On Interval Valued Generalized Difference Classes Defined by Orlicz Function

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1 Tuih oua of Aayi ad Numbe Theoy, 03, Vo., No., Avaiabe oie a hp://pub.ciepub.com/ja///0 Sciece ad ducaio Pubihig DOI:0.69/ja---0 O Ieva Vaued Geeaized Diffeece Cae Defied by Oicz Fucio Ayha i, Bipa Hazaia,* Depame of ahemaic, Facuy of Sciece ad A, Adiyama Uiveiy, Adiyama, Tuey Depame of ahemaic, Gadhi Uiveiy, Roo Hi, Auacha Padeh, Idia *Coepodig auho: bh_gu@yahoo.co.i Received Ocobe 0, 03; Revied Novembe 04, 03; Acceped Novembe, 03 Abac I hi pape, uig he diffeece opeao ad Oicz fucio, we ioduce ad examie ome geeaized diffeece equece pace of ieva umbe. We pove compeee popeie of hee pace. Fuhe, we iveigae ome icuio eaio eaed o hee pace. eywod: equece pace, ieva umbe, diffeece equece, compeee Cie Thi Aice: Ayha i, ad Bipa Hazaia, O Ieva Vaued Geeaized Diffeece Cae Defied by Oicz Fucio. Tuih oua of Aayi ad Numbe Theoy, o. (03): doi: 0.69/ja Ioducio The wo of ieva aihmeic wa oigiay ioduced by Dwye [3] i 95. The deveopme of ieva aihmeic a a foma yem ad evidece of i vaue a a compuaioa device wa povided by ooe [5] ad ooe ad Yag [6]. Fuhemoe, ooe ad ohe [3]; [4]; [0] ad [7] have deveoped appicaio o diffeeia equaio. Chiao i [] ioduced equece of ieva umbe ad defied uua covegece of equece of ieva umbe. Şegöü ad yimaz i [0] ioduced ad udied bouded ad covege equece pace of ieva umbe. Recey i udied ogy λ-ad ogy amo λ-covege equece pace of he ieva umbe i [5], epecivey. Ao, i udied ome ew ype equece pace of he ieva umbe i [6,7] ad acuay equece pace fo ieva umbe i [8]. I Hazaia [] ioduced he oio of λ-idea covege ieva vaued di eece cae defied by uiea-oicz fucio. izmaz [] ioduced he oio of di eece equece pace a foow: ( ) = { = : ( ) } X x x x X fo X =, c ad c 0. Lae o, he oio wa geeaized by ad Çoa [9] a foow: ( ) = { = : ( ) } X x x x X fo X =, c ad c 0, whee 0 x = x = x x x = x ad ao ( + ) hi geeaized diffeece oio ha he foowig biomia epeeaio:, i x = ( ) x + i fo a. i i= 0 Reca i [8], [3] ha a Oicz fucio i coiuou, covex, o-deceaig fucio defie fo 0 0 = 0 ad ( x ) > 0 fo x > 0 ad x > uch ha ( x) 0 a x. If covexiy of Oicz fucio i epaced by ( x+ y) ( x) + ( y) he hi fucio i caed he moduu fucio ad chaaceized by Ruce [9]. A Oicz fucio i aid o aify codiio fo a vaue u, if hee exi > 0 uch u u, u 0. Subequey, he oio of ha Oicz fucio wa ued o defied equece pace by Ai e. a., [], Tipahy ad ahaa [], Tipahy e. a., [], Tipahy ad Sama [3] ad may ohe.. Peimiaie A e coiig of a coed ieva of ea umbe x uch ha a x b i caed a ieva umbe. A ea ieva ca ao be coideed a a e. Thu we ca iveigae ome popeie of ieva umbe, fo iace aihmeic popeie o aayi popeie. We deoe he e of a ea vaued coed ieva by IR. Ay eeme of IR i caed coed ieva ad deoed by x. Tha i x = { x : a x b}. A ieva umbe x i a coed ube of ea umbe (ee []). Le x ad x be fi ad a poi of he ieva umbe x, epecivey. Fo x = x,, x y = x, x I R, we have x = y x = x, x = x,

2 Tuih oua of Aayi ad Numbe Theoy 49 { } x + y = x : x + x x x + x, ad if α 0, he ad if α < 0, he { : } α α α x = x x x x { } αx = x : αx x αx, x : x y mi { x x, x x, x x, x x, } x. max { x x, x x, x x, x x, } The e of a ieva umbe IR i a compee meic pace ude he meic d defied by (, ) max {, } d x y x x x x = [5]. I he pecia cae x= [ aa, ] ad y [ bb] =,, we obai uua meic of. Le u defie afomaio f : by f = x = ( x ). The x ( x ) = i caed equece h of ieva umbe ad x i caed em of he ieva umbe equece x = ( x ). The e of a equece of he ieva umbe deoed by w cf. []. = of ieva umbe i aid o be A equece x x covege o he ieva umbe x 0 if fo each ε > 0 hee exi a poiive iege 0 uch ha d( x, x0 ) < ε fo a 0 ad we deoe i by im x = x0. Thu, im x = x0 im x = x0 ad x = x0 []. im A equece pace i aid o be oid (o oma) if αx α x x = x fo a = ( ) wheeve ( ) equece α ( α ) = of caa wih α fo a. A equece pace i aid o be ymmeic if = impie x( π ) whee π i a x x pemuaio of. A equece pace i aid o be moooe if coai he caoica pe-image of a i ep pace. = < < < ad be a equece Le pace. A -ep e of i a ca of equece λ = {( x ) : ( x) }. A caoica pe-image of a equece a foow: x λ i a equece y = ( y ) defied x, if ; y 0, ohewie. A caoica pe-image of a ep e λ i a e of caoica pe-image of a eeme i i i caoica pe-image λ, i.e. = y y λ if ad oy caoica pe-image of ome x = ( x ) λ. y = y i A equece pace i aid o be equece ageba if x y = ( x y) wheeve x = ( x ), y = ( y ). A equece pace i aid o be covegece fee if y y x = x ad y = 0 = ( ) wheeve ( ) x =, whee 0 [ 0,0] wheeve 0 = i he zeo eeme. Rema.. A equece pace i oid impie i moooe. 3. ai Reu I hi pape we ioduce ad examie ome geeaized diffeece equece of ieva umbe uig he Oicz fucio. Defiiio 3.. Le x = ( x ) be a equece of ieva umbe ad be a Oicz fucio. We defie he foowig equece pace: (, 0 ) x ( x ) : im 0, = = c(, ), fo ome > 0 ad x0 IR ( x,0) d x ( x ) : im 0, = = c0 (, ), fo ome > 0 ( x,0) d x ( x ) : up, = < (, ), fo ome > 0 whee i x = ( ) x + i. i i= 0 Thoughou he pape, X wi deoe ay oe of he oaio cc, 0 ad. Theoem 3.. (, ) ad c(, ) compee meic pace wih he meic ρ ( x, y) = d( x, y) = (, ) d x y + if 0 : up >. ae Poof. Le ( x ) be ay Cauchy equece i (, ) whee x = ( x ) = ( x, x, x, ) (, ) fo each. The fo give ε > 0. Fo a fixed X 0 > 0 ad chooe a > 0 uch ha The hee exi 0 uch ha ax 0.

3 50 Tuih oua of Aayi ad Numbe Theoy ρ ( x, x ) = d( x, x) = (, ) ε + if 0 : up > <, ax0 fo, 0. Hece ( ) d x, x < ε, fo, 0. = ( ) ε d x, x <, fo, 0. (3.) The ( x ) i a Cauchy equece i IR ad o ( x ) i a covege equece i IR. Le im x = x. Agai fom (3.) (, ) up fo, 0. (, ) ρ ( x, x ) a0 fo, 0, ad N. ( ) d x, x <, fo, 0. Hece ( x ) ad o ( x ) fo a. Le im ε i a Cauchy equece i IR fo a Fo =, we have i a covege equece i IR x x = fo a. i im x = im ( ) x+ i = x. i i = 0 Simiay we have x = x+ i = x i i 0 fo a =,,,. i im im = Thu im x + exi. Le im x+ = x+. Poceedig i hi way iducivey we cocude ha im x = x fo a. Uig coiuiy of, we have (, ) up fo 0. (, ) d x x if > 0 : up < ε, fo 0. Thu fo a 0, we obai ha (, ) (, ) d x x d x x + if > 0 : up < ε. = Tha i ρ x, x < ε, i. e., x x a. The he iequaiy (,0) (, x x x ) + ( x,0), fo 0 ρ ρ ρ impie ha x ( ),. Thi compee he poof. Theoem 3.. The cae of ieva umbe of equece c0 (, ) ad c(, ) ube of (, ). ae owhee dee Poof. Fom Theoem 3.. we have c0 ( ) ad c0 (, ) pace (, ). Ao c0 (, ) ad c(, ) ae coed ube of he compee meic ae pope ube which foow fom he foowig exampe. xampe 3.. Le = ad ( x) = x. Coide he ieva equece x = ( x ) defied a foow: ad x 0, fo eve; +, fo odd, + fo eve; + x +, fo odd. Thu ( x ) c(, ) c0 (, ) bu ( x ) ( ) Hece he eu. Theoem 3.3. (, X ) X (, ),. fo X= cc, 0, ad he icuio ae ic. Poof. We give he poof fo he iequaiy, c c, oy. The e of he eu foow imia way. Le x ( x ) c ome > 0, we have (, 0 ) =,. The fo im = 0 fo ome x0 I Sice x = x x+ + x x. R. (3.)

4 The fom he equaio (3.) ad he coiuiy of, he eu foow fom he foowig eaio (, 0 ) (, 0) ( +, 0) d x x d x x +. Thi how ha ( x ) c( ),. To how ha he icuio ae ic, coide he foowig exampe. x = x ad =. Coide he xampe 3.. Le equece of ieva umbe x = ( x ) defied by x =, + fo a i.e. x a ad x 0 a. Thu x x c x = x c Hece he icuio = 0, bu i ic. 0. xampe 3.3. Le ( x) = x ad. equece of ieva umbe x = ( x ) defied by The x. x =, + fo a. = Thu x ( x ) c. Tuih oua of Aayi ad Numbe Theoy 5 = Coide he = = Hece he icuio i ic. Theoem 3.4. Le ad be wo Oicz fucio. The (i) X (, ) X (, ) X ( ) X ( ) X ( + ) (ii),,,, fo X = c, c0, Poof. (i) We pove he eu fo X = c ad he e of he cae wi foow imiay. Le ( ) ( ) x = x = c,. The fo > 0 we have (, 0 ) im = 0 fo ome x0 I Le 0< ε < ad wih 0 fo 0 < <. We wie The fo R. (3.3) < < uch ha (, 0 ) D = N:, (, 0 ) D = N: >. < ε we have (, 0 ) > (, 0 ) (, 0 ) < (, 0 ) < + whee D ad a deoe he iege pa of a. Give ε > 0 by he defiiio of Oicz fucio fo (, 0 ) > we have (, 0 ) (, 0 ) + (, 0 ) < fo D ad, uig (3.3). Agai fo we have (, 0 ) (, 0 ) < ε, fo D ad, uig (3.3). > max, we have Thu fo { } (, 0 ) Hece x ( x ) c( ) < ε. = =,. Thu ( ) ( ) c, c,. ε

5 5 Tuih oua of Aayi ad Numbe Theoy (ii) I wi foow fom he foowig iequaiy d( x, x 0 ) ( + ) (, 0) (, 0) d x A +. The poof of he foowig eu i ao ouie wo. Theoem 3.5. Le ad be wo Oicz fucio aifyig codiio ( ). If β = im, he ( ) X, = X,, whee X= cc, 0. Theoem 3.6. The cae of equece of ieva umbe c(, ) ad (, ) ae o equece ageba i geea. Poof. The eu foow fom he foowig exampe. xampe 3.4. Le = ad ( x) = x. Coide he wo equece of ieva umbe x= ( x ), y= ( y ) defied by [ + ] = [ ] x=,, y,. Theefoe fo a, we have x =, y =. Thu x = ( x) y ( y) c( ) ( ) Now, we have ( x y ) ( ), =,,. = ( ), ( + ),( + )( + ) = +,, i.e. ( x y) (, )( c(, )) Thi compee he poof. Theoem 3.7. The cae of ieva umbe of c,, c0,, ae o equece ( ) ( ) ad covegece fee. Poof. Le = ad ( x) x. equece x = ( x ) defied a foow: ad = Coide he ieva 0 = 0,0, fo = i, i ; x 0, ohewie,0, fo i, i ; = + x 0, fo = i, i >, i. Hece x 0 a. Thu ( ) x = x c0, c,,. Le y = ( y ) defied a foow: ad [ 0, ] 0 = 0,0, fo = i, i ; y ohewie +,0, fo = i, i ; y [ 0, ], fo = i, i >, i ; ( +, ), ohewie. Thu y = ( y ) ( ) ( c( ) c0 ( ) ),,,. Theefoe he cae of ieva umbe (, ), 0 (, ) ad (, ) c c ae o covegece fee. Theoem 3.8. The cae of ieva umbe (, ), 0 (, ) ad (, ) c c moooe o oid. Poof. Le = ad ( x) x. equece x = ( x ) defied by: ad x =, + fo a ae eihe = Coide he ieva x = 0, fo a ( + ) i.e. x 0 a. Thu ( ) x = x c0, c,. Le = { : = i, i } e c0 (, ) ep e c0 (, ) of c ( ) N be a ube of ad be he caoica pe-image of he - ( ) 0 (, ) ( ) 0 (, ) 0,, defied a foow: y = y c i he caoica pe-image of x = x c impie ad Now y x fo ; 0 fo, + fo ; y = 0 fo, + fo ; y +, fo + Thu y ( y ) c(, ) ( c0 (, ) ). he cae of ieva umbe c(, ) = Theefoe ad

6 ( ) c0, pace ae o oid. ae o moooe. By he Rema., hee Now e defie he equece ad x [, ] Tuih oua of Aayi ad Numbe Theoy 53 x = x by x = + fo a =, hu x = ( x ) ( ) Le = { : = i, i } e (, ) e (, ) of y = ( y ) (, ) x = ( x ) (, ) impie ad,. be a ube of ad be he caoica pe-image of he -ep Now,, defied a foow: i he caoica pe-image of x, fo ; y 0 fo., + fo ; y 0 fo, + fo ; y +, + fo. Theefoe y = ( y ) (, ) ad (, ) i o moooe. By he Rema., hi pace i o oid. Theoem 3.9. The cae of ieva umbe (, ), 0 (, ) ad (, ) c c ae o ymmeic. Poof. The eu foow fom he foowig exampe. xampe 3.5. Le = ad ( x) = x. Coide he ieva equece x = ( x ) defied by ad x =. x =, + fo a Thu x = ( x ) ( ),. Le he equece of ieva umbe y = ( y ) be a eaageme of he equece of ieva umbe x = ( x ) defied a foow: i.e. x, x, x4, x3, x9, x5, x6, y = ( y ) x6, x5, x7, x36, x8, x49, x, fo a odd; + y x, fo a eve ad m+ m aifie m( m ) < m( m +. ) The fo a odd ad m ; aifyig + m( m ) < m( m+, ) we have + + y = m+, m+ + Fom he a wo equaio, i i cea ha ( B ) i ubouded, hu y = ( y ) ( ) ca (, ) i o ymmeic. Refeece,. Theefoe he [] Y. Ai,. ad B.C. Tipahy, The equece pace N p (,,q,) o emiomed pace, Appied ahemaic ad Compuaio, 54(004), [] uo-pig Chiao, Fudamea popeie of ieva veco maxom, Tamui Oxfod oua of ahemaic, 8()(00), [3] P.S. Dwye, Liea Compuaio, New Yo, Wiey, 95. [4] P.S. Dwye, o of maix compuaio, imuaeou equaio ad eige-vaue, Naioa Bueu of Sada, Appied ahemaic Seie, 9 (953), [5] A. i, λ-equece pace of ieva umbe, (ubmied). [6] A. i, Sogy amo λ-covegece ad aiicay amo λ- covegece of ieva umbe, Scieia aga, 7() (0), 7-. [7] A. i, A ew ca of ieva umbe, oua of Qafqaz Uiveiy, 3 (0), [8] A. i, Lacuay equece pace of ieva umbe, Thai oua of ahemaic, 0() (0) [9]., R. Çoa, O geeaized di eece equece pace, Soochow. ah., (4) (995), [0] P.S. Fiche, Auomaic popagaed ad oud-off eo aayi, pape peeed a he 3 h Naioa eeig of he Aociaio of Compuig achiay, ue 958. [] B. Hazaia, O λ-idea covege ieva vaued diffeece cae defied by uiea-oicz fucio, Aca ahemaica Vieamica, (Acceped fo pubicaio). [] H. izmaz, O ceai equece pace, Caad. ah. Bu., 4()(98), [3].A. aoei ad Y.B. Ruicii, Covex fucio ad Oicz pace, Goige, Nedead, 96. [4] S. aov, Quaiiea pace ad hei eaio o veco pace, ecoic oua o ah-emaic of Compuaio, ()(005). [5] R.. ooe, Auomaic o Aayi i Digia Compuaio, LSD-484, Locheed iie ad Space Compay, 959. [6] R.. ooe ad C.T. Yag, Ieva Aayi I, LSD-85875, Locheed iie ad Space Compay, 96. [7] R.. ooe ad C.T. Yag, Theoy of a ieva ageba ad i appicaio o umeic aayi, RAAG emoie II, Gauuu Bue Fueyu-ai, Toyo, 958. [8] H. Naao, Cocave modua,. ah.soc.apa, 5(953), [9] W.H. Ruce, F-pace i which he equece of coodiae veco i bouded, Caad..ah.,5(973), [0]. Segöü ad A. yimaz, O he equece pace of ieva umbe, Thai oua of ahemaic, 8(3)(00), [] B.C. Tipahy ad S. ahaa, O a ca of geeaized acuay diffeece equece pace defied by Oicz fucio, Aca ahemaica Appicaa Siica, 0()(004), [] B.C. Tipahy, Y. Ai ad., Geeaized di eece equece pace o emiomed pace defied by Oicz fucio, ahemaica Sovaca, 58(3)(008), [3] B.C. Tipahy ad B. Sama, Doube equece pace of fuzzy umbe defied by Oicz fucio, Aca ahemaica Scieia, 3B()(0),