RESEARCH PAPERS FACULTY OF MATERIALS SCIENCE AND TECHNOLOGY IN TRNAVA SLOVAK UNIVERSITY OF TECHNOLOGY IN BRATISLAVA

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1 RESEARCH PAPERS FACULTY OF MATERALS SCENCE AND TECHNOLOGY N TRNAVA SLOVAK UNVERSTY OF TECHNOLOGY N BRATSLAVA Numer 8 SNGULARLY PERTURBED LNEAR NEUMANN PROBLEM WTH THE CHARACTERSTC ROOTS ON THE MAGNARY AXS Ľudmil VACULÍKOVÁ Vldimír LŠKA Arc We iveie e prolem o exiece d mpoic evior o oluio or e iulrl perured lier Neum prolem > <. << Our pproc relie o e li o ierl equio equivle o e prolem ove. Ke wor iulrl perured ODE Neum prolem oudr codiio crceriic roo roducio i pper we will ud e iulrl perured lier prolem 3 > < << C ( ) (.) wi Neum oudr codiio. (.) Ľudmil Vculíová RNDr. Vldimír Liš MSc. - Deprme o Memic iue o Applied ormic Auomio d Memic Fcul o Meril Sciece d Tecolo i Trv Slov Uiveri o Tecolo Brilv Hjdóczo 97 4 Trv Slov Repulic e-mil: ludmil.vculiov@u. vldimir.li@u. 63

2 We c view i equio memicl model o e dmicl em wi i-peed eedc. Te iuio coidered ere i compliced e c crceriic equio o i diereil equio roo o e imir xi i.e. e em e o perolic. For perolic oe e dmic cloe criicl miold i wellow ( ee e.. [] [3-] ) u or e o-perolic em e prolem o exiece d mpoic eviour i ope i eerl d le o e uil ecicl diiculie i olier ce []. Te coiderio elow m e irucive or ee oe. We prove ere exi iiiel m equece { } uc covere uiorml o u o were i oluio o prolem (.) (.) d u i oluio o reduced prolem ( we we pu We will coider or e prmeer e e J ol J i (.) ) u i.e.... ( ) π λ π λ u. were λ > e rirril mll u ixed co wic uree e exiece d uiquee o e oluio o (.) (.). Exmple. Coider e lier prolem e d i oluio > < <<. e ( ) e ( ) e. ( ) i ( ) Hece or ever equece { } J e oluio o coidered prolem e O( ) e covere uiorml or o e oluio u o e reduced prolem o. Te mi reul o i ricle i e ollowi oe. 3 Teorem. For ll C ( ) Mi reul d or ever equece { } J uique oluio o prolem (.) (.) ii u uiorml o or ere exi 64

3 More preciel u O( ) o. Proo. Firl we ow ( ) ( ) i ( ) i ( ) i oluio o (.) (.). Diereii (.) wice i io coiderio d H ( ) H H ( ) d we oi i i ( ) co ( ) ( ) ( ) (.) (.) i ( ) co ( ) ( ) ( ) i From (.3) d (.) er lile leric rreme we e. (.3) ( ) i.e. i oluio o diereil equio (.) d rom (.) i i e o veri i oluio iie (.). Le e rirr u ixed. Deoe d e ierl 65

4 66 i co Te i co. eri d pr we oi i co i i co i. co co Alo. co i i co Now we eime. We oi

5 67. co i i λ (.4) Te ierl i (.4) covere o zero or. J deed wi repec o umpio o we m iere pr i (.4). Tu co co i.5 co d i i co i.6 were. up up d Suiui (.5) d (.6) io (.4) we oi priori eime o oluio o (.) (.) or ll o e orm.7 i λ Becue e ri ide o e iequli (.7) i idepede o e coverece i uiorml o. Teorem ol. Remr. A remr we coclude i e ce e coverece re i J O ollow rom (.7).

6 Acowledme Ti reerc w uppored e Slov Gr Aec Miir o Educio o Slov Repulic uder r umer /68/8. Cocluio our coriuio we deermied coverece re o e oluio o ceri cl o e iulrl perured diereil equio ujec o Neum oudr codiio o e oluio o reduced prolem mll prmeer ie derivive e o zero. Reerece: [] Crioper K.R.T. JONES. Geomeric Siulr Perurio Teor C..M.E. Lecure Moecii Terme Jue 994 Lecure Noe i Memic 69 Sprier-Verl Heideler 995. [] VRÁBEĽ R. Siulrl perured emilier Neum prolem wi o-ormll perolic criicl miold. E.J.Quliive Teor o Di. Equio No. 9 pp. -. [3] VRÁBEĽ R. Upper d lower oluio or iulrl perured emilier Neum prolem. Memic Boemic 997 No. pp [4] VRÁBEĽ R. Tree poi oudr vlue prolem or iulrl perured emilier diereil equio. E.J.Quliive Teor o Di. Equio 9 No.7 pp. -4. [5] VRÁBEĽ R. O e oluio o diereil equio ( ) p wi rirril lre zero umer. Jourl o Compuiol Ali d Applicio 4 Vol. 6 No. pp [6] VRÁBEĽ R. Siulrl perued rmoic quric poeil ocillor prolem. Zeicri ür ewde Memi ud Pi ZAMP 4 Vol. 55 pp [7] VRÁBEĽ R. Semilier iulr perurio. Nolier Ali TMA 995 Vol. 5 No. pp [8] VRÁBEĽ R. Quilier d qudric iulrl perured periodic oudr vlue prolem. Arcivum Memicum Vol. 36 No. pp. -7. [9] VRÁBEĽ R. Quilier d qudric iulrl perured Neum` prolem. Memic Boemic 998 Vol. 3 No. 4 pp [] VRÁBEĽ R. Upper d lower oluio or iulrl perured emilier eum` prolem. Memic Boemic 997 Vol. No pp [] VRÁBEĽ R. Ampoic evior o T-periodic oluio o iulrl perured ecod-order diereil equio. Memic Boemic 996 Vol. No. pp Reviewer: Pvel Híc Aoc. Proeor PD. - Deprme o Memic d ormic Fcul o Educio Uiveri i Trv Róer Vráeľ Aoc. Proeor PD. - Deprme o Memic iue o Applied ormic Auomio d Memic Fcul o Meril Sciece d Tecolo i Trv Slov Uiveri o Tecolo Brilv 68