UNIFORM BLOWUP PROFILES FOR DIFFUSION EQUATIONS WITH NONLOCAL SOURCE AND NONLOCAL BOUNDARY

Long time behavior of solutions to semilinear parabolic equations with nonlocal nonlinear source ut - △u = ∫Ω g(u)dx inΩ× (0, T) and with nonlocal boundary condition u(x, t) = ∫Ω f(x, y)u(y, t)dy on(e) Ω× (0, T) is studied. The authors establish local existence, global existence and nonexistence of solutions and discuss the blowup properties of solutions. Moveover, they derive the uniform blowup estimates for g(s) = sp(p ＞ 1) and g(s) = es under the assumption fΩ f(x, y)dy ＜ 1 for x ∈(e)Ω.     

A weak condition for secant method to solve systems of nonlinear equations

In this paper, a new weak condition for the convergence of secant method to solve the systems of nonlinear equations is proposed. A convergence ball with the center x0 is replaced by that with xl, the first approximation generated by the secant method with the initial data x-1 and x0. Under the bounded conditions of the divided difference, a convergence theorem is obtained and two examples to illustrate the weakness of convergence conditions are provided. Moreover, the secant method is applied to a system of nonlinear equations to demonstrate the viability and effectiveness of the results in the paper.     

一类差分系统周期解的存在性

This paper studies the nonautonomous nonlinear system of difference equations △x(n) = A(n)x(n) f(n, x(n)), n ∈ Z, (*)where x(n) ∈ RN,A(n) = (aij(n))N×N is an N × N matrix, with aij∈ C(R,R) for i,j =ω, z) = f(t, z) for any t ∈ R, (t, z) ∈ R × RN and ω is a positive integer. Sufficient conditions for the existence of ω-periodic solutions to equations (*) are obtained.     

Solutions and Multiple Solutions for p（x）-Laplacian Equations with Nonlinear Boundary Condition

The authors study the p（x）-Laplacian equations with nonlinear boundary condition. By using the variational method, under appropriate assumptions on the perturbation terms f1 （x, u）, f2（x, u） and h1（x）, h2（x）, such that the associated functional satisfies the ＂mountain pass lemma＂ and ＂fountain theorem＂ respectively, the existence and multiplicity of solutions are obtained. The discussion is based on the theory of variable exponent Lebesgue and Sobolev spaces.     

Extension of Floquet's theory to nonlinear quasiperiodic differential equations

In this paper, we consider the following autonomous system of differential equations: x = Ax f(x,θ), θ = ω, where θ∈Rm, ω = (ω1,…,ωm) ∈ Rm, x ∈ Rn, A ∈ Rn×n is a constant matrix and is hyperbolic, f is a C∞ function in both variables and 2π-periodic in each component of the vector e which satisfies f = O(||x||2) as x → 0. We study the normal form of this system and prove that under some proper conditions this system can be transformed to an autonomous system: x = Ax g(x), θ = ω. Additionally, the proof of this paper naturally implies the extension of Chen's theory in the quasi-periodic case.     

中立型非线性泛函差分方程三个正周期解的存在性（英文）

This paper is concerned with the nonlinear neutral functional difference equations△x(n) =-a(n)x(n) +h(n)f(n,x(n-T(n)),△x(n-δ(n))),where a,h and f are nonnegative sequences.Sufficient conditions for the existence of at least three positive T-periodic solutions are established by using a fixed point theorem due to Avery and Peterson.     

OSCILLATION CRITERIA FOR CERTAIN NONLINEAR DIFFERENCE EQUATIONS

1 IntroductionIn a number of recent papers [l-61 oscil1ation of solutions of second ordernonlinear difference equations has been investigated. ln this paper, we studythe oscil1atory behavior of solutions of the second order nonlinear differenceequationA(r.--lAx.--l) p.f(x.) = 0, n = l, 2,'' (1)where the difference operator a is defined by Ax. = xn 1 -- xn) n = 1, 2,' ')and the functions r, p, f are to be specified in the following text. By a solutionof Eq.(1), we mean a real sequence x = …     

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 …     

Positive solutions for some 1-dimensional boundary value problems of Laplace-type

This paper deals with the existence of triple positive solutions for the 1-dimensional equation of Laplace-type(φ(x'(t)))' q(t)f(t,x(t),x'(t)) = 0, t ∈ (0, 1),subject to the following boundary condition:a1φ(x(0)) - a2φ(x'(0)) = 0, a3φ(x(1)) a4φ(x'(1)) = 0,where φ is an odd increasing homogeneous homeomorphism. By using a new fixed point theorem,sufficient conditions are obtained that guarantee the existence of at least three positive solutions. The emphasis here is that the nonlinear term f is involved with the first order derivative explicitly.     

BLOW-UP RATE OF POSITIVE SOLUTION OF UNIFORMLY PARABOLIC EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

This paper deals with the blow-up of positive solutions of the uniformly pa-rabolic equations ut = Lu + a(x)f(u) subject to nonlinear Neumann boundary conditions . Under suitable assumptions on nonlinear functi-ons f, g and initial data U0(x), the blow-up of the solutions in a finite time is proved by the maximum principles. Moreover, the bounds of "blow-up time" and blow-up rate are obtained.     

Hartman定理的推广

本文研究微分方程全局拓扑线性化,在某些弱条件下得到系统χ’=Aχ+f(χ)的 全局线性化,推广了Hartman的结果.     

临界情形下的全局拓扑线性化

非线性系ｘ＇＝Ａｘ＝ｆ（ｘ）线性化的基本条件是Ａ的特征根实部异于零．经典线性化的结论只限于原点小邻域［１］．后改进为全局性的［２,３］,但要求ｆ（ｘ）有界．在文［４］中我们去掉了ｆ（ｘ）有界的限制．本文将进一步去掉Ａ的特征根实部异于零的限制,证明了,只要ｆ（ｘ）有适当结构,全局线性化仍是可能的．     

在满足广义指数型二分性条件下的线性化

本文将Ｈａｒｔｍａｎ线性化定理和Ｐａｌｍｅｒ线性化定理推广到具有广义指数型二分性的系统,证明了等价函数的强一致连续性;作为应用,在例子中给出了一类系 统的稳定性的判别方法．     

全局拓扑线性化理论的推广

微分方程dx/dt=Ax f(x)(其中A的特征根实部异于零)拓扑线性化的经典结论是由Hartman与 Grobman给出的,但是他们的结论都是局部拓扑线性化,即要求同胚函数限制在原点的小邻域内．如 果要延伸到全局上的话,必须f(x)有界．本文研究了系统(1．3),证明当此系统满足适当的条件时可全 局线性化．     

Generalized exponential dichotomy and global linearization

Hartman's linearization theorem says that if all eigenvalues of matrix A have no zero real part and f(x) is small Lipschitzian, then nonlinear system x^′=Ax+f(x) and its linear system x^′=Ax are topologically equivalent. In 1970s Palmer extended the theorem to nonautonomous systems. In this paper we extend Hartman's theorem to the systems with generalized exponential dichotomy.     

一个非线性常微分方程周期解的存在定理

该文通过应用Miranda定理,证明了在R~n空间某有界区域的边界上一些不等式成立时,非线性微分方程x=f(x,t)在此区域内存在周期解的定理。     

PERIODIC SOLUTIONS OF FORCED LIENARD EQUATIONS

The existence of periodic solutions of the nonlinear system $[\mathop x\limits^{..} + f(x)\mathop x\limits^. + g(t,x) = c(t)\]$ is studied by using the theory of nonhomogeneous linear periodic systems and the Schauder fixed point theorem.     

Solutions and Multiple Solutions for p(x)-Laplacian Equations with Nonlinear Boundary Condition

The authors study the p(x)-Laplacian equations with nonlinear boundary condition.By using the variational method,under appropriate assumptions on the perturbation terms f1(x,u),f2(x,u) and h1(x),h2(x),such that the associated functional satisfies the "mountain pass lemma" and "fountain theorem" respectively,the existence and multiplicity of solutions are obtained.The discussion is based on the theory of variable exponent Lebesgue and Sobolev spaces.     

一般退化时滞差分系统解的存在性及通解

本文研究退化时滞差分系统Ex(k+ 1)= Ax(k)+ ∑li= 1Bix(k- i)+ f(k) (k= 0,1,2,…),x(k)= φ(k) (k= 0,- 1,- 2,…,- l),其中E、A、Bi∈Rm ×n,x(k)∈Rn,f(k)∈Rm ,rank(E)< n.给出了上述系统解的存在性条件及通解表达式.     

一类非线性薛定谔方程组无穷多解的存在性

讨论了一类非线性薛定谔方程组无穷多解的存在性.在假设的V(x),b(x),f(x)条件下,通过减弱喷泉定理的条件,运用变形的喷泉定理,证明了相关方程组的无穷多解的存在性.较扰动方法更加简捷.