Journal of Quality Measurement and Analysis JQMA 7(1) 2011, Jurnal Pengukuran Kualiti dan Analisis

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1 Jornl o Qliy Mesremen n Anlysis JQMA 7 7- Jrnl Pengrn Klii n Anlisis A NON-OA BOUNDARY VAUE PROBEM WIH INEGRA ONDIIONS OR A SEOND ORDER HYPERBOI EQUAION S Mslh Nili Sempn -Seemp engn Syr Kmirn bgi S Persmn Hiperboli Pering Ke Y.. MEHRAIYEV & E.I. AZIZBEKOV ABSRA In his pper he clssic solion o one-imensionl bonry vle problem or hyperbolic eion wih non-clssic bonry coniions is invesige. or h he se problem is rece o he no-sel-join bonry vle problem wih eivlen bonry coniion. hen g he meho o seprion o vribles by mens o he nown specrl problem he given no sel-join bonry vle problem is rece o n inegrl eion. he eisence n nieness o he inegrl eion re prove by mens o he conrcion mppings principle n i is shown h his solion is nie or no-join bonry vle problem. inlly g he eivlence he heorem on he eisence n nieness o non-locl bonry vle problem wih inegrl coniion is prove. Keywors: Mie problem conrce mppings ie poin hyperbolic eion ABSRAK Dlm mlh ini penyele lsi bgi mslh nili sempn mr s n persmn hiperboli engn syr sempn -lsi iji. Un i mslh erseb irnn ep mslh nili sempn -swmpingn engn syr sempn yng ser. Dengn menggnn eh pemishn pemboleh bh melli mslh sperm yng iehi mslh nili sempn yng -swmpingn erseb irnn ep s persmn mirn. Kewjn n enin bgi persmn mirn erseb ibin engn menggnn prinsip pemen pengecn n injn bhw penyele ini lh ni bgi mslh nili sempn -mpingn. Ahir seli engn menggnn esern eorem ewjn n enin bgi mslh nili sempn -seemp ibin. K nci: Mslh cmprn pemen mengec ii ep persmn hiperboli. Inrocion onsier he eion in he omin D { : } n se or i problem wih iniil coniions

2 Y.. Mehrliyev & E.I. Azizbeov n non-locl coniions where is given nmber re he given ncions is esire ncion. Erlier he bonry vle problems wih non-locl inegrl eions were consiere in he ppers by Beilin Bozini 997 n Plin. Here or we hve n Ionin ype bonry coniion Ionin 977. Deiniion. Uner he clssic solion o problem - we nersn he ncion coninos in close omin D ogeher wih ll is erivives conine in eion n sisying ll coniions - in he orinry sense. he ollowing lemm is prove similrly Mehrliyev & Ysiov 9. emm. e [ D [ n he ollowing greemen coniions be lille:. 5 hen he problem on ining he clssic solion o problem - is eivlen o he problem on eining o he ncion rom - n. 6. Ailiry cs Now in orer o invesige problem - 6 we cie some nown cs. onsier he ollowing specrl problem Ionin 977 Ksmov & Mirzoyev 7:

3 A non-locl bonry vle problem wih inegrl coniions or secon orer hyperbolic eion Bonry vle problem 7 8 is no sel-join. he problem Y Y 9 Y Y Y Y will be conjge problem. We enoe he sysem o eigen n join ncions o problem in he ollowing wy Ksmov & Mirzoyev 7: b... b... where... / b /. We choose he sysem o eigen n join ncions o he conjge problem s ollows Ksmov & Mirzoyev 7: Y... Y Y b... I is irecly veriie h he biorhogonliy coniions i Yj i Yj ij re lille. Here ij is he Kronecer symbol. he ollowing heorem is vli. heorem Mehrliyev & Ysiov 9. he sysem o ncions orms Riesz bsis in he spce n he esimes r g g R g where g g Y g Y... 9

4 Y.. Mehrliyev & E.I. Azizbeov r b b b c R 8 b re vli or ny ncion Uner he ssmpions g. i g g i s s s s g g g g s i i we esblish he vliiy o he esimes: [ i i g g 5 i i i g g b ig. 6 rher ner he ssmpions g i [ i g g g s s s s g g i s i. We prove he vliiy o he esimes: i i g g 7 i i i g g b i g. 8 Now enoe by B ncions o he orm Khveriyev & Azizbeov n ggrege o ll he

5 A non-locl bonry vle problem wih inegrl coniions or secon orer hyperbolic eion consiere in [ n D where ech o he ncions rom... is coninos on [ [ [ J where. he norm in his se is eine s ollows: B J. I is nown h B is Bnch spce.. Eisence n Unieness o he Solion o he Bonry Vle Problem Since he sysem orms Riesz bsis in n sysems orm sysem o ncions biorhogonl in ech solion o problem - 6 hs he orm: 9 where Y... Moreover n Y re eine by relions n respecively. Applying he meho o seprion o vribles or eermining he esire ncions... rom we hve: where

6 Y.. Mehrliyev & E.I. Azizbeov Y Y Y.... Solving problem - we hve: where.... Aer sbsiion o epressions o respecively in 9 we hve:

7 A non-locl bonry vle problem wih inegrl coniions or secon orer hyperbolic eion. 8 Now proceeing rom einiion o he solion o problem - 6 similr o Khveriyev & Azizbeov he ollowing lemm is prove. emm. I is ny solion o problem - 6 he ncions... sisy sysem 5-7. heorem. e [ [ [ D D. hen problem - 6 ner smll vles o hs nie clssic solion. Proo. Denoing P P where P P P el he righ hn sies o respecively n we wrie eion 8 in he orm: P. 9 We will sy eion 9 in he spce I is esy o see h B. P [ [ [.

8 Y.. Mehrliyev & E.I. Azizbeov P [ [ [ P [ [ [ [ [. Here llowing or 5-8 we hve: P [ D [ B [ P D [ B P [ 8 b

9 A non-locl bonry vle problem wih inegrl coniions or secon orer hyperbolic eion 8 b 8 8 b 8 D 8 D [ B Now consier he operor P in he sphere where K KR A rom A B B. D D 8 b 8 b 8 b. D I is seen rom - h or ny KR he esimes : P B A B B P P 5 B B B where B 6 [ re vli. hen i ollows rom esimes 5 h ner sicienly smll vles o he operor P cs in he sphere K KR rom B n i is conrcive. hereore in he sphere K KR he operor P hs nie ie poin h is solion o eion 9. 5

10 Y.. Mehrliyev & E.I. Azizbeov 6 he ncion s n elemen o he spce B is coninos n hs coninos erivives on D. Now prove h n re coninos in D. rom 5-7 we hve: Here by 5-8 we hve: [ D 7 [ [ 6 6

11 A non-locl bonry vle problem wih inegrl coniions or secon orer hyperbolic eion 7 6 D 8 [ b b D b 8 D. 9 [ [ [ [ D [ 8 b 8 b 8 D b [ 8 8 D b 6 8 D 6 D.

12 Y.. Mehrliyev & E.I. Azizbeov 8 I ollows rom esimes 7-9 h is coninos in D n rom he esimes - h is coninos in D. rher i ollows rom sysems 5-7 n 5-7 h ce by he given heorem n he more so. hs coniions re lille. I is obvios h coniions is lille or he ncion. I is esy o see h [... [. Now i we se sysems 5-7 n 5-7 eliy es he orm:

13 A non-locl bonry vle problem wih inegrl coniions or secon orer hyperbolic eion where he ncions... re eermine by relion n.... Uner he coniions o he heorem i is obvios h. 5 hen i ollows rom 5 h or ny ie [ : hs relions n 6 yiel [. 6 onseenly he ncion sisies eion everywhere in D. So is solion o problem - 6 n by lemm i is nie. he heorem is prove. By mens o lemm we prove he ollowing: heorem. e ll he coniions o heorem n greemen coniions 5 be lille. hen or sicienly smll vles o problem - hs nie clssic solion.. onclsion he ollowing resls hve been obine: he eisence o he solion o no sel-join bonry vle problem or secon orer hyperbolic eion hs been prove he nieness o he solion o no sel-join bonry vle problem or secon orer hyperbolic eion hs been shown he eisence o he clssic solion o non-clssic bonry vle problem wih inegrl bonry or secon orer hyperbolic eion hs been prove he nieness o he clssic solion o non-clssic bonry vle problem wih inegrl bonry or secon orer hyperbolic eion hs been shown. 9

14 Y.. Mehrliyev & E.I. Azizbeov Reerences Beilin S.. Eisence o solions or one-imensionl wve eions wih nonlocl coniion. Elecronic J. o Dier. E. 76: -8. Bozini A Solion ore n probleme mie vec coniions non locles por ne clsse eions hyperbolies. Bllein e l lsse es Sciences. Acemie Royle e Belgie 8: 5-7. Ionin N. I Solions o bonry vle problem in he concions heory wih nonlocl bonry coniions. Dierens. Urvn. : 9-. Ksmov.B. & Mirzoyev V.S. 7. On one generlision o Ionin s emple. he bsrcs o scieniic conerence evoe o h nniversry o he honore scienis c. A.I.Hseynov: B. Khveriyev K.I. & Azizbeov E.I.. Invesigion o clssicl solion o one-imensionl no seljoin mie problem or clss o semi-liner pseohyperbolic eions o orh orer. Vesni Binsogo Gosrsvennogo Universie phys.-mh. ser. : -. Mehrliyev Y.. & Ysiov M.R. 9. he solion o bonry vle problem or secon orer prbolic eion wih inegrl coniions. Proceeings o IMM NAS o Azerb. : 9-. Plin.S.. Non locl problem wih inegrl coniions or hyperbolic eion. Di. Urvnen. 7: Mechnics-Mhemics cly B Se Universiy B AZERBAIJAN E-mil: zel_zerbijn@mil.r