Turkish Journal of. Analysis and Number Theory. Volume 3, Number 6,

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1 ISSN (Pri) : - ISSN (Olie) : - Volue, Nuber 6, 5 hp://jahuedur hp://wwwciepubco/joural/ja Turih Joural of Aalyi ad Nuber Theory Sciece ad Educaio Publihig Haa Kalyocu Uiveriy Sca o view hi joural o your obile device

2 Turih Joural of Aalyi ad Nuber Theory Ower o behalf of Haa Kalyocu Uiveriy: Profeor Taer Yilaz, Recor Correpodece addre: Sciece ad Educaio Publihig Depare of Ecooic, Faculy of Ecooic, Adiiraive ad Social Sciece, TR-74 aziaep, Turey Web addre: hp://jahuedur hp://wwwciepubco/joural/tjant Publicaio ype: Biohly

3 Turih Joural of Aalyi ad Nuber Theory ISSN (Pri): - ISSN (Olie): - hp://wwwciepubco/joural/tjant Mehe Acigoz Feg Qi Ceap Özel Sera Araci Erdoğa Şe Edior-i-Chief Uiveriy of aziaep, Turey Hea Polyechic Uiveriy, Chia Douz Eylül Uiveriy, Turey Aia Edior Haa Kalyocu Uiveriy, Turey Nai Keal Uiveriy, Turey Hoorary Edior R P Agarwal Kigville, TX, Uied Sae M E H Iail Uiveriy of Ceral Florida, Uied Sae Taer Yilaz Haa Kalyocu Uiveriy, Turey H M Srivaava Vicoria, BC, Caada Edior Hery W ould We Virgiia Uiveriy, Uied Sae Toa Diagaa Howard Uiveriy, Uied Sae Abdelejid Bayad Uiveriéd'éry Val d'eoe, Frace Haa Jolay Uiveriéde Lille, Frace Ivá Mező Najig Uiveriy of Iforaio Sciece ad Techology, Chia C S Ryoo Haa Uiveriy, Souh Korea Jueag Choi Doggu Uiveriy, Souh Korea Dae Sa Ki Sogag Uiveriy, Souh Korea Taeyu Ki Kwagwoo Uiveriy, Souh Korea uoao Wag Shaxi Noral Uiveriy, Chia Yua He Kuig Uiveriy of Sciece ad Techology, Chia Aleadar Ivıc Kaedra Maeaie RF-A Uiveriea U Beogradu, Serbia Criiel Morici Valahia Uiveriy of Targovie, Roaia Nai Çağa Uiveriy of azioapaa, Turey Üal Ufuepe Izir Uiveriy of Ecooic, Turey Ceil Tuc Yuzucu Yil Uiveriy, Turey Abdullah Özbeler Aili Uiveriy, Turey Doal O'Rega Naioal Uiveriy of Irelad, Irelad S A Mohiuddie Kig Abdulaziz Uiveriy, Saudi Arabia Duiru Baleau Çaaya Uiveriy, Turey Ahe Sia CEVIK Selcu Uiveriy, Turey Erol Yılaz Aba Izze Bayal Uiveriy, Turey Hüar Kayha Aba Izze Bayal Uiveriy, Turey Yaar Soze Haceepe Uiveriy, Turey I Naci Cagul Uludag Uiveriy, Turey İlay Arla üve Uiveriy of aziaep, Turey Sera Kaya Nura Uiveriy of Uşa, Turey Ayha Ei Adiyaa Uiveriy, Turey M Taer Koa ebze Iiue of Techology, Turey Haifa Zeraoui Ou-El-Bouaghi Uiveriy, Algeria Siraj Uddi Uiveriy of Malaya, Malayia Rabha W Ibrahi Uiveriy of Malaya, Malayia Ade Kilica Uiveriy Pura Malayia, Malayia Are Bagdaarya Ruia Acadey of Sciece, Mocow, Ruia Viorica Mariela Ugureau Uiveriy Coai Bracui, Roaia Valeia Eilia Bala Aurel Vlaicu Uiveriy of Arad, Roaia RK Raia MP Uiv of Agriculure ad Techology, Idia M Muralee Aligarh Muli Uiveriy, Idia Vijay upa Neaji Subha Iiue of Techology, Idia Hee Dua auhai Uiveriy, Idia Abar Aza COMSATS Iiue of Iforaio Techology, Paia Moiz-ud-di Kha COMSATS Iiue of Iforaio Techology, Paia Robero B Corcio Cebu Noral Uiveriy, Philippie

4 Turih Joural of Aalyi ad Nuber Theory, 5, Vol, No 6, Available olie a hp://pubciepubco/ja//6/ Sciece ad Educaio Publihig DOI:69/ja--6- O Sei-yeric Para Keou Maifold T Sayaarayaa, K L Sai Praad, Depare of Maheaic, Pragahi Egieerig College, Surapale, Adhra Pradeh, Idia Depare of Maheaic, ayari Vidya Parihad College of Egieerig for Woe, Viahapaa, Adhra Pradeh, Idia Correpodig auhor: lpraad@yahooco Received Augu 5, 5; Acceped Ocober, 5 Abrac I hi paper we udy oe rearable properie of para Keou (briefly p -Keou) aifold aifyig he codiio R( X, Y) R =, R( X, Y) P = ad P( X, Y) R =, where R(X, Y) i he Rieaia curvaure eor ad P(X, Y) i he Weyl projecive curvaure eor of he aifold I i how ha a eiyeric p -Keou aifold ( M, g ) i of coa curvaure ad hece i a p -Keou aifold Alo, we obai he eceary ad ufficie codiio for a p -Keou aifold o be Weyl projecive ei-yeric ad how ha he Weyl projecive ei-yeric p -Keou aifold i projecively fla Fially we prove ha if he codiio P( X, Y) R = i aified o a p -Keou aifold he i calar curvaure i coa Keyword: para Keou aifold, curvaure eor, projecive curvaure eor, calar curvaure Cie Thi Aricle: T Sayaarayaa, ad K L Sai Praad, O Sei-yeric Para Keou Maifold Turih Joural of Aalyi ad Nuber Theory, vol, o 6 (5): doi: 69/ja--6- Iroducio The oio of a alo para-coac Rieaia aifold wa iroduced by Sao [7] i 976 Afer ha, T Adai ad K Mauoo [] defied ad udied p- Saaia ad p-saaia aifold which are regarded a a pecial id of a alo coac Rieaia aifold Before Sao, Keou [6] defied a cla of alo coac Rieaia aifold I 995, Siha ad Sai Praad [9] have defied a cla of alo para-coac eric aifold aely para-keou (briefly p- Keou) ad pecial para Keou (briefly p- Keou) aifold I a rece paper, he auhor Sayaarayaa ad Sai Praad [8] udied coforally yeric p -Keou aifold, ha i he p-keou aifold aifyig he codiio R( X, Y) C =, ad hey prove ha uch a aifold i coforally fla ad hece i a p-keou aifold, where R i he Rieaia curvaure ad C i he coforal curvaure eor defied by C( X, Y ) Z g( Y, Z) QX g( X, Z) QY = R( X, Y ) Z S( Y, Z) X S( X, Z) Y r [ g( Y, Z) X g( X, Z) Y ] ( )( ) () Here S i he Ricci eor, r i he calar curvaure ad Q i he yeric edoorphi of he age pace a each poi correpodig o he Ricci eor S [] ie, g( QX, Y) = S( X, Y ) () A Rieaia aifold M i locally yeric if i curvaure eor R aifie R =, where i Levi- Civia coecio of he Rieaia eric [4] A a geeralizaio of locally yeric pace, ay geoeer have coidered ei-yeric pace ad i ur heir geeralizaio A Rieaia aifold M i aid o be ei-yeric if i curvaure eor R aifie R( X, Y) R = where RXY (, ) ac o R a derivaio [] Locally yeric ad ei-yeric p-saaia aifold are widely udied by ay geoeer [,5] I hi udy, we coider he p-keou aifold aifyig he codiio R( X, Y) R =, ow a eiyeric p-keou aifold, where RXY (, ) i coidered a a derivaio of eor algebra a each poi of he aifold for age vecor X ad Y ad he p- Keou aifold (M, g) ( > ) aifyig he codiio R( X, Y) P =, where P deoe he Weyl projecive curvaure eor [] defied by g( Y, Z) QX P( X, Y) Z = R( X, Y) Z g( X, Z) QY () Here we coider he p-keou aifold M for > ; a if for =, he projecive curvaure eor ideically vaihe I ecio, i i how ha a ei-yeric p- Keou aifold (M, g) of coa curvaure i a p-keou aifold I he ex ecio we obai he eceary ad ufficie codiio for a p-keou aifold o be Weyl projecive ei-yeric ad how ha he Weyl projecive ei-yeric p - Keou aifold i projecively fla Fially we prove ha if he codiio P( X, Y) R = i aified o a p - Keou aifold he i calar curvaure i coa p-keou Maifold

5 46 Turih Joural of Aalyi ad Nuber Theory Le M be a -dieioal differeiable aifold equipped wih rucure eor (, ξ, η) where i a eor of ype (,), ξ i a vecor field, η i a -for uch ha ( ) = () ( X ) = X ( X ) ; X = X () The M i called a alo para coac aifold Le g be he Rieaia eric i a -dieioal alo para-coac aifold M uch ha g( X, ) = ( X) () =, ( X) =, ra = (4) g( X, Y) = g( X, Y) ( X ) ( Y) (5) for all vecor field X ad Y o M The he aifold M [7] i aid o adi a alo para-coac Rieaia rucure (, ξ, η, g) ad he aifold i called a alo para-coac Rieaia aifold A aifold of dieio ' wih Rieaia eric 'g adiig a eor field ' of ype (, ), a vecor field ' ad a -for ' aifyig (), () alog wih ( X) Y( Y) X = (6) ( XY) Z = [ g( X, Z) ( X ) ( Z)] ( Y) [ g( X, Y) ( X ) ( Y)] ( Z) X = X = X ( X ) (7) (8) ( X ) Y = g( X, Y) ( Y) X (9) i called a para-keou aifold or briefly p - Keou aifold [9] A p -Keou aifold adiig a -for ' aifyig ( X ) Y = g ( X, Y ) ( X ) ( Y ) () g( X, ) = ( X )ad ( X) Y = ( X, Y), where i a aociae of, () i called a pecial p -Keou aifold or briefly p - Keou aifold [9] I i ow ha [9] i a p -Keou aifold he followig relaio hold: S( X, ) = ( ) ( X) where g( QX, Y) = S( X, Y) () g[ R( X, Y) Z, ] = [ R( X, Y, Z)] = g( X, Z) ( Y) g( Y, Z) ( X ) () R(, X ) Y = ( Y) X g( X, Y) (4) R( X, Y, ) = ( X ) Y ( Y) X; whe X iorhogoal o (5) where S i he Ricci eor ad R i he Rieaia curvaure Moreover, i i alo ow ha if a p -Keou aifold i projecively fla he i i a Eiei aifold ad he calar curvaure ha a egaive coa value ( ) Epecially, if a p -Keou aifold i of coa curvaure, he calar curvaure ha a egaive coa value ( ) [9] I hi cae, ad hece S( Y, Z) = ( ) g( Y, Z) (6) S( Y, Z) = S( Y, Z) ( ) ( Y) ( Z) (7) Alo, if a p -Keou aifold i of coa curvaure, we have S( Y, Z) g( X, P) ' R( X, Y, Z, P) = ( ) S( X, Z) g( Y, P) (8) The above reul will be ued furher i he ex ecio p-keou Maifold Saifyig R( X, Y) R = I hi ecio, we coider ei-yeric p - Keou aifold, ie, p -Keou aifold aifyig he codiio R( X, Y) R = where RXY (, ) i coidered a a derivaio of eor algebra a each poi of he aifold for age vecor X ad Y Now ( R( X, Y) R)( U, V ) W = R( X, Y) R( U, V ) W R( R( X, Y) U, V ) W R( U, R( X, Y) V ) W R( U, V ) R( X, Y) W () Puig X = i (), ad o uig he codiio R( X, Y) R =, we ge g( R(, Y) R( U, V ) W, ) g( R( R(, Y) U, V ) W, ) g( R( U, R(, Y) V ) W, ) g( R( U, V ) R(, Y) W, ) () = By uig he equaio () ad (4), fro () we ge ' R( U, V, W, Y) ( Y) ( R( U, V ) W ) ( U) ( R( Y, V ) W ) ( V ) ( R( U, Y) W ) ( W ) ( R( U, V ) Y) g( Y, U) ( R(, V ) W ) g( Y, V ) ( R( U, ) W ) g( Y, W ) ( R( U, V ) ) = where ' R( U, V, W, Y) = g( R( U, V) W, Y ) O puig Y = U i (), we ge ' R( U, V, W, U) ( V ) ( R( U, U) W ) ( W ) ( R( U, V ) U) g( U, U) ( R(, V ) W) g( U, V ) ( R( U, ) W ) g( U, W) ( R( U, V ) ) = () (4) Now puig U = e i, where { ei }, i =,, i a orhogoal bai of he age pace a ay poi, ad aig he uaio of (4) over i, i, we ge (6) Alo, uig he equaio (), (6) ad () we ge (8), how ha he aifold i of coa curvaure Thu we ae he followig reul

6 Turih Joural of Aalyi ad Nuber Theory 47 Theore : A ei-yeric p -Keou aifold i of coa curvaure Now, fro (6) ad (8) we have ' R( X, Y, Z, P) = g( X, Z) g( Y, P) g( Y, Z) g( X, P), (5) ad fro equaio (6) ad (5), we have S( X, Y) = ( )[ g( X, Y) ( X) ( Y)] (6) O coracio of (6) wih covaria eor ( X, Y) = g( X, Y), we ge ( X, Y) = g( X, Y) ( X) ( Y), how ha he aifold i a p -Keou oe Thu, we ae he followig heore Theore : If a ei-yeric p -Keou aifold ( M, g ) i of coa curvaure, he aifold i a p -Keou oe 4 p-keou Maifold Saifyig R( X, Y) P = I hi ecio, we coider Weyl projecive eiyeric p-keou aifold, ie, p-keou aifold aifyig he codiio R( X, Y) P = Now ( R( X, Y) P)( U, V ) W = R( X, Y) P( U, V ) W P( R( X, Y) U, V ) W P( U, R( X, Y) V ) W P( U, V ) R( X, Y) W (4) Pu X = i (4) The he codiio R( X, Y) P = iplie ha g( R(, Y) P( U, V ) W, ) g( P( R(, Y) U, V ) W, ) g( P( U, R(, Y) V ) W, ) g( P( U, V ) R(, Y) W, ) (4) = The o uig equaio (), () ad (), we ge ( P( X, Y) Z) = (4) O he oher had, by uig (), (4), ad (4), we ge g( R(, Y) P( U, V) W, ) = g( P( U, V) W, Y) (44) The fro equaio (4) ad (4), he lef had ide of (44) i zero, give ha g( P( U, V) W, Y ) = for all U, V, W ad Y ad hece PXY (, ) = Thi lead o he followig heore: Theore 4: A Weyl projecive ei-yeric p - Keou aifold i projecively fla Bu i i ow ha [], a projecively fla Rieaia aifold i of coa curvaure Alo i ca be eaily ee ha a aifold of coa curvaure i projecively fal Hece we have he followig heore Theore 4: A p -Keou aifold i Weyl projecive ei-yeric if ad oly if he aifold i of coa curvaure Alo i i ow ha a p -Keou aifold of coa curvaure i a p -Keou aifold [8] Hece we coclude he followig reul: Theore 4: A Weyl projecive ei-yeric p - Keou aifold i of coa curvaure ad hece i a p -Keou aifold I i rivial ha i cae of a projecive yeric Rieaia aifold he codiio R( X, Y) P = hold good 5 p-keou Maifold Saifyig P( X, Y) R = I i ow ha he codiio R( X, Y) P = doe o iply P( X, Y) R = I hi ecio, we udy he rearable propery of p -Keou aifold aifyig he codiio P( X, Y) R = Now, we have ( P( X, Y) R)( U, V ) W = P( X, Y) R( U, V ) W R( P( X, Y) U, V ) W R( U, P( X, Y) V ) W R( U, V ) P( X, Y) W (5) Pu X = i (5) The he codiio P( X, Y) R = iplie ha g( P(, Y) R( U, V ) W, ) g( R( P(, Y) U, V ) W, ) g( R( U, P(, Y) V ) W, ) g( R( U, V ) P(, Y) W ), (5) = Puig X =, Z = U i () ad o uig () ad (), we ge ( R( P(, Y) U, V )) W = ( U)[ ( R( Y, V ) W ) ( R( QY, V ) W )] ( ) (5) Siilarly, by puig X =, Z = V i () ad o uig () ad (), we ge ( R( U, P(, Y) V ) W (54) = ( V )[ ( R( U, Y) W ) ( R( U, QY ) W)] ( ) I iilar by puig X =, Z = W i () ad o uig () ad (), we ge ( R( U, V ) P(, Y) W ) = ( W )[ ( R( U, V ) Y) ( R( U, V ) QY )] ( ) (55) O uig (4), (5), (54) ad (55), we ge fro eq (5) ha ( U )[ ( R( Y, V ) W ) ( R( QY, V ) W )] ( ) ( V )[ ( R( U, Y) W ) ( R( U, QY ) W )] ( ) ( W )[ ( R( U, V ) Y) ( R( U, V ) QY )] = ( ) (56)

7 48 Turih Joural of Aalyi ad Nuber Theory By puig Y = U i eq (56), we ge ( U )[ ( R( U, V ) W ) ( R( QU, V ) W )] ( ) ( V )[ ( R( U, U ) W ) ( R( U, QU ) W )] ( ) ( W )[ ( R( U, V ) U ) ( R( U, V ) QU )] = ( ) The o uig () ad (), we ge (57) g( U, W ) ( V ) g( V, W ) ( U) ( W ) = (58) [ S( U, U) ( V ) S( V, U) ( U)] ( ) Now puig U = e i, where i=,, ad aig he uaio of (58) over i, i, we ge r = ( ), ice ( V), how ha he calar curvaure i coa Hece we have he followig heore Theore 5: If a p -Keou aifold aifie he codiio P( X, Y) R = he i calar curvaure i coa Acowledgee The auhor acowledge Prof Kalpaa, Baara Hidu Uiveriy ad Dr B Sayaarayaa of Nagarjua Uiveriy for heir valuable uggeio i preparaio of he aucrip They are alo haful o he referee for hi valuable coe i he iprovee of hi paper Coflic of Iere The auhor declare ha here i o coflic of iere regardig he publicaio of hi paper Referece [] Adai,T ad Mauoo, K, O coforally recurre ad coforally yeric p-saaia aifold, TRU Mah,, 5-, 977 [] Adai, T ad Miyazawa, T, O P-Saaia aifold aifyig cerai codiio, Teor (NS),, 7-78, 979 [] Bihop, R L ad oldberg, S I, O coforally fla pace wih couig curvaure ad Ricci raforaio, Caad J Mah, 4(5), , 97 [4] Cara, E Sur ue clae rearquable d epace de Riea, Bull Soc Mah Frace, 54, 4-6, 96 [5] De, U C, Ciha Ozgur, Kadri Arla, Cegizha Muraha ad Ahe Yildiz, O a ype of P-Saaia aifold, Mah Balaica (NS),, 5-6, 8 [6] Keou, K, A cla of alo coac Rieaia aifold, Tohou Mah Joural, 4, 9-, 97 [7] Sao, I, O a rucure iilar o he alo coac rucure, Teor (NS),, 9-4, 976 [8] Sayaarayaa, T ad Sai Praad, K L, O a ype of Para Keou Maifold, Pure Maheaical Sciece, (4), 65 7, [9] Siha, B B ad Sai Praad, K L, A cla of alo para coac eric Maifold, Bullei of he Calcua Maheaical Sociey, 87, 7-, 995 [] Szabo, Z I, Srucure heore o Rieaia pace aifyig R(X,Y)R =, I The local verio J Diff eo, 7, 5-58, 98 [] Yao, K, Iegral forula i Rieaia eoery, Pure ad Applied Maheaic,, Marcel Deer, Ic, New Yor, 97 [] Yao, K ad Ko, M, Srucure o Maifold, Serie i Pure Maheaic,, World Scieific Publihig Co, Sigapore, 984

8 Turih Joural of Aalyi ad Nuber Theory, 5, Vol, No 6, 49-5 Available olie a hp://pubciepubco/ja//6/ Sciece ad Educaio Publihig DOI:69/ja--6- Upper Boud of Parial Su Deeried by Marix Theory Rabha W Ibrahi Iiue of Maheaical Sciece, Uiveriy Malaya, Malayia Correpodig auhor: rabhaibrahi@yahooco Received Sepeber 6, 5; Acceped Noveber 4, 5 Abrac Oe of he ajor proble i he geoeric fucio heory i he coefficie boud for fucioal ad parial u The ipora ehod, for hi purpoe, i he Hael arix Our ai i o iroduce a ew ehod o deerie he coefficie boud, baed o he arix heory We uilize variou id of arice, uch a Hilber, Hurwiz ad Tura We illurae ew clae of aalyic fucio i he ui di, depedig o he coefficie of a paricular ype of parial u Thi ehod how he effecivee of he ew clae Our reul are applied o he well ow clae uch a arlie ad covex Oe ca illurae he ae ehod o oher clae Keyword: aalyic fucio, uivale fucio, ui di, parial u, coefficie boud Cie Thi Aricle: Rabha W Ibrahi, Upper Boud of Parial Su Deeried by Marix Theory Turih Joural of Aalyi ad Nuber Theory, vol, o 6 (5): 49-5 doi: 69/ja--6- Iroducio The Hael deeria repree a ajor par i he heory of igulariie [,] I addiio, i uilize i he iveigaio of power erie wih iegral coefficie [] Alo, i appear i he udy of eroorphic fucio [4], ad variou properie of hee deeria ca be foud i [5] I i well ow ha he Feee-Szego fucioal a a H Thi fucioal i furher geeralized a a a for oe (real or coplex) Feee ad Szego iroduced harp boud of a a for real of uivale fucio I i a very ipora cobiaio of he wo coefficie which decribe he area proble poed earlier by rowall Furherore, reearcher coidered he fucioal 4 a a a (ee [6]) Babalola [7] deeried he Hael deeria H for oe ubclae of aalyic fucio Ibrahi [8] copued he Hael deeria for fracioal differeial operaor i he ope ui di Parial u are udied widely i he uivale fucio heory Szeg [9] proved ha if he fucio f z z a z i arlie, he i parial u are arlie for z / 4 f z i covex, he i parial u f z f z z a z Moreover, if i covex for z / 8 Laer Owa [] ipoed he arliee ad covexiy for pecial cae of f z z az I addiio, Daru ad Ibrahi [] pecified he aupio, which idicaed ha he parial u of fucio of bouded urig are alo of bouded urig Recely, Daru ad Ibrahi [] coidered he Ceáro parial u, i ha bee how ha hi ype of parial u preerve he properie of he aalyic fucio i he ope ui di I hi wor, we deal wih he parial u of he for f z z a / z, We iroduce oe clae of aalyic fucio defied by i parial u The abiliy of hee clae i udied by uilizig Hurwiz arice covoluig he wih Hilber arix (a pecial ype of Hael arix) Moreover, we dicu oe parial u forulaed uder Tura deeria The upper boud a well a he lower boud of he coefficie a Thi ew proce iclude oe well ow reul Our oucoe deped o copuaioal reul of differe order of he Tura deeria We how ha oe geoeric properie, of he ew clae are eablihed by copuig he Tura deeria uch a arliee ad covexiy Proceig Le be he cla of aalyic fucio f z z a z i U z : z oralized by he codiio f f parial u of he for ad For a f z a z a z az az, covolued wih he Hilber arix elee i he fi order, we obai he parial u

9 5 Turih Joural of Aalyi ad Nuber Theory a a a f z z z z az,, a For he above parial u f z, we le a a a f z z z z az, a, z U : b z b z b z b z b, b a, b () The ior of Hurwiz arix for (), are defied by f b f b b b b b b b5 f b b b4 b b Defiiio For z U, he polyoial f z i called able, aypoically able ad uable if ad oly if j, j, j, for all j,,,, repecively Fro (), we defie he parial u a g z z z : We proceed o coruc ew clae baed o A copuaio iplie z z zg P z, g a z : a w, w z g z Thu for,,4,, we have he followig clae: P w a w () P w a w, () We call he above clae he coefficie a ad hey deoed by coefficie a -covex, which deoed by a follow: -arlie S a Siilarly, we defie he, a z z zg Q z a z g : a w, w z Thu for,,, we have he followig clae: Q w a w Q w a w, (4) (5) I he ae aer of he above clae, oe ca coruc a -cla uch a cloe o covex, uiforly clae ad cocave Baed o hee clae, we ca udy he abiliy of arliee a well a covexiy Moreover, relaio cocerig hee clae ca be forulaed uch a H, Oucoe H, We have he followig abiliy reul for he clae S a ad a : Theore Coider P S a a The a, polyoial of degree i arlie able, while of degree i o able Proof By eployig P, i Eq (), polyoial of degree ad ca be expreed repecively a follow: ad ad a a, p w w w : b b w b w a a a, p w w w w : b b w b w b w Le a, hu we obai a p, p a p, Theore Coider Q a, a The a polyoial of degree i covex able, while of degree i o able Q a a The polyoial Proof Coider, of degree ad ca be forulaed repecively a follow:

10 Turih Joural of Aalyi ad Nuber Theory 5 ad ad a a, q w w w : b b w b w a a a, q w w w w : b b w b w b w Le a, hu we obai q q a, q a 6, Coider p S a equece We deal wih polyoial p S a, (parial u) aifyig he recurre relaio ad,, p wp w p w p w, p w a w Defie he Tura deeria a follow: (6) w p w p w p w, (7) We hall prove iequaliy of he for,, c w C w c C (8) Theore Aue p w aifie (6) The p p, Proof By (6), we have hi yield ha p w p w w p p p Coequely, we obai p (9) By he defiiio of w, we coclude ha w w p w, () he uig (9) ad (), we arrive a he deired aerio Thi coplee he proof Theore 4 Le he be icreaig equece If w, wu \, Proof I uffice o how ha w By he proof of Theore ad he fac ha p ad p, we coclude ha w p p p p Therefore, by he aupio of he heore, we have w Hece by iducio we obai w, g w : p p g w Defie a fucio he aifie he followig propery : Propoiio For we have wg w g w g w Proof A calculaio iplie ha p w p w w g w g w wp w p w p w wg w wp p w p w wg Theore 5 For we have, w w g w g w g w where g w p w p w, Proof We oberve ha ad g w p w p w wp, () g w g w p w p w p w p w w p p () Subracig () fro (), we coclude he deired aerio Theore 6 Coider ha p achieve (6) wih Le, be icreaig uch ha / ad The, ()

11 5 Turih Joural of Aalyi ad Nuber Theory w c, c, wu, Proof Clearly ha () i equivale o beig icreaig Defie he forula A w : g w g w g w Sice, herefore, i view of Theore 5, we obai w w, w (4) By Propoiio, we have he followig expreio : wg w g w A w g w g w Muliplyig Eq(5) by we arrive o (5) ad replacig by, A w A w g w Coequely, we coclude ha By ieraig he quaiie A A A ad A, we aai i A w A Bu by uilizig Eq(5) ad Eq(6), we fid A w g g g g Therefore, (4) becoe (6) A w : c Hece he proof Theore 7 Coider ha p achieve (6) wih p Le p, be decreaig uch ha / ad The, w C, C, wu, (7) Proof By leig :, wih he followig properie: li The la propery i valid by he oooiciy of i (7) Defie a polyoial P by uilizig a follow: for where aifyig p, U, P w w w Obviouly, P aifie :,,, P P P (8) li P P w P w Thi iplie ha P w, w U i uiforly bouded o a copac e for By he defiiio of he Tura deeria, we obai where uch ha w P w P w P w li We coclude ha here exi a coa C, uch ha,, w C w U Rear If i Theore 6 ad 7, we obai ha he coefficie a For exaple, if (arlie cla), he /, Thu, a Theore 7 iplie ha a w Moreover, he above reul ca be coidered for a equece of polyoial q a, 4 Applicaio I hi ecio, we uilize he Tura deeria o fied he coefficie boud of he clae S a ad a We have he followig propoiio Propoiio 4 Coider he clae S a The a ad a,, S a ad

12 Turih Joural of Aalyi ad Nuber Theory 5 Proof By uilizig ad repecively for fidig he upper boud of a ad a A copuaio iplie ha ad a a w w w w a w a w a w a w I view of Rear, we coclude ha a a w w w 4 4 a a, w 4 4, whe a Siilarly for a I he iilar aer of Propoiio 4, we have he followig reul: Propoiio 4 Coider he clae a Proof By uilizig q w q wq w obai a The :, we w a w a w, which iplie ha w 5 Cocluio whe a We ipoed a ew echique for fidig he coefficie boud Thi ehod baed o everal ype of arice The ajor ype wa he Tura i he ope ui di We proved he boudede of hi arix fro below a well a fro above We defied clae of aalyic fucio, depedig o oe coefficie, calculaig by oe pecial ype of parial u The abiliy of hee clae i coidered by uilizig he Hurwiz arix We illuraed oe applicaio of hi ehod for wo well defied clae (arlie ad covex) The above ehod ca be eployed o oher clae uch a uifor, cocave ec Coflic of Iere The auhor declare ha here i o coflic of iere regardig he publicaio of hi aricle Referece [] P Diee, The Taylor Serie Dover, New Yor (957) [] A Edrei, Sur le deria rcurre e le igulari due focio doe por o dveloppee de Taylor Copo Mah 7, -88 (94) [] D Caor, Power erie wih iegral coefficie Bull A Mah Soc 69, 6-66 (96) [4] R Wilo, Deeriaal crieria for eroorphic fucio Proc Lod Mah Soc 4, (954) [5] R Vei, P Dale, Deeria ad Their Applicaio i Maheaical Phyic Applied Maheaical Sciece, vol 4 Spriger, New Yor (999) [6] D Baal, Upper boud of ecod Hael deeria for a ew cla of aalyic fucio Appl Mah Le 6(), -7 () [7] K O Babalola, O H() Hael deeria for oe clae of uivale fucio Iequal Theory Appl 6, -7 (7) [8] R W Ibrahi, Bouded oliear fucioal derived by he geeralized Srivaava-Owa fracioal differeial operaor Ieraioal Joural of Aalyi, -7 () [9] Szego, Zur heorie der chliche abbiluge Mah A, 88- (98) [] S Owa, Parial u of cerai aalyic fucio I J Mah Mah Sci 5(), () [] M Daru, R W Ibrahi, Parial u of aalyic fucio of bouded urig wih applicaio Copu Appl Mah 9(), 8-88 () [] R W Ibrahi, M Daru, Ceáro parial u of cerai aalyic fucio, Joural of Iequaliie ad Applicaio, 5, -9 ()

13 Turih Joural of Aalyi ad Nuber Theory, 5, Vol, No 6, Available olie a hp://pubciepubco/ja//6/ Sciece ad Educaio Publihig DOI:69/ja--6- The Soluio of Iiial Value Proble for Secod-order Iegro-differeial Equaio wih Delayed Argue i Baach Space Tigig ua School of Maheaic ad Copuer Sciece, Shaxi Noral Uiveriy, Life, Shaxi 44, P R Chia Correpodig auhor: guaigig985@6co Received Sepeber 5, 5; Acceped Noveber, 5 Abrac By uig he parial order ehod ad oe ew copario reul, he axial or iial oluio of he iiial value proble for oliear ecod order iegro-differeial equaio wih delayed argue i Baach pace are iveigaed I hi paper, we require oly a lower oluio or a upper oluio ad oe weaer codiio preeed here, ad we exed ad iprove oe rece reul (ee [-]) Keyword: ecod-order iegro-differeial equaio, delayed argue, eaure of o-copace, oluio, oooe ieraive echique Cie Thi Aricle: Tigig ua, The Soluio of Iiial Value Proble for Secod-order Iegrodiffereial Equaio wih Delayed Argue i Baach Space Turih Joural of Aalyi ad Nuber Theory, vol, o 6 (5): doi: 69/ja--6- Iroducio The heory of differeial equaio wih deviaed argue i very ipora ad igifica brach of oliear aalyi I i worhwhile eioig ha differeial equaio wih deviaed argue appear ofe i iveigaio coeced wih aheaical phyic, echaic, egieerig, ecooic ad o o (cf [,,], for exaple) Oe of he baic proble coidered i he heory of differeial equaio wih deviaed argue i o eablih coveie codiio guaraeeig he exiece of oluio of hoe equaio, we refer o oe rece paper [,4,5,6,7] ad referece Le E be a real Baach pace wih ad le P be a coe i E The parial order i iroduced by coe P, ie, x, y E, x y if ad oly if y x P A coe P i aid o be oral if here exi a coa NP uch ha x, y E, x y iplie x NP y ; N P i called he oral coa of P Recall ha a coe P i aid o be regular if every icreaig ad bouded i order equece i E ha a lii, ie, x x x y iplie x x a for oe x E The regulariy of P iplie he oraliy of P Le E be he dual pace of E, P E x, x P i called he dual coe Obviouly, x P if ad oly if x, for all P Le P u C J, E : u for all J, C where J, a (a > ) ad, C J E deoe he Baach pace of all coiuou appig u : J E wih he or u C ax u : u J I i clear ha P C i a coe of he CJ, E ad o i defie a parial orderig i CJ, E Obviouly, he oraliy of P iplie he oraliy of P C ad he oral coa of P C ad P are he ae For furher deail o coe heory, oe ca refer o [,8,9] Le J, E u : J E u ecod C J, E u : J E u coiuouly differeiable, C - order coiuouly differeiable I hi paper, we coider he oluio for he followig iiial value proble (IVP) of oliear ecod-order iegro-differeial equaio of ixed ype i ordered Baach pace E,,,, u, u, u f Fu, u Tu Su u x, u x, where J, x, x E, F C J E E E E E, E, ad () CJ J u u T, u d, S h, u d a,,

14 Turih Joural of Aalyi ad Nuber Theory 55 Le, C D, R, h, C D, R, D, R a, D, R, J J, R, For ay ax,, D, h ax h,, D B C J, E, J, le B u u B, TB Tu u B, SB Su u B The oluio for iiial value proble (IVP) of oliear fir-order iegro-differeial equaio of ixed ype i ordered Baach pace have ade coiderable headway i rece year (ee [,6]) Bu here ha bee lile dicuio for he oluio of (IVP) () I he pecial cae where f doe o coai u ad u, he oluio for iiial value proble (IVP) () i Baach pace have oe reul (ee [,5]) I aoher pecial cae where f doe o coai u, i [4], Su obaied oe ew reul by uig Möch fixed poi heore ad ew copario reul I hi paper, we fir eablih a ew copario heore, ad he, by requirig oly a lower oluio or a upper oluio ad oe weaer codiio,we iveigae he exiece of he iial or axial oluio of he (IVP) (), where f coai u, Su ad delayed argue u are ore exeive ha hoe i [,5] Several Lea uder he codiio which The followig copario reul ad lea play a ipora role i hi paper Lea (Copario heore) Aue ha E i a Baach pace, P i a coe i E, o J, ad, u u C J E aifie, u, u Mu Ku Nu L Tu, () u where M, K, N, L are o-egaive coa, ad provided oe of he followig wo codiio hold (i) M K a 6N La a 6, N Na Na L e M K N e Na Na (ii), L N ae a e M K Na N The u, u, J Proof For ay he, P, le p u J p u, p u, p u, Tp Tu, J Thu, by () we have ha, p p Mp Kp Np L Tp, J, p Le p p he p C J R, p p d Hece, we have ha,, ad p M L, rdr p d K p d Np, J, () p Now, we hall prove ha I he cae of codiio (i), if p, J p i o rue, he here i a, a uch ha ax p :, he If, he p,, (), we have p,, So, icreaig i,, we have p which coradic p If, he here exi a, p Fro (), we have p Le The, by p i, uch ha p M L d K N L M K N,, Thu, we have ha p p p d a L M K N d M K a La Na 6 The, by p, we have which coradic (i) I he cae of codiio (ii) holdig, le N p e M K a a 6, 6N La

15 56 Turih Joural of Aalyi ad Nuber Theory ad applyig i o (), by a iilar proce, we ca obai, J, ad o p, J Therefore, p, J, which iplie ha p p, J By he arbirarily of J we have u, u, Lea i proved Lea [] Le B CJ, E bouded, he B P, be couable ad L J, R, ad : J u( ) d u B B d Lea [] Le B CJ, E equicoiuou, le B coiuou o J ad J be couable ad, J, he () i Bd B d J Lea 4 [,6] Aue ha C J, R aifie a M d M d M d, J where M, M, M are coa The, J, provided oe of he followig wo codiio hold a (i) MaM am e M am, (ii) a M am am Mai Reul We li for coveiece he followig aupio (H ): (i) There exi u C J E J aifyig, u Fu, J, u x, u x (ii) There exi v C J E, aifyig v Fv, J, v x, v x (H ): (i) Wheever J ad,, ui vi i C J, E u, u, ui vi u v,,,,,,,,,,,, M u v K u v N u v LT u v f u u u Tu Su f v v v Tv Sv, (ii) Wheever J ad,, u v i Q C J, E u, u, ui vi, u v,,,,,,,,,,, M u v K u v N u v LT u v f u u u Tu Su f v v v Tv Sv, where M, K, N, L are o-egaive coa ad aify (i) or (ii) i Lea i i (H ): (i) There exi h CJ E ad J, aifyig Fu h (ii) There exi g CJ E J, aifyig Fu g,, for ay u,, for ay u Q ad (H 4 ): For ay couable bouded equicoiuou e B u C J, E ad J,,,,,, f B B B TB SB c B c B c B c TB c SB 4 5 where,,,5 c i are o-egaive coa i aifyig oe of he followig wo codiio: (i) ac h e a a c c c M K N al ac4 5 c i M K N al ac4, i (ii) aa c i 4M K N i al ac4 ac5h E Theore Le P be a oral coe ad o J Aue ha codiio H i, H i, H i ad H 4 hold, he IVP() ha a iial oluio u i Moreover, here exi oooe icreaig ieraive equece u uch ha u u aifyig u u x x uiforly o J, where, u, u, u, Tu, Su f M u u d, () K u u N u u LT u u,, Proof Fir, for ay u C J, E, ha () ha a uique oluio u CJ E Nex, by(), we have i i eay o prove,, u, u, u, Tu, Su f M u u u x K u u d, () N u u LT u u u x,,,,

16 Turih Joural of Aalyi ad Nuber Theory 57,,,,,, u x,,, u u u f u Tu Su M u u K u u N u u LT u u By () ad (H )(i), we have u u M u u K u u N u u LT u u, u u u u, u u u u, () ad by Lea, we ca obai u u, uu, J Tha i u u, u u Suppoe u, u, u u, u u, by () ad H i we have, u u, u u, u u M u u K u u N u u LT u u ad o, by Lea, we have u u, u u, J Tha i u u, ad u u u Fro he above, by iducio, i i o difficul o prove ha u u u u, (4) u u u u (5) By (), (4) ad (H )(i), we ow u u x x hd v, (6) J, ad o, by (), (5) ad (6), we have u u x h d, J (7) The, le B u : N, B u N :, by he oraliy of P ad (6) (7), we ow ha u, u are bouded equece i CJ, E For ay u-, by (H )(i) ad (H )(i), i i eay o ow ha f, u, u, u, Tu, Su i bouded A he ae, by () ad (), i i o difficul o how ha u, u are equicoiuou o J, Le B, B, J, ad by he uifor boudede of B() ad uifor,, h,, i i eay o how ha coiuiy of (TB)(), (SB)() are bouded ad equicoiuou Therefore, by Lea, we have a TB, r Brdr r dr, (8) SB h, r Brdr h r dr, (9) he, fro (), (), (8), (9), (H 4 ), Lea ad, C J, R, ad Lea, we ow B,,, B, B, f B TB SB a MB KB d NB LTB c c B c B c TB c SB a d 4 5 4a M K B d 4aN Bd 4aL TBd a cc d ac d ac4 d ac5h d 4a M K d 4aN d 4aL d a c c M K d ac 4aN d ac4 4aL d ac5h d Siilarly, we have B c c M K d c 4N d c4 4L d c5h d () ()

17 58 Turih Joural of Aalyi ad Nuber Theory ge Le r ax,, by (), (), we ca a r M r d M r d M r d, J, where M a ci M K N, i M a c4 L, M a ch Therefore, by Lea 4 ad he codiio (i) or (ii) i,, J (H 4 ), we ee r Ad o Hece B, B The BB, are relaively copac e i CJ, E Accordig o (4), (5) ad he oraliy of P, we ow u, u are coverge equece repecively i CJ, E Hece, here exi a u C J, E ha aifie u u, u u, By aig lii i () a, we have, u, u, u x x f u, Tu, d, J, Su o, u i a oluio of (IVP)() i If here exi a v ad v i alo a oluio of (IVP)() i, he v u, u u ad, v, v, v f, v, Tv, Sv v x, v x () By (), () ad (H )(i), uig iducio, we ca afely obai, u v u v,,, () Leig i () ad uig he oraliy of P, we have u u v v, Tha i, oluio of (IVP)() i The proof of he heore i coplee Theore Le P E u i a iial be a oral coe ad o J Aue ha codiio (H)(ii), (H)(ii), (H)(ii) ad (H4) hold, he IVP() ha a axial oluio v i Q Moreover, here exi oooe decreaig ieraive equece v Q uch ha v uiforly o J, where v v aifyig, v, v, f v, Tv, Sv M v, v d (4) K v v N v v LT v v v x x,, Proof The proof of Theore i alo he ae a ha of Theore, o we oi i Theore Le P E be a regular coe ad o J Aue ha codiio (H)(i), (H)(i) ad (H)(i) hold, he he reul i Theore hold Proof Accordig o he proof of Theore, we have (4), (5), by he regulariy of P, we ca obai ha u v, u v, uiforly o J, he re of he proof i iilar o he proof of Theore Theore 4 Le P E be a regular coe ad o J Aue ha codiio (H) (ii), (H) (ii) ad (H)(ii) hold, he he reul i Theore hold Proof By uig he iilar ehod of he proof of Theore, we ca ge he correpodig cocluio Rear I (IVP)(), if f doe o coai he delayed argue u, u ad he differeial argue u he Theore iplie he ai reul of [,6], bu he codiio i hi paper are ore exeive ha hoe of [,6] So he reul preeed i hi paper geeralize ad uify he reul of [,6] Rear I paper [], he auhor dicued he proble (IVP)() i which f doe o coai u, u ad aue he icreae of Tu Obviouly i hi paper, i he geeral cae, we coider he ecod-order iegrodiffereial equaio i which f coai u, u ad weae he icreae of u, u, Tu, Su ad we obai he iial ad axial oluio ad he ieraio equece of (IVP) () Moreover, he codiio (H4) i hi paper are ore exeive ha hoe i [] Therefore Theore iprove ad geeralize he reul i [] Rear We ca ee ha Theore i uiable for ay eaure of o-copace which i equal o he Kuraowi eaure of o-copace fro he proof of Theore Acowledge The auhor ha he referee for hi\her careful readig of he aucrip ad ueful uggeio Suppor Thi wor i uppored by he NNSF of Chia (No54) ad he Scieific ad Techological

18 Turih Joural of Aalyi ad Nuber Theory 59 Iovaio Progra of Higher Educaio Iiuio i Shaxi (No45) Referece [] D J uo, Iiial value proble for ecod-order iegrodiffereial equaio i Baach pace, Noliear Aalyi, 7(999): 89- [] Liha Liu, Ieraive ehod for oluio ad coupled quaioluio of oliear iegro-differeial equaio of ixed ype i Baach pace, Noliear Aalyi, 4(): [] D J uo, V Lahiaha, X Z Liu, Noliear iegral equaio i abrac pace, Kluwer Acadeic Publiher, Dordrech, 996 [4] Hua Su, Liha Liu, Xiaoya Zhag, Yoghog Wu, lobal oluio of iiial value proble for oliear ecod-order iegro-differeial equaio of ixed ype i Baach pace, JMah>Appl, (7): 9-5 [5] Fagqi Che, Yuhu Che, O oooe ieraive-ehod for iiial value proble of oliear ecod-order iegrodiffereial equaio i Baach pace, Appl Mah Mech, (5)(): [6] Liha Liu, The oluio of oliear iegro-differeial equaio of ixed ype i Baach pace, Aca Mah Siica, 8(6)(995): 7-7 (i Chiee) [7] S W Du, V Lahiaha, Moooe ieraive echique for differeial equaio i Baach pace, J Mah Aal Appl, 87(98): [8] D J uo, V Lahiaha, Noliear proble i abrac coe, Acadeic Pre,Boo ad New Yor, 988 [9] Daju uo, Noliear Fucioal Aalyi, d edio, Sciece ad Techology, Jia, [] RP Agarwal, D ORega, PJY Wog, Poiive Soluio of Differeial, Differece ad Iegral Equaio, Kluwer Acadeic Publiher, Dordrech, 999 [] KDeilig, Noliear Fucioal Aalyi, Spriger-Verlag, Berli, 985 [] TA Buro, Differeial iequaliie for iegral ad delay differeial equaio, i: Xizhi Liu, David Siegel (Ed), Copario Mehod ad Sabiliy Theory, i: Lecure Noe i Pure ad Appl Mah, Deer, New Yor, 994 [] Wag, L Zhag, Sog, Iegral boudary value proble for fir order iegro-differeial equaio wih deviaig argue, J Copu Appl Mah, 5 (9) 6-6 [4] Wag, Boudary value proble for ye of oliear iegro-differeial equaio wih deviaig argue, J Copu Appl Mah, 4 () 56-6 [5] Wag, L Zhag, Sog, Sye of fir order ipulive fucioal differeial equaio wih deviaig argue ad oliear boudary codiio, Noliear Aalyi,74 () [6] Wag, Moooe ieraive echique for boudary value proble of a oliear fracioal differeial equaio wih deviaig argue, J Copu Appl Mah, 6 () 45-4 [7] Wag, SK Nouya, L Zhag, Exiece of uliple poiive oluio of a oliear arbirary order boudary value proble wih advaced argue, Elecroic Joural of Qualiaive Theory of Differeial Equaio 5 () -

19 Turih Joural of Aalyi ad Nuber Theory, 5, Vol, No 6, 6-64 Available olie a hp://pubciepubco/ja//6/4 Sciece ad Educaio Publihig DOI:69/ja--6-4 D e ad Srucure-Preervig Map Jori N Buloro, Robero B Corcio,, Lora S Alocera, Michael P Baldado Jr Maheaic Depare, Cebu Noral Uiveriy, Cebu Ciy, Philippie 6 Sciece Cluer, Uiveriy of he Philippie - Cebu Maheaic Depare, Negro Orieal Sae Uiveriy Correpodig auhor: rcorcio@yahooco Received Sepeber 9, 5; Acceped Deceber, 5 Abrac Thi paper iveigae D e of group i relaio o rucure-preervig ap I how coecio bewee o-ivoluio of group ad he cocep of D e I paricular, we prove ha he exiece of a eigroup ioorphi bewee he failie of D e of wo group i equivale o a exiece of a pecial ype of bijecio bewee he ube coaiig all elee of order greaer ha wo of he group Keyword: D e, o-ivoluio, orphi Cie Thi Aricle: Jori N Buloro, Robero B Corcio, Lora S Alocera, ad Michael P Baldado Jr, D e ad Srucure-Preervig Map Turih Joural of Aalyi ad Nuber Theory, vol, o 6 (5): 6-64 doi: 69/ja--6-4 Iroducio The elee of a group of order wo play a very ipora role o oly i group heory bu i oher brache of aheaic, hey are ow a ivoluio We call elee of order greaer ha wo a oivoluio i hi paper The rucure called D e i coruced wih he cocep of ivere ad reveal oe properie relaed o ivoluio [7] I fac, a group ha oly oe D e if ad oly if i i a eleeary abelia -group A ube D of a group i a D e wheever every elee of o i D ha i ivere i D Thi paper how reul ha would lead o he copario of he uber of o-ivoluio of wo arbirary group We udy coecio of rucural-preervig appig bewee group ad heir correpodig D e failie We borrow cocep ad oaio fro e heory [5] X \ Y x X x Y i he Le X ad Y be e, he coplee of Y i X If f : X Y i a fucio wih A X he f A f a a A, called he iage of A i f The cardialiy of a e X i deoed by X We deoe he e of all ivoluio of a group ogeher wih he ideiy elee by S ; ha i, S x x e A D e D of group i a iiu D e if ad oly if he ivere of each x D \ S i o i D [] Noe ha for a fiie group, hi idea coicide wih he iiu D e eioed i [6] We wrie T a he faily of all D e of a group ad T i he ube coaiig all iiu D e [] I wa how i [7] ha T i a eigroup wih repec o uio of e We deviae a lile o dicu he oivaio of D e ad oe relaed lieraure The defiiio of D e i V, E be a baed o doiaig e of graph Le graph ad D V D i aid o doiae if for ay u V \ D, here exi v D uch ha u, v E (ee []) A eioed i [], a pecial ype of graph coruced fro a group wa iroduced by Kadaay ad Saradache [4] i 9 A ideiy graph of a orivial group i a udireced graph fored by adjoiig every o-ideiy elee o he ideiy e of ad x, y are coeced wheever xy e I view of ideiy graph of fiie group, he poi coaied i a iiu D e for a pecial ype of iduced ubgraph called ar [] Hece, we ca view T i a a faily of ar relaed o he group Reul We ar by howig how T ca be geeraed fro he correpodig T i Propoiio Le be a group The T i geerae T a a eigroup Moreover, if T, where T T i T X Y X, Y T i i

20 Turih Joural of Aalyi ad Nuber Theory 6 Proof: Le D be i T If D \ S he D S ad we oly have oe D e i hi cae Tha i, T Ti Aue D \ S ad deoe x D S x D \ Coider a oepy ube A of uch ha, for each x A, x \ A We oberve ha D ca be expreed a \ \ \ D D A D A where D \ A ad D \ \ A are i T i Thu, T i geerae T We rear ha a iiu D e cao be wrie a a uio of wo diic D e Le x be i The we wrie ad T x D T x D T i x D T i x D The followig lea i [7] give a cerai characerizaio of he ivoluio i Lea [7] Le x be a o-ideiy elee of a group The x i a ivoluio if ad oly if T x T The followig propoiio i a refiee of Lea Propoiio Le be a orivial group A o-ideiy elee x i i a ivoluio if ad oly if Ti x Ti Proof: Le x be a ivoluio i Sice T x T, he Ti x Ti Suppoe Ti x Ti Le D T ad by Propoiio, D D D where D ad D ad D are elee of T i By aupio, D are boh i T i x Hece, X D Thi ea ha T x T, ad by Lea, x i a ivoluio The propoiio below prove ha a ioorphi of failie of D e preerve he iialiy propery Propoiio Le ad H be group ad : T T H be a eigroup ioorphi The D i a iiu D e D i a iiu D e of H of if ad oly if Proof: The cae Aue i S i rivial Suppoe S D i i o The here D i iiu while exi a lea oe pair proof of Propoiio, here exi uch ha Di Dj D y, y boh i D i A i he D j ad D i T i H wih y Dj ad y D I follow ha here exi diic D j ad ha Dj Dj ad D D Di Dj D Dj D D i T uch Bu hi iplie ha ad o Di Dj D Thi i a coradicio o a rear followig Propoiio For he covere, uppoe D i i a iiu while D i i o There exi diic where D D D D j ad D i T i i j Hece, Di Dj D Dj D where Dj D, hi i aburd Propoiio 4 Le ad H be group ad : T T H be a eigroup ioorphi If D T i ad x \ D he D x D y where y H \ D Proof: Suppoe D i i T i o coaiig a elee x of The D x i a elee of T where xx, i he oly pair of ivere i hi D e A i he proof of Propoiio, D x D D \ x x where D \ x x hooorphic propery of iplie ha i alo i T i The D \ x D D \ x x where D, D \ x x Propoiio Furher, here u exi H \ S H where (WLO) i T i H by y, y D ad y D \ x x y i Suppoe here exi aoher elee z which hare he ae characeriic wih y We ay aue ha y ad y ad z are i D x x of he above argue, D z are i D \ x \ while A a coequece ca be expreed a D \ z z D \ x x D x D y y where he hree facor are diic elee of T i H By he urjecive propery of ad Propoiio, here exi D i ad Thi ea ha T uch ha D j i i Di D \ y y Dj D z z ad \

21 6 Turih Joural of Aalyi ad Nuber Theory i j D x D D D \ x x By he properie of, we have ad o i j \ D x D D D x x i j D x D D D \ x x ( ) Sice he hree facor o he righ hadide of equaio ( ) are diic elee of T i, we ge a lea wo pair of ivere Bu we oly have x ad x fro he lef hadide of ( ), hi i aburd Hece, y ad u be he oly pair of ivere i D D \ x x ad o D x D y y Le u ow ae ad prove he ai reul of hi paper Theore Le ad H be group wih \ S The T i ioorphic o T H if ad oly if here : \ S H \ S uch ha exi a bijecio x x for ay x i \ S Proof: Le : T T H be a ioorphi We for he : \ S H \ S uch ha bijecio x x for ay x i \ S Firly, we chooe a fix D i T i Le x be i \ S, he eiher x D or x D If x D, he he pair x ad H H x uique i D x By Propoiio 4, here exi a uique pair y y i \ H H S where D x D y We ca ow for y ad x y x x i ad we proceed a i he fir cae If x D he x D Therefore, if x \ S he here exi a uique y H \ SH uch ha x y ad x y We how ha i a ijecio by way of coradicio Suppoe ha a b i \ S uch ha ad a o a b Sice a i apped o a a we for where a H \ S H, he D j ad D i T i : If a D he D D; j If a D he D D \ a a ; If b D he D D; j If b D he D D \ b b Hece, we have he followig cae: Cae : a D ad b D b a Now, Dj a D D \ a a D b D D \ b b D b D D b b Dj a D D \ a a Cae : a D ad b D j \ \ D \ a D D \ a a D b D b b D D b D \ b b D Cae : a D ad b D Dj a D \ a a D D b D D \ b b j D a D \ a a D Cae 4: a D ad b D j D a D \ a a D \ D b D b b D Noe ha i ay of he cae above, Dj a D D D b D D ad, for oe D D T D, \ i Now, he oly pair of ivere i Dj a i a a while oly b b i D b Le y a, a \ D Sice a b, he y D \ D \ Hece, y D D Dj a D D D y D D D b Sice i ijecive, we have Dj a = D b ad Thi furher iplie ha a ad a are boh i D, hi i a coradicio To how ha i i urjecive, aue a elee y H \ S H Uig D i T H, eiher y D or y D If y D, he y D ad he pair y y i uique i D y Furher, D y D D \ y y

22 where D \ y y Turih Joural of Aalyi ad Nuber Theory 6 i a iiu D e of H By Propoiio 4 ad he ioorphi : T T, we have D y uique pair x x However, i H T which coai a D y D D \ y y D y D D \ y y where D \ y y i i i we ay have x D ad x D \ y y x T WLO, Thu, we ae y ad x y x i which D x D D \ y y \ D x D D y y D x D y O he oher had, give ha y D, he y D We proceed a above owig ha y y i he oly pair of ivere i D y followig he ae paer of reaoig, we will ill obai a uique pair x ad x fro \ S i which we ca wrie y ad x y x Hece, x i urjecive Suig up, we have he required bijecio For he covere, uppoe here exi a bijecio S H S : \ \ uch ha x x for H ay x i \ S We for a eigroup ioorphi : T TH Le D be i T, he D S X where X \ S We defie by D S X H where X i he iage of X wih repec o The verificaio ha i a ioorphi i a rouie We prove ore properie ivolvig orphi ad D e Propoiio 5 Le : H be a ooorphi of group ad H The D i a D e of H he D i a D e of ; i If By ii If D i a iiu D e of H he D i a iiu D e of D be a D e of H ad x \ D Sice Proof: (i) Le i ijecive, he x u o be i D By aupio, ha x D x (ii) Suppoe D (i), D i a D e of If x i i D Thi iplie i a iiu D e of H By par x D S he x D \ By aupio, x x D Thu, x ad hi prove our clai We oberve ha if T a igleo eigroup (ha i, \ S ) he he followig hold rue vacuouly Lea Le : H be a appig of group ad D H where \ S If a ioorphi : T T H ha he propery ha D y D y, for D T i ad y \ D, he y D Proof: Le D Ti, y \ D, ad : T T H be a ioorphi uch ha D y D y wih a above Fro he proof of Propoiio 4, - i H - - D y D D y y where D D y y i T Now we have D y D D y y, iplyig ha cao be i D Oherwie, we y - will ge D D D y y which i aburd Theore Le : H be a ooorphi of group ad H where \ S The here exi a uch ha ioorphi : T T H D y D y, for every D T i ad y \ D, if ad oly if S \ H \ SH Proof: Suppoe : T T H i a ioorphi uch ha D y D y, for every D T i ad y \ D If z S he z x for oe \ x S \ Thu, z x x z z Auig ha would iply x x which ea x x ice i ijecive Thi i a coradicio Hece, z u be i H \ S H Now, if z H \ SH he chooe a iiu D e of H, ay D o coaiig z By Propoiio 4 ad, D z D y for oe y D \ By propery of, D z D y D y

23 64 Turih Joural of Aalyi ad Nuber Theory I i ow evide ha A i Lea, y D \ For he covere, aue ha S z S \ H \ SH We ow have a bijecio : \ S H \ SH uch ha x x for all x \ S By Theore, we have he ioorphi : T T H defied by where D D S X H T wih Le D be i i D S X where X \ S D S X for oe X \ S T ad y \ D Suppoe The D y S X y SH X y Bu we have X y X y Coequely, D y SH X y S X y D y H Referece [] J N Buloro ad J M P Balaceda Cojugaio acio o he faily of iiu D e of a group MS Thei, Uiveriy of he Philippie - Dilia, 5 [] T Haye, S Hedeiei ad P Slaer Fudaeal of doiaio i graph (Marcel Deer, Ic, New Yor, 998) [] T W Hugerford Algebra (Spriger-Verlag, Ic, New Yor, 976) [4] V Kadaay ad F Saradache roup a graph (Ediura CuAr, Roaia, 9) [5] D C Kurz Foudaio of abrac aheaic (Sigapore: Mcraw-Hill Ic, 99) [6] C J S Roero, J M Oola, J N Buloro ad M P Baldado, Jr O he D e of fiie group Ieraioal Joural of Algebra 8 (4), 6-68 [7] C J S Roero, J M Oola, J N Buloro ad M P Baldado, Jr Soe properie ofd e of a group Ieraioal Maheaical Foru 9 (4), 5-4

24 Turih Joural of Aalyi ad Nuber Theory, 5, Vol, No 6, Available olie a hp://pubciepubco/ja//6/5 Sciece ad Educaio Publihig DOI:69/ja--6-5 Soe Fixed Poi Theore of Iegral Type Coracio i Coe b-eric Space Rahi Shah, Abar Zada, Ihfaq Kha Depare of Maheaic, Uiveriy of Pehawar, Pehawar, Paia Correpodig auhor: afeer_rahi@yahooco Received Ocober 6, 5; Acceped Deceber, 5 Abrac I he pree paper, we iroduce he cocep of iegral ype coracio wih repec o coe b-eric pace Alo we proved oe fixed poi reul of iegral ype coracive appig i coe b-eric pace We give a exaple o uppor our ai reul Keyword: coe b-eric pace, fixed poi, iegral ype coracive appig Cie Thi Aricle: Rahi Shah, Abar Zada, ad Ihfaq Kha, Soe Fixed Poi Theore of Iegral Type Coracio i Coe b-eric Space Turih Joural of Aalyi ad Nuber Theory, vol, o 6 (5): doi: 69/ja--6-5 Iroducio The udy of fixed poi heory play a ipora role i applicaio of ay brache of aheaic Fidig a fixed poi of coracive appig becoe he ceer of rog reearch aciviy There are oe reearcher who have wored abou he fixed poi of coracive appig ee [4,] I 9, Baach [4] preeed a ipora reul regardig a coracio appig, ow a he Baach coracio priciple Bahi i [] iroduced he cocep of b-eric pace a a geeralizaio of eric pace He proved he coracio appig priciple i b-eric pace ha geeralized he faou Baach coracio priciple i eric pace The cocep of coe eric pace wa preeed by Haug ad Zhag [5] i 7 They replace a ordered Baach pace for he real uber ad proved oe fixed poi heore of coracive appig i coe eric pace Huai ad Shah give he cocep of coe b-eric pace a a geeralizaio of b-eric pace ad coe eric pace i [6] Alo hey iproved oe rece reul abou KKM appig i coe b-eric pace I, Braciari [8] iroduced he oio of iegral ype coracive appig i coplee eric pace ad udy he exiece of fixed poi for appig which are defied o coplee eric pace aifyig iegral ype coracio Recely F Khojaeh e al [9], preeed he cocep of iegral ype coracio i coe eric pace ad proved oe fixed poi heore i uch pace May reearcher udie variou coracio ad a lo of fixed poi heore are proved i differe pace; ee [-7,9,,,,,7,8,] I he ai ecio of hi paper we preeed oe fixed poi heore of Iegral ype coracive appig i eig of coe b-eric pace Moreover, we pree uiable exaple ha uppor our ai reul Preliiarie The followig defiiio ad reul will be eeded i hi paper Defiiio [5] Le be a real Baach pace ad be a ube of The i called coe if ad oly if: ; (i) i cloed, oepy ad (ii) cp dq for all pq, where cd, are oegaive real uber; (iii) Defiiio [5] Suppoe be a coe i real Baach pace, we defie a parial orderig wih repec o by p q q p We hall wrie p q o idicae ha p q bu p q, while p q will ad for q p i Defiiio [5] The coe i called oral if here i uber K uch ha for all pq,, p q iplie p K q The lea poiive uber K aifyig he above iequaliy i called he oral coa of coe Throughou hi paper we alway uppoe ha i a real Baach pace, i a coe i wih i ad i parial orderig wr coe Defiiio 4 [5] Le Y be a o-epy e Suppoe ha he appig d : Y Y aifie: d u, v for all u, v Y wih u v; (d) (d) d u, v if ad oly if u v; (d) d u, v d v, u for all u, v Y (d4) d u, v d u, w d w, v ; for all u, v, w Y The d i called a coe eric o Y ad (Y, d) i called a coe eric pace

25 66 Turih Joural of Aalyi ad Nuber Theory Exaple 5 [5] Suppoe R, u, v u, v R, Y R ad d : Y Y uch ha d u, v u v, u v, where i a coa The Yd, i coe eric pace Defiiio 6 [6] Le Y be a o-epy e ad be a give real uber A appig d : Y Y i aid o be coe b-eric if ad oly if, for all u, v, w i Y, he followig codiio are aified: d u, v for all u, v Y wih u v; (i) (ii) d u, v if ad oly if u v; (iii) d u, v d v, u for all u, v Y (iv) d u, v d u, w d w, v ; for all u, v, w Y The d i called a coe b-eric o Y ad (Y, d) i called a coe b-eric pace Exaple 7 [4] Le d : Y Y R, u, v u, v R, p p uch ha d u v u v u v where ad,,, p are coa The, Yd i coe b-eric pace Lea 8 [5] Le Yd, be a coe eric pace ad a oral coe wih oral coa K Le u N be a equece i Y The coverge o u if ad oly if u N li d u, u Lea 9 [5] Le Yd, be a coe eric pace ad a oral coe wih oral coa K Le u N be a equece i Y The i a Cauchy equece if ad oly if, u N li d u, u Lea [5] Le Yd, be a coe eric pace ad a equece i Y If i coverge, u N u N he i i a Cauchy equece Lea [5] Le Yd, be a coe eric pace ad be a oral coe wih oral coa K Le u ad v be wo equece i Y ad u u, v v a The d u, v d u, v a I, Braciari i [8] iroduced a geeral coracive codiio of iegral ype a follow Theore [8] Le Yd, be a coplee eric pace,,, ad f : Y Y i a appig uch ha for all x, y Y, d d d f x, f y d x, y where :,, i oegaive ad Lebegueiegrable appig which i uable (ie, wih fiie iegral) o each copac ube of, uch ha for each, d, he f ha a uique fixed poi a Y, uch ha for each x Y, li f x a I [9], Khojaeh e al defied ew cocep of iegral wih repec o a coe ad iroduce he Braciari reul i coe eric pace We recall heir idea o ha he paper will be elf coaied Defiiio Suppoe ha i a oral coe i Le a, b E ad a b We defie a, b: x : x b a, for oe,, a, b : x : x b a, for oe, Defiiio 4 The e a x, x,, x b i called a pariio for ab, if ad oly if he e xi, x) are pairwie dijoi ad i i ) Defiiio 5 For each pariio P of, icreaig fucio : ab,, a, b x, x b i ab ad each we defie coe lower uaio ad coe upper uaio a Co L, P : xi xi xi i Co U, P : xi xi xi i repecively Defiiio 6 Suppoe ha i a oral coe i : ab, ab, i called a iegrable fucio o wih repec o coe or o ipliciy, Coe iegrable fucio, if ad oly if for all pariio P of ab, where Co Co Co li L, P S li U, P Co S u be uique We how he coo value b Co S by a Le,, fucio x d x iply by x d ab deoe he e of all coe iegreble Lea 7 [9] Le f g a b b a,,, The followig wo aee hold () If a, b a, c, he b c f d f d a a, f a b,, b b b () for f g d f d gd a a for, a Defiiio 8 [9] The fucio : i called ubaddiive coe iegrable fucio if ad oly if for all cd,