一类带权函数的拟线性椭圆方程

该文利用变分方法讨论了方程 -△_p u=λa(x)(u⁺)^p-1-μa(x)(u^-)^p-1+f(x, u), u∈W₀^1,p(\Omega)在(λ, μ)\not\in ∑_p和(λ, μ) ∈ ∑_p 两种情况下的可解性, 其中\Omega是 R^N(N≥3)中的有界光滑区域, ∑_p为方程 -△_p u=α a(x)(u⁺)^p-1-βa(x)(u^-)^p-1, u∈ W₀^1,p(\Omega)的Fucik谱, 权重函数a(x)∈ L^r(\Omega) (r≥ N/p)$且a(x)>0 a.e.于\Omega, f满足一定的条件.     

带权的 p-Laplacian 非线性特征值问题解的存在性

文利用变分方法讨论了方程-△p^u=λ a(x)(u^{+})^q-1-μ a(x)(u^-)^q-1+f(x,u), u∈W₀^1,p(Ω), 当 p≠q时的可解性. 其中Ω是 R^N(N≥ 3)中的有界光滑区域,权重函数a(x)∈ L^r(Ω), (r≥Np/Np-Nq+pq)且a(x)>0, a.e.于Ω, f满足某些条件.     

一类非线性波方程初边值问题解的爆破

该文研究如下具有非线性阻尼项和非线性源项的波方程的初边值问题 u_tt -u_xxt -u_xx -(σ(u²_x)u_x)_x+δ|u_t|^p-1u_t=μ|u|^q-1u, 0 < x <1, 0≤ t ≤T, (0.1) u(0, t)=u(1, t)=0, 0≤t≤ T, (0.2) u(x, 0)=u₀(x), u_t(x, 0)=u₁(x),0≤x≤1.(0.3) 文章将给出问题(0.1)--(0.3)的解在有限时刻爆破的充分条件, 同时将证明问题的局部广义解和局部古典解的存在性和唯一性.     

偶数阶Sturm-Liouville边值问题的多个正解

该文讨论了偶数阶边值问题 (-1)^m y^(2m)₌f(t,y), 0≤t≤1,a_i+1y⁽²ⁱ⁾ (0)-β_i+1y ⁽²ⁱ⁺¹⁾ (0)=0, γ_i+1y⁽²ⁱ⁾ (1)+δ_i+1y⁽²ⁱ⁺¹⁾ (1)=0,0≤i ≤m-1正解的存在性.借助于Leggett-Williams 不动点定理,建立了该问题存在三个及任意奇数个正解的充分条件.     

一类非线性高阶波动方程的初边值问题

该文研究一类非线性高阶波动方程u_tt－a_1^Uxx+a_₂u_x4+a₃u_x4_tt=φ(u_{x ⁾}x+f(u,u_x,u_xxu_xxx,u_x4)的初边值问题.证明整体古典解的存在唯一性并给出古典解爆破的充分条件.     

四阶椭圆方程基态解的集中性态

该文分析了四阶椭圆方程△^２u=|x|^au^p-1,x∈Ω; u=\Delta u=0 , x ∈аΩ, (Ω表示Rⁿ中以原点为中心的球)基态解的集中性态,并证明了当p趋近于 2^*=\frac{2n}{n-4} (n>4)时基态解u_p集中在Ω的边界附近.     

RIGHT FOCAL BOUNDARY VALUE PROBLEM WITH SINGULARITY

This article proves existence results for singular problem (-1)^n-px⁽ⁿ⁾(t)=f(t, x(t), ..., x^(n-1)(t)), for 0(i)(0)=0, i=1, 2, ..., p-1, x⁽ⁱ⁾(1)=0, i=p, p+1, ... , n-1. Here the positive Carathedory function f may be singular at the zero value of all its phase variables. The interesting point is that the degrees of some variables in the nonlinear term f(t, x₀, x₁, ..., x_n-1) are allowable to be greater than 1. Proofs are based on the Leray-Schauder degree theory and Vitali's convergence theorem. The emphasis in this article is that f depends on all higher-order derivatives. Examples are given to illustrate the main results of this article.     

脉冲泛函微分方程的渐近性态

该文讨论脉冲泛函微分方程$\left\{\begin{array}{ll}x^,(t)=f(t,x_t), t≥ t₀,△x=I_k(t,x(t^-)), t=t_k,k∈ Z⁺,给出了方程零解渐近稳定性和一致渐近稳定性的充分条件,指出这些条件推广或改进了文献[7--9]的相应结论.     

关于单位圆内解析系数的线性微分方程的复振荡理论

该文研究了线性微分方程L(f)=f^(k)+A_k-1(z)f^(k-1)+ ^… +A₀(z)f=F(z) (k∈ N)的复振荡理论, 其中系数A_j(z) (j=0, ^… , k-1)和F(z)是单位圆△={z:|z|<1}内的解析函数. 作者得到了几个关于微分方程解的超级, 零点的超收敛指数以及不动点的精确估计的定理.     

关于赋权图中重圈的一个范型定理

设G=(V, E; w)为赋权图,定义G中点v的权度d_G^w(v)为G中与v相关联的所有边的权和.该文证明了下述定理: 假设G为满足下列条件的2 -连通赋权图: (i) 对G中任何导出路xyz都有w(xy)=w(yz); (ii)对G中每一个与K_1,3或K_1,3+e同构的导出子图T, T中所有边的权都相等并且min{max{d_G^w(x), d^w_G(y)}:d(x,y)=2,x,y∈ V(T)}≥ c/2. 那么, G中存在哈密尔顿圈或者存在权和至少为 c 的圈. 该结论分别推广了Fan^[5], Bedrossian等人^[2]和Zhang等人^[7]的相关定理     

POSITIVE SOLUTION TO FOURTH-ORDER IMPULSIVE DIFFERENTIAL EQUATIONS WITH P-LAPLACIAN

In order to study three-point BVPs for fourth-order impulsive differential equation of the form(\phi_p(u'(t)))'- f(t,u(t))=0, t≠ t_i,△ u(t_i)=-I_i(u(t_i)), i=1, 2, ..., k,△u'(t_i)=-L_i(u(t_i)), i=1, 2, ..., k,(\star)with the following boundary conditionsu'(0)=u(1)=0, u'(0)=0=u'(1)-\phi_q(α)u'(η),the authors translate the fourth-order impulsive differential equations with p-Laplacian (\star) into three-point BVPs for second-order differential equation without impulses and two-point BVPs for second-order impulsive differential equation by a variable transform. Based on it, existence theorems of positive solutions for the boundary value problems (\star) are obtained.     

Smooth solution of one boundary-value problem

We study the boundary value problem for the quasilinear equation u _u ? u_xx=F[u, u_t], u(x, 0)= u(x, π)=0, u(x+w, t)=u(x, t), x ε ®, t ε [0, π], and establish conditions under which a theorem on the uniqueness of a smooth solution is true.     

非线性常微分方程初值问题的间断有限元

该文用m次间断有限元求解非线性常微分方程初值问题u'=f(x,u),u(0)=u₀,用单元正交投影及正交性质证明了当m≥1时,m次间断有限元在节点x_j的左极限U(x_j-0)有超收敛估计(u-U(x_j-0)=O(h^2m+1),在每个单元内的m+1阶特征点x_ji上有高一阶的超收敛性(u-U)(x_ji)=O(h^m+2).     

A note on discrete heat conduction

If the longitudinal line method is applied to the Cauchy problem u_t = u_xx, u(0, x) = u₀(x) with a bounded function u₀, one is led to a linear initial value problem v￠(t)=A v(t), v(0)=wv'(t)=A v(t),\, v(0)=w in l^￥ (\Bbb Z)l^\infty (\Bbb Z). Using Banach limit techniques we study the asymptotic behaviour of the solutions of these problems as t tends to infinity.     

Solutions to Second-order Three-point Problems on Time Scales

In the first part of the paper, we establish the existence of multiple positive solutions to the nonlinear second-order three-point boundary value problem on time scales, u ^?? (t)+f(t,u(t))=0, u(0)=0, 𝛂u(𝛈)=u(T) for t∈[0,T]?╥, where ╥ is a time scale, 𝛂>0, η∈(0,p(T)?╥, and 𝛂η<T. We employ the Leggett-Williams fixed-point theorem in an appropriate cone to guarantee the existence of at least three positive solutions to this nonlinear problem. In the second part, we establish the existence of at least one positive solution to the related problem u ^??(t)+a(t)f(u(t))=0, u(0)=0, 𝛂u(η)=u(T), again using a fixed-point theorem for operators.     

用椭圆描述的四阶边值问题的两参数非共振条件

该文讨论四阶常微分方程边值问题u(4)=f(t,u,u″),0≤t≤1,u(0)=u(1)=u″(0)=u″(1)=0解的存在性,其中f:[0,1]×R×R→R连续.文中提出了一个保证该问题解存在的两参数非共振条件,该条件是用椭圆描述的.