Proeedigs of he World Cogress o Egieerig Vol I WCE July 6-8 Lodo U.K. Coo Soluio of Noliear Fuioal Equaios via Ieraios Muhaad Arshad Akbar Aza ad Pasquale Vero Absra We obai oo fied ois ad ois of oiidee of a air of aigs saisfyig a geeralized oraive ye odiio i oe eri saes. Our resuls geeralize soe well-kow ree resuls i he lieraure. Ide ers-- We obai oo fied ois ad ois of oiidee of a air of aigs saisfyig a geeralized oraive ye odiio i oe eri saes. Our resuls geeralize soe well-kow ree resuls i he lieraure. I. INRODUCION AND PRELIMINARIES A large variey of he robles of aalysis ad alied aheais redue o fidig soluios of o-liear fuioal equaios whih a be forulaed i ers of fidig he fied ois of a oliear aig. I fa fied oi heores are very iora ools for rovig he eisee ad uiqueess of he soluios o various aheaial odels (differeial iegral ad arial differeial equaios ad variaioal iequaliies e. rereseig heoea arisig i differe fields suh as seady sae eeraure disribuio heial equaios euro rasor heory eooi heories fiaial aalysis eideis bioedial researh ad flow of fluids. hey are also used o sudy he robles of oial orol relaed o hese syses []. Fied oi heory oered wih ordered Baah saes hels us i fidig ea or aroiae soluios of boudary value robles []. I 96 S. Ghaler geeralized he idea of eri sae ad irodued -eri sae whih was followed by a uber of aers dealig wih his geeralized sae. A lo of aerials are available i oher geeralized eri saes suh as sei eri saes quasi sei eri saes ad D- eri saes. Huag ad Zhag [6] irodued he oe of oe eri sae ad esablished soe fied oi heores for oraive ye aigs i a oe eri sae. Subsequely soe oher auhors [ 4 5 7 8 ] sudied he eisee of fied ois ois of oiidee ad oo fied ois of aigs Mausri reeived Marh 5 aeed Marh. his work was suored by Higher Eduaio Coissio (HEC Pakisa. M. Arshad is wih he Deare of Mahs ad Sas Ieraioal Islai Uiversiy H- Islaabad 44 Pakisa (Phoe: +9-5984 Fa: +9-5-6877 e-ail: arshad_zia@yahoo.o. A. Aza is wih he Deare of Maheais COMSAS Isiue of Iforaio ehology Chak Shahzad Islaabad 44 Pakisa (eail: akbaraza@yahoo.o. P. Vero is wih he Uiversià degli Sudi di PaleroDiarieo di Maeaia ed AliazioiVia Arhirafi 4 9 Palero Ialy (e-ail: vero@ah.uia.i. saisfyig a oraive ye odiio i oe eri saes. I his aer we obai ois of oiidee ad oo fied ois for a air of aigs saisfyig a ore geeral oraive ye odiio. Our resuls irove ad geeralize soe sigifia ree resuls. A subse P of a real Baah sae E is alled a oe if i has he followig roeries: (i P is o-ey losed ad (ii a b R ad P {}; y P a by P; (iii P ( P = {}. For a give oe P E we a defie a arial orderig o E wih rese o P by y if ad oly if y P. We shall wrie < y if y ad y while << y will sads for y i P where i P deoes he ierior of P. he oe P is alled oral if here is a uber κ suh ha for all y E y κ y. ( he leas uber κ saisfyig ( is alled he oral osa of P. I he followig we always suose ha E is a real Baah sae ad P is a oe i E wih i P ad is a arial orderig wih rese o P. Defiiio Le X be a oey se. Suose ha he aig d : X X E saisfies: (i y for all y X ad d ( y = if ad oly if y ; (ii d ( y = y for all y X ; (iii d ( y z + z y y z X. he d is alled a oe eri o X ad ( d for all X is alled a oe eri sae. Le be a sequee i X ad X. If for ISBN: 978-988-8-6-5 ISSN: 78-958 (Pri; ISSN: 78-966 (Olie WCE
Proeedigs of he World Cogress o Egieerig Vol I WCE July 6-8 Lodo U.K. eah << here is N suh ha for all << he is said o be overge or { } overges o ad is alled he lii of. We deoe his by li = or as. If for eah << here is N suh ha for all << he is alled a Cauhy sequee i X. If every Cauhy sequee is overge i X he X is alled a olee oe eri sae. Le us reall [5] ha if P is a oral oe he X overges o X if ad oly if d as. Furherore X is a Cauhy sequee if ad oly if d ( as. Lea Le ( X d be a oe eri sae P a be a be a oe. Le { } be a sequee i X ad { } sequee i P overgig o. If a for every N wih is a Cauhy sequee. Proof. Fi << ad hoose I( δ = { E : < δ} suh ha > he { } + I( δ IP. Sie a here eiss N be suh ha a I for every. Fro a IP we dedue a << for every ad hee is a Cauhy sequee. Reark Le A B C D E be o egaive real ubers wih A + B + C + D + E < B = C or D E. If F = ( A + B + D( C D ad G = ( A + C + E( he FG <. I fa if B = C he A + B + D A + C + E FG = C D A + C + D A + B + E = < C D ad if D = E A + B + D A + C + E FG = C D A + B + E A + C + D = <. C D A PAIR ( f OF SELF-MAPPINGS ON X ARE SAID O BE WEAKLY COMPAIBLE IF HEY COMMUE A HEIR COINCIDENCE POIN (I.E. f = WHENEVER =. A POIN y X IS CALLED POIN OF COINCIDENCE OF AND f IF HERE EXISS A POIN X SUCH HA y = =. II. MAIN RESULS he followig heore iroves/geeralizes he resuls [ heores...4] [4 heores ] [6 heores 4] [7 heore.8] [ heores..6.7.8] ad [ heores Corollary ]. heore 4 Le ( X d be a olee oe eri sae P be a oe ad be osiive iegers. Assue ha he aigs f : X X saisfy: y α fy + β[ + fy y] + γ[ y + fy ] for all y X where α β γ are o egaive real ubers wih α + β + γ <. If ( X f ( X ad f (X is a olee subsae of X he ad f have a uique oo oi of oiidee. f f are weakly Moreover if ( ad ( oaible he ad f have a uique oo fied oi. Huag ad Zhag [6] roved he above resul by resriig ha (a P is oral (b f = I X ( = = (d oe of he followig is saisfied : i. α < β = γ = ( [6 heores ] ii. β < α = γ = ( [6 heores ] iii. γ < α = β = ( [6 heores ]. Abbas ad Jugk [] eeded he resuls of Huag ad Zhag [6] by reovig resriio (b ad obai oo fied ois ad ois of oiidee of aigs f. Meawhile Rezaour ad Halbarai [] iroved he resuls of [6] by oiig he assuio (a. Vero [] reoved resriio (b ad relaed (d by obiig (i ad (ii. Aza Arshad ad Beg [4] ad Jugk e al [7] ISBN: 978-988-8-6-5 ISSN: 78-958 (Pri; ISSN: 78-966 (Olie WCE
Proeedigs of he World Cogress o Egieerig Vol I WCE July 6-8 Lodo U.K. eeded hese resuls o a geeralized oraive odiio by oiig he resriios (a (b. he followig heore is a furher geeralizaio of heore 4 whih reoves resriios (a (b ( ad relaes (d wih a ore geeralized oraive odiio. heore 5 Le ( X d be a olee oe eri sae P be a oe ad be osiive iegers. If he aigs f : X X saisfy: y A fy + B + C fy y + D y + E fy for all y X where A B C D E are o egaive real ubers wih A + B + C + D + E < B = C or D = E. If ( X f ( X ad f (X or (X is a olee subsae of X he ad f have a uique oo oi of oiidee. Moreover if ( f ad ( f are weakly oaible he ad f have a uique oo fied oi. Proof. Le be a arbirary oi i X. Choose a oi i X suh ha =. his a be doe sie ( X f ( X. Siilarly hoose a oi i X suh ha =. Coiuig his roess havig hose i X we obai i X suh ha k. he d ( + + + = d ( + Ad ( + + Bd ( + Cd ( + + + Dd ( + + Ed ( + [ A + B] d ( + + Cd ( + + + D d ( + [ A + B + D] d ( + [ C + D] d ( + + +. I ilies ha [ C D] ha is where + [ A + B + D] + + d Fd A + B + D F =. C D Siilarly we obai = + whih ilies wih + + [ A + C + E] + [ B + E] + +. + + + d ( k + + G + + A + C + E G =. k = Now by iduio we obai for eah + ( FG + F( FG F k F( FG ad G + + + + q F i By Reark < q we have + q+ + + + + + k+ ( FG. i i ( FG + ( FG + F( FG ( FG + FG FG + = + + FG ( + F FG q + + + 4 i= + q+ q. ISBN: 978-988-8-6-5 ISSN: 78-958 (Pri; ISSN: 78-966 (Olie WCE
Proeedigs of he World Cogress o Egieerig Vol I WCE July 6-8 Lodo U.K. I aalogous way we dedue FG q+ ( + F FG d ad d FG q ( + F FG ( FG + q ( + F. FG ( Hee for < < a FG where a = ( + F FG wih he ieger ar of /. Fi << ad hoose I( δ = { E : < δ} suh ha + I( δ IP. Sie a as by Lea we dedue ha { } is a Cauhy sequee. If f (X is a olee subsae of X here eis u v X suh ha v = fu (his holds also if X is olee wih v (X. Fi ad hoose N be suh ha v << k << v k for all where + D k = a Now fu So << k A + E C. u fu + ( + D fu + ( A + E fu + C + ( B + E fu u. u fu Hee u k fu + k fu + k << + + = fu u << for every N. Fro fu u IP beig P losed as fu u P ad so fu u = ilies ha fu = u. we dedue. his Siilarly by usig he iequaliy fu u fu + + + u we a show ha fu = u whih i ur ilies ha v is a oo oi of oiidee of ad f ha is v = fu = u = u. Now we show ha f ad have a uique oo oi of oiidee. For his assue ha here eiss aoher oi v i X suh ha v = fu = u = u for soe u i X. Fro v v we obai ha. = u u A fu fu + B fu u + C fu u + D fu u + E fu u ( A + D + E v v v = v f ad ( f are weakly oaible he v = fu = Moreover ( u = fv ad v = fu = f u = fv whih ilies v = v = fv = w (say. he w is a oo oi of oiidee of herefore v = w by uiqueess. hus v is a uique oo fied oi of f ad f. { } R ad f Eale 6 Le X = E = ad P = { y E : y ( }. ISBN: 978-988-8-6-5 ISSN: 78-958 (Pri; ISSN: 78-966 (Olie WCE
Proeedigs of he World Cogress o Egieerig Vol I WCE July 6-8 Lodo U.K. Defie : R d X X as follows: y = 5 ( 7 4 ( 4 8 ( 7 Defie he aigs if = y if y ad y X {} if y ad y X {} if y ad y X {}. f : X X as follows: f = ( if ( = if =. Noe ha ( = for eah X 5 d ( ( ( =. 7 he if α + β + γ < we have 4α + 9β + 7γ 8α + 8β + 4γ 7 5α + β + γ α + β + γ < 7 5( α + β + γ ( α + β + γ 7 5 < = ( (. 7 his ilies αd + γ = αd + γ f ( ( f ( f ( + β + f ( f ( ( + f ( ( + β[ + ] [ + ] ( ( [ (] 5 < = ( ( 7 for all α β γ [ wih α + β + γ <. herefore heore 4 ad is orollaries ( heores...4] [4 heores ] [6 heore.8] [ heores..6.7.8] ad [ heore Corollary ] are o aliable. Fro if y y = 5 if y = ad ( 7 A fy + B + C fy y 5 + D y + E fy = 7 5 for y = ad A = B = C = D = E = 7 i follows ha all odiios of heore 5are saisfied for 5 A = B = C = D = E = 7 ad so ad f have a uique oo oi of oiidee ad a uique oo fied oi. Corollary 7 Le ( X d be a olee oe eri sae P be a oe ad be osiive iegers. If a aig : X X saisfies: + C y y A y + B y + D y + E y for all y X where ABCDE are o egaive real ubers wih A + B + C + D + E < B = C or D = E. he has a uique fied oi. Proof. By heore 5 we ge X suh ha = =. he resul he follows fro he fa ha = + D = A + B + E + C A + B + C + D + E = ( A + D + E whih ilies =. Eale (Aliaios 8 Le X = C([] R E = R a > ad y = su [ ] for every y X ad ( y( [ ] ( u v : u }. a su ( y( P = { R v I is easily see ha ( X d is a olee oe eri sae. Defie : X X by For y X ( ( 4 + ( u = u ( + u e du. ISBN: 978-988-8-6-5 ISSN: 78-958 (Pri; ISSN: 78-966 (Olie WCE
Proeedigs of he World Cogress o Egieerig Vol I WCE July 6-8 Lodo U.K. su [ ] y = a su [ ] ( y( ( y( su ( ( u y( u e du [ ] a su ( ( u y( u e du [ ] = e y. Siilarly y e y.! Noe ha 9 if = 987 if = 4 e =!. if = 7.5 if = 8. hus for A = B = C = D = E = = all.5 = 8 odiios of Corollary 7 are saisfied ad so has a uique fied oi whih is he uique soluio of he iegral equaio: ( 4 + ( u = or he differeial equaio: u ( + u e du. ( = ( + e [] ( = 4. Hee he use of Corollary 7 is a delighful way of showig he eisee ad uiqueess of soluios for he followig lass of iegral equaios: b + K( ( u u du = ( C([ a b] R a. eri saes J. Mah. Aal. Al. 4 (8 46-- 4. [] H. Aa Fied oi Equaios ad o-liear eige-value robles i ordered Baah saes SIAM Review 8 (976 6--79. [] M. Arshad A. Aza ad P. Vero Soe oo fied oi resuls i oe eri saes Fied Poi heory Al. 9 (9. Arile ID 49965. [4] A. Aza M. Arshad ad I. Beg Coo fied ois of wo as i oe eri saes Red. Cir. Ma. Palero 57 (8 4--44. [5] C. Di Bari ad P. Vero Weakly -airs ad oo fied ois i oe eri saes Red. Cir. Ma. Palero 58 (9 5--. [6] L.-G. Huag ad X. Zhag Coe eri saes ad fied oi heores of oraive aigs J. Mah. Aal. Al. (7 468--476. [7] G. Jugk S. Radeovi S. Radojevi ad V. Rakoevi Coo fied oi heores for weakly oaible airs o oe eri saes Fied Poi heory Al. 9 (9. Arile ID 6484. [8] D. Kli. Wardowski Dyai roesses ad fied ois of se-valued oliear oraios i oe eri saes Noliear Aalysis (9 doi:.6/j.a.9.4.. [9] V.Poa A geeral fied oi heore for weakly oaible aigs i oa eri saesurk J. Mah.5 ( 465-474. [] S. Rezaour ad R. Halbarai Soe oes o aer "Coe eri saes ad fied oi heores of oraive aigs." J. Mah. Aal. Al. 45 (8 79--74. [] N. Shahzad Fied ois for geeralized oraios ad aliaios o orol heory Noliear Aalysis 68 (8 (8 867--876. [] P. Vero Coo fied ois i oe eri saes Red. Cir. Ma. Palero 56 (7 464--468. [] D. Wardowski Edois ad fied ois of sevalued oraios i oe eri saes Noliear Aalysis 7 (9 5--56. REFERENCES [] M. Abbas ad G. Jugk Coo fied oi resuls for o ouig aigs wihou oiuiy i oe ISBN: 978-988-8-6-5 ISSN: 78-958 (Pri; ISSN: 78-966 (Olie WCE