On Eigenvalues of Nonlinear Operator Pencils with Many Parameters

Transcription

1 Ope Scece Joual of Matheatc ad Applcato 5; 3(4): 96- Publhed ole Jue 5 ( O Egevalue of Nolea Opeato Pecl wth May Paaete Rakhhada Dhabaadeh Guay Salaova Depatet of Fuctoal Aaly Ittte of Matheatc Aa Mechac of NAS of Aebaa Baku Aebaa Depatet of Algeba ad Geoety Gaa State Uvetety Gaa Aebaa Eal adde akhhadadhabaade@ableu (R Dhabaadeh) guay-alaova@alu (G Salaova) o cte th atcle Rakhhada Dhabaadeh Guay Salaova O Egevalue of Nolea Opeato Pecl wth May Paaete Ope Scece Joual of Matheatc ad Applcato Vol 3 No 4 5 pp 96- Abtact he autho gve the eceay ad uffcet codto of the extece of the coo egevalue of the olea eveal opeato pecl wth ay paaete he opeato pecl cota alo the poduct of thee paaete fte degee he ube of equato thee yte ay be oe tha the ube of paaete I the poof the autho eetally ue the eult of ultpaaete pectal theoy ad the oto of the aalog eultat of two ad eveal opeato pecl ay paaete Keywod Reultat Opeato Multpaaete Syte Egevalue Itoducto Spectal theoy of opeato oe of the potat decto of fuctoal aaly he developet of phycal cece becog oe ad oe challege to atheatca I patcula the eoluto of the poble aocated wth the phycal pocee ad coequetly the tudy of patal dffeetal equato ad atheatcal phyc equato equed a ew appoach he ethod of epaato of vaable ay cae tued out to be the oly acceptable ce t educe fdg a oluto of a coplex equato wth ay vaable to fd a oluto of a yte of oday dffeetal equato whch ae uch eae to tudy Neceay Defto ad Reak Gve oe defto ad cocept fo the theoy of ultpaaete opeato yte eceay fo udetadg of the futhe codeato Let be the lea ultpaaete yte the fo: B ( ) x = ( B B ) x = () k k k k k k = whe opeato B k act the Hlbet pace Defto [] = ( ) C a egevalue of the yte () f thee ae o-eo eleet x H = uch that () atfy ad decopoable teo x = x x x called the egevecto coepodg to egevalue = ( ) C Defto he opeato B B actg the pace H duced by a opeato H to the teo pace H = H H f o each decopoable teo x = x x of teo poduct paceh = H H we B x = x x B x x x ad o all have the othe eleet of H = H H the opeato defed o leaty ad cotuty Defto 3 ([5] [6]) Let x = x x x be a egevecto of the yte () coepodg to t egevalue = ( ) ; x - aocated vecto (ee[4]) to a the B

2 97 Rakhhada Dhabaadeh ad Guay Salaova: O Egevalue of Nolea Opeato Pecl wth May Paaete egevecto x of the yte () f thee a et of vecto { } x H H atfyg to codto B ( ) x B x B x = x = whe < () = = Idce eleet ( x ) H H thee ae vaou aageet fo et of tege o wth = Defto 4 I [3] fo the yte () aalogue of the Cae deteat whe the ube of equato equal to the ube of vaable defed a follow: o decopoable teo x = x x opeato ae defed wth help the atce α α α α Bx Bx Bx Bx B x B x B x B x α x = = (3) B3x3 B3x 3 B3x3 B3x3 B x B x B B whee α α α ae abtay coplex ube ude the expao of the deteat ea t foal expao whe the eleet x = x x x the teo poduct of eleet x x x If αk = α = k the ght de of () equal to x whee k x = x x x O all the othe eleet of the pace H opeato ae defed by leaty ad cotuty E ( = ) the detty opeato of the pace H Suppoe that fo all x ( x x) δ ( x x) δ > ad all B k ae elfadot opeato the pace H Ie poduct [] defed a follow; f x = x x x ad y = y y y ae decopoable teo the [ x y] = ( x y) whee ( x y ) the e poduct the pace H O all the othe eleet of the paceh the e poduct defed o leaty ad cotuty I pace H wth uch a etc all opeato Γ = ae elfadot Defto5( [7][8]) Let be two opeato pecl depedg o the ae paaete ad actg geeally peakg vaou Hlbet pace A( ) = A A A A B( ) = B B B B Opeato Re ( A( ) B( )) peeted by the atx A E A E A E A E A E A E E B E B E B E B E B E B whch act the ( H H ) - dect u of cope of the pace H H I a atx (4) ube of ow wth opeato A equal to leadg degee of the paaete pecl B( ) ad the ube of ow wth B equal to the leadg degee of paaete A( ) Noto of abtact aalog of eultat of two opeato pecl codeed the[7] fo the cae of the ae leadg degee of the paaete both pecl ad the []fo geeally peakg dffeet degee of the paaete the opeato pecl heoe [78] [Let all opeato ae bouded coepodg Hlbet pace oe of opeato A o B ha bouded vee he opeato pecl A( ) ad B( ) have a coo pot of pecta f ad oly f { } Ke Re ( A( ) B( )) ϑ Reak If the Hlbet pace H ad H ae the fte deoal pace the a coo pot of pecta of opeato pecl A( ) ad B( ) ae the coo egevalue (ee [6] [7]) Code budle depedg o the ae paaete k { k B ( ) = B B B = B ( ) - opeato budle actg a fte deoal Hlbet paceh coepodgly Wthout lo of geealty k k we adopt k k k I the pace H (the dect u of k k teo poduct H = H H of pace H H H ) ae toduced the opeato R ( = ) wth the help of opeatoal atce (3) Let B ( ) be the opeatoal budle actg a fte deoal Hlbet paceh R B B Bk B B Bk Bk B B B = B B Bk B B B k k B B Bk (4)

3 Ope Scece Joual of Matheatc ad Applcato 5; 3(4): = 3 he ube of ow wth opeato B = k the atx R equal to k ad the ube of ow wth opeato B = k equal to k We degate p ( B ( )) σ the et of egevalue of a opeato B ( )Fo [5] we have the eult: heoe [9] σ ( B ( )) { θ} f ad oly f { θ} KeBk KeR ( = { θ}) p 3 he Syte of Opeato Pecl May Vaable Code the yte k k ( ) ( = = A x = A A A A = he paaete ete the yte olealy ad the yte (4) cota alo the poduct of thee paaete Dvde the yte of equato (5) to goup of each goup If oe equato ea outde thee equato we add by othe opeato fo the yte (4) Each goup cota opeato ad wll be codeed epaately I (5) the coeffcet of the paaete k = ae the opeato A whch act the pace H dex dcate o the paaete dexk - o the degee of the paaete Itoduce the otato: (5) = k k k k = (6) Futhe ueate the dffeet poduct of vaable the yte (5) o ceag of the degee of the paaete Let the ube of te wth the poduct of the paaete ae equal to Itoduce the otato = ( ) t = ɶ k k k t t whee t the ube whch coepod the ultple at the odeg of ultple of paaete the yte (5) So ew otato to the poduct coepod the paaeteɶ k k k t t ( ɶ k k k t t ) aсcоdgly opeato A = D = ; = k ; k k k = k = ax k = k (7) A ; = D t = ;; = k k k k k k t whe the ube of dffeet poduct of paaete eteg the yte(5) I ew otato the yte (5) the teo poduct of pace H H H cota k k k paaete ad equato Let k k k = k he k k k k k k k k k t k t = k = k = k [ ɶ D ] x [ ɶ D ] x = k = ; k = ; = Add the yte (8) wth help of ew equato o ae that the coecto betwee the paaete followg fo the equato of the yte (5) atfy Itoduce the opeato actg the fte deoal pace R ad defg wth help of the atce = = = = = ( ) k = (8) = he ube tad o the dagoal eleet of the ft ow of the atx ; dagoal eleet of the ow k of the atx equal alo to ad o o Bede all atce k ( ) have the ode Add the yte (8) by the followg equato (9)

4 99 Rakhhada Dhabaadeh ad Guay Salaova: O Egevalue of Nolea Opeato Pecl wth May Paaete ( ɶ ɶ ) x = ( ɶ ɶ k k k k k k k k ɶ ) x = k k k k k k ( ɶ ɶ ) x = k k k k k k k k ( ɶ ɶ ɶ ) x = k k k k k k x R > ( ɶ ɶ ɶ t t k t k k t ɶ ) x = k ( ) t ( ) t () t = Deote ɶ k ( ) the ultple of the paaete eteg the yte (5) havg the coeffcet A Syte ((5)()) fo the lea ultpaaete ( ) yte cotag k k k equato ad k k k paaete o th yte we ay apply all eult gve the begg of th pape heoe [4] Let the followg codto: а) opeatoak t A k k k ; t the paceh ae bouded at the all eag ad k b) opeato ext ad bouded atfy: he the yte of ege ad aocated vecto of (5) cocde wth the yte of ege ad aocated vecto of each opeato Γ ( = ) Gve two equato fo () Let be equato ( ) x = ( ) x = 3 () Let и x = ( α β) the copoet of the egevecto of the yte ((5)()) We have α β = ( ) β α = β α = ; = Futhe fo the codto x = ( α β ) t follow β 3α = β α = ad coequetly ɶɶ 3 = ɶ Eale we poved that ɶ = Coequetly ɶ 3 3 = O aalogy fo othe paaete of ((5)()): f ( ɶ ɶ ɶ k k k ) the egevalue of the yte -((5) 4 k ()) the = = ɶ = 4 k k k k = ; = k o each ultple of paaete ( ɶ ɶ ɶ k ) t = ɶ k t ; t t coepoded the equato k ( ɶ ɶ ɶ t k k t k k t k k k k t ɶ ) x = k ( ) t ( ) t t k k t Code the lat equato whch k k ɶ ( ) t ɶ = = = = Opeato defg wth help the atce act pace R O k t k t k t k t egevecto ( α α ) R equato ( ɶ ɶ k k ) ɶ α = k = ɶ k t ( t atfy ) ɶ α

5 Ope Scece Joual of Matheatc ad Applcato 5; 3(4): 96- Coequetly ɶ α = α ɶ α = ɶ α k t ɶ α = α ɶ α = α k k ɶ α = α k k Lat ea = k ; Fo the obtaed lea ultpaaete yte we cotuct opeato o ule (3) he codto Ke { ϑ} = ea that opeato Γ = ae pa coute[] So opeato Γ act fte have ot the deoal pace H ad opeato Γk k k eo egevalue the fo the ay egevalue ( ) of the yte((5) ()) k k k = thee uch ege eleet k k k o that the equalte Γ = ; = k k k ; atfy Fo aalogy codto we obta the aalogy eult fo all goup We have the eveal yte of opeato polyoal oe paaete he yte ha the fo l = = = Whee = k k k ; the ube of goup We apply the eult of [9](theoe th pape) heoe 3 Let the codto of the theoe fulflled All opeato o have vee he the yte (4) ha the coo egevalue f ad oly f Ke t = = ( ) k k k t t We apply the eult of [9](theoe th pape) Opeato t ae duced to the paceh Hl by the opeato t coepodgly heoe 3 Let the codto of the theoe fulflled All opeato o have vee he the yte(4) ha the coo egevalue f ad oly f Refeece Ke ( ) k k k o [] Atko F V Multpaaete pectal theoy BullAeMathSoc [] Bowe PJ Multpaaete pectal theoy Idaa Uv Math J [3] Sleea B D Multpaaete pectal theoy Hlbet pace Pta Pe Lodo 978 p8 [4] Dhabaadeh RM Spectal theoy of ultpaaete yte of opeato Hlbet pace aacto of NAS of Aebaa [5] Dhabaadeh R M Salaova G H Multtpaaete yte of opeato ot lealy depedg o paaete Aeca Joual of Matheatc ad Matheatcal Scece vol No- p39-45 [6] Dhabaadeh RM Spectal theoy of two paaete yte fte-deoal pace aacto of NAS of Aebaa v p-8 [7] Balk AI (Балинский) Geeato of oto of Обобщение понятия Beutat ad Reultat DAN of Uk SSR eph-ath ad tech of cece 98 ( Rua) [8] Khayq (Хайниг Г) Abtact aalog of Reultat fo two polyoal budle Fuctoal aalye ad t applcato 977 p94-95 [9] Dhabaadeh RM O extece of coo ege value of oe opeato-budle that deped polyoal o paaete Baku Iteatoal opology cofeece 3-9 oct 987 e Baku 987 p93