Nagumo條件下的奇攝動(dòng)邊值問(wèn)題研究及其改進(jìn).pdf

1、福建師范大學(xué)李曉琴碩士學(xué)位論文摘要本文主要利用微分不等式的技巧(或稱(chēng)為上下解方法)，在一定條件下證明幾類(lèi)不帶小參數(shù)的非線性微分方程邊值問(wèn)題解的存在性，在此基礎(chǔ)上通過(guò)構(gòu)造高階漸近解得出解的誤差估計(jì)來(lái)研究幾類(lèi)帶有小參數(shù)的奇攝動(dòng)邊值問(wèn)題，同時(shí)提出了一個(gè)Nagumo條件的替代條件，并研究了幾類(lèi)不滿足Nagumo條件的二階非線性邊值問(wèn)題的微分不等式理論本文主要分為四章：第一章，首先，簡(jiǎn)單介紹了奇異攝動(dòng)理論的背景和前人的一些工作接著，給出上下解的概

2、念和Nagumo條件，并且給出了二階微分不等式的一些基本結(jié)果，及后面會(huì)用到的基本引理第二章，研究一類(lèi)帶有小參數(shù)的二階非線性n點(diǎn)邊值問(wèn)題，分別在強(qiáng)穩(wěn)定和弱穩(wěn)定的條件下證明了解的存在性并給出了解的一致有效估計(jì)第三章，處理一類(lèi)奇攝動(dòng)三階非線性微分方程的兩點(diǎn)邊值問(wèn)題，利用邊界層函數(shù)法，得到了解的高階漸近展開(kāi)式，再利用壓縮映像原理，證明了解的存在性并得到了解的誤差估計(jì)第四章，提出了一個(gè)Nagumo條件的替代條件，研究幾類(lèi)二階非線性微分方程在右端函

3、數(shù)不滿足Nagumo條件下的微分不等式理論，得到其存在性定理關(guān)鍵詞：非線性微分方程；邊值問(wèn)題；微分不等式；奇攝動(dòng)；高階展開(kāi)；存在性；Banach壓縮映像；Nagumo條件tI(、福建師范大學(xué)李曉琴碩士學(xué)位論文AbstractInthispapertakingadvantageofdifferentialinequalities(upperandlowers0_lutionsmethod)，weprovetheexistenceresul

4、tsofsomeclassesofboundaryvalueproblemsfornonlineardifferentialequationswithoutsmallparameterFurthermore，weapplytheresultstosingularlyperturbedboundaryvalueproblemswhichinvolvesmallparameterbyconstructingthehigherorderasy

5、mptoticsolutionandgettingtheerrorestimatebetweenasymptoticsolutionandexactsolutionWealsoposeaconditionwhichsubstitutesforNagumocondition，andstudythedifferentialinequalityofsomekindsoftwoordernonlineartwopointandn—pointbo

6、undaryvalueproblemswhichdon’tsatis6rtheNagumoconditionThispaperisdividedintofourchaptersInchapter1，firstlyweintroducesimplelythebackgroundofsingularlyperturbationtheoryandsomeimportantresultswhichformerscholarshavegetSec

7、ondlytheconceptofuppersolution，lowersolutionandNagumoconditionareintroduced，thensomeprincipalresultsofsecondorderandthirdorderdifferentialinequalityareshowed，andsomeprincipallemmalsareintroducedwhichwillbeusedinthefollow

8、ingchaptersInchapter2，westudyaclassoftwoordernonlinearnpointboundaryvalueproblemswithsmallparameter，provetheexistenceofsolutionandobtaintheerrorestimateofthesolutionundertheconditionsofstrongstabilityandweakstabilityresp

9、ectivelyInchapter3，weconsideraclassoftwopointboundaryvalueproblemwithsingularperturbationforthird—ordernonlineardifferentialequationByusingtheboundarylayerfunctionmethodatfirstthehigherorderasymptoticsolutionisconstructe

10、d，thenweprovetheexistenceofsolutionandobtaintheerrorestimateofthesolutionb‘yusingBanachcontracionmappingprincipleInchapter4，weposeaconditionwhichsubstitutesforNagumocondition，studythedifferentialinequalitiesofsometwoorde