THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 16, Number 2/2015, pp

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1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Seies A, OF THE ROMANIAN ACADEMY Volume 6, Numbe 2/205, pp 2 29 ON I -STATISTICAL CONVERGENCE OF ORDER IN INTUITIONISTIC FUZZY NORMED SPACES Eem SAVAŞ Istabul Commece Uivesity, Depatmet of Matematics, Sutluce-Istabul, Tuey eemsavas@yaoocom I tis pape, followig a vey ecet ad ew appoac, we itoduce te otios of I -statistical covegece ad I -statistical covegece of ode, wee 0<, wit espect to te ituitioistic fuzzy om ( μ,v ad we maily ivestigate tei elatiosip, ad also mae some obsevatios about tese classes Key wods: I -statistical covegece, I -statistical covegece, ode, ituitioistic fuzzy om INTRODUCTION Amog vaious developmets of te teoy of fuzzy sets [2] a pogessive developmet as bee made to fid te fuzzy aalogues of te classical set teoy I fact te fuzzy teoy as become a aea of active eseac fo te last 0 yeas It as a wide age applicatios i te field of sciece ad egieeig, eg, populatio dyamics, caos cotol, compute pogammig, oliea dyamical systems, fuzzy pysics, fuzzy topology, etc Recetly fuzzy topology poves to be a vey useful tool to deal wit suc situatio wee teuse of classical teoies beas dow Te teoy of ituitioistic fuzzy sets was itoduced by Ataassov []; it as bee extesively used i decisio-maig poblems [2] Te cocept of a ituitioistic fuzzy metic space was itoduced by Pa [6] Futemoe, Saadati ad Pa [7] gave te otio of a ituitioistic fuzzy omed space Statistical covegece is a geealizatio of usual covegece ad is itoduced by Fast [7] I last few yeas, tis idea as also bee adapted to te sequeces of fuzzy umbes Te cedit goes to Nuay ad Savas [5], wo fist defied statistical covegece ad statistical Caucy sequeces of fuzzy umbes A sequece x= { x } of eal umbes is said to be statistically coveget to L if fo abitay > 0, te set K( = { : x L } as atual desity zeo By a lacuay = ( ; = 0,,2, wee 0 = 0, we sall mea a iceasig sequece of oegative iteges wit as Te itevals detemied by will be deoted by I = (, ] ad = Te atio will be deoted by q I aote diectio, i [9] a ew type of covegece called lacuay statistical covegece was itoduced a follows: A sequece x= { x } of eal umbes is said to be lacuay statistically coveget to L (o, S -coveget to L if fo ay > 0, lim { I : x L } = 0, wee A deotes te cadiality of A

2 22 Eem Savaş 2 Moe ivestigatios i tis diectio ad moe applicatios of lacuay statistical covegece ca be foud i [2, 3, ] Recetly i [6] we used ideals to itoduce te cocepts of I -statistical covegece, I -lacuay statistical covegece ad ivestigated tei popeties O te ote ad i ([3], [] a diffeet diectio was give to te study of tese impotat summability metods wee te otios of statistical covegece of ode ad λ -statistical covegece of ode wee itoduced ad studied I [], P Kostyo et al itoduced te cocept of I -covegece of sequeces i a metic space ad studied some popeties of suc covegece Note tat I -covegece is a iteestig geealizatio of statistical covegece Moe ivestigatios i tis diectio ad moe applicatios of ideals ca be foud i [5, 6, 8, 9, 20, 2, 22] I tis pape we sall itoduce te cocept of I -lacuay statistical covegece of ode wit espect to te ituitioistic fuzzy omed space ( μ,v, ad ivestigate some of its cosequeces Tougout te pape, will deote te set of all atual umbes We ow ecall some otatio ad basic defiitios used i te pape Defiitio [23] A biay opeatio :[0,] [0,] [0,] is said to be a cotiuous t -coom if it satisfies te followig coditios: (i is associate ad commutative, (ii is cotiuous, (iii a 0 = a fo all a [0,], (iv a b c d weeve a c ad b d fo eac abcd,,, [0,] Fo example, we ca give a b ab, a b mi a, b, a} b mi a b, a} b= max a, b fo all ab, [0,] Saadati ad Pa [7] as ecetly itoduced te cocept of ituitioistic fuzzy omed space as follows Defiitio [7] 2 Te five-tuple ( X,, v,,} = = { } = { + } ad { } μ is said to be a ituitioistic fuzzy omed space (fo sot, IFNS if X is a vecto space, is a cotiuous t -om, } is a cotiuous t -coom, ad μ, v ae fuzzy sets o X ( 0, satisfyig te followig coditios fo evey, [( a ] μ ( xt, + v( xt,, [( ] b ( xt c ( xt [( ] μ, > 0, μ, = if ad oly if x = 0, t, = μ x, μ x, t μ y, s μ x+ y, t+ s, [( d ] μ ( xt [( ] fo eac 0, e ( ( ( f ( x, :( 0, [ 0,] g lim μ ( xt, = ad μ ( xt = [( ] [( ] [( ] [( ] μ is cotiuous, t ( i ( v x, t <, lim, 0, t 0 v x, t = 0 if ad oly if x = 0, x y X, ad s, t > 0:

3 3 O I -statistical covegece of ode 23 t v x, t = μ x, v x, t v y, s v x+ y, t+ s, [( j ] ( [( ] fo eac 0, ( ( ( l (,:( 0, [ 0,] m lim v( x, t = 0 ad v( x t [( ] [( ] v x is cotiuous, t lim, = t 0 I tis case ( μ,v is called a ituitioistic fuzzy om Te obseve tat ( X,, v,,} μ is a ituitioistic fuzzy omed space We also ecall tat te cocept of covegece i a ituitioistic fuzzy omed space is studied i [0, 7] Defiitio [7] 3 Let ( X, μ, v,, be a IFNS Te, a sequece x = { x } to L X wit espect to te ituitioistic fuzzy om ( μ,v if fo evey 0 suc tat μ ( x L, t > ad v( x L, t < fo all 0 It is deoted by ( 0 v x L as is said to be coveget > ad t > 0, tee exists μ, v lim x= L o 2 I -STATISTICAL CONVERGENCE ON IFNS I tis sectio we deal wit some mai defiitios ad teoems Befoe poceedig fute, we sould ecall some otatio o te I -statistical covegece ad ideal covegece Te family I 2 Y of subsets a oempty set Y is said to be a ideal i Y if ( i I ; ( ii AB, I imply A B I ; ( iii A I, B A imply B I, wile a admissible x I fo eac x Y If I is a ideal i Y te te collectio ideal I of Y fute satisfies { } c ( = { : } F I M Y M I foms a filte i Y wic is called te filte associated wit I Let I 2 be a x otivial ideal i Te a sequece { } te set A( = { : x x } belogs to I (see [] Defiitio Let ( X, μ, v,, be a IFNS Te, a sequece { } i X is said to be I -coveget to x X, if fo eac > 0 x = x is said to be I -statistically coveget of ode to L X, wee 0 <, wit espect to te ituitioistic fuzzy omed space μ,v, if fo evey > 0, ad evey δ > 0 ad t > 0, ( I tis case we wite : { : μ( x, o v( x, } δ I x ( μ, v ( ( ( μ, v L S I Te class of all I -statistically coveget of ode sequeces wit espect to te ituitioistic fuzzy omed space ( μ,v will be deoted by simply ( ( v S μ I Rema Fo I = I, S ( I -covegece coicides wit statistical covegece of ode, wit fi espect to te ituitioistic fuzzy omed space ( μ,v Fo a abitay ideal I ad fo = it coicides wit I -statistical covegece, wit espect to te ituitioistic fuzzy omed space ( μ, v, [22] We I = I ad = it becomes oly statistical covegece [0] wit espect to te ituitioistic fuzzy fi omed space ( μ,v,

4 2 Eem Savaş x x Defiitio 5 Let ( X, μ, v,, be a IFNS A sequece { } = is said to be I -statistically coveget of ode o S ( I -coveget to L wit espect to te ituitioistic fuzzy omed space ( v μ,, if fo evey > 0, δ > 0 ad t > 0, I tis case we wite : { I : μ( x, o v( x, } δ I ( μ, v S ( I limx= L o x v L( S ( I Te class of all I -statistically coveget of ode sequeces wit espect to te ituitioistic fuzzy omed space ( μ,v will be deoted by simply S v ( I ( I ad ( ( μ,v μ, v We sall deote by S ( N I te collectios of all I -lacuay statistical coveget of ode ad I -coveget of ode sequeces wit espect to te ituitioistic fuzzy omed N space ( μ,v espectively We ae eady to pove te followig esult wic gives a topological caacteizatio of v v S ( I ad S ( I spaces As te lie of te poofs fo bot ae simila we give detailed poof fo te class v S ( I THEOREM Let ( X, μ, v,, be a IFNS Te ( ( μ, v v S l stads fo te space of all bouded sequeces of ituitioistic fuzzy om (,v wee l v Poof Suppose tat { } ( ( μ, v v ( x l μ,v I is a closed subset of μ ( l μ,v, x S I l is a coveget sequece ad tat it coveges to We eed to pove tat ( ( μ, v v x S I l positive stictly deceasig sequece { } Assume tat to 0 Coose a positive itege suc tat x x sup { v ( x x, t } ad x L S μ (,v ( ( I, Tae a wee = fo a give > 0 Clealy { } 2 coveges = < Let 0< δ < Te Aμ, v(, t = : I : μ( x L, t o v( x L, t < δ belogs to F ( I belogs to F ( I Aμ, v, t Bμ, v, t F Te + I : μ ( x L+, t o Bμ, v(, t = : δ < + v( x L+, t Sice ( ( ( I ad F ( I, we ca coose A (, t B (, t : (, o (, Im μ x L t v x L t 2 δ < + ( + ( + μ x L+, t o v x L+, t μ, v μ, v

5 5 O I -statistical covegece of ode 25 Hece tee must exist a I fo wic we ave simultaeously, μ ( x L, t > o + v( x L, t < ad (, + μ x L+ t > o (, v x L+ t < Fo a give > 0 coose suc 2 tat > 2 2 ad < Te it follows tat 2 2 t + t v L x, v L+ x, < ad + t + t v( x x, t sup v x x, sup v x x, < Hece we ave t + t + t v( L L+, t v L x, v x L+, v x x, < ad similaly μ ( L, L+ t > Tis implies tat { L } is a Caucy sequece i ad let L L as suc tat ad similaly it follows tat ( We sall pove tat ( v <, sup v( x x, t fo ay give 0 x L S I Fo ay > 0 ad t > 0, coose <, μ ( L L, t > o v( L L, t < Now sice { I : v( x L, t } t t t I : v x x, v x L, v L L, t I : v x L, 3 2 { : (, } t I μ x L t < I : μ x L,, 3 2 : { I : μ( x, o v( x, } δ t t : I : μ x L, o v x L, δ, ( δ > Hece we ave ( v x L S I Tis completes te poof of te teoem I te followig, we pove a esult egadig I -lacuay statistical covegece of ode wit μ,v espect to te ituitioistic fuzzy om ( Defiitio 6 Let be a lacuay sequece Te x = { x } L X wit espect to te ituitioistic fuzzy om (,v δ > 0, : ( x Lt, o v( x Lt, μ I I I μ is said to be N ( (,v I -coveget to μ if, fo ay > 0, t > 0 ad

6 26 Eem Savaş 6 ( Tis covegece is deoted by ( ( μ,v simply by ( ( μ v N x I, x L N I, ad te class of suc sequeces will be deoted THEOREM 2 Let = { } be a lacuay sequece ad ( X, μ, v,, be a IFNS Te ( ( ( ( μ,v L N I te ( ( ( μ,v x L S I, ad ( b ( ( μ N,v I is a pope subset of ( ( μ, v S I (,v Poof ( i ( a If > 0 ad x ( L N μ ( I, we ca wite ( μ( x, o v( x, ( μ( x, o v( x, ad so, μ(, v( x L t I I x L t o, { I : ( x L, t o v( x L, t } μ ( ( x L, t o v( x L, t { I : ( x L, t o v( x L, t } μ μ I Te, fo ay δ > 0 ad t > 0 : { I : μ ( x, o v( x, } δ : ( x Lt, ( o v( x Lt, μ δ δ I I I Tis poves te esult ( b I ode to establis tat te iclusio ( ( μ, v N S ( ( μ, v a If I I is pope, let be give, ad defie x to be,2,, fo te fist iteges i I ad x = 0 otewise, fo all =,2, Te, fo ay > 0 ad t > 0 { I : μ( x 0, t o v( x 0, t }, ad fo ay δ > 0 we get : { I : μ ( x 0, t o v( x 0, t } δ : δ ( μ, v Sice te set o te igt-ad side is a fiite set ad so belogs to I, it follows tat x 0 ( S ( I O te ote ad, ( + ( μ ( x 0, t o v( x 0, t = I 2 Te ( + : μ ( x 0, t o v( x 0, t : = = I 2 I = mm, +, m + 2,, { }

7 7 O I -statistical covegece of ode 27 ( fo some wic belogs to F ( I, sice I is admissible So ( v x N I te Rema 2 I Teoem 2 of [22] it was fute poved tat: ( ii If x (,v L( N μ ( I ; ( iii μ, v μ, v μ, v v I = I ( ( ( ( ( S l N l 0 ( x l μ,v ad x (,v ( ( L S μ I, Howeve wete tese esults emai tue fo 0 < < is ot clea ad we leave tem as ope poblems We will ow ivestigate te elatiosip betwee I -statistical ad I -lacuay statistical covegece of ode THEOREM 3 Fo ay lacuay sequece, I -statistical covegece of ode wit espect to te ituitioistic fuzzy om ( μ,v implies I -lacuay statistical covegece of ode wit espect to te ituitioistic fuzzy om ( μ,v if limif q > Poof Suppose fist tat limif q > Te tee exists > 0 suc tat q + fo sufficietly ( lage, wic implies tat Sice ( ( μ,v x L S I +, fo evey > 0, t > 0, ad fo sufficietly lage, we ave { : μ ( x L, t o v( x L, t } { I : μ( x L, t o v( x L, t } { I : μ( x L, t o v( x L, t } + Te, fo ay δ > 0, we get : { I : μ( x L, t o v( x L, t } δ δ : { : μ( x L, t o v( x L, t } I ( + Tis completes te poof Rema 5 Te covese of tis esult is tue fo = Howeve fo < it is ot clea ad we leave it as a ope poblem Fo te ext teoem we assume tat te lacuay sequece satisfies te coditio tat fo ay set C F( I, { : < <, C } F ( I THEOREM Fo a lacuay sequece satisfyig te above coditio, I -lacuay statistical μ,v implies I -statistical covegece of ode wit espect to te ituitioistic fuzzy om ( covegece of ode wit espect to te ituitioistic fuzzy om (,v i+ sup = B < i= 0 ( ( μ, v μ, 0 < <, if Poof Suppose tat x L( S ( I ad fo, δ, δ > 0 defie te sets C = { : { I : μ ( x L, t o v( x L, t } < δ} ad T = { : { : μ ( x L, t o v( x L, t } < δ}

8 28 Eem Savaş 8 It is obvious fom ou assumptio tat C F( I, te filte associated wit te ideal I Fute obseve tat A { : (, o (, } j = I j μ x L t v x L t < δ fo all j C Let be suc j tat < < fo some C Now { : μ( x L, t o v( x L, t } { : μ( x L, t o v( x L, t } = { I : μ x L, t o v x L, t } + + { I : μ x L, t o v x L, t } ( ( ( ( ( { I : μ( x L, t o v x L, 2 ( t } + { I2 : μ ( x L, t 2 ( o v( x L, t } { I : μ( x L, t o v( x L, t } ( 2 ( A A2 A ( i+ i sup Aj sup Bδ j C i= 0 = + + = < δ Coosig δ = ad i view of te fact tat { :, C} T B < < wee C F( I it follows fom ou assumptio o tat te set T also belogs to F( I ad tis completes te poof of te teoem REFERENCES KT Ataassov, Ituitioistic fuzzy sets, Fuzzy Sets ad Systems, 20, (986 2 K Ataassov, G Pasi, R Yage, Ituitioistic fuzzy itepetatios of multi-peso multiciteia decisio maig, Poceedigs of 2002 Fist Iteatioal IEEE Symposium Itelliget Systems,, 5 9 ( R Cola, Statistical covegece of ode, Mode metods i Aalysis ad its Applicatios, New Deli, Idia, Aamaya Pub, 200, pp 2 29 Cola, C A Betas, λ -statistical covegece of ode, Acta Mat Scietia, 3B, 3, (20 5 P Das ad E Savas, O I-statistical ad I-lacuay statistical covegece of ode alpa, Bulleti of te Iaia Matematical Society, 0, 2, (20 6 P Das, E Savaş, SK Gosal, O geealizatios of cetai summability metods usig ideals, Appl Mat Lett, 2, (20 7 H Fast, Su la covegece statistique, Colloq Mat, 2, 2 2 (95 8 JA Fidy, O statistical covegece, Aalysis, 5, (985 9 J A Fidy ad C Oa, Lacuay statistical covegece, Pacific J Mat, 60, 3 5 (993 0 S Kaaus, K Demici, O Duma, Statistical covegece o ituitioistic fuzzy omed spaces, Caos Solitos Factals 35, (2008 Kostyo, T Salat, W Wilczyi, I -covegece, Real Aal Excage, 26, 2, ( S A Moiuddie, ad E Savaş, Lacuay statistically coveget double sequeces i pobabilistic omed spaces, A Uiv Feaa, Sez VII, Sci Mat, 58, 2, (202 3 S A Moiuddie, M Aiyub, Lacuay statistical covegece i adom 2 -omed spaces, Appl Mat If Sci, 6, 3, (202 M Musalee ad SA Moiuddie, O lacuay statistical covegece wit espect to te ituitioistic fuzzy omed space, Joual of Computatioal ad Applied Matematics, 233, 2, 2 9 ( F Nuay ad E Savaş, Statistical covegece of sequeces of fuzzy umbes, Matematica Slovaca, 99, 3, (995 6 JH Pa, Ituitioistic fuzzy metic spaces, Caos Solitos Factals, 22, (200 7 R Saadati, JH Pa, O te ituitioistic fuzzy topologicial spaces, Caos Solitos Factals, 27, 33 3 ( E Savaş, Patulaada Das, A geealized statistical covegece via ideals, Appl Mat Lett, 2, (20

9 9 O I -statistical covegece of ode 29 m 9 E Savaş, -stogly summable sequeces spaces i 2-omed spaces defied by ideal covegece ad a Olicz fuctio, App Mat Comp, 27, ( E Savaş, A-sequece spaces i 2-omed space defied by ideal covegece ad a Olicz fuctio, Abst Appl Aaly, Aticle ID 7382 (20 2 E Savaş, O some ew sequece spaces i 2-omed spaces usig ideal covegece ad a Olicz fuctio, J Ieq Appl Aticle Numbe: 82392, DOI: 055/200/82392 Publised 22 E Savaş, M Güdal, Cetai summability metods i ituitioistic fuzzy omed spaces, Joual of iteliget ad fuzzy systems, 27, (20 23 B Scweize, A Sla, Statistical metic spaces, Pacific JMat, 0, (960 2 LA Zade, Fuzzy sets, Ifom Cotol, 8, (965 Received Febuay 25, 202