About solutions of nonlinear algebraic system with two variables

Transcription

1 Pue a pple Matheatc Joual 3; ( : 3-37 Publhe ole Febuay 3( o:.648/j.paj.3.5 bout oluto of olea algebac yte wth two vaable Rahhaa Dzhabazaeh İttute Matheatc a Mechac of NN of zebaja au Eal ae: ahhaa.zhabazae@able.u (R. Dzhabazaeh To cte th atcle: Rahhaa Dzhabazaeh. bout Soluto of Nolea lgebac Syte wth Two Vaable. Pue a pple Matheatc Joual. Vol. No. 3 pp o:.648/j.paj.3.5 btact: Fo olea algebac yte wth two vaable uffcet coto of extece of oluto ae gve. The poof of thee tateet eceve a a coollay of oe coo evewg coee th pape. I patcula th wo the extece of ultple bae o ege a aocate vecto of a two paaete yte of opeato fte-eoal pace pove. Defto of the aocate vecto ultple copletee of ege a aocate vecto of two-paaete ot elfajot yte olealy epeg o pectal paaete ae touce. t the poof of thee eult we eetally ue the oto of the aalog of a eultat of two polyoal bule. Keywo: lgebac Spectal Reultat Nolea. Itoucto The pectal theoy of opeato oe of eetal ecto of a fuctoal aaly. May patal equato a the equato of atheatcal phyc coecte wth phycal pocee eae the ew appoach fo a oluto of thee poble. The etho of a epaato of vaable fo a oluto of the equato wth patal evatve ofte appea the ot copeheble a euce oluto of the coplcate equato wth ay vaable to a oluto of yte of the oay ffeetal equato whch eeach uch eae. So fo exaple wth help of ultpaaete theoy poble of a quatu echac [756] theoe of a ffacto [9] theoe of elatc evelope calculato of uclea eacto [4] equlbu pocee of ffuo type [3] a owa oto [3] bouay value poble fo the equato of ellptc-paabolc type a Cauchy poble fo the ulta paabolc equato [] etc. ae olve. Depte of a ugecy a pecpto of thee eeache the pectal theoy of ultpaaete yte tue uffcetly. valable outcoe th aea utl ecetly ae obtae oly fo ultpaaete elfajot yte of the opeato lealy epeg o pectal paaete. The foue of eeache of pectal poble of the ultpaaete elfajot yte wa F.V. to []. Stue the outcoe whch ae avalable fo ultpaaete yetcal ffeetal yte to ha cotucte the pectal theoy of elfajot ultpaaete yte fte-eoal pace. Futhe by ea of paage to the lt to ha geealze the eceve outcoe o a cae of ultpaaete yte wth the elfajot copact opeato fte-eoal Hlbet pace. I the futhe t ha appeae poble to bul the cotucto of to fte-eoal pace. It ha allowe cotuct the pectal theoy of ultpaaete yte Hlbet pace [3 8] wth help a eg touce by to fo tuy of ultpaaete yte fte-eoal pace ut ufotuately the techque of eeach thee wo eetally ue oly elfajot opeato eteg to ultpaaete yte of opeato. Fo ot elfajot ultpaaete yte the vetgate techque oe ot allow to olve the plet poble of the pectal theoy of opeato. The autho offe the ew appoach to eeach of ultpaaete poble. I wo [5 6] cocept of bae ultple bae eetal to the pectal theoy of ultpaaete opeatoal yte of cocept of the aocate vecto copletee a ultply copletee of yte of ege a aocate

2 Pue a pple Matheatc Joual 3 ( : vecto ae touce.. Multpaaete Spectal Theoy a It pplcato to Nolea lgebac Syte Fo the tateet of extece of a oluto of a( x y a a x... a x a y... µ a y ` b ( x y b b x... b x b y b y... ` we eetally ue outcoe of autho coceg the ultpaaete yte publhe wo [4-6]. I the gve atcle the ultply bae popety of yte of ege a aocate vecto of two paaete yte ( µ x (... µ... µ x ` ( µ y (... µ... µ y ` vetgate. elow we hall euce thee outcoe a they have epeet eag. etablhe teo pouct H of fte-eoal paceh a H Deo of pace H the pouct of eo of pace H a H. I ( lea opeato (... act fte-eoal pace H ; a lea opeato (... act fte-eoal paceh. If f f H a g g H H the e pouct of thee eleet pace H H efe f f g g ] f g ( g g [ by ea of the foulae ( H H H H Th efto pea to othe eleet of teo pouct pace o leaty. Let' euce a ee of ow poto coceg the pectal theoy of ultpaaete yte. Defto. ([ 3] ( µ C a ege value of the yte ( f thee ae ozeo eleet x H y H uch that ( fulflle. teo z x y ae a egevecto of yte (. Defto. opeato E (accogly E whee E (accogly E etcal opeato H (accogly H ae opeato uce H H by opeato (accogly. ( ( Defto3.[5] teo ocate vecto to a egevecto g coto ae atfe z ae ( - the a- z x y f follow- ( µ z!! µ ( µ z (3!! µ ; ;. ( aageet fo et of the whole oegatve ube o wth poble ecug a zeo. Defto4. Syte of eleet { x } (... of fte-eoal pace H H fo -fol bae th pace f ay of eleet f... f of paceh H ca be pea out ee f c x (... wth coeffcet ot epeet o a ex of vecto f... f If yte{ x } (... ae cotucte o{ x }. wth by the ceta ule equalte f c x (... ea extece of -fol bae o yte of ege a aocate vecto of the yte (. Defto5.[5]. Let be two polyoal bule... ( (... epeg o the ae paaete a actg geeally peag vaou Hlbet pace H H accogly. The opeato Re ( ( ( ue cotucto (4 ( ( (5 [7] ae abtact aalog of a Reultat fo polyoal bule (4. I efto of a Reultat (5 of bule (4 le wth opeato epeate of te a le wth оpeato epeate exactly of te. thee ae the hghet oe of paaete bule of ( ( ( accogly. Thu the Reultat a opeato actg pace ( H H that thee a ect u

3 34 Rahhaa Dzhabazaeh: bout oluto of olea algebac yte wth two vaable of cope of teo pouct pace H. H Value of Re ( ( ( equal to t foal expao whe each te of th expao teo pouct of opeato. Let opeato o vetble. If the hghet oe of paaete bule of ( a ( coce (ee [7] o f the hghet oe of paaete bule of ( a ( geeally peag ca ot coce (ee [] bule (4 have a coo pot of pecta the oly cae whe the Reultat of thee bule ha ozeo eel. It atual that cae of whe bule act fte-eoal pace th coo pot of pecta of thee bule a coo ege value of thee bule. Theoe. Let opeato (... alo (... act fte-eoal pace H a H accogly a oe of thee followg coto fulflle: a ax( Ke { θ } Ke { θ} ; ae elfajot opeato eveyoe the pace b ax( Ke θ Ke { } { θ} opeato ae elfajot eveyoe the pace c Ke( ( { } θ ae the elfajot opeato actg eveyoe t fte-eoal pace. The the ax( -fol bae o yte of ege a aocate vecto of (4 tae place. Poof. It fxe oe of paaete (4. Fo a eteacy we hall uppoe that t paaete a.the t eceve two bule each of whch epe o oe paaete µ. ( µ... µ... µ ` ` ( µ... µ... µ ` We touce label (... (6 (... The a Reultat of bule ` ( µ ( µ... µ ` ( µ ( µ... µ epeg o oe paaete µ loo le: ( µ ( ( ( ( ( ( ( ` I (7 le wth opeato ae epeate of te a le wth opeato ae epeate of te. Thu the Reultat ( µ ( a opeato pace ( H that thee a ect u of cope of teo pouct pace H H. Fo outcoe of wo [] follow that pecta of opeato ` ( µ... µ... µ ` ( µ... µ... µ ` teect f a oly f ( ( { θ} ( o KeR. opeato ( µ ( µ act fte-eoal pace th coo pot of the pecta ca be oly coo ege value of thee bule. The lat ea that the two paaete yte ( ha a ege value ( µ (. If the eleet... z ( x x x ( H belog to the eel of a Reultat Re ( ( µ ( the ( x x... x ( x x... x 3 ( x ( x x... x x... x ( x ( x x... x 3 x... x y equetal elato of eleet x H (7 (8

4 Pue a pple Matheatc Joual 3 ( : x H H fo yte of equalte (8 we coe to oe equato teo pouct paceh H. To the la equato we coe by epeetato of value of Reultat Re ( ( (. y vtue of bule of the eceve equato opeatoal coeffcet at egee we hall egate though C (...ax(. Let' touce a label ax(.thu we coe to oe equato C( x C x C x... C x actg a teo H H. the paaete pouct pace fxe abtaly we coe a bule C( x C x C x... C x (9 fte-eoal paceh H. Fo outcoe of the pectal theoy of opeato follow that the eceve bule (9 fte-eoal pace ha -fol bae o yte of t ege a aocate vecto. Really at ealzato of coto Theoe ae atfe all coto of the Theoe of Kelyh [] coceg ultply copletee of yte of ege a aocate vecto of polyoal bule C ( C C... C The theoe of Kelyh followg: let T elfajot copletely cotuou opeato of the fte oe ( σ { θ} KeT opeato p TT (... copletely cotuou opeato the -fol copletee of yte of ege a aocate vecto of the opeatoal bule T ( T T... T actg a Hlbet pace tae place. Let' e that ue a ege value of a polyoal bule T ( uch the coplex ube uetoo that fo t ext uch ozeo vectox that T ( x x -th aocate vecto x to a egevecto x the vecto atfyg to coto x T ( x T ( x... T ( x ;...! Let{ x } be a yte of ege a aocate vecto of a bule T (the evatve yte of vecto o yte of ege a aocate vecto of polyoal bule T ( ae cotucte by ule x j t j e (x t t t x... x!! j t... T I cae of fte-eoal pace all coto of the Theoe of Kelyh ae fulflle coequetly yte of ege a aocate vecto of a bule C ( C C... C fo -fol bae pace H H. Really at ealzato of a coto a of the Theoe coeffcet at a opeato C. It eay to ee that t elfajot opeato a Ke ( { θ} ; t ealzato of coto b of the Theoe t ha that Ke { θ} opeato elfajot a at lat at ealzato of coto c of the Theoe the opeato ( elfajot a Ke( { } ( θ alo. I all thee thee cae at coeffcet at of bule C ( elfajot vetble opeato. 3 Rea to pove cocuece of yte of ege a aocate vecto of a two-paaete poble ( a yte of ege a aocate vecto of the polyoal bule (9 eceve by the above-tate way teo pouct paceh H. Letx be a egevecto of the equato (9. Suppog x x a ug that fact that (9 thee a expao of a Reultat (4 we etee vecto x x... x 3. ee fo (8 t ha that a eleet ( x x x... 3 x ( H ete to the eel of a Reultat ( µ ( of bule ( µ a ( µ. I wo [5 6] the ubequet copoet of the eleet ( x x x... x eteg to the eel of Reultat ( ( ae fo the ft te well eough tue. I patculate cae of whe opeatoal bule act fte-eoal pace a epe o the ae paaete the apect of the ft copoet x of the eleet x x 3... x ( H H ete- ( x g to the eel of a Reultat Re ( ( µ ( cotucte by the followg ae : x ha the fo lea cobato of eleet U V U V... U V whee U...U (accogly V...V a etcte cha of ege a aocate vecto of the bule ( µ (accogly ( µ at the fxe value of paaete. The paaete µ beg oe coo ege value µ both of opeato ( ( µ a ( µ. U...U (accogly V...V yte of ege a aocate vecto of a opeato ( µ (accogly ( µ a equatg to the coo ege value µ (. The eco a followg copoet of the eleet

5 36 Rahhaa Dzhabazaeh: bout oluto of olea algebac yte wth two vaable ( x x x... x 3 coce wth the eleet of evatve yte cotucte o the ft copoet U V U V... U V. Ege value ( µ ( of twopaaete yte ( ca be eveal a all they have the ft cooate ube. The fo of the ft copoet of eleet eteg the eel of the eultat of a two polyoal bule fo oe paaete actg fte-eoal pace aouce []. Fo two-paaete yte Hlbet pace at ealzato of oe coto the apect of the ft cooate x of a eleet ( x x x 3... x ( H eteg to the eel of a Reultat Re ( ( µ ( aouce [5]. We hall how that each egevecto of the equato C ( x the ft copoet of a eleet eteg to the eel of Reultat of bule ( µ a ( µ. x x Really coeg we have that the left pat of the equato C( x epeet expao of a Reultat (7 o a vectox. Covetg to efto of ege a aocate vecto of two-paaete yte fo (9 we cove that the eleet x x ca be a egevecto of two-paaete yte of opeato (3 a equatg to a ege value ( µ ( o the aocate vecto of th yte (3 a equatg to oe ege value µ ( whch ef- ( to thee o evato o the paaete. Such aocate vecto of two-paaete yte woul be expeet fo ag the aocate vecto of two-paaete yte a ecto µ. Slaly aocate vecto whch efto thee o evato o paaete µ we hall ae the aocate vecto of two-paaete yte a ecto. Thu all egevecto of the equato (9 coepog to ege value ae ethe ege vecto o aocate vecto o ectoµ of the twopaaete yte coepog to a ege value ( µ (. Rea to pove that all aocate vecto of (9 alo be the aocate vecto of a two-paaete poble (. y thee a ft aocate vecto to a egevecto Let x of the equato (9 coepog to t ege value. The we have C( ɶ C( ɶ xɶ (Cɶ C ɶ... C ɶ Cɶ x ɶ... Cɶ x ɶ (ax( ( C( x Cx Cx... C x ( Equato ( ea that thee ozeo eleet... ( x x x H eteg to the eel of Reultat of bule µ µ µ ` ( ` µ µ µ ( ` uch that (8 fulflle. It clea that all ege vecto of (6 coce wth the -t copoet of eleet fo the Ke Re ( ( µ ( µ.we have the copletee of ege a aocate vecto of the Re ( ( µ ( pace( H H If( (... y y y y H the -t aocate vecto to ege vecto... ( x x x H of the opeato Re ( ( µ ( the we have that the followg expeo ( a equalte (8 ae fulflle ɶ ( ɶ y ɶ... ɶ ( ɶ xɶ ɶ ( ɶ y ɶ... 3 ɶ ( ɶ xɶ ɶ ( ɶ y ɶ... ɶ ( ɶ xɶ ɶ ( ɶ y ɶ... ɶ ( ɶ xɶ ɶ ( ɶ y ɶ... ɶ ( ɶ xɶ ( ɶ ( xɶ ɶ x ɶ... ɶ xɶ (3 ( x x... x a

6 Pue a pple Matheatc Joual 3 ( : ( xɶ ɶ x ɶ... ɶ xɶ x x... x ( The lat ea that y thee the aocate vecto of x two-paaete yte coepog to egevecto wth ege value ( µ (. Fo (8 we obta (. Fo ( a (3 we obta (.y aalogy the age t poble to pove that the aocate vecto of othe oe of the equato (9 coepog to a ege value alo ae the aocate vecto of two-paaete yte (. t each ege value of the equato (9 coepog to ege a aocate vecto of (9 coce wth ege a aocate vecto of the two-paaete yte ( coepog oe ege value ( µ ( of yte (. Ege vecto of the equato (9 egevecto of ( o t aocate vecto o ecto µ. ocate vecto of the equato (9coepog to ege value ae alo aocate vecto of the yte (. Eale we pove that the yte of all ege a aocate vecto of the equato (9 fo the ax( -fol bae of pace H. The the yte of ege a aocate vecto of two-paaete yte ( alo fo ax( -fol bae paceh H. The Theoe pove. I the pecal cae whe H H R a opeato (... ;... ae eal ube. If to coe vaablex a y paaete the the algebac yte ( a pecal cae of ultpaaete yte (. Coollay. Let oe of thee followg coto fulflle: a ax( a b ; b ax( b a c a ( b a b The the algebac yte ( ha the oluto. 3. Cocluo I th pape t pove the extece of oluto of oe olea algebac yte.fo the poof of th t eetally ue the eult of autho obtae fo the ultpaaete yte of opeato the fte eoal pace Refeece [] to F.V. Multpaaete pectal theoy. ull.e.math.soc [] al.i Geeato of oto of ezutat a Reultat DN of U SSR e.ph.-ath a tech. of cece98 pp.3-6( Rua. [3] owe P.J. Multpaaete pectal theoy. Iaa Uv. Math. J [4] Dzhabazaeh R.M. O extece of coo ege value of oe opeato-bule that epe polyoal o paaete. Iteatoal cofeece of topology 3-9 oct. 987 au. Tez. pat page 99 [5] Dzhabazaeh R.M. Spectal theoy of two paaete yte fte eoal pace. Taacto of zebaja Natoal caey of Scece v.xviii3-4pp.-8 [6] Dzhabazaeh R.M. Multpaaete pectal theoy. Labet caec Publhg at pp.8 [7] Pugoveĉu E. Quatu echac Hlbet pace. caec Pe New Yo Loo 97 [8] Sleea.D. Multpaaete pectal theoy Hlbet pace. Pta Pe Loo 978 pp.8. [9] Hagave.. Sleea.D. The uecal oluto of two-paaete egevalue to the poble of ffacto by a plae agula ecto. J. It.Math. pplc [] Voytovch N.N. Kateelebau.Z. Svov.N.Geealz a etho of chaactetc ocllato the theoy of a ffacto - М.: publhg "Scece" pp. 46. [] Gechev Т.Q. (Генчев bout the ultapaabolc equato. DS of USSR 963 т.5 p ( Rua [] Kelyh М.V. bout copletee of egefucto of oe clae of elf-cojugate lea opeato. UMN 97 т.7 ue 4 p.5-4.( Rua [3] Lev P. (Леви. Stochatc pocee a a owa oto. М.: publhg "Scece" 979 pp 375 ( Rua. [4] MachuG.I. (Марчук. Metoy of calculato of uclea eacto. М.: "State to Publhg"96 ( Rua. [5] Rchahe R (Рихмахер. Ptpy of oe atheatcal phyc. M. publhg "M" 98p. 486 ( Rua. [6] Fo V.. (Фок. The beqg of a quatu echac. М.: Publhg "Scece" 976 pp376. [7] Khayq Q (Хайниг. btact aalog of a elat of two polyoal bule. The Fuctoal aaly a t applcato 977 ue 3 pp ( Rua