一类二阶隐式微分方程的可解性

一类二阶隐式微分方程的可解性 应用数学 MATHEMATICAAPPLlCATA 2007.20(3):473,477 SolvabilityofaSecond--orderImplicit DifferentialEquation FENGYu—qiang()~育强) (SchoolofScience,WuhanUniversityofScienceandTechnology,Wuhan430065,China) Abstract:Weareconcernedwiththesecond—orderimplicitdifferentialequationasfollows: ,(t,u(f),"(f))一0,a.e.t?(0,1) withboundarycondition"(O)="(1)一0.Loweranduppersolutionsmethodanditerativetech— niqueareemployedtOstudythesolvabilityofthisproblemandsomeexistenceresultsareobta ined. Keywords:Second—orderimplicitdifferentialequation;Loweranduppersolutions method;Iterativetechnique;Existence CLCNumber:0175.08AMS(2000)Su~ectClassification:34A09;34B15 Documentcode:AArticleID:1001—9847(2007)03—0473—05 1.Introduction Inthispaper,weinvestigatetheexistenceofsolutionstothefollowingboundaryvalue problemofsecondorderimplicitdifferentialequation 』f(… t,u(t.'a.e?(.'D, (P)1(O)一 (1)一0.… Theexplicitsecond—orderdifferentialequation (,)+f(t,())一0 withkindsofboundaryconditionshaveattractedconsiderableattention,andmanytechniques forsuchproblemshaveappeared(see[-1,6]andreferencestherein). Butwhenitcomestoimplicitequation,themethodsforexplicitequationcanbehardly extendedanddirectlyapplied. Inthispaper,weshallgivesomeresultsconcerningtheexistenceofsolutionsfortheim— plicitProblem(P)bytheuseofupperandlowersolutionsmethodanditeativetechnique. Thispaperisorganizedasfollows:InSection2somenotationsandpreliminariesarein— troduced.Themainresultandanexamplewhichillustratestheapplicationofthemainresult isgiveninSection3.Theproofofthemaintheoremisestablishedinthelastsection. Receiveddate:October8,2006 Foundationitem:SupportedbyChinaNationalScienceFoundationGrant(60574075)andScienceFoun— dationofEducationOfficeofHubeiProvinceGrant(Q2OO61101) Biography:FENGYu-qiang?male,Han,Shanxi,associateprofessor.engagedinnonlinearanalysis 474MATHEMATICAAPPLICATA2007 2.Preliminaries Throughoutthispaper,weassumethefollowingconditionshold: (H1)厂:[O,1]×R×.R—RisaCaratheodoryfunction,i.e. (i)forf1.e.t?[O,1],f(t,?,?):R×R—Riscontinuous: (ii)forevery(,)?R×R,厂(?,,):[O,1]一Rismeasurable. (iii)foreveryk>0,thereisareal—valuedfunction?L[O,1]suchthat ff(t,,)f?(,), forf1.e.t?[O,13andJ/2,J?k,JJ?五. (Hz)thereexistsM>0suchthatforany1,2,1,2?R,1?,1?anda. e.t? [O,1]holds: f(t,Ul,v1)一f(t,u2,v2)?M(v1,v2). Let X=L[O,1],W[O,1]一(z(,)?XIz"?U-o,13}, w5[O,1]一(z?W2,1[O,1]fz(O)一z(1)一O}. LetK一(z?XIz(,)?0,a.e.t?X),thenKisaregularconeofX.ForVz?X, letl1.27ll—IIz(,)Id,.Jo ForVz,Y?X,definez?yify--z?K,thenXisflpartialorderspaceendowedwith thepartialorder?.Denote[z,3,]=((,)?XIz(,)?(,)?(,)). Definition2.1If口,?w5[O,1]satisfy ff(t,口(,),口(,)?0,a.e.t?(O,1), 1f(t,fl(t),(,)?0,a.e.t?(o,1), then口,J9arecalledthelowerandtheuppersolutionsof(P),respectively. 3.MainResultsandApplication Themainresultofthispaperisthefollowing: Theorem3.1If(P)hasanuppersolutionandfllowersolution口whichsatisfy 口(,)?()a.e.t?(o,1), thenthereexistiterativesequencesin[口,whichconvergetothesolutionsof(P). Corollary3.11)Ifthefollowingconditionshold:? (i)f0ra.e.t?(O,1),f(t,0,O)?0; (ii)thereexistsc>0suchthatforf1.e.t?(O,1)and?[0,c/43,f(t,,一2c)?0. Then(P)hasflsolutionsatisfying0?(,)?c(t—t). 2)Ifthefollowingconditionshold: (i)forf1.e.t?(0,1),f(t,0,O)?0; (ii)thereexistsc>0suchthatforf1.e.t?(O,1)and?[一c/4,O],f(t,,2c)?o. Then(P)hasflsolutionsatisfying — c(t一)?u(,)?0. 3)Ifthereexistsc>0suchthat (i)forf1.e.t?(0,1)and?[O,c/4],f(t,,一2c)?0; (ii)fora.e.t?(O,1)and?[一c/4,O],f(t,",2c)?o. Then(P)hasflsolutionsuchthatllll?c/3. No.3FENGYu—qiang:SolvabilityofaSecond—orderImplicitDifferentialEquation475 WegivethefollowingexampletOillustratetheapplicationofourresults. Example3.1Let f(t,M,)一e"+()(2v+e-), w 0,to,~ 卜 3N Q Q , ' andQdenotesthesetofrationalnumbers. ItiseasytOcheckthatfsatisfies(H1). ItcanbeverifiedthatforanyMl,M2,1,712?R,M1?M2,1?2anda.e.t?E0,1] f(t,M1,1)一f(t,M2,"02)?2(1一2). Hence(H2)holds. Notingthatf(t,0,O)>0,f(t,",一2)<0fora.e.t?(O,1)and0?M?1/4,weclaim thatthefollowingboundaryvalueproblem 』"', +''M()+e一一)==o,a?e??(o,), (P)1 u(0)一M(1)一0… hasasolutionM?w3[O,1]satisfying0?M()?t(1一)invirtueofCorollary3.21). 4.ProofoftheMainResult ProofofTheorem3.1DuetO口()?()fora.e.t?(o,1)anda(0)一a(1)一fl(o) =卢(1)一0,wecaneasilygeta()?()fora.e.t?(O,1). Inwhatfollows,weshallproveTheorem3.1byfivesteps: Step1Converttheboundaryvalueproblem(P)intoanintegralequation.Infact.prob— lem(P)canberewrjttenaS f—M")一.,),M"(一M"(a.e.?(o,1), lu(0)一M(1)一0. (P) Let()一一M().NotingthatM(O)一M(1)一0,thenisasolutionofthefollowngin— tegralequation: 'r1 — ks(,,J.G(",一))+( r1 impliesM()一lG(t,s)(s)dsisasolutionofproblem(P),whered0 (Q) G二 ci. ? 1 , . 1 兰:;:二;二: (iii),(,G(,s)(),一z())+z()??r(,G(,)(),一())+(,); ciV111兰:;:二:二' 476MATHEMATICAAPPLICATA2007 Step3Constructtwosequences{z(,)),{(,))asfollows: z计(,)一(,,G(,,)z()d,一z(,))+z(,),=.,1,2,…, lzo()=z(); 计- ()一厂(f,G(,,)()d,一(,))+(,), 【Yo(,)=(,). =0,l,2,…, Weshallprovethat{z()),{())convergetosomez,Y?X.Infact,bvcondi— tion(H2)and(i),(iii)inStep2,wecanobtain zo(,)?zl(,)?z2(,)?…?z(,)?…?Y(,)…?Y2(,)?l(,)?Yo(,). Since.TO(,),Yo(,)?X,wehave{z(,)),{Y(,))cX.Duetotheregularityof(P) andthemonotonicityof{-T(,)),{Y(,)),thereexist-T,Y?Xsuchthat{z(,))con— vergesto-T'and((,)}convergestoY. Inaddition,wehave.33o?-T?z?Y?Y?Yo,一1,2,…. Step4z,YarethesolutionsofIntegralequation(Q). SincefisaCaratheodoryfunctionandlimx一z,thenweget rlr1 limlG(t,s)x(s)ds—IG(t,s)x.()d, and 厂(,dl0G(),,,)1_/,J0一G()z)一z))'p\,lJ, byCarathodoryconditionandLebesgusDominantConvergenceTheorem. Notethat z井-(,)=厂(,,G(,,)z()d,一z(,))十z(,), thenwehave … lim?一『(,G(t,s)x.(s)ds,--x.(t))+)], whichimplies z.(,)一l,(,,G(,,s)z()ds,一z.(,))+z(,), i.e.zisflsolutionofIntegralequation(Q). Inflsimilarway,wecanproveYisflsolutionofIntegralequation(Q),tOO. Step5Let .-- f>(,,)z()d,==J_G(,,s)()ds, then 一 (.)"(,)一z(,),.(0)一.(1)=0, (一)(,)一.(,),.(0)一(1)=0. Byabovediscussion,wecaneasilygetthatUandUarethesolutionsof(P)inEot. Thiscompletestheproof. ProofofCorollary3.1Undercondition(1),let z(,)三0,v(,):c(t—t2). No.3FENGYu-qiang:SolvabilityofaSecond-orderImplicitDifferentialEquation477 Undercondition(2),let z():一c(t—t),Y();0. Andwhilecondition(3)holds,let z()一一f(t—t),v()一c(t—t). Itiseasytocheckthatz,aretheloweranduppersolutionsof(P)undercondition(i),i= 1,2,3,respectively.Theorem3.1assertstheexistenceofsolution"of(P)undercondition (i),i一1,2,3. References: [1]WangH.Ontheexistenceofpositivesolutionsforsemilinearellpticequationsintheannals[J].J.Differ— entialEquations,l994,109:l,7. [2]DalmassoR.Positivesolutionsofsingularboundaryvalueproblems[J].NonlinearAnalysis(TMA), 1996.27:645,652. [3]JohnnyHenderson,ThompsonHB.Multiplesymmetricpositivesolutionsforasecondorderboundary valueproblem[J].Proc.Amer.Math.Soc.,2000,128:2373~2379. [4]BomammoG.Existenceofthreesolutionsforatwo-pointboundaryvalueproblem[J].AppliedMath. Lett.,2000,13:53,57. [5]HeXiaoming,GeWeigao.Existenceofthreesolutionsforaquasilineartwo-piontboundar yvalueproblem [J].Compt.Math.App1.,2003,45:765,769. [61FengYuqiang,LiuSanyang.Existence,multiplicityanduniqunessresultsforasecondord erM-point boundaryvalueproblem[J].Bul1.KoreanMath.Soc.,2004,41:483,492. 一 类二阶隐式微分方程的可解性 冯育强 (武汉科技大学理学院,湖北武汉430065) 摘要:本文关注如下的二阶隐式微分方程 f(t,"(),"())一0,a.e.t?(O,1), 边值条件为"(O)一"(1)一0.利用上下解 方法 快递客服问题件处理详细方法山木方法pdf计算方法pdf华与华方法下载八字理论方法下载 和迭代技巧研究了该问 题 快递公司问题件快递公司问题件货款处理关于圆的周长面积重点题型关于解方程组的题及答案关于南海问题 的可解性并得到了一 些解的存在性结果. 关键词:二阶隐式微分方程;上下解方法;迭代方法;存在性