Absence of eigenvalues for integro-differential operators

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1 Poceedigs of the 4th EUOPEAN COMPUTING CONFEENCE Absece of eigealues fo itego-diffeetial oeatos Maius-Maiel Stăescu uitu olcu Sabi izescu Paul ideu eatet of Alied Matheatics Uiesity of Caioa 396 Caioa oaia Faculty of Mechaics Uiesity of Caioa 6 Caioa oaia dbolcu@yahooco; sabi_izescu@yahooco; ideu@cetaluco Abstact: - I this ae we obtai the sufficiet coditios (but i soe sese ee the otiu fo the absece of eigealues of the oeatos (i geeal oselfadoits geeated by itegodiffeetial exessios The ai esults ae obtaied usig a abstact schee Keywods: - Sectal theoy o-selfadoit oeatos elatiely bouded etubatios diffeetial ad itego-diffeetial oeatos eigealue Itoductio The oble of the absece of eigealues of itego-diffeetial oeatos ust as othe oes iolig sectal oeties of a itegodiffeetial oeato aose fo the actical ecessities of lasa oscillatios theoy (i this esect we ote the wos of oh ad E Gose [] N G Va Kae [] ad K M Case [3] of atheatical theoy of scatteig of eutos (see fo istace J ehe ad G M Wig [4] ad see also [5] ad of othe icile situatio fo quatu hysics ad echaics We also ote the wos [6-8] (as well as the efeeces theei i which atheatical odels iolig itegodiffeetial oeatos ca be foud as well A at of the esults of the eset ae was aouced without ay oof i ou aticle [9] The Hilbet saces ae deoted by Η Η the ie oducts ad the os i those Hilbet ad The set of saces ae deoted by ( Η liea oeatos closed ad desely defied o Η with alues i Η is deoted by C ( Η Η ( Η Η stads fo the aach sace of all bouded liea oeatos defied o Η with alues i Η ad ( Η Η the subsace of ( Η Η cosistig of all coact oeatos defied o Η with alues i Η Fo eey oeato A i Η the doai the age the esolet set ad the sectu ae deoted by A A A σ A esectiely The ( ( ρ ( ad ( Η oit sectu of the oeato A (the set of all σ A eigealues of A is deoted by ( The absece of eigealues of soe itego-diffeetial oeatos These esults ae siilaly with the esults which efe to the diffeetial oeatos ad with the othe esults which efe to the Wiee-Hof-tye oeatos (see [] We ealize the followig schee (see [] et H be a oeato defied o the Hilbet sace H of the fo H = A ( whee has the fo = S T = ad the oeatos A S T ( = satisfy the followig coditios: (i A is closed ad desely defied; (ii the colex ube λ is ot a eigealue of λ A ; the oeato A that is σ ( (iii the oeatos S T ( = act i the Η sace with the oeties A S ( ( ( ( Η( T = As well as i the ae [] i the Η sace we coside the oeatos faily ( with oety: ISSN: ISN:

2 Poceedigs of the 4th EUOPEAN COMPUTING CONFEENCE (i = ( e ad = I ( I is the idetical oeato i the Η sace O the doai we defie = ( of the oeato ( the o u u ( u = Η Such as esult we obtai the oalized sace Η (i geeal ucolete Clealy Η = Η Moeoe we suose that the followig oeties ae fulfilled: ( thee is such that if Η ad ( A λi( = S T Η ad (i if the ( A I ST γ λ Η γ( A λi ST a ( < a < ; γ = = u u whee ( A λ I = S T the u= γ( A λi ST c ' ( c = cost; '; γ = ea Thoughout this ae we coside oly the situatio o which the etubatio oeato is subodiated to uetubed oeato A I this coectio we assue additioally that the fuctio ( x y q ad esectiely the eels eithe fo = o = ae idetical equal to zeo Thus i the su deteiig eithe = ad = o = ad = ea et be the oeato H of the fo ( If the coditios (i-(i ae fulfilled the λ is ot a eigealue of the oeato H Poof Suose o the cotay let λ be a eigealue of the oeato H the thee is u u Η such that Hu = λu ( et Η be a Hilbet sace foed lie a diect saces Η aely su fo ( Η= = Η ; We coside the oeato : Η Η such as = ( ( u ( = = the oeato T : Η Η ( = ( Η T = T = = = ad we deotes the oeato S : Η Η thus S = S ( = ( Η ( S ; = = = We ea that ST = S T = ad i accodace with (3 it esults that Au STu= λu o u ( A λ I STu = (3 We deote = u ecause λ σ ( A it esults that If = the u = ( = ad we obtai the cotadictio Au = λu ecause ( A ( ( = o the basis of (4 it esults that u ( ad we obtai ( A λ I ST= (4 The faily of the oeatos ( fo the sace Η is i coesodece with the faily of the oeatos ( fo the sace Η whee = ( = = ; ; ; = ( ( = I accodace with coditio ( the coditio (5 ioles the followig elatio = ( = A λi ST a ( < a< ; Theefoe a ( < a< ; (5 Peiously we etioed that ad o the base of (i it esults that < ecause o the base of (6 it esults a we obtai a cotadictio with < a < Thus the lea is oof The itego-diffeetial oeato who s the sectu is studied i this ae has the fo ISSN: ISN:

3 Poceedigs of the 4th EUOPEAN COMPUTING CONFEENCE H = = M (6 whee M ( = ae of the tye ( M u( x = a u( x q ( x u( x ( x y u( y dy ad ( = ; x sace ( ad they act i the We coside the oeato H with its axial doai of defiitio ie with the doai cosistig of all fuctios u W ( ( W ( deotes the Sobole sace of ode oe such that u belog to the doai of M fo each = ad M W ( I aticula if q ( = ae cotiuous ad bouded fuctios togethe with thei deiates of the ( ode o the sei-axis ad the eels ( x y( = ae such that ( the itegal oeatos with eels x y ( = ; = ae x bouded o ( cosideed to be the Sobole sace W ( Hee a ( = ; a ubes q ( = the eels ( x y( = the the doai of H is ae colex ad esectiely ae sooth as it will be ecessay colex-alued fuctios d deotes the diffeetial oeato = i dx with the doai of defiitio deteied by the set of all fuctios u ( which ae absolutely cotiuous o eey bouded iteal of the ositie sei-axis ad u' ( Moe ( = u It suoses that the oeato H acts i the sace ( et coside the oeato u x = q x u x ( ( ( ( ( x y u( y dy ( = ad the the oeato H is eeseted lie a etubed oeato H = A whee ad A= = a = = It is ow that the esolet set ρ ( of the oeato coicides with the oe ue half-lai ad so the sectu of the oeato cosists of all oits of the closed lowe half- σ o lai Moeoe the oit sectu ( the eal axis is abset Sectal oeties of the oeato A ae well ow ad i aticula ifoatio o the sectu of A ca be deied fo istace fo [] Hee let us oly ae soe eas that will be ecessay fo ou futhe discussios et λ be ξ = a colex ube ad deote by ( the oots of the olyoial A ( ξ λ The A λ I = a ( ξ I (7 = The esolet set of the oeato coicides with the oe ue half-lae = { z C / Iz > } the oe lowe half-lae is fillig with the oit sectu of ad the eal axis beig i the cotiuous at of the sectu is fee fo the oit sectu of A ie σ ad σ ( = Φ We ote that c ( fo Iz > oe has x ( zi u( x = i ex( iz( y x u( y dy (8 The eesetatio (8 fo Iz = is also holds but the it is ecessay to coside u fo the age of the oeato zi Now fo the equality (7 it is easy to coclude that λ σ( A if ad oly if the oots of the olyoial A ξ ae cotaied i the closed lowe half-lae Iξ Moeoe a oit λ is ot a eigealue ( λ A ( ξ λ has of A oided that the olyoial its zeos oly i Iξ Next let λ be ot a eigealue of the uetubed oeato A Accodig to what has ISSN: ISN:

4 Poceedigs of the 4th EUOPEAN COMPUTING CONFEENCE aleady bee said the olyoial eeseted as follows ( λ = ( ξ ξ A( = A ( ξ λ ca be Aξ ξ (9 whee ( = ubes ( = ad ( ξ ξ ae eal ai wise distict A is a olyoial the oots of which belog to the ue half-lae Iξ > We let = fo the case i which the olyoial the eal axis Theoe et H be a oeato of tye ( the colex which acts i the sace ( A ( ξ λ has o zeos o ube λ is ot a eigealue of the oeato A the olyoial A ( ξ λ ca be eeset of the = ax / = fo (8 { } If ( x q( x ( ( > ; = ad if ( x y = fo x> y ad the itegal oeatos with eels x x y = ; > ( ( ( ae bouded i ( the λ is ot a eigealue of the oeato H Poof We eify the coditios (i-(i The Hilbet sace Η is foed lie a diect su fo saces ( ( We deote by Η= ( = = S ad = cosideed equal with ( = T is et us coside the oeatos ( u( x = ( x u( x ( u ( o( Clealy the coditios (i-(i ae fulfilled ad eai to eify the coditios ( ad (i ecause the oeato γ ( A λi ( γ = o his defiitio doai is a liea cobiatio of the oeatos of the tye ( µ I l (whee l = ad Iµ it is sufficiet to estiate the o ( µ I ( ( ; = ; l= ; l Fo this it uses the followig Hady iequality (see [3] u c µ I u > ( ( ( ( whee c ( whe This iequality is alied of l ties successiely We hae two cases: Iµ = ad Iµ > I the ioled case Iµ = i accodace with eious iequality (see the eious Hady iequality (9 it obtais ( I a( ( N µ whee ( [4] Theefoe a whe (see lea of ( µ I a( ( = ; whee ( a whe I cocodace with the coditios iosed of the fuctios q ad ( = it is obtai c ( c is the costat which does ot deed o ; = ; ; > µ I a I coclusio ( ( ( = ; whee a ( whe If Iµ > i cocodace with lea of [4] it is tue that l ( µ I a( ε ( = ; ; ε> a whe Thus we obtai c( = ; ; > whee ( ε ε whee c is the costat which does ot deed o ; = ; ; ε > Theefoe l ( µ I a( ( = ; ; ε > ISSN: ISN:

5 Poceedigs of the 4th EUOPEAN COMPUTING CONFEENCE whee ( a whe The oety ( is oof fo both of cases Fo oof of the coditio (i we stad the sae situatios as well as i oety ( If Iµ = it obtais ( µ I a( ( = ; ε; ε > a whe whee ( Thus ε c ε ε ( = ; ε; > c is the costat which does ot deed o ad theefoe ( µ I c ε ( = ; ε; > ; c is costat Iµ the ( I a( If > whee ( [4] Theefoe µ a whe ( see lea of ε c( = ; > ε; ε > whee c is the costat which does ot deed o ad thus ( µ I c ε ( = ; > ε; ε > c is costat Thus the oety (i is tue The theoe is oof If the olyoial A ( ξ λ has also ull solutios (let be fo exale ξ with = > theoe ca be efied oe exactly the followig theoe is tue Theoe et H be a oeato of tye ( which acts i the sace ( ad let be q ( x = ( x ad ( x y = ( x y fo eey = with < η (η a itege fixed ube <η < A ( ξ λ ca be eeseted o the et be that fo (8 whee we coside that ξ = ad = ax{ η } If ( x q( x ( ( > ; = ; withη < < ad if ( x y = fo x> y ad the itegal oeato with eels x x y ( ( ( > ; = ; withη < < ae bouded i ( the λ is ot a eigealue of the oeato H Poof The oeato ca be eeseted of tye = η< < ad the sybol of the oeato A is A ( ξ = ξ ( ξ ξ A( ξ = We estiate the exessios of the fo γ = = γ ( ξ ( q ( x K ( ξ ( q( x K = whee η < < ; γ = ; ( ad this ca be eeseted lie a liea cobiatio of the fo l ( I ( q ( x K µ ( l = ; η < < 3 Alicatios 3 I a sace ( ( < coside the itego-diffeetial oeato d u du ( Hu( x = q( x q( x u( x dx dx ( x y u( y dy ( < x< ; u W ( whee ( = q ae the easuable fuctios o the ositie sei-axis ad ted to zeo whe x ted to ifiite ad the itegal oeato K with eel ( x y is suosed bouded i ( The defiitio doai ( H of the oeato H is cosideed the set of all u deiaties fuctios o ositie sei-axis with deiatie u ' absolutely cotiuous o eey bouded iteal of the ositie sei-axis ad thee = ISSN: ISN:

6 Poceedigs of the 4th EUOPEAN COMPUTING CONFEENCE is the deiatie of the two ode u '' thee is o alost all sei-axis u'' ( (i the sese of distibutios ad u ( = I cocodace with the theoe we obtai the followig esult Coollay et be x q x = ( ( ( ( ad let be that the eel ( x y oeato K is such that ( x y = of the itegal fo x> y (close o all sei-axis ad the itegal oeato with eel ( x ( x y the sace ( is bouded o If > the the oeato H has ot eigealues o the ositie sei-axis ad if > the the oit λ = also is ot a eigealue of the oeato H We etio that the oeato H cosideed i the exale 3 with q ( x ad ( x y = ( x ( y x whee ( x = a x e ( a ; > it is studied i the ae [5] i coectio with the obles about the diffusio theoy of eutos i otos 3 et coside the oeato ( Hu( x = q ( x d dx du dy u du dx ( ( x y dy < x< ; u W ( H acts i the sace ( I cocodace with the theoe the followig affiatio is tue Coollay If ( x q ( x ( ( > x y = fo x> y ad if the itegal oeato ( with eel ( ( x y( > ( x is bouded o the the oit sectu of the oeato H o the ositie sei-axis (iclusie the oit λ = is abset Siila esults ca be foulated fo istace fo the itego-diffeetial oeatos which ca be obtaied as a esult of the Schödige tye oeatos (i geeal oselfadoits see fo istace [6] etubed with itegal oeatos efeeces: [] oh E P Goss Theoy of lasa oscillatios A oigi of ediu lie behaio Phys e Vol [] N G a Kae The theoy of statioay waes i lasa A Phys Vol [3] K M Case Plasa oscillatios A Phys Vol [4] J ehe G M Wig Solutio of liiaized oltza tasot equatio fo slab geoety ue J Vol [5] E A Catchole A itego-diffeetial oeato J odo Math Soc Vol6 No [6] K M Case P F Zweifel iea tasot theoy Addiso-Wesley Publ Co 967 [7] S E Cheeshatse Sectal aalysis of o-selfadoit diffeetial oeatos aisig i the oe-diesioal scatteig oble of the ow aticles Mat Sb Vol9 No (ussia [8] aiso Neuto Tasot Theoy Oxfod 96 [9] P A Couhai M Staescu O the sectu of soe itego-diffeetial oeatos ull Acad de St a M Math Vol3 No [] PA Couhai O the oit sectu of the etubed itegal Wiee-Hof oeato Mat Zaeti Vol5 No 99-3 [] PA Couhai O the sectu of sigula oselfadoit diffeetial oeatos Oeato Theoy: Adaces ad Alicatios ihäuse Velag asel Vol [] M A Naia iea iffeetial Oeatos Naua M 969 (ussia [3] S Palo Of the oselfadoit Schödige oeato Pob Mat Fizii- GU Vol [4] PA Couhai O the sectu of a etubed Wiee-Hof oeato Mat Issled Vol (ussia [5] EA Catchole A itego-diffeetial oeato J odo Soc 6 Vol [6] SE Ceesate The sectal aalisys of oselfadoit oeatos Math Vol9 No ISSN: ISN:

Structure and Some Geometric Properties of Nakano Difference Sequence Space

Structure and Some Geometric Properties of Nakano Difference Sequence Space Stuctue ad Soe Geoetic Poeties of Naao Diffeece Sequece Sace N Faied ad AA Baey Deatet of Matheatics, Faculty of Sciece, Ai Shas Uivesity, Caio, Egyt awad_baey@yahooco Abstact: I this ae, we exted the