NON-CLASSICAL GENERALIZED SOLUTIONS FOR SOME HYPERBOLIC SYSTEMS

1.IntroductionThemostspecialfeatureofconservationlawsofhyperbolicsystemsisthattheirsolutionsalwayshavediscolltinuities.ThiskindofdiscontinuityrepresentstheshoCkwavephenomena.Sopeopleintroducetheideaofgeneralizedsolutionwhichsatisfiesequationsinthesenseofdistribution.Underthisdefinition,alotofachievementshavebeenobtainedsince1950'sinthisfield.However,inrecelltyears,manyauthors(see[l--6])discoveredthatthereisnoexistenceanduniqllenessofsolutionstoRiemannproblemsforsomeconservationlaws.Bynumeric…     

ON THE EXISTENCE OF MEROMORPHIC SOLUTIONS OF ALGEBRAIC DIFFERENTIAL EQUATIONS

ＯＮＴＨＥＥＸＩＳＴＥＮＣＥＯＦＭＥＲＯＭＯＲＰＨＩＣＳＯＬＵＴＩＯＮＳＯＦＡＬＧＥＢＲＡＩＣＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳ￥ＹＵＡＮＷｅｎｊｕｎ（ＤｅｐａｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＸｉｎｊｉａｎｇＮｏｏｒｍａｌＵｎｉｖｅｒｓ...     

ON THE EXISTENCE AND UNIQUENESS OF PERIODIC SOLUTIONS OF VOLTERRA INTEGRAL DIFFERENTIAL EQUATIONS

In this paper,we use the contraction principle and the Leray-Schauder principleto deal with the existence of periodic solutions of the Volterra integral differential equation.Some new existence criteria and unique existence criteria are obtained.     

THE GENERALIZED SOLUTIONS OF PBVP FOR INTEGRO-DIFFERENTIAL EQUATIONS

ＴＨＥＧＥＮＥＲＡＬＩＺＥＤＳＯＬＵＴＩＯＮＳＯＦＰＢＶＰＦＯＲＩＮＴＥＧＲＯ－ＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＺＨＵＡＮＧＷａｎ；ＣＨＥＮＹｕｂｏ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｈａｎｄｏｎｇＮｏｒｍａｌＵｎｉｖｅｒｓ...     

FIXED POINTS OF THE lTH POWER OF DIFFERENTIAL POLYNOMIALS GENERATED BY SOLUTIONS OF DIFFERENTIAL EQUATIONS

In this paper, we study the problem on the fixed points of the lth power of linear differential polynomials generated by second order linear differential equations. Because of the control of differential equation, we can obtain some precise estimate of their fixed points.     

INTERPOLATED FINITE ELEMENT METHODS FOR SECOND ORDER HYPERBOLIC EQUATIONS AND THEIR GLOBAL SUPERCONVERGENCE

We develop the interpolated finite element method to solve second-order hy-perbolic equations. The standard linear finite element solution is used to generate a newsolution by quadratic interpolation over adjacent elements. We prove that this interpo-lated finite element solution has superconvergence. This method can easily be applied togenerating more accurate gradient either locally or globally, depending on the applications.This method is also completely vectorizable and parallelizable to take the advantages ofmodern computer structures. Several numerical examples are presented to confirm ourtheoretical analysis.     

GLOBAL CLASSICAL SOLUTIONS AND EXPLICITEXACT SOLUTIONS TO A KIND OF QUASILINEARPARTIAL DIFFERENTIAL EQUATIONS

1.IntroductionandMainResultTherearemanyexamplesconcerningtheformationofsingularitiesinthesoluti0nsofnon-linearpartialdifferefitialequations.Thus,althoughasolutionmaybesmoothinitially,itmaybecomesingularatsomelatertime(see[1-6]).However,T.Nishidahaspr0vedthatiftheCln0rmoftheinitialdataissufficientlysmall,thentheCauchyproblemadmitsauniqueglobalsmooths0lutionont2Otoareduciblequasilinearhyperbolicsystemwithfirst(lower)orderdissipativeterm(see[7]).Thereafter,manyauthorshavegeneralizedNishida'sr…     

POSITIVE SOLUTIONS FOR DIRICHLET PROBLEMS OF SINGULAR SEMILINEAR ELLIPTIC EQUATIONS

ＰＯＳＩＴＩＶＥＳＯＬＵＴＩＯＮＳＦＯＲＤＩＲＩＣＨＬＥＴＰＲＯＢＬＥＭＳＯＦＳＩＮＧＵＬＡＲＳＥＭＩＬＩＮＥＡＲＥＬＬＩＰＴＩＣＥＱＵＡＴＩＯＮＳ￥ＣＵＩＳｈａｎｇｂｉｎ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＬａｎｚｈｏｕＵｎｉｖｅｒｓ...     

ON THE NONEXISTENCE OF PERIODIC SOLUTIONS FOR LIENARD-TYPE EQUATIONS

1.IntroductionConsiderthefollowinggeneralizedLi6nardtypeequationwhereshallowing[1],thisequationwillbecalledasaLi6nard-typeequationofordern.Whenn=1,theequation(1.1)canbewrittenasthefollowingformClearly)whenfi(x)=j(x),f200=0,theequation(1.3)isreducedtothewellknownLi6nardequationx j(x)x g(x)=0.(1.4)Recently,theequation(1.3)hasalreadybeenstudiedwithparticularemphasisontheex-istenceanduniquenessofperiodicsolutionsbyseveralauthors.Forexample,needmanandKuangll],GuidorizzilZ],ZhouJinI3],andJiangJ…     

MULTILEVEL CORRECTION FOR COLLOCATION SOLUTIONS OF VOLTERRA NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS

1. IntroductionWe consider a Volterra integro-differential equationwith initial condition u(0) = not where f, P, k are continuous on their respective domainsJ, JxR and axs (n:~ {(t,s): 05s5t5T}) such that (1.l) possesses a unique solutionu E C'(J).It has been shown in [1] that the collocation approximation for (1.1) by discolltinuous piece--wise polynomial spline collocation at the Gauss points restore optimal local superconvergenceat the knots but does not yield global superconvergence on…     

A STAGE-STRUCTURED SI ECO-EPIDEMIOLOGICAL MODEL WITH TIME DELAY AND IMPULSIVE CONTROLLING*

This paper formulates a robust stage-structured SI eco-epidemiological model with periodic constant pulse releasing of infectious pests with pathogens. The authors show that the conditions for global attractivity of the ＇pest-eradication＇ periodic solution and permanence of the system depend on time delay, hence, the authors call it ＂profitless＂. Further, the authors present a pest management strategy in which the pest population is kept under the economic threshold level （ETL） when the pest population is uniformly persistent. By numerical analysis, the authors also show that constant maturation time delay for the susceptible pests and pulse releasing of the infectious pests can bring obvious effects on the dynamics of system.     

AN EXISTENCE THEOREM FOR RADIAL POSITIVE SOLUTIONS OF NONLINEAR DEGENERATE ELLIPTIC EQUATIONS

ＡＮＥＸＩＳＴＥＮＣＥＴＨＥＯＲＥＭＦＯＲＲＡＤＩＡＬＰＯＳＩＴＩＶＥＳＯＬＵＴＩＯＮＳＯＦＮＯＮＬＩＮＥＡＲＤＥＧＥＮＥＲＡＴＥＥＬＬＩＰＴＩＣＥＱＵＡＴＩＯＮＳＬＩＵＨａｉｌｉａｎｇ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨｅｎａｎＮｏ...     

EXISTENCE OF MULTIPLE SOLUTIONS AND BIFURCATIONFOR CRITICAL SEMILINEAR BIHARMONIC EQUATIONS

ＥＸＩＳＴＥＮＣＥＯＦＭＵＬＴＩＰＬＥＳＯＬＵＴＩＯＮＳＡＮＤＢＩＦＵＲＣＡＴＩＯＮＦＯＲＣＲＩＴＩＣＡＬＳＥＭＩＬＩＮＥＡＲＢＩＨＡＲＭＯＮＩＣＥＱＵＡＴＩＯＮＳＥＸＩＳＴＥＮＣＥＯＦＭＵＬＴＩＰＬＥＳＯＬＵＴＩＯＮＳＡＮＤＢＩＦＵＲＣＡＴＩＯＮＦ...     

SOME STABILITY THEOREMS FOR DIFFERENCE EQUATIONS WITH TIME DELAY

1.IntroductionDifferellceequationswithtimedelayhavebeenanimportantclassofmathematicallllodels.Fol'instance,inbiology,systemsofdifferenceequationshavebeedmuchusedtomodeltheinteractiollsofspecieswithlion-overlappinggeneration,butthemodelwillberelatedtodifferenceequationswithtimedelaywhenollemustconsidertheinfluenceoftimedelayeffectforspecies11limber.Thereisextensiveresearchondifferentialequationswithtimedelaybutthereisoillylittleworkdoneondifferenceequationswithtimedelay.Ouraimistoestablishstabi…     

A LIOUVILLE THEOREM FOR GLOBAL SOLUTIONS OF HOMOGENEOUS CAUCHY PROBLEM OF MRABOLIC EQUATIONS

ＡＬＩＯＵＶＩＬＬＥＴＨＥＯＲＥＭＦＯＲＧＬＯＢＡＬＳＯＬＵＴＩＯＮＳＯＦＨＯＭＯＧＥＮＥＯＵＳＣＡＵＣＨＹＰＲＯＢＬＥＭＯＦＭＲＡＢＯＬＩＣＥＱＵＡＴＩＯＮＳ￥ＭＩＮＴａｉｓｈａｎ（ＢｅｉｊｉｎｇＬｉｇｈｔＩｎｄｕｓｔｒｙＩｎｓｔｉｔｕｔｅ，Ｂｅｉ...     

BLOW-UP SOLUTIONS FOR A CLASS OF NONLINEAR PARABOLIC EQUATIONS WITH MIXED BOUNDARY CONDITIONS

The type of problem under consideration is where D is a smooth bounded domain of RN, q = |(?)u|2. By constructing an auxiliary function and using Hopf's maximum principles on it, existence theorems of blow-up solutions, upper bound of "blow-up time" and upper estimates of "blow-up rate" are given under suitable assumptions on a, b, f, g, σand initial date U0(X). The obtained results are applied to some examples in which a, b, f, g and σare power functions or exponential functions.     

CONVERGENCE PROPERTIES OF DFP METHOD WITH MODIFIED QUASI-NEWTON EQUATIONS

1 IntroductionWe know that in order to obtain a superlinearly convergent method it is necessary to approximate the Newton step asymptotically (see [ll). How can we do this without actually evaluatingthe Hessian matrix by ally approximate to the Hessian matrix at every iteration? The answerwas discovered by Davidonl2] and was subsequently developed and popularized by Fletcher andPowell[3l. It consists of starting with any approximation to the Hessian matrir, and at eachiteration, updating th…     

BOUNDARY SOLUTIONS OF OPERATOR EQUATIONS IN PARIAL ORDERING SETS WITH APPLICATIONS TO SYSTEMS OF FUNCTIONALEQUATIONS

1.IntroductionThepaPers[l-5]studythecoupledfixedp0intsofmixedmonotoneoperatorsinBanachspaces;theyshowusgeneralizationsofthefixedpoillttheoremsonincreasingoperatorsanddecreasingoperators.Applicationsoftheseresultsmaybefoundinmanyfieldsofmathematicssuchasnonlineaxdifferentialequations,illtegralequationsanddynamicprogrammingtheories.Inthispaper,westudytheekistenceofsolutionsandboundarysolutionstoasystem0foperatorequationsinpartialorderingsets.Wegeneralizeandimproveonsomeresultsaboutthecoupledfix…     

A PRIOR ESTIMATE AND EXISTENCE OF POSITIVE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATION WITH THE THIRD BOUNDARY VALUE PROBLEM

1 IntroductionLet n be a bounded open domain with boundary aa, n c R", N 3 2. We consider a priorestimate and ekistence of positive solution of the fOllowing equationsWhere operatorL conforms to the Sum Ru1es, that is: for a repeated i or j, it mean to sum them up from 1 toN, ai;(x), ai(x), a(x) are bounded measurable in a. FOr operator L, we alWays assume: thereealst positive constants po and U1) such thatf(u) is a nonlinear function which grows in the form of Up with p 2 1, n is the out…     

ON A PROPERTY OF SOLUTIONS OF M-MATRIXEQUATIONS

1.IntroductionConsideringthefollowingsystemoflinearequationsAx~b(l.1)whereAisannxnM-matrix,bisanndimensionalvectorandbeRang(A).Theresearchonthesolutionofequation(1.1)isveryimportantinLeontiefmodelofinput-outputanalysis(see[1,2]).G.St..k.ma[Zjstudiestheabo…