BOUNDEDNESS OF SOLUTIONS FOR SUPERLINEAR REVERSIBLE SYSTEMS

51. IntroductionThe boundedness problem of solutions of the following nonlinear scalar differelltial equa-nons of second order..x f(x, t)ic g(x, t) = 0 (1'1)has been widely investigated by many authors since the 1940's, where f(x, t 1) = j(x, t),g(x, t 1) = g(x, t).When j(x, t) = 0, Equation (1.1) is a conservative system and takes the following form..x g(x, t) ~ 0. (1'2)The first contribution of the boundedness of all the solutions of Equation (1.2) is due toMorrislll, whO proved …     

Solitary waves and their bifurcations of KdV like equation with higher order nonlinearity

We investigate the KdV like equation with higher order nonlinearity ut + a(1 +bun)unux + uxxx = 0with n ≥ 1, a, b ∈ R and α≠ 0. The bifurcations and explicit expressions of solitary wave solutions for theequation are discussed by using the bifurcation method and qualitative theory of dynamical systems. Thebifurcation diagrams, existence and number of the solitary waves are given.     

一类四阶两点边值问题正解的存在性

运用上下解方法及拓扑度理论讨论了非齐次边界条件下四阶两点边值问题u″″(t)=f(u(t)),t∈(0,1),u(0)=u″(0)=u″(1)=0,u(1)=λ,其中λ＞0为参数,f∈C([0,+∞),[0,+∞)).在非线性项满足一定的增长条件下,获得了上述问题存在正解时λ的取值范围.     

Sign-Changing Solutions and Multiple Solutions Theorems for Semilinear Elliptic Boundary Value Problems with a Reaction Term Nonzero at Zero

In this paper, we use the ordinary differential equation theory of Banach spaces and minimax theory, in particular, the local mountain pass lemma to study asymptotically linear and superlinear elliptic boundary value problems with a reaction term nonzero at zero, and get new multiple solutions and sign-changing solutions theorems. Moreover, the sign-changing solutions change sign exactly once under some conditions.     

Sign-Changing and Multiple Solutions Theorems for Semilinear Elliptic Boundary Value Problems with Jumping Nonlinearities

In this paper, we use the ordinary differential equation theory of Banach spaces and minimax theory, and in particular, the relative mountain pass lemma to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity, and get new multiple solutions and sign-changing solutions theorems, at last we get up to six nontrivial solutions. Received April 21, 1998, Revised November 2, 1998, Accepted January 14, 1999     

Energy methods for stochastic differential equations

This article is devoted to the existence of strong solutions to stochastic differential equations (SDEs). Compared with Ito's theory, we relax the assumptions on the volatility term and replace the global Lipschitz continuity condition with a local Lipschitz continuity condition and a Hoelder continuity condition. In particular, our general SDE covers the Cox–Ingersoll–Ross SDE as a special case. We note that the general weak existence theory presumably extends to our general SDE (although the explicit time dependence of the drift term and the volatility term might require some extra considerations). However, avoiding weak existence theory we prove the existence of a strong solution directly using a priori estimates (the so-called energy estimates) derived from the SDE. The benefit of this approach is that the argument only requires some basic knowledge about stochastic and functional analysis. Moreover, the underlying principle has developed to become one of the cornerstones of the modern theory of partial differential equations (PDEs). In this sense, the general goal of this article is not just to establish the existence of a strong solution to the SDE under consideration but rather to introduce a new principle in the context of SDEs that has already proven to be successful in the context of PDEs.     

二阶两点边值问题的多解存在性

本文讨论一类二阶两点边值问题$x^{\prime\prime}(t)+f(t,x(t),x^{\prime}(t))=0, t\in (0, 1)$, $a x(0)-b x^\prime(0)=0, ~~c x(1)+d x^\prime(1)=0$,~~其中 $f:[0,1]\times R^2\longrightarrow R$ 是连续的, $ a>0,b\ge 0,c>0,d\ge 0$. 通过运用上下解方法和 Leray-Schauder 度理论,得到了三个解的存在性结果.     

Periodic and subharmonic solutions of a class of superquadratic Hamiltonian systems

Periodic solutions and infinitely distinct subharmonic solutions are obtained for a class of nonconvex and nonautonomous superquadratic Hamiltonian systems by using the minimax methods in critical point theory.     

奇异非线性Sturm-Liouville问题的正解

本文研究奇异非线性Sturm-Liouville问题其中(Lφ)(x)=(p(x)φ′(x))′+g(x)φ(x),并且允许h(x)在x=0和x=1奇异。应用锥理论和不动点指数方法,在与相应的线性算子第一特征值有关的条件下获得了正解的存在性结果,本质地推广和改进了文献[1-9]中的主要结论。     

Existence and Uniqueness of Solutions of a Class of Singular Parabolic Equations

We will establish an existence and regularity theory for weak solutions of a class of singular parabolic equations associated with Dirichlet data, whose prototype is u_t - div(|∇u|^{p-2}∇u) = 0 (1 < p < 2)     

一类非线性发展方程的对称分析及级数解

基于李对称理论分析了广义Burgers方程的推广方程,获得其有限维李对称.进一步,研究向量场的伴随表示构造优化系统.最终基于对称约化,获得了方程的约化系统及包含级数解在内的群不变解.     

W versus C local minimizers and multiplicity results for quasilinear elliptic equations

In this paper we show that the local minimizers of a class of functionals in the C¹-topology are still their local minimizers in . Using this fact, we study the multiplicity of solutions for a class of quasilinear elliptic equations via critical point theory.     

Existence of quasiperiodic solutions and Littlewood’s boundedness problem of super-linear impact oscillators

So far most application of Kolmogorov-Arnold-Moser (KAM) theory has been restricted to smooth dynamical systems. In this paper, it is shown by a series of transformations that how KAM theory can be used to analyze the dynamical behavior of Duffing-type equations with impact. The analysis is carried out for the example

(0.1)     

非线性三点共轭边值问题正解存在性的特征值准则

本文提出了三点边值问题-v″(t)=b(t)f(v(t)),满足v′(0)=0及v(1)=αv(η)的共轭问题-u″(t)=b(t)f(u(t)),u′(0)=u(1)=0及u′_+(η)-u′_-(η)=αu′(1),得到了相应的Green函数．将其转化为Hammertein型积分方程,借助于其相应线性问题的第一特征值,利用锥上的不动点指数理论,给出了共轭问题单个正解及多个正解存在的特征值准则．     

关于高阶常系数线性中立型方程周期解的讨论

本文讨论高阶常系数线性中立型方程的周期解问题,作者利用Fourier级数理论给出周期解存在,唯一的充分必要条件,所得结果包含和推广了文献[1]中的结果。     

Eigenvalues for Radially Symmetric Fully Nonlinear Operators

In this paper we present an elementary theory about the existence of eigenvalues for fully nonlinear radially symmetric 1-homogeneous operators. A general theory for first eigenvalues and eigenfunctions of 1-homogeneous fully nonlinear operators exists in the framework of viscosity solutions. Here we want to show that for the radially symmetric operators or in the one dimensional case a much simpler theory, based on ode and degree theory arguments, can be established. We obtain the complete set of eigenvalues and eigenfunctions characterized by the number of zeroes.     

Symmetry reductions and traveling wave solutions for the Krichever–Novikov equation

In this paper, we study the Krichever–Novikov equation from the point of view of the theory of symmetry reductions in PDEs. By using this theory, we find that for the Krichever–Novikov equation some similarity solutions are solutions with physical interest: solitons, kinks, antikinks, and compactons. Copyright © 2012 John Wiley & Sons, Ltd.     

Existence of periodic and subharmonic solutions for second order functional difference equations

In this paper, by using the critical point theory, the existence of periodic and subharmonic solutions to a class of second order functional difference equations is obtained. The main approach used in our paper is variational technique and the Saddle Point Theorem. The problem is to solve the existence of periodic and subharmonic solutions of second order forward and backward difference equations.     

Critical points and positive solutions of singular elliptic boundary value problems

Usually we do not think there is variational structure for singular elliptic boundary value problems, so it cannot be considered by using critical points theory. In this paper, we use critical theory on certain convex closed sets to solve positive solutions for singular elliptic boundary value problems, especially use the ordinary differential equation theory of Banach spaces to obtain new results on the existence of multiple positive solutions. The method is useful for other singular problems.     

双层浅水波模型柯西问题的经典解

主要研究浅水理论中双层浅水波模型柯西问题的经典解.在适当的初值条件下,得到整体经典解存在唯一性的充要条件.同时对单层浅水波模型也得到相应结果.