IJST () A: 47-55 Iaia Joual of Sciece & Techology htt://.hiazu.ac.i/e O alot tatitical covegece of e tye of geealized diffeece equece of fuzzy ube B. C. Tiathy A. Bauah M. t 3 * ad M. Gugo 4 Matheatical Sciece Diviio Ititute of Advaced Study i Sciece ad Techology Pachi Boagao Gachu Guahati-7835 Idia Deatet of Matheatic Noth Gauhati College College Naga Guahati-783 Idia 3 Deatet of Matheatic ıat Uiveity 39 lazig Tuey 4 Deatet of cooetic Iou Uiveity Malatya Tuey -ail: tiathybc@yahoo.co achuta_ath@ediffail.co iailet68@gail.co 3 & gugo44@gail.co 4 Abtact I thi ae e itoduce a e tye of alot tatitical covegece of geealized diffeece equece of fuzzy ube. We give the elatio betee the togly alot Ceào tye covegece ad alot tatitical covegece i thee ace. utheoe e tudy oe of thei oetie lie coletee olidity yeticity etc. We alo give oe icluio elatio elated to thee clae. eyod: Alot tatitical covegece; diffeece equece; fuzzy ube; olide; yeticity; covegece fee. Itoductio The otio of tatitical covegece a itoduced by at [] ad Schoebeg [] ideedetly. Ove the yea ad ude diffeet ae tatitical covegece ha bee dicued i the theoy of ouie aalyi egodic theoy ad ube theoy. Late it a futhe ivetigated fo the equece ace oit of vie ad lied ith uability theoy by Coo [3] idy [4] Mualee et al. ([5] [6]) Šalát [7] Tiathy [8] ad ay othe. I ecet yea geealizatio of tatitical covegece have aeaed i the tudy of tog itegal uability ad the tuctue of ideal of bouded cotiuou fuctio o locally coact ace. Statitical covegece ad it geealizatio ae alo coected ith ubet of the Stoe-Cech coactificatio of the atual ube. Moeove tatitical covegece i cloely elated to the cocet of covegece i obability. The exitig liteatue o alot tatitical covegece ad togly alot covegece aea to have bee eticted to eal o colex equece but Altıo et al. [9] exteded the idea to aly to equece of fuzzy ube ad alo Altı et al. ([] []) t et al. ([] [3]) Başaı ad Mualee [4] Çola et al. [5] Göha et al. *Coeodig autho Received: 9 July / Acceted: 7 Setebe [6] Nuay [7] Savaş [8] Tiathy et al. ([9] [] []) Talo ad Başa [] tudied the equece of fuzzy ube. I the eet ae e itoduce ad exaie the cocet of alot tatitical covegece ad togly alot covegece of geealized diffeece equece of fuzzy ube. I ectio e give a bief ovevie about tatitical covegece fuzzy ube ad uig the geealized diffeece oeato ad the equece. We defie the cocet of alot tatitical covegece ad togly alot covegece of equece of fuzzy ube. I ectio 3 e etablih oe icluio elatio betee ad S betee S ad S.. Defiitio ad eliiaie The defiitio of tatitical covegece ad togly Ceào covegece of a equece of eal ube ee itoduced i the liteatue ideedet of oe aothe ad have folloed diffeet lie of develoet ice thei fit aeaace. It tu out hoeve that the to
IJST () A: 47-55 48 defiitio ca be ily elated to oe aothe i geeal ad ae equivalet fo bouded equece. The idea of tatitical covegece deed o the deity of ubet of the et N of atual ube. The deity of a ubet of N i defied by () li () ovided the liit exit hee i the chaacteitic fuctio of. It i clea that ay fiite ubet of N ha zeo atual deity ad c. A equece x of colex ube i aid to be tatitically coveget to if fo evey > N : x. I thi cae e ite tat li x o S li x. uzzy et ae coideed ith eect to a oety bae et of eleet of iteet. The eetial idea i that each eleet x i aiged a ebehi gade u (x) a value i [] ith u (x) coeodig to oebehi < u ( x) < to atial ebehi ad u (x) to full ebehi. Accodig to Zadeh [3] a fuzzy ubet of i a oety ubet {( x u( x)) : x } of fo oe fuctio u :. The fuctio u itelf i ofte ued fo the fuzzy et. Let C( R ) deote the faily of all oety coact covex ubet of R. The ace CR ha liea tuctue iduced by the oeatio A B a b : a A b B ad A a : a A fo A BCR ad R. If R ad A B C( R ) o (A B) A B ( )A ( A) A A ad if the ( ) A A A. The ditace betee A ad B i defied by the Haudoff etic (A B) ax{u if a b u if a b } aa bb bb aa hee. deote the uual uclidea o i R. It i ell o that ( C( R ) ) i a colete etic ace. Deote L( ) {u : u atifie ( i) ( iv ) belo} hee i ) u i oal that i thee exit a x R uch that u ( x ) ; ii ) u i fuzzy covex that i fo x y R ad u( x ( ) y) i[ u( x) u( y)]; iii ) u i ue eicotiuou; iv ) the cloue of { x R : u( x) > } deoted by [ u ] i coact. If u L( R ) the u i called a fuzzy ube ad L( R ) i aid to be a fuzzy ube ace. o < the -level et [u] of u L( R ) i defied by u {x R :u(x) }. The fo ( i) ( iv) it follo that the -level et [u] ae i the ace C( R ). o the additio ad cala ultilicatio i L( R ) e have uv [u] [v] [ u ] [ u] hee u v L( R ) R. The aitetic oeatio fo -level et ae defied a follo: Let u v L( R ) ith the level et be a b u a a b The e have u. v v a. a u v a a b b a u v a b b a a i ai b ax ai b. i i Defie fo each q < q d(uv) q ([u][v]) d /q
49 IJST () A: 47-55 ad d ( u v) u ([ u] [ v] ) hee i the Haudoff etic. Clealy d ( u v) li d ( u v) q ith dq d if q ([4] [5]). o ilicity i otatio thoughout the ae d ill deote the otatio d q ith q. The geealized de la Vallée-Pouio ea i defied by t x x I hee i a o-deceaig equece of oitive ube uch that a ad I. A equece x x i aid to be V uable to a ube if t x a. V uability educe to C uability he fo all N. A equece ( ) of fuzzy ube i a fuctio fo the et N of all atual ube ito L( R ). Thu a equece of fuzzy ube i a coeodece fo the et of atual ube to a et of fuzzy ube i.e. to each atual ube thee coeod a fuzzy ube (). It i oe coo to ite athe tha () ad to deote the equece by ( ) athe tha. The fuzzy ube i called the th te of the equece. be a equece of fuzzy ube. Let The equece of fuzzy ube i aid to be bouded if the et : N of fuzzy ube i bouded ad coveget to the fuzzy ube itte a li if fo evey > thee exit a oitive itege uch that d < fo >. Let ad c deote the et of all bouded equece ad all coveget equece of fuzzy ube eectively [6]. The faou ace ĉ of all alot coveget equece a itoduced by Loetz [7] ad a equece x x i aid to be togly alot coveget to a ube (ee Maddox [8]) if li i x uifoly i. i q The diffeece equece ace c ad c coitig of all eal valued equece x x uch that x x x i the equece ace c ad c ee defied by ızaz [9]. The idea of diffeece equece i geealized by t ad Çola [3] Başa ad Altay [3] Mualee [3] Tiathy et al. ([9] [33]) ad ay othe. Let be the et of all equece of fuzzy ube. The oeato : i defied by ad ( ) x fo all N. Thoughout the ae ill deote ay oitive itege ad fo coveiece e ill ite itead of. Defiitio.. Let be a equece of fuzzy ube. The the equece i bouded if the et aid to be : N of fuzzy ube i bouded ad coveget to the fuzzy ube itte a li if fo evey > thee exit a uch that oitive itege d < fo all >. By ad c e deote the et of all bouded equece ad all coveget equece of fuzzy ube eectively. be a o-deceaig equece of oitive ube uch that a ad be a equece of fuzzy ube. The the equece of fuzzy ube i aid to be alot tatitically coveget to the fuzzy ube if fo evey > Defiitio.. Let i li I : d uifoly i i N. The et of all alot tatitically coveget equece of fuzzy ube i deoted by S. S. I thi cae e ite I the ecial cae
IJST () A: 47-55 5 fo all N itead of S. Defiitio.3. Let e hall ite S be a o-deceaig equece of oitive ube uch that a be a equece of fuzzy ube ad be ay equece of tictly oitive eal ube. We defie the folloig et : li d i uifoly i i N I : li d i I uifoly i i N : u d i < uifoly i i N i I hee ( t ) If togly alot t (...) otheie. e ay that i Ceao coveget to the fuzzy ube ad i itte a. We get the folloig equece ace fo the above equece ace givig aticula value to ad. i) ad he fo all N ii) If fo all N the ad iii) If the ad. A equece ace i aid to be oal (o olid) if Y i uch that Y d ad d ilie Y. A equece ace i aid to be ootoe if cotai the caoical e-iage of all it te : < N ad ace. Let N be a equece ace. A te ace of i a equece ace :. A caoical e-iage of a equece i a equece Y if ì ï Î í ï otheie ïî Y defied a A caoical e-iage of a te ace i a et of caoical e-iage of all eleet i i.e. Y i i caoical e-iage if ad oly if Y i caoical e-iage of oe. Rea: If a equece ace i olid the i ootoe. A equece ace i aid to be yetic if heeve ( ) Î hee i a eutatio of N. A equece ace i aid to be covegece fee if ( Y ) Î heeve ad ilie Y. The equece ace ad S cotai oe ubouded equece of fuzzy ube hich ae diveget too. To ho that let ad fo all N. The the equece belog to but the equece i diveget ad i ot bouded. o the claical et x covege to hich ilie x covege to but thi cae doe ot hold fo the equece of fuzzy ube. xale. Let fo all N ad. Coide the equece a follo:
5 IJST () A: 47-55 t if t t 3 3 t if t otheie i coveget to the fuzzy ube L hee the the equece t if t L t t 3 if t 3. otheie We fid the level et of ad follo eectively: ad 3 a 4 4. The e have L hee L. 3. Mai eult I thi ectio e give oe icluio elatio betee ad S betee S ad S. Theoe 3.. Let the equece be bouded. The ad the icluio ae tict. i obviou. So e ill oly ho that. Let. The e have Poof: The icluio D D d i d i d I I I D H d i Dax ud I H hee u H ad D ax( ). Thu e get. To ho that the icluio i tict coide the folloig exale: Let fo all N ad. Coide the equece of fuzzy ube a follo: t t t t if 3... t otheie otheie. The fo e have level et of ad ad a follo: if 3... otheie if 3... if 3... otheie. No it i eay to ee that < < fo ad all N hee. Thu the equece of fuzzy ube i bouded but i ot coveget. We give the folloig thee theoe ithout oof. Theoe 3.. The ace ad S ae cloed ude the oeatio of additio ad cala ultilicatio. q Theoe 3.3. Let < q ad q. bouded. The Theoe 3.4. If liif / > S S. be the
IJST () A: 47-55 5 Theoe 3.5. The ace ad the etic ae colete etic ace ith Y d Y u d Y i i i I hee axu. Poof: We hall ove oly fo. The othe ca be teated iilaly. Let be a Cauchy equece i hee... fo each N. The t t t d u d i i a t. i I t Theefoe t d I i d ad a t i fo each fixed i N. No fo d d d... d t t t t t d a t fo each e have N. Theefoe... Cauchy equece i LR. Sice colete it i coveget i a li ay fo each N. equece fo each > uch that Hece fo each Sice L R i i a Cauchy thee exit t < fo all t. i N e get t t i i I d < ad d < t. So e have li t t d d < ad t d d < t i i i i I I li fo all i N ad. Thi ilie that < fo all that i a hee. Sice d i d i d i i I I I D i i i D d d I e obtai. i a colete etic ace. Theefoe Theoe 3.6. If liif > ad i togly alot Ceao coveget to the fuzzy ube the uiquely. Poof: Suoe that liif >. Let ad. The ad d i li I li i The e have I d uifoly i i. i D d d I I D d i uifoly i i I Hece li d I ad o a cotadictio. Thu the liit i uique.
53 IJST () A: 47-55 be a equece of fuzzy ube the e have Theoe 3.7. Let. Poof: Let cotat > uch that. The thee exit a d u d fo all i i i I ad o e have. Coveely let. a cotat > all ad o I The thee exit uch that d i I fo d fo all ad i. Thu. Theoe 3.8. Let be a equece of fuzzy ube ad < h if u H the S Poof: The oof follo fo the folloig iequality d d I. i i I d i I h H :d i.i be a bouded equece of fuzzy ube the S. Theoe 3.9. Let Poof: Suoe that S. Sice thee i a cotat T > uch that d T. Give > e have d i axt T I I I d d < i i h H h H T T I : d ax i h H Hece ax.. Theoe 3.. The equece ace ad ae olid ad hece ootoe but the equece ace S ad Poof: Let ad ae ot olid. Y be uch that d Y d fo each N. The e get d Y d. I I Hece i olid ad hece ootoe. The ace i ot olid. Thi follo fo the folloig exale: Let fo all N ad. Let u coide the equece 3... ad if i odd if i eve the 46.... i ot olid. Hece Theoe 3.. Let deote ay of the equece ace S ad. The the folloig tateet hold: a ) i ot yetic b ) i ot covegece fee. Poof: Sice the oof ca be obtaied fo the ace S i a iila ay e coide the oly the ace. ) a Let fo all N ad. 3... Coide the equece
IJST () A: 47-55 54 Y 3546... by a eaageet of ( ). The Y. b ) Let fo all N ad 3. Suoe that the I. Defie the equece t by t. Defie t t The e have t t otheie. t 3t 8 3 t 8 3 3 3 t t3t83 t 8 3 3 otheie. Theefoe t defied by Hece t li 3 t t Y defied by hee t t t t otheie. t. Y t At thi tage t t t t otheie. t i No coide t t 3 3 3 t 3 3 t 3Y It i clea that t 3 otheie. Y t. ho that the ace covegece fee. 4. Cocluio Thi i ot The equece of fuzzy ube ee itoduced ad tudied by Matloa [6] ad the fit diffeece equece of fuzzy ube ee tudied by Savaş [8] Talo ad Başa [34]. No i thi ae e th tudy the diffeece equece of fuzzy ube fo oe equece clae. The eult obtaied i thi tudy ae uch oe geeal tha thoe obtaied by othe. To do thi oe faily ide clae of equece of fuzzy ube uig the geealized diffeece oeato ad a odeceaig equece of oitive eal ube uch that a have bee itoduced. utheoe uig thee cocet e etablih oe icluio elatio betee ad S betee S ad S ad ho that the equece ace ad etic ace ith uitable etic. ae colete Refeece [] at H. (95). Su la covegece tatitique. Colloquiu Math 4-44. [] Schoebeg I. J. (959). The itegability of cetai fuctio ad elated uability ethod Ae. Math. Mothly 66 36-375. [3] Coo J. S. (988). The tatitical ad tog Ceào covegece of equece. Aalyi 8(- ) 47-63. [4] idy J. A. (985). O tatitical covegece. Aalyi 5(4) 3-33. [5] Mualee M. & Mohiuddie S. A. (9). Statitical covegece of double equece i ituitioitic fuzzy oed ace. Chao Solito actal 4(5) 44-4. [6] Mualee M. & dely O. H. H. (9). O the ivaiat ea ad tatitical covegece. Al. Math. Lett. () 7-74.
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