PERIODICAL SOLUTION OF SOME DIFFERENTIAL EQUATIONS UDC 517.9(045)=20. Julka Knežević-Miljanović

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1 FCT UNIVESITTIS Sri: chanic uomaic Conrol and oboic Vol. N o PEIODICL SOLUTION OF SOE DIFFEENTIL EQUTIONS UDC 57.95= Julka Knžvić-iljanović Facul o ahmaic Univri o lgrad knzvic@oincar.ma.bg.ac.u brac. In hi ar w inviga h xinc o riodical oluion o h ir ordr dirnial uaion d = d Th obaind rul ar alid o a dirnial uaion whr h righ hand id conain a olnomial o h ourh dgr wih rc o. K word: Priodical oluion ir-ordr nonlinar ordinar dirnial uaion. W will inviga h dirnial uaion o h ir ordr d =. d Diniion: Th righ hand id o h dirnial uaion aii i h ollowing condiion ar ulilld: Funcion i dind in h inrval and in D whr D i a clod ara o h lan ; and ar - riodical wih rc o and d ; or [] whr i a nonngaiv ummarabl uncion on []; i a maurabl on []; 5 or D [] whr i a nongaiv ummarbl uncion on []; civd arch Suord b inir o cinc hnolog and dvlomn o ublic o Srbia rojc "Srucur o uncional anali and dirnial uaion" No 856 S ubjc claiicaion: Primar. Scondar C5.

2 88 JULK KNEŽEVIĆ-ILJNOVIĆ 6 For ach ri {x n } or vr n and a uncion x uch ha i x n x n han xn x d n whr x = u x. Th oluion o h uaion i abolu-coninuou uncion = on a ini inrval which aii h uaion. I i a oluion o h uaion hn i i a o how ha d d = d C. I C = hn = and i an riodical oluion. W will inviga h oraor F o ha [ ] d d F = d Th ollowing horm i a o how o b ru []. Thorm. uncion = i an riodical oluion o h uaion i and onl i i i a ixd oin o h oraor F. um ha = i a ixd oin o h oraor F. Thn i = w hav or Thn and h oraor F i = d d d d d d d = d. d d = d d d d d d F = d d [ d ].

3 Priodical Soluion o Som Dirnial Euaion 89 Thorm. L h condiion xi. I d d d d d hn hr xi a la on riodical oluion o h uaion. Proo. I i oibl o rov ha h oraor F i coninuou a. Oraor ranorm h hr o il. For vr coninuou uncion w hav d d F d d d d d = d d d d d d d = d d d d d ccording o h condiion o h horm whr i a corronding non-ngaiv valu xi a la on riodical oluion o ha h horm i rovd. Thorm. L h condiion b aiid and whr i a nongaiv ummabl uncion on []. I d d d d 5 hn in hr xi a la on riodical oluion o h uaion. Proo. Th oraor i coninuou in or [] and w will rov ha i ranorm h hr in il. W hav d d F d d d d d

4 9 JULK KNEŽEVIĆ-ILJNOVIĆ d d d d d d d d d bcau o h condiion 5. Thror in h hr xi a la on - riodical oluion b uing h rul o h Thorm and h Thorm. Now w conidr h uaion d d = 6 which i alo conidrd in [][]. L h uncion i i = b -riodical wih rc o ummabl or [] and i or a nongaiv ummabl uncion on []. Din h oraor = d d d 7 whr =. 7 L. Than and. For -riodical oluion aii h condiion d d d d L alo ai d d d 8 Thn w can xr h ollowing arion: Thorm. L h uncion b dind b 7 and l i ai h condiion or and or [] aii h condiion 8. I d whr > hn in xi a la on -riodical oluion o h uaion 6.

5 Priodical Soluion o Som Dirnial Euaion 9 Proo. Undr h condiion h coninuou oraor 7 ranorm hr ino il. W hav = d d d d d d d d d d d d d d d d d d d d d d d d d = and h horm i rovd. EFEENCES. P. Harman: "Ordinar dirnial uaion" 97.. Л.А. Люстерник В.И. Соболев: "Елементы функционального анализа" Лебедева В..: "О количестве периодических решений дифференциального уравнения первого порядка с полиномиальной правой частю" Дифференциальные уравнения т. н Лебедева В.М.: "О числе периодических решений дифференциального уравнения первого поряедка правой честю в виде неполного многочлена " Дифференциальные уравнения т У н PEIODIČN EŠENJ NEKIH DIFEENCIJLNIH JEDNČIN Julka Knžvić-iljanović U radu razmaraju riodična ršnja dirncijaln jdnačin rvog rda d d =. Dobijni rzula rimnjuj na dirncijalnu jdnačinu čija j dna rana olinom čvrog na u odnou na. Ključn rči: riodična ršnja nlinarna dirncijalna jdnačina