2T-PERIODIC SOLUTION FOR m ORDER NEUTRAL TYPE DIFFERENTIAL EQUATIONS WITH TIME DELAYS

ＩｎｔｒｏｄｕｃｔｉｏｎＩｔｉｓｗｅｌｌ＿ｋｎｏｗｎｔｈａｔｎｅｕｔｒａｌｔｙｐｅｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎａｒｅｗｉｄｅｌｙｕｓｅｄｉｎｂｉｏｌｏｇｙ ,ｐｈｙｓｉｃｓａｎｄｃｈｅｍｉｓｔｒｙ .Ｔｈｅｓｔｕｄｙｏｆｔｈｅｎｅｕｔｒａｌｅｑｕａｔｉｏｎｈａｓｉｍｐｏｒｔａｎｔｓｉｇｎｉｆｉｃａｎｃｅｉｎｂｏｔｈｔｈｅｏｒｙａｎｄａｐｐｌｉｃａｔｉｏｎ .Ｉｎｔｈｉｓｐａｐｅｒ,ｗｅｃｏｎｓｉｄｅｒｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｏｆｎｅｕｔｒａｌｅｑｕａｔｉｏｎａｓ∑ｍｉ=0[ａｉｘ(ｍ-ｉ…     

Optimal Control of Turbulent Flow as Measure Solutions

Abstract

We use the concept of relaxed or measure-valued solutions for control problems of turbulent flow related to Navicr-Stokes equation. Sufficient conditions guaranteeing the existence of measure solutions arc presented. Results on existence of optimal controls for Blow up time and Bolza problems for such systems are also presented. New results on relaxed necessary conditions of optimality are proved. Further it is shown that the relaxed necessary conditions reduce to classical Pontryagin type necessary conditions if measure solutions degenerate into Dirac structure. The paper is concluded with an algorithm based on the new necessary conditions for computing optimal controls.     

非线性电报方程组的3个双周期正解问题

This paper studies existence of at least three positive doubly periodic solutions of a coupled nonlinear telegraph system with doubly periodic boundary conditions. First, by using the Green function and maximum principle, existence of solutions of a nonlinear telegraph system is equivalent to existence of fixed points of an operator. By imposing growth conditions on the nonlinearities, existence of at least three fixed points in cone is obtained by using the Leggett-Williams fixed point theorem to cones in ordered Banach spaces. In other words, there exist at least three positive doubly periodic solutions of nonlinear telegraph system.     

Global dynamical behavior of a three-body system with flexible connection in the gravitational field

Summary In this paper, the global behavior of relative equilibrium states of a three-body satellite with flexible connection under the action of the gravitational torque is studied. With geometric method, the conditions of existence of nontrivial solutions to the relative equilibrium equations are determined. By using reduction method and singularity theory, the conditions of occurrence of bifurcation from trivial solutions are derived, which agree with the existence conditions of nontrivial solutions, and the bifurcation is proved to be pitchfork-bifurcation. The Liapunov stability of each equilibrium state is considered, and a stability diagram in terms of system parameters is presented. Received 10 March 1998; accepted for publication 21 July 1998     

Kink wave determined by parabola solution of a nonlinear ordinary differential equation

By finding a parabola solution connecting two equilibrium points of a planar dynamical system,the existence of the kink wave solution for 6 classes of nonlinear wave equations is shown.Some exact explicit parametric representations of kink wave solutions are given.Explicit parameter conditions to guarantee the existence of kink wave solutions are determined.     

Weakly nonlinear boundary-value problems for differential equations in a Banach space in the critical case

We obtain necessary and sufficient conditions for the existence of solutions of weakly nonlinear boundary-value problems for differential equations in a Banach space. A convergent iterative procedure is proposed for the determination of solutions. We also establish a relationship between necessary and sufficient conditions.     

EXISTENCE OF BOUNDED SOLUTIONS ON THE REAL LINE FOR LIENARD SYSTEM

The existence of monotone and non-monotone solutions of boundary value problem on the real line for Lienard equation is studied. Applying the theory of planar dynamical systems and the comparison method of vector fields defined by Lienard system and the system given by symmetric transformation or quasi-symmetric transformation, the invariant regions of the system are constructed. The existence of connecting orbits can be proved. A lot of sufficient conditions to guarantee the existence of solutions of the boundary value problem are obtained. Especially, when the source function is bi-stable, the existence of infinitely many monotone solusion is obtained.     

冲击动力系统的鲁棒稳定性分析

考虑冲击动力系统的ｋ－ｐ周期运动的鲁棒稳定性问题。首先，根据微分方程的解、冲击条件和衔接条件，应用迭代法给出了系统存在ｋ－ｐ周期运动的充分必要条件，并利用稳定性的等价原理，通过周期运动的扰动差分方程导出其稳定条件；然后，着重对含有不确定参数的冲击动力系统的ｋ－ｐ周期运动的稳定性进行了分析，得出了鲁棒稳定的充分条件，文末给出了用于阐明理论结果的算例。     

Pseudo Almost Periodic Solutions to a Class of Semilinear Differential Equations

This paper is concerned with the existence and uniqueness of pseudo almost periodic solutions to a class of semilinear differential equations involving the algebraic sum of two (possibly noncommuting) densely defined closed linear operators acting on a Hilbert space. Sufficient conditions for the existence and uniqueness of pseudo almost periodic solutions to those semilinear equations are obtained. An erratum to this article is available at .     

Existence of bounded solutions on the real line for Liénard system

The existence of monotone and non-monotone solutions of boundary value problem on the real line for Liénard equation is studied. Applying the theory of planar dynamical systems and the comparison method of vector fields defined by Liénard system and the system given by symmetric transformation or quasi-symmetric transformation, the invariant regions of the system are constructed. The existence of connecting orbits can be proved. A lot of sufficient conditions to guarantee the existence of solutions of the boundary value problem are obtained. Especially, when the source function is bi-stable, the existence of infinitely many monotone solusion is obtained.     

Equilibrium Configurations of an Inflated Cylindrical Membrane

We study the inflation of a cylindrical elastic membrane under conditions where more than one equilibrium radius is possible at the same pressure. In particular, we investigate the existence of solutions connecting two sections of different radii. The existence of such solutions is found to be linked to a Maxwell equal area rule.     

Dynamical behavior of traveling wave solutions of ion acoustic plasma equations

By using the theory of planar dynamical systems to the ion acoustic plasma equations, we obtain the existence of the solutions of the smooth and non-smooth solitary waves and the uncountably infinite smooth and non-smooth periodic waves. Under the given parametric conditions, we present the sufficient conditions to guarantee the existence of the above solutions.     

Oscillatory behavior for high order functional differential equations

I.IntroductionAstheoscillatorybehaviorandasymptoticbehavioroffunctionaldifferentialequations'solutionsareveryimportantbothintheoryandpractice,lotsofreseachresultshavecomeoutinthisfieldinrecentyears.Indocument[l1,theoscillationandasymptoticclassificationof…     

On the boundedness and periodicity of the solutions of a certain vector differential equation of third-order

IntroductionWeshallbeconcernedherewiththird-ordersystemofnon-lineardifferentialequationsoftheformX F(X,X)X Ba H(X)=P(t,X,X,X),(l)whereX6R",R"denotestherealn-dimensionalEuclldeanspaceRxRx'xR(nfactors),R=(--ac,ac),Fisannxn-matrixfunction,Bisarealnxn-constantsymmetricmatrix,H:R"~R"1andp:RxR"xR"xR"~R".ThedotsindicatCdifferentiationwithrespeCttot.ThefunctionsF,HandParecontinuousforallvaluesoftheirrespeCtivearguments.Moreover,theexistenceandtheuniquenessofthesolutionsof(l)willbeassu…     

Existence of Traveling-Wave Solutions for Hyperbolic Systems of Balance Laws

This paper is concerned with traveling-wave solutions for hyperbolic systems of balance laws satisfying a stability condition and a Kawashima-like condition. We are interested in the case where the traveling-wave equations have a singularity, which is absent for 2 × 2 systems satisfying the two conditions. To deal with the singularity, we reduce the problem to a parametrized one without singularity by using the center manifold theorem. For the parametrized problem, we prove the existence of solutions by modifying an existing argument in the literature. In this way, we show the existence of traveling-wave solutions.     

EXISTENCE OF POSITIVE RADIAL SOLUTIONS FOR SOME SEMILINEAR ELLIPTIC EQUATIONS IN ANNULUS

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｓｔｕｄｉｅｓｏｆｐｏｓｉｔｉｖｅｒａｄｉａｌｓｏｌｕｔｉｏｎｓｆｏｒｆｏｌｌｏｗｉｎｇｓｅｍｉｌｉｎｅａｒｅｌｌｉｐｔｉｃｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍ(Ｐ)　Δｕ(Ｘ) +ｇ( ｜Ｘ｜)ｆ(ｕ(Ｘ) ) =0 ,　　Ｒ1<｜Ｘ｜<Ｒ2 ,ｕ(Ｘ) ｜｜Ｘ｜ =Ｒ1 =ｕ(Ｘ) ｜｜Ｘ｜ =Ｒ2 =0(ｗｈｅｒｅＲ1>0 ,Ｘ ∈Ｒｎ,ｎ ≥ 2 )ａｒｅｂｅｉｎｇｃｏｎｔｉｎｕｅｄｆｏｒｒｅｃｅｎｔ 2 0ｙｅａｒｓｗｉｔｈｏｕｔｉｎｔｅｒｒｕｐｔｉｏｎ[1- 11],ｂｅｃａｕｓｅｔｈｅｐｒｏｂｌｅｍ (Ｐ)ｈａｓｗｉ…     

Travelling wave solutions for a second order wave equation of KdV type

The theory of planar dynamical systems is used to study the dynamical behaviours of travelling wave solutions of a nonlinear wave equations of KdV type.In different regions of the parametric space,sufficient conditions to guarantee the existence of solitary wave solutions,periodic wave solutions,kink and anti-kink wave solutions are given.All possible exact explicit parametric representations are obtained for these waves.     

Investigation of the structure of the set of continuous solutions of systems of linear functional difference equations

We establish conditions for the existence of continuous solutions of systems of linear functional difference equations with linearly transformed argument and develop a method for the construction of these solutions.     

Bifurcations of travelling wave solutions for a coupled nonlinear wave system

Introduction FangShaomeiandGuoBoling[1]consideredthefollowingtimeperiodicproblemof dampedcouplednonlinearwaveequations:ut f(u)x-αuxx βuxxx 2vvx=G1(u,v) h1(x),vt-γvxx 2(uv)x g(v)x=G2(u,v) h2(x),(1)whereα,β,γareconstants,andγ>0,β≠0.Undertheperiodicboundaryconditions,the authorsobtainedtheuniqueexistenceofstrongsolutionsfortheabovesystem.InthispaperweshallconsiderbifurcationbehaviorofthetravellingwavesolutionsofEq.(1)inthecaseGi(u,v)≡0,hi(u,v)≡0(i=1,2).Letξ=x-ct,u=u(x-ct),where cis…     

Asymptotic behavior of solutions of a nonlinear ordinary differential equation of order <Emphasis Type="Italic">n</Emphasis>

Asymptotic representations of a certain class of regular nonoscillating solutions of a differential equation with exponential nonlinearity are obtained. Necessary and sufficient conditions for the existence of these solutions are established.