一类双曲型方程反问题的存在性与唯一性

针对双曲型方程定解问题{utt=a2uxx+f(t),0xπ,a∈R且a≠0,u(0,t)=v1(t),u(π,t)=v2(t),t0,u(x,0)=g(x),ut(x,0)=h(x),0≤x≤π研究了可以唯一决定未知函数组{v1(t),v2(t),f(t)}的基本条件,提出了该定解问题的反问题,并且讨论了此反问题的存在性与唯一性.     

Banach空间中非线性奇异两点边值问题的多重正解

本文研究Banach空间E中非线性奇异边值问题-x'=f(t,x), t∈(0,1), a₁x(0)-a₂x'(0)=θ, b₁x(0)-b₂x'(1)=θ.其中θ是E中的零元素, f({t,x})在端点t=0和t=1处具有奇性. 利用不动点定理获得了该问题至少有两个正解的结果.     

Improvement of Hartman''''s linearization theorem

Hartman's linearization theorem tells us that if matrix A has no zero real part and f(x) is bounded and satisfies Lipchitz condition with small Lipchitzian constant, then there exists a homeomorphism of Rn sending the solutions of nonlinear system x' = Ax + f(x) onto the solutions of linear system x = Ax. In this paper, some components of the nonlinear item f(x) are permitted to be unbounded and we prove the result of global topological linearization without any special limitation and adding any condition. Thus, Hartman's linearization theorem is improved essentially.     

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.     

Almost Automorphic Solutions for Hyperbolic Semilinear Evolution Equations

In the present paper, we deal with mild solutions for the semilinear evolution equation \begin{displaymath} \frac{d}{dt}x(t)=Ax(t)+f(t,x(t)),\qquad t\in \R, \end{displaymath} under the sectoriality of $A$, a linear operator with not necessarily dense domain, in a Banach space $X$ and $\sigma(A)\cap i\R=\emptyset$. We prove the existence and uniqueness of an almost automorphic solution in an intermediate space $X_{\alpha}$, when the function $f\colon \ \R\times X_{\alpha}\longrightarrow X$ is almost automorphic. An example illustrating the obtained result is given.     

Existence and Uniqueness of Almost Automorphic Mild Solutions to Some Semilinear Abstract Differential Equations

We consider the semilinear differential equation x(t) = Ax(t) + f(t, x(t)), t in a Banach space X, where A is the in.nitesimal generator of an exponentially stable C₀-semigroup. Under suitable conditions on f, we prove the existence and uniqueness of an almost automorphic mild solution to the equation.     

S_n+W_n的邻点强可区别全色数

设G(V,E)是阶数不小于3的简单连通图,k是自然数,f是从V(G)∪E(G)到1,2,…,k的映射,满足:对任意的uv∈E(G),f(u)≠f(v),f(u)≠f(uv)≠f(v);对任意的uv,uw∈E(G)(v≠w),f(uv)≠f(uw);对任意的uv∈E(G),C(u)≠C(v),其中C(u)={f(u)}∪{f(v)uv∈E(G)}∪{f(uv)uv∈E(G)},则称f是图G的一个邻点强可区别的全染色法,简记作k-AVSDTC,且称χast(G)=min{k G的所有k-AVSDTC}为G的邻点强可区别的全色数.得到了星与轮联图的邻点强可区别的全色数.     

Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations

We prove a local existence and uniqueness theorem for abstract parabolic problems of the type when the nonlinearity satisfies certain critical conditions. We apply this abstract result to the Navier-Stokes and heat equations.

     

R2中非局部椭圆型方程基态解的存在性

本文考虑了一类非局部椭圆型方程-△u+V(x)u=(1/|x|μ*Q(x)F(u)/|x|β)Q(x)f(u)|x|β,x∈R^x,其中V是正的连续位势函数,0<μ<2,0≤β<1/2,2β+μ≤2,F(s)是f(s)的原函数.假设非线性项f(s)满足Trudinger-Moser型次临界指数增长,利用变分方法证明了该方程基态解的存在性.     

PERIODIC SOLUTIONS OF LINEAR NEUTRAL INTEGRO-DIFFERENTIAL EQUATIONS

Consider the linear neutral FDEd/dt[x(t) Ax(t - r)] = R [dL(s)]x(t s) f(t)where x and / are ra-dimensional vectors; A is an n x n constant matrix and L(s) is an n x n matrix function with bounded total variation. Some necessary and sufficient conditions are given which guarantee the existence and uniqueness of periodic solutions to the above equation.     

Solutions of a system of diffusion equations

We study existence and multiplicity of homoclinic type solutions to the following system of diffusion equations on \mathbbR ×W{\mathbb{R}} \times \Omega :

一类具有非局部项和临界项椭圆方程的正解

本文研究下列具有临界项的Kirchhoff型方程(a+b∫_R~3[|▽u|~2+V/(x)u~2]dx).[-△u+V(x)u]=μf(x,u)+K(x)u~5,x∈R~3,其中a,b,μ0,位势函数V,K满足一些恰当的条件,非线性项f满足超三次或超线性增长性条件.利用山路定理,得到三个存在性结果.     

一类不满足(AR)条件的p-Laplace方程的无穷多解

利用变分方法,得到以下p-Laplace方程-Δ_pu+V(x)|u|^p-2u=f(x,u),x∈R^N,(1)有无穷多高能量.其中1N上无界函数,非线性项f(x,u)不满足(AR)条件.     

对称Cantor集自乘积集的Hausdorff中心测度

设Cλ是由迭代函数系统(IFS){f1,f2}生成的对称Cantor集,其中f1(x)=λx, f2(x)=1-λ+λx,0<λ<1/2,x∈[0,1]．在压缩比λ满足一定条件时,本文得到了Cλ与其自身的笛卡尔乘积Cλ×Cλ的Hausdorff中心测度的计算公式．     

On Initial-boundary-value Problems for a Class of Systems of Quasi-linear Evolution Equations

In this paper the initial-boundary-value problems for pseudo-hyperbolic system of quasi-linear equations: {(-1)^Mu_{tt} + A(x, t, U, V)u_x^{2M}_{tt} = B(x, t, U, V)u_x^{2M}_{t} + C(x, t, U, V)u_x^{2M} + f(x, t, U, V) u_x^k(0,t) = ψ_{0k}(t), \quad u_x^k(l,t) = ψ_{lk}(t), \quad k = 0,1,…,M - 1 -u(x,0) = φ_0(x), \quad u_t(x,0) = φ_1(x) is studied, where U = (u_1, u_x,…,u_x^{2M - 1}) V = (u_t, u_{xt},…,u_x^{2M - 1_t}), A, B, C are m × m matrices, u, f, ψ_{0k}, ψ_{1k}, ψ_0, ψ_1 are m-dimensional vector functions. The existence and uniqueness of the generalized solution (in H² (0, T; H^{2M} (0, 1))) of the problems are proved.     

带有非紧条件的拟线性Schrodinger-Poisson系统非平凡解的存在性

本文研究了如下带有非紧条件的拟线性Schrodinger-Poisson系统{-△u+V(x)u+Фu+k/2u△u2=λ|u|^p-2u+f(u),x ∈R^3,-ΔФ=u^2,x∈R^3, 其中κ<0,λ>0,p≥12,f∈C(R,R),V∈C(R3,R).文中首先构造截断函数,利用集中紧性原理和逼近的方法,得到了截断后系统非平凡解的存在性;然后利用Moser迭代技巧,讨论上述系统非平凡解的存在性.     

On a Class of Infinite-Dimensional Hamiltonian Systems with Asymptotically Periodic Nonlinearities

The authors study the existence of homoclinic type solutions for the following system of diffusion equations on R × RN:{■tu-xu + b ·▽xu + au + V(t,x)v = Hv(t,x,u,v),-■tv-xv-b·▽xv + av + V(t,x)u = Hu(t,x,u,v),where z =(u,v):R × RN → Rm × Rm,a > 0,b =(b1,···,bN) is a constant vector and V ∈ C(R × RN,R),H ∈ C1(R × RN × R2m,R).Under suitable conditions on V(t,x) and the nonlinearity for H(t,x,z),at least one non-stationary homoclinic solution with least energy is obtained.     

对非线性椭圆边值问题解的存在性的研究

利用非线性增生映射值域的扰动定理 ,研究了非线性椭圆边值问题 ( @)在 L2 (Ω )中解的存在性 .( @) -△pu +g( x,u) =f a.e.在Ω中-〈v,| u|p- 2 u〉∈βx( u( x) ) a.e.在Γ上其中 f∈ L2 (Ω )给定 ,Ω RN,N 1 ,△ pu=div( | u|p- 2 u)为 P拉普拉斯算子 ,1 2 NN +1 ,v为 Γ的外法向导数 ,g:Ω× R→ R满足 Caratheodory条件 ,对 x∈ Γ,βx是正常、凸、下半连续函数 φx=φ( x,· )的次微分 ,其中 φ:Γ×R→ R.     

On ground state solutions for superlinear Hamiltonian elliptic systems

In this paper, we study the following Hamiltonian elliptic systems $$\left\{\begin{array}{ll}-\Delta u+V(x)u= g(x,v),\quad {\rm in }\, \mathbb{R}^N,\\-\Delta v+V(x)v= f(x,u),\quad {\rm in } \, \mathbb{R}^N.\end{array}\right.$$ where ${V(x)\in C(\mathbb R^N), f(x,t), g(x,t)\in C(\mathbb{R}^N\times \mathbb{R})}$ are superlinear in t at infinity. Without Ambrosetti–Rabinowtitz condition, the existences of ground state solutions are obtained via the combination of generalized linking theorem and monotonicity method.     

图K_(3,4)∨K_t的点可区别正常边染色

设f是图G的一个正常边染色.对任意x∈V(G),令S(x)表示与点x相关联的边的颜色所构成的集合.若对任意u,v∈V(G),u≠v,有S(u)≠S(v),则称f是图G的一个点可区别正常边染色.对一个图G进行点可区别正常边染色所需的最少的颜色的数目称为G的点可区别正常边色数,记为χ_s'(G).讨论了图K_(3,4)∨K_t的点可区别正常边染色及其色数,利用正多边形的对称性构造染色以及组合分析的方法,确定了图K_(3,4)∨K_t的点可区别正常边色数,得到了当t是大于等于2的偶数以及t是奇数且3≤t≤25时,χ_s'(K_(3,4)∨K_t)=t+7;当t是奇数且t≥27时,χ_s'(K_(3,4)∨K_t)=t+8.