变系数Sine-Gordon方程的反散射变换

In this paper, the variable coefficient Sine-Gordon type equation uxt=a(t)sinu β(t)uxx k(t)(xux)x is discussed. It is related to the eigenvalue problem Vx=QV.The structure equation and the evolution laws of scattering data for the second equation are derived and the inverse scattering solution of the first equation is obtained.     

Blow-up Solutions for Mixed Nonlinear Schrodinger Equations

This paper is concerned with the initial boundary-value problem for the following nonlinear evolution equation: Фt=iαФxx βФ^2Ф^-x γ|Ф|^2Фx ig(|Ф|^2)Ф Under certain conditions on the initial data and the function g(s),we study the existence and nonexistence of global solution for this equation.The blow-up solution and the blow-up time are also investigated.     

THE NONLOCAL INITIAL PROBLEMS OF A SEMILINEAR EVOLUTION EQUATION

The purpose of this paper is to investigate the existence of solutions to a nonlocal Cauchy problem for an evolution equation. The methods used here include the semigroup methods in proper spaces and Schauder's theorem. And the results are applied to a system of nonlinear partial differential equations with nonlinear boundary conditions.     

SCALE-INVARIANT SOLUTION FOR FRACTIONAL ANOMALOUS DIFFUSION EOUATION

Fractional diffusion equation for transport phenomena in fractal media is an integro-partial differential equation. For solving the problem of this kind of equation the invariants of the group of scaling transformations are given and then used for deriving the integro-ordinary differential equation for the scale-invariant solution. With the help of Mellin transform the scale-invariant solution is obtained in terms of Fox functions. And the series and asymptotic representations for the solution are considered.     

脉冲发展方程初值问题的正解

This paper deals with the existence of e-positive mild solutions（see Definition 1）for the initial value problem of impulsive evolution equation with noncompact semigroup u（t）＋ Au（t）= f（t,u（t））,t ∈ [0,＋∞）,t = tk,u-t=tk = Ik（u（tk））,k = 1,2,...,u（0）= x0 in an ordered Banach space E.By using operator semigroup theory and monotonic iterative technique,without any hypothesis on the impulsive functions,an existence result of e-positive mild solutions is obtained under weaker measure of noncompactness condition on nonlinearity of f.Particularly,an existence result without using measure of noncompaceness condition is presented in ordered and weakly sequentially complete Banach spaces,which is very convenient for application.An example is given to illustrate that our results are more valuable.     

正负系数中立型差分方程的全局吸引性

A neutral difference equation with positive and negative coefficients△(xn-cnxn-k) pnxn-l-qnxn-r=0,n=0,1,2.…，is considered and a sufficient condition for the global attractivity of the zero solution of this equation is obtained, which improves and extends the all known results in the literature.     

EXISTENCE OF GLOBAL ATTRACTORS FOR A NONLINEAR EVOLUTION EQUATION IN SOBOLEV SPACE Hk

In this paper we prove that the initial-boundary value problem for the nonlinear evolution equation ut = △u ＋ λu - u^3 possesses a global attractor in Sobolev space H^k for all k≥0, which attracts any bounded domain of H^k（Ω） in the H^k-norm. This result is established by using an iteration technique and regularity estimates for linear semigroup of operator, which extends the classical result from the case k ∈ [0, 1] to the case k∈ [0, ∞）.     

GLOBAL EXISTENCE OF SOLUTIONS OF CAUCHY PROBLEM FOR GENERALIZED BBM-BURGERS EQUATION

In this paper, the existence and the uniqueness of the local generalized solution and the local classical solution of the Cauchy problem for the generalized BBM-Burgers equationare proved. The existence and the uniqueness of the global generalized solution and the global classical solution for the Cauchy problem of equation （1） are proved when n = 3, 2, 1. Moreover, the decay property of the solution is discussed.     

UNIQUENESS OF THE MILD SOLUTION OF SEMILINEAR STOCHASTIC EVOLUTION EQUATION IN HILBERT SPACE

In this paper, we will consider following initial value problem of semilinear stochastic evolution equation in Hilbert Space: where W(t) is a wiener process in H, H and Y are two real separable Hilbert Spaces, A is an infinitesimal generator of a strongly continuous semigroup s(t) on Y,f(t, y): [0, T]×Y→Y, and G(t, y): [0, T]×Y→L(H, Y), y_0: Ω→Y is a ramdom variable of square integrable. We apply theory of the semigroup and obtain two conclusions of uniqueness of the mild solution of (1) which include the corresponding results in [4].     

Differential Evolution Hemivariational Inequalities with Anti-periodic Conditions

The goal of this paper is to deal with a new dynamic system called a differential evolution hemivariational inequality(DEHVI) which couples an abstract parabolic evolution hemivariational inequality and a nonlinear differential equation in a Banach space.First,by apply ing surjectivity result for pseudomonotone multivalued mappins and the properties of Clarke’s subgradient,we show the nonempty of the solution set for the parabolic hemivariational inequality.Then,some topological properties of th...     

发展方程xttt+iCx=f的解结构

The construction of solution for three-order evolution equation xttt=A3x is skillfully obtained and the semigroup of equation operator is theoretically proved,then the solution for three-order evolution equation xttt iCx=f is constructed from the appropriate transformation,and the necessary and sufficient conditions of its unitary semigroup are presented.     

正整数的一类三分拆的应用

利用正整数n的一类特殊的3分拆n=n1+n2+n3,n1>n2>n3≥1,且n2+n3>n1的Ferrers图将不定方程4x1+3x2+2x3=n(n≥9)的正整数解与这种分拆联系起来,从而得到了该不定方程的正整数解数公式;同时也给出了正整数n的一类4分拆的计数公式.此外,还给出了周长为n的整边三角形的计数公式的一个简单证明.     

Pseudosymmetric equations in a free monoid

In 2011, Makanin obtained the general solution of the symmetric equation $$x_1 x_2 \cdots x_{n - 1} x_n = x_n x_{n - 1} \cdots x_2 x_1$$ in a free monoid. In the present paper, a generalization of this result is given. Namely, we shall describe the general solution of the so-called pseudosymmetric equations in a free monoid which are obtained from the symmetric equations by transposing one (any) pair of adjacent variables.     

Differentiability of the age-dependent population system based on parity progression

In this work, we study the differentiability of the stationary age-dependent population system based on parity progression. By using the semigroup approach, we prove that the solution semigroup of the population evolution equation is eventually differentiable.     

抽象半线性发展方程初值问题解的存在性

本文研究Banach空间E中具有非紧半群的半线性发展方程初值问题u′(t)+Au(t)=f(t,u(t)),t≥0;u(0)=x_0解的存在性,其中-A为E中等度连续C_0-半群的生成元,f:[0,∞)×E→E连续。在f满足较弱的非紧性测度条件下,获得了该问题饱和mild解的存在性。特别,当E为有序弱序列完备Banach空间时,我们获得了一个不需要非紧性测度条件的便于应用的存在性结果。     

Nonlinear volterra equations with infinite delay

This paper is concerned with the existence and stability of nonlinear Volterra equations which have infinite delay and are of the form: $$x (\varphi ) (t) = W (t, \tau ) \varphi (0) + \int\limits_\tau ^t {W (t, s)} F(s,x_s (\varphi )) ds, x_\tau (\varphi ) = \varphi \in C_u .$$ Here,X denotes a Banach space;W(t, s) is a linear evolution operator mappingX toX; C _u is the space of uniformly continuous functions endowed with the supremum norm; andF(·,·) is a continuous mapping ofR×C _u toX. The autonomous version of the preceding equation is also considered. A nonlinear semigroup is associated with its solutions and the infinitesimal generator of the semigroup is characterized. The generator is then used to represent and approximate solutions to the autonomous equation.     

Nontrivial solutions of a higher-order rational difference equation

We prove that, for every k ∈ ?, the following generalization of the Putnam difference equation $$ x_{n + 1} = \frac{{x_n + x_{n - 1} + \cdots + x_{n - (k - 1)} + x_{n - k} x_{n - (k + 1)} }} {{x_n x_{n - 1} + x_{n - 2} + \cdots + x_{n - (k + 1)} }}, n \in \mathbb{N}_0 , $$ has a positive solution with the following asymptotics $$ x_n = 1 + (k + 1)e^{ - \lambda ^n } + (k + 1)e^{ - c\lambda ^n } + o(e^{ - c\lambda ^n } ) $$ for some c > 1 depending on k, and where λ is the root of the polynomial P(λ) = λ ^k+2 ? λ ? 1 belonging to the interval (1, 2). Using this result, we prove that the equation has a positive solution which is not eventually equal to 1. Also, for the case k = 1, we find all positive eventually equal to unity solutions to the equation.     

三阶线性变系数差分方程的Mikusiski算符解法（Ⅲ）

本文在［３］，［４］工作的基础上，利用变数算符的思想以及Ｍｉｋｕｓｉｎｓｋｉ算符域中移动算符和变系数移动算符级数的有关结果，解决了一般的三阶线性变系数差分方程的求解问题，并且绘出了一些特殊的三阶线性变系数差分方程的更好的解式；此外，还试图为实现更高阶线性变系数差分方程的求解提供思想方法。     

一非线性发展方程的反问题

研究一非线性发展方程的未知源项的反演问题.首先,把所考虑的初边值问题化成一等价非线性发展方程的Cauchy问题;然后,利用半群理论,论证反问题解的存在性和唯一性;最后,利用压缩映射不动点方法,得到反问题的可解性.