THE FINITE DIMENSION BEHAVIOR OF THE SCHRDINGER EQUATION WITH NONLOCAL INTEGRAL NONLINEARITY

1.IntroductionInthispaperweconsiderthefollowinginitialproblem,lUt box OIUlpU 7U["III'dX IkU~gi(1.l)j--colul'dx iku~g,(1.l)nit=0~u',(1.2)whereP,7andkarerealnumber,k>0,theexponentpsatisfiesPZ0,ifP50,05P<4,ifP>0.(l.3)ForthenonlinearSchrodingerequation.lat aam P(lul')u ikm=g,(1.4)itslongtimebehaviorhasbeenobservedbynumericalmethodandphysicalargUmefits,(1)chaoticattractorexists;(2)afinitedimensional"space"confinestheattractors(see[l,2]),ajndithasalsobeenprovedbymathematicalmethodthatthereex…     

SECOND BOUNDARY VALUE PROBLEM FOR NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

1IntroductionandMainResultConsiderthefollowingnonautonomoussecondboundaryvalueproblem(BVP)on[0,1]:y" f(x,y)=0,y(0)=y,(1)=0,(1)wheref:[0,l]xR"-R"hascontinuouspartialderivatives.Systemy" f(x,y)=0canbeinterpretedphysicallyasthegeneralizedNewtonianequation.Ouraimistostudytheexistenceanduniquenessofproblem(l).Investigatingsuchaboundaryvalueproblemisfewerthantheotherone.Duringthepasttwentyyears,therehasbeenagreatamountofworkinthetheoryofboundaryvalueproblemforsecondorderordinarydifferentialequat…     

THE EXTREME POINTS OF SEVERAL CLASSES OF ANALYTIC FUNCTIONS

1Intr0ducti0nLetAden0tethesetofallfunctionsanalyticinA={z:Izl<1}.LetB={W:WEAandIW(z)l51}.Aisalocallyconvexlineaztop0l0gicalspacewithrespecttothetopologyofuniformconvergenceon`c0mpact8ubsetsofA-LetTh(c1,'tc.-1)={p(z):p(z)EA,Rop(z)>0,p(z)=1 clz czzz ' c.-lz"-l 4z" ',wherecl,',cn-1areforedcomplexconstants}.LetTh,.(b,,-..,b,-,)={p(z):P(z)'EAwithReP(z)>Oandp(z)=1 blz ' b.-lz"-l 4z" '-,wherebl,-'-jbu-1areffeedrealconstantsanddkarerealnumbersf0rk=n,n 1,'--}-LetTu(l1,'i'tI.-1)={…     

A Note on the Bloch Function in SCV

51,Intr0ductionThroughoutthispaper,Cwilldenotethecomplexfield,andC"willbethecartesianl)r0cl-uctofncopiesofClherenisanypositiveinterger.TheP0intsofC"arethusordered)I-tul>lesz=(zl,...,z`),whereeachzieC.ThainnerproductinC"is(z,w)=Zzjwj(z,weC"),j=1andtheassociatednormlzI=(z,z)1l2(zeC")makeC"intoann-dimensionalHilbertspacewhoseopenunitballwillbedenotedbyB..ForapairofpointszandwinB,,thehyperdi8tancebetweenzandwisdenotedbyp(z,w)=oplog(the),wllereh(z,w)=(lz-wl= Izwl'~lzl2It`)l2)1/2.['I1-w5l]-'…     

ON GLOBAL EXISTENCE AND NONEXISTENCE FOR SOME NONLINEAR PARABOLIC EQUATIONS

0IntroductionInthispaper,weconsidertheinitial-boundaryvalueproblemforthefailliliarequationwherefiisaboundeddomaininR"withsmoothboulldaryoff,p22isacollstantalldlp(z,u)l5of'(tl" 'forsomea20andc>0.FOrp(x,ti)=Itll"'u,theauthorsofpaper[4,7llolwereillterestedillllollllegativesolutionandhadobtainedfollowillgresults(alsosee[12]).1)If25a 2相似文献   

IMPASSE POINTS AND REDUCTION OF QUASILINEAR DAES WITH INDEX-K

1Introduction(lollsi(l(ir(flllasililloa,l'(hfF(31,')rll,ial()(f[llal.ionfw[1(ireA:UI'-L(It")atldG:U"-It"ar(3ofCZfullctionsoilU".l./"isa,coml)a':t.sllbs')1,of'it".if,isofmor(3intereststoscholars.B(?causco.rill(:I.allk,it,(:al['tl)(1lie(luc(3(l10(lailya,rlOI)li:sviatileiltnl)licitfuTlctiollill'3()r'ltll,wlliclld(?t,(31.llflill'3(1bombssp(i(:ialfcaturos.II'irstly,itImlyhavcrl'l,uni(lll(}solllt,iorll,()IVI'f;r(0)=blotj?(0)=po,it'(x,,p,)=0,if'ilcquil.essolTlecol'Tlf)a,1,il)ilit,y'StJI)l,o…     

A NEW FORM OF DISCRIMINANT FOR HILL'S EQUATION

1IntroductionHill'sequationoftenhastwoforms.OneistheotheriswhereaandAareconstants,p(t)satisfiesp(t)30pECIorjorp(t)dt>0(see[2]).Obviously,thesetwoformscanbeinterChangedtoeachother.Letgi(t)andyZ(t)betwosolutionsofEq.(1.1)(orEq.(1.2))whichsatisfyinitialvalueconditions:yi(0)=1,yi(0)=0andy2(0)=0,yi(0)=l,respectively.WriteA=yi(r) yi(r).(l.3)ThenaiscalledthediscriminantofEq.(l.1)(orEq.(l.2)).ItiswellknownthatEq.(l.l)(orEq.(1.2))isstableifandonlyifIII<2orlal=2andyi(r)=yi(r)=0.Thetraditionalmet…     

直线方程四则运算的意义和价值

1　直线方程的加减运算1 1　意义已知两条直线l1 :A1 x B1 y C1 =0 ,l2 :A2 x B2 y C2 =0 .我们来分析l3:(A1 A2 )x (B1 B2 )y C1 C2 =0和l1 、l2 有什么关系 .( 1 )当l1 ∥l2 时 ,l3也和它们平行 .因为l1 ∥l2 ,有 ,A1 A2 =B1 B2,则 A1 A2A2 =B2 B2B2,所以l3∥l2 .( 2 )当l1 和l2 相交时 .记两直线的交点为P(x0 ,y0 ) ,那么 ,A1 x0 B1 y0 C1 =0和A2 x0 B2 0 C2 =0 ,因此 ,(A1 A2 )x0 (B1 B2 )y0 C1 C2 =0也成立 .所以l3也过点P .我们还可以推广到一般的情况 :直线A1 x B1 y C1 λ(A2 x B2 C2 ) =0…     

BIFURCATION SOLUTION BRANCHESAND THEIR NUMERICAL APPROXIMATIONS OF A SEMI-LINEAR ELLIPTIC PROBLEM WITH TWO PARAMETERS

1.IntroductionConsidertheequationdependingontheparametersA,pER,wheref'R-Rands:R-RaresmoothoddfunctionandLetS'u(x)-u(T--x),r={S,I}.Then(l.l)isr-equivalent.Theequality(l.Za)isjustanormalizationoffatx=0.WeintroduceaSobolevspaceX:=Ha(0,7),anddefineamappingT'gEL'(0,T)u'=TaEXimplicitly'Aweakformof(1.1)inXxRZisDuetof(0)~0,theproblem(1.3)(resp.(1))hasatrivialsolutioncurveIfwerestrictp=0,then(l.3)reducestoaproblemwithsingleparameteranditsbifurcationsonthetrivialsolutioncurveCOarewellknown,…     

POSITIVE SOLUTIONS OF FOURTH-ORDER TWO-POINT BOUNDARY VALUE PROBLEMS

1IntroductionInthispaper,weconsidertheexistenceofpositivesolutionsoftheequationwithoneofthefollowingsetsofboundaryvalueconditionsWhenthefunctionf(x,y)p)inequation(l.1)isoftheformf(x,y,p)Hy,theboundaryvalueproblem(l.1)-(1.2)describesthedeformationsofanelasticbeambothofwhoseendssimplysupportedat0and1,andtheboundaryvalueproblem(1.l)-(l.3)describesanelasticbeamwithoneofitsendsimplysupportedandtheotherendclampedbyslidingclamps.In[11,Aftabizadehshowedtheexistenceofasolutionto(l.l)-(l.2)undertherest…