0, ϵ > 0 ad δ > 0, thee eit 0 N uch that fo all 0. µ( l, t) > 1 α ad v( l, t) < α, Theefoe, fo all 0 the et, ). The fo eve α (0, 1), > 0, δ > 0 ad t > 0 the followig tatemet ae equivalet:

5 544 N. Kowa & P. Debath / Soglaaai J. Sci. Techol. 40 (3), , 2018 { N : if {t > 0 : µ( l, t) 1 α o ν ( l, t) α} ϵ} ha at mot fiitel ma tem. Thu it follow that, α o ν ( { N : 1 { : if {t > 0 : µ( Theoem 3.6 l, t) α} ϵ} δ} I. Thu S(I, Let =( ) lim = l., ). If S(µ,ν ) lim =l, the S(I, l, t) 1 ) be a equece i a IFNLS (X, µ, ν, ) lim =l. Poof. Let S(µ,ν) lim = l. The α (0, 1), t > 0, ϵ > 0 ad δ > 0 thee eit 0 N uch that = 0, fo all 0. δ ({ N : µ( l, t) 1 α o ν ( l, t) α}) Thi implie, δ ({ N : µ( α}) = 0, fo all 0. α o ν( So we have, l, t) 1 α o ν( { N : 1 { : if {t > 0 : µ( Theoem 3.7 l, t) α} ϵ } δ} I. Thu S (I, ) lim = l. l, t) l, t) 1 Theoem 3.8 Let = ( ) be a equece i a IFNLS (X, µ, ν,, ). Let I be a o-tivial ideal of N. If =( tatiticall covegece i X ad = ( X uch that { N : i alo ) i a ) i a equece i, fo ome } I, the -tatiticall covegece to the ame limit. Poof. Let α (0, 1), t > 0 ad S(I, α} ϵ } δ} I. { N : 1 { : if {t > 0 : µ( Now, { N : 1 { : if {t > 0 : µ( α} ϵ } δ} { N : N : 1 { : if {t > 0 : µ( ) lim = l. The - l, t) 1 l, t) 1, fo ome } { l, t) 1α} ϵ} δ}. But both the et i the ight-had ide of the above icluio elatio belog to I. Thu { N : 1 α} ϵ } δ} I. ϵ } δ} I. limit. { : if {t > 0 : µ( Similal, we ca pove that { N : 1 Theefoe S (I, Hece i { : if {t > 0 : v( ) lim = l. l, t) 1 l, t) α} -tatiticall coveget to the ame Let = ( ) be a equece i a IFNLS (X, µ, ν,, ). The S(I, ) lim = l if ad ol if thee eit a iceaig ide equece K={} of atual umbe uch that fo K ad α (0, 1), α ad ν ( lim K = l. { N : 1 { : if {t > 0 : µ( l, t) > 1 l, t) < α} < ϵ } < δ} F (I) ad (µ, ν ) 4. - Covegece i IFNLS Hee we ae goig to peet the cocept of - covegece i IFNLS ad etablih it elatio with covegece. I -

6 N. Kowa & P. Debath / Soglaaai J. Sci. Techol. 40 (3), , Defiitio 4.1 Let = ( ) be a equece i a IFNLS (X, µ, ν,, Fom the followig theoem we ca coclude that - covegece i toge tha I -covegece. ). The = ( ) i aid to be -coveget to l X with epect to the ituitioitic fuzz om (µ, ν) if, fo eve t > 0, α (0, 1), ϵ > 0 ad N we have, Theoem 4.2 { : if {t > 0 : µ( l, t) > 1 α ad ν( l, t) < α} < ϵ }. It i deoted b (µ, ν) lim = l. Let = { ). The (µ, ν) lim i uique, if = { } be a equece i a IFNLS (X, µ, ν,, with epect to the ituitioitic fuzz om (µ, ν ). } i -coveget Poof. Let u coide (µ, ν) lim = l1 ad (µ, ν) lim = l2 (l1 l2 ). Tae a fied α (0, 1), fo which we chooe γ (0, 1) uch that (1 γ ) (1 γ ) > 1 α ad γ γ < α. Now, fo eve t > 0, ϵ > 0, N, we have, ( ( {1 : if {t > 0 : µ( l1, t) < α} < ϵ. Ad {2 : if {t > 0 : µ( l2, t) < α} < ϵ. l1, t) > 1 α ad ν l2, t) > 1 α ad ν Coide = ma {1, 2}. The fo, we will get a p N uch that p µ( p l1, t 2 ) > µ( l2, t ) > µ( l2, t ) > 1 γ. 2 2 Thu we have l1, t ) > 1 γ ad µ( 2 µ(l1 l2, t) µ( p l1, t ) µ( 2 p l2, t ) 2 > (1 γ) (1 γ) > 1 α. Sice α > 0 i abita, we have µ(l1l2, t) = 1 fo all t > 0, which implie that l1 = l2. Similal we ca how that, v(l1l2, t) < α, fo all t > 0 ad abita α > 0, ad thu l1 = l2. Hece (µ, ν) lim i uique. Theoem 4.3 Let = ( ν,, ). If (µ, ν) lim = l, the ) be a equece i a IFNLS (X, µ, lim = l. Poof. Let u coide (µ, ν) lim = l. The fo eve t > 0, α (0, 1) ad ϵ > 0 we have { : if {t > 0 : µ( ( l, t) < α} < ϵ }. α ad ν( Implie, fo eve δ > 0, { N : 1 { : if {t > 0 : µ( 1 α o ν( lim = l. l, t) < α} < ϵ } < δ}. So, we have, l, t) > 1 α ad ν A = { N : 1 { : if {t > 0 : µ( Theoem 4.4 l, t) > 1 l, t) α} ϵ } δ} {1, 2,..., 1}. A I beig admiible, o we have A I. Hece Let ={ l, t) } be a equece i a IFNLS (X, µ, ν,, ). If (µ, ν ) lim = l, the thee eit a ubequece { m } of =( ) uch that (µ, ν) lim = l. Poof. Let u coide (µ, ν) lim = l. The fo eve t > 0, α (0, 1), ϵ > 0 ad N we have { : if {t > 0 : µ( l, t) > 1 α ad ν( l, t) < α} < ϵ }. Cleal, we ca elect a m uch that µ( m l, t) > µ( l, t) > 1 α ad ν( m l, t) < ν( l, t) < α. It follow that (µ, ν) lim m m = l.

7 546 N. Kowa & P. Debath / Soglaaai J. Sci. Techol. 40 (3), , 2018 Theoem 4.5 Let I be a otivial ideal of N ad = ( equece i a IFNLS (X, µ, ν,, ). If ={ coveget i X ad ={ ) be a } i - } i a equece i X uch Defiitio 5.1 ). The = ( Let = ( ) be a equece i a IFNLS (X, µ, ν,, ) i aid to be -Cauch equece with epect to the ituitioitic fuzz om (µ, ν) if, that { N : fo ome } I, the i alo fo eve t > 0, α (0, 1), ϵ > 0, thee eit 0, m N with m -coveget to the ame limit. Poof. Let u coide that = ( Fo α (0, 1), t > 0 ad ϵ > 0 we have, { : if {t > 0 : µ( ) i -covegece i X. l, t) > 1 α ad atifig, ( { : if {t > 0 : µ( m, t) < α} < ϵ}. m,t) > 1 α ad ν ν( l, t) < α} < ϵ}. Thi implie, fo all δ > 0, { N : 1 { : if {t > 0 : µ( l, t) > 1 Defiitio 5.2 Let = ( ). The = ( ) be a equece i a IFNLS (X, µ, ν,, ) i aid to be -tatiticall Cauch α ad ν( ad ν( o ν( l, t) < α} < ϵ } < δ}i. Thi implie, { N : 1 { : if {t > 0 : µ( l, t) α} ϵ } δ} I. Theefoe, we have { N : 1 { : if {t > 0 : µ( l, t) α} ϵ } δ} { N : fo ome } { N : 1 { : if {t > 0 : µ( 1 α o ν( l, t) α} ϵ } δ}. l, t) 1 α l, t) 1 ε, l, t) A both the ight-had ide membe of the above equatio ae i I, theefoe we have, { N : 1 { : if {t > 0 : µ( α o ν ( l, t) α} ϵ } δ} I. Hece i -covegece at the ame limit. l, t) 1 equece with epect to the ituitioitic fuzz om (µ, ν) if, fo eve t > 0, ϵ > 0, δ > 0 ad α (0, 1) thee eit m N atifig, 1α ad ν( { N : 1 { : if {t > 0 : µ( Defiitio 5.3 Let = ( ). The = ( m, t) < α} < ϵ} < δ} F (I). m, t) > ) be a equece i a IFNLS (X, µ, ν,, ) i aid to be * I -tatiticall Cauch equece with epect to the ituitioitic fuzz om (µ, ν) if, thee eit a et M ={m1 < m2 <... < m <...} N uch that the et M = { N : m } F (I) ad the ubequece ( m ) of = ( ituitioitic fuzz om (µ, ν). ) i a Cauch equece with epect to the 5. called -Statiticall Cauch Sequece i IFNLS Hee we itoduce a ew fom of Cauch equece -tatiticall Cauch equece ad fid ome eult. Theoem 5.4 Let = ( ). If = ( ) i ) be a equece i a IFNLS (X, µ, ν,,

8 N. Kowa & P. Debath / Soglaaai J. Sci. Techol. 40 (3), , tatiticall coveget with epect to the ituitioitic fuzz om (µ, ν), the it i -tatiticall Cauch with epect to the ituitioitic fuzz om (µ, ν). Poof. Suppoe that = ( ) be a -tatiticall coveget equece which covege to l. Fo a give α > 0, chooe γ > 0 uch that (1 γ ) (1 γ ) > 1 α ad γ γ < α. The fo a t > 0, ϵ > 0 ad δ > 0, we have, Kµ (γ, t) = { N : 1 { : if {t > 0 : µ( l, t ) > 1 γ } < ϵ} < δ} 2 ad t 2 Kν (γ, t) = { N : 1 { : if {t > 0 : ν( l, ) < γ} < ϵ} < δ}. The Kµ (γ, t) F (I ) ad Kν (γ, t) F (I ). Let K (γ, t) = Kµ (γ, t) Kν (γ, t). The K (γ, t) F (I ). If K (γ, t) ad we chooe a fied m K (γ, t), the µ( which implie, < δ. m,t) µ( l, t 2 ) µ( > (1 γ) (1 γ) > 1 α. Thi cleal implie that if {t > 0 : µ( 1 { : if {t > 0 : µ( Alo, ν( Thi implie that, m, t) > 1 α} < ϵ. m m l, t 2 ), t) > 1 α} < ϵ} m, t) ν( l, t ) ν( l, t ) 2 2 < γ γ < α. if {t > 0 : ν( m, t) < α} < ϵ, which agai implie, < δ. 1 { : if {t > 0 : ν( m, t) < α} < ϵ} m 1α ad ν( Rema 5.5 Theefoe { N : 1 { : if {t > 0 : µ( m Hece = ( ) i, t) < α} < ϵ} < δ} F(I). -tatiticall Cauch. m, t) > Thi till emai a ope poblem if the covee of Theoem 6.4 i tue, i.e., whethe eve Cauch equece i -tatiticall -tatiticall coveget (o it become -tatiticall coveget ude cetai ew coditio). The completee of IFNLS with epect to ome otio of covegece would help the eeache to ivetigate ma aalogou eult of claical Fuctioal Aali ad Fied Poit Theo i the ettig of a IFNLS. 6. Cocluio I thi pape we have itoduced the cocept of tatiticall covegece i IFNLS ad etablihed ome ew eult. The eteded eult give u a ew idea about tatitical covegece i IFNLS. A ew tpe of Cauch equece i.e. - Cauch equece ha alo bee itoduced i thi pape. Some eitig eult ae geealized a well a eteded ad ome ew eult ae icopoated. The eult obtaied i thi pape ae moe geeal tha the coepodig eult fo claical ad fuzz omed pace. The covee of Theoem 6.4 would be a ve good topic fo futue tud, becaue if the covee ca be poved to be tue, the the IFNLS become -tatiticall complete. A - -tatiticall complete IFNLS would, i tu, ope up a ew aea of eeach o fied poit theo ad oliea fuctioal aali i it.

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