The Global Solution and “Blow up" Phenomenon for a Class of System of Nonlinear Schrodinger Equations with the Magnetic Field Effect

In this paper, the auther considers following initial value problem for the system of nonlinear Schrodinger equation with the magnetic field effect $i\varepsilon _i-\Delta \varepsilon +\beta q(|\varepsilon |^2)\varepsilon +\eta \varepsilon \times (\varepsilon \times \varepsilon )=0$(1.1) $\varepsilon |t=0=\varepsilon _0(x),x\in R^2,$(1.2) where\beta,\eta are real constants, \varepsilon = (\varepsilon ^1, \varepsilon ^2, \varepsilon ^3). First, the existence of the global solution for problem (1.1), (1.2) is established by means of the method of integral estimates, and then the “blow up” theorem is obtained nuder some conditions.     

一类带p-Laplacian的Sturm-Liouville型2m点边值问题三个正解的存在性

该文考虑了下面的具一维$p$\,-Laplacian算子的多点边值问题 $ \left\{ \begin{array}{rl} &;\disp (\phi_{p}(x'(t)))'+h(t)f(t,x(t),x'(t))=0,\hspace{3mm}01,~\alpha_{i}>0,~\beta_{i}>0,~0<\sum\limits_{i=1}^{m-1}\alpha_{i}\xi_{i}\leq1,~ 0<\sum\limits_{i=1}^{m-1}\beta_{i}(1-\eta_{i})\leq1,~0=\xi_{0} <\xi_{1}<\xi_{2}<\cdots<\xi_{m-1}<\eta_{1}<\eta_{2}<\cdots<\eta_{m-1}<\eta_{m}=1,~i=1,2,\cdots,m-1.$ 通过运用锥上的不动点定理, 该文得到了至少三个正解的存在性. 有趣的是文中的边界条件是一个新型的Sturm-Liouville型边界条件, 这类边值问题到目前为止还很少被研究.     

具有脉冲的Dirichlet边值问题的Lyapunov不等式及其应用

该文首先研究具有脉冲的线性Dirichlet边值问题 $\left\{ \begin{array}{ll} x'(t)+a(t)x(t)=0, t\neq \tau_{k}, \ \Delta x(\tau_{k})=c_{k}x(\tau_{k}),\ \Delta x'(\tau_{k})=d_{k}x(\tau_{k}), \ x(0)=x(T)=0, \end{array} \right. (k=1,2\cdots,m) $ 给出该Dirichlet边值问题仅有零解的两个充分条件, 其中$a:[0,T]\rightarrow R$, $c_{k}, d_{k}, k=1,2,$ $\cdots,m$是常数, 该文首先研究具有脉冲的线性Dirichlet边值问题 $$\left\{ \begin{array}{ll} x'(t)+a(t)x(t)=0, t\neq \tau_{k}, \ \Delta x(\tau_{k})=c_{k}x(\tau_{k}),\ \Delta x'(\tau_{k})=d_{k}x(\tau_{k}), \ x(0)=x(T)=0, \end{array} \right. (k=1,2\cdots,m) $$ 给出该Dirichlet边值问题仅有零解的两个充分条件, 其中$a:[0,T]\rightarrow R$, $c_{k}, d_{k}, k=1,2,$ $\cdots,m$是常数, $0<\tau_{1}<\tau_{2}\cdots<\tau_{m}<T$为脉冲时刻. 其次利用上面的线性边值问题仅有零解这个性质和Leray-Schauder度理论, 研究具有脉冲的非线性Dirichlet边值问题 $$\left\{ \begin{array}{ll} x'(t)+f(t,x(t))=0, t\neq \tau_{k}, \ \Delta x(\tau_{k})=I_{k}(x(\tau_{k})), \ \Delta x'(\tau_{k})=M_{k}(x(\tau_{k})), \ x(0)=x(T)=0 \end{array} \right. (k=1,2\cdots,m) $$ 解的存在性和唯一性, 其中 $f\in C([0,T]\times R,R)$, $I_{k},M_{k}\in C(R, R),k=1,2,\cdots,m$. 该文主要定理的一个推论将经典的Lyaponov不等式比较完美地推广到脉冲系统.     

一类具梯度项和 源的抛物方程有界弱解的存在性

我们考虑了一类原型为$$\begin{cases}u_t-\Delta u=\overrightarrow{b}(x,t)\cdot\nabla u+\gamma|\nabla u|^2-\text{div}{\overrightarrow{F}(x,t)}+f(x,t), &(x,t)\in \Omega_T,\\ u(x,t)=0,&(x,t)\in\Gamma_T,\\ u(x,0)=u_0(x), &x\in\Omega,\end{cases}$$的一类抛物方程. 其中, 函数$|\overrightarrow{b}(x,t)|^2,|\overrightarrow{F}(x,t)|^2,f(x,t)$位于空间$L^r{(0,T;L^q(\Omega))}$, $\gamma$是一个正常数. 在源项和梯度的系数项在空间$L^r{(0,T;L^q(\Omega))}$具有合适的可积条件下, 本文的目的在于证明先验的$L^\infty$估计以及方程存在有界解. 主要的方法包括通过正则化建立扰动问题, 用非线性的检验函数实现Stampacchia迭代技术以及极限过程中的紧性论断.     

亚纯函数微分多项式f^kQ[f]+P[f]的零点分布

研究了超越亚纯函数$f$的微分多项式$f^kQ[f]+P[f]$的零点分布. 给出了以下结果:对于满足$\delta(\infty,f)\geq1-\alpha>0$ ($\alpha$为常数, $0\leq \alpha<1$ )的超越亚纯函数$f(z)$, 若$T(r,f)=O((\log r)^2)$,则微分多项式$f^kQ[f]+P[f]$ ($Q[f]\not\equiv 0,\ P[f] \not\equiv 0$)在 可数个圆盘并集之外有无穷多个零点,其中$k>\frac{1+\Gamma_{P}+\gamma_{P}+\alpha(1+\Gamma_Q+\Gamma_{P}-\gamma_{P})} {1-\alpha }$, $\Gamma_{Q}$是$Q[f]$的权, $\Gamma_{P}$和$\gamma_{P}$是$P[f]$的权和次数.     

广义线性回归极大似然估计的强相合性

设有该文第1节所描述的广义线性回归模型,以$\underline{\lambda}_n$和$\overline{\lambda}_n$分别记$\sum\limits_{i=1}^{n}Z_iZ_i^{\prime}$的最小和最大特征根,$\hat{\beta}_n$记$\beta_0$的极大似然估计.在文献[1]中,当\{$Z_i,i\ge1$\}有界时得到$\hat{\beta}_n$强相合的充分条件,在自然联系和非自然联系下分别为$\underline{\lambda}_n\rightarrow\infty$, $(\overline{\lambda}_n)^{1/2+\delta}=O(\underline{\lambda}_n)$（对某$\delta>0$)以及$\underline{\lambda}_n\rightarrow\infty$, $\overline{\lambda}_n=O(\underline{\lambda}_n)$.作者将后一结果改进为只要求$(\overline{\lambda}_n)^{1/2+\delta}=O(\underline{\lambda}_n)$,从而与自然联系情况下的条件达到一致.     

On Applications of Vector Measure to the Optimal Control Theory for Distributed Parameter Systems

Let X and Z be two reflexive Banach spaces, U\in Z and b(\cdot,\cdot):[t_0,T]*U\rightarrow X continuous. Suppose $x(t)\equiv x(t,u(\cdot))$ is a function from [t_0, T] into X , satisfying the distrbnted parameter system $dx(t)\dt=A(t)x(t)+b(t,u(t)),t_0+\int_t_0^T {+r(t,u(t))dt}$. We have proved the following theorem. Theorem. Suppose u^*(\cdot) is the optimal control function, $x^*(t)=x(t,u^*(\cdot))$ and $\psi (t)=-U'(T,t)Q_1x^*(T)-\int_t^T{U'(\sigma,t)Q(\sigma)x^*(\sigma)d\sigma}$, then the maximum principle $<\psi(t),b(t,u^*(t))>-1/2r(t,u^*(t))=\mathop {\max }\limits_{u \in U} {\psi (t),b(t,u)>-1/2r(t,u)}$ (16) holds for almost all t on [t_0, T ].     

一类分数阶微分方程多点边值问题解的存在性

本文讨论下面一类分数阶微分方程多点边值问题 $$\align &D^{\alpha}_{0+}u(t) = f(t, u(t),~D^{\alpha-1}_{0+}u(t), D^{\alpha-2}_{0+}u(t), D^{\alpha-3}_{0+}u(t)),~~t\in(0,1), \\&I^{4-\alpha}_{0+}u(0) = 0, ~D^{\alpha-1}_{0+}u(0)=\displaystyle{\sum_{i=1}^{m}}\alpha_{i}D^{\alpha-1}_{0+}u(\xi_{i}),\\&D^{\alpha-2}_{0+}u(1)=\sum\limits_ {j=1}^{n}\beta_{j} D^{\alpha-2}_{0+}u(\eta_{j}),~D^{\alpha-3}_{0+}u(1)-D^{\alpha-3}_{0+}u(0)=D^{\alpha-2}_{0+}u(\frac{1}{2}),\endalign$$其中$3<\alpha \leq 4$是一个实数.通过应用Mawhin重合度理论和构建适当的算子,得到了该边值问题解的存在性结果.     

Global Multi-Holder Estimate of Solutions to Elliptic Equations of Higher Order

In this paper the global multi-Holder estimate of solutions to general boundary value problem of elliptic equations of higher order is discussed. Let м be the solution of Pu=f of m-th order elliptic equation with Dirichlet conditions $D_n^iu=f_j,0\leq j \leq m/2-1$ where f\inC^r,\delta(\Omega),g_j\in C^{m-j+r,\delta}(\partial \Omega) with {0<\gamma =0,\delta>1} or {\gamma =1,\delta \leq 0}.Then u\inC^{m+[\tilde \gamma],[\tilde \delta]},where ([\tilde \gamma],[\tilde \delta])=(\gamma,\delta) if 0<\gamma <1 and \delta \in R^1,([\tilde \gamma],[\tilde \delta])=(\gamma,\delta -1) if \gamma=0,\delta >1 or \gamma =1,\delta \leq 0.Moreover,in the case \gamma =0 and 0\leq \delta <1,u\in C^(m-1)+1,\delta -1.     

THE SUFFICIENT AND NECESSARY CONDITIONS FOR NOETHER’S SOLVABILITY OF SINGULAR INTEGRAL EQUATIONS WITH TWO CARLEMAN’S SHIFTS

In this paper we consider the problem of solvability of singular integral equtions with two Carleman's shifts \[\begin{gathered} (\mathcal{K}\varphi )(t) \equiv {a_0}(t)\varphi (t) + {a_1}(t)\varphi [\alpha (t)] + {a_2}(t)\varphi [\beta (t)] + {a_3}(t)\varphi [\gamma (t)] \hfill \ + \frac{{{b_0}(t)}}{{\pi i}}\int_\Gamma {\frac{{\varphi (\tau )}}{{\tau - t}}} d\tau + \frac{{{b_1}(t)}}{{\pi i}}\int_\Gamma {\frac{{\varphi (\tau )}}{{\tau - \alpha (t)}}} d\tau + \frac{{{b_2}(t)}}{{\pi i}}\int_\Gamma {\frac{{\varphi (\tau )}}{{\tau - \beta (t)}}d\tau } \hfill \ + \frac{{{b_s}(t)}}{{\pi i}}\int_\Gamma {\frac{{\varphi (\tau )}}{{\tau - \gamma (t)}}} d\tau + \int_\Gamma {K(t,\tau )\varphi (\tau )d\tau = g(t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (1,1)} \hfill \\ \end{gathered} \] Suppose that Г is a closed simple Lyapunoff's curve and \[\alpha (t)\], \[\beta (t)\] which satisfy Carleman's. conditions and \[\alpha [\beta (t)] = \beta [\alpha (t)]\] are two different homeomorphisms of Г onto itself, and that \[{a_k}(t),{b_k}(t)\], k = 0, 1, 2, 3 belong to the,space \[{H_\mu }(\Gamma ),g(t)\] belongs to the space \[{L_p}(\Gamma ),p > 1\]), p>l and \[K(t,\tau )\] has only weak singularity. The following main results are obtained: 1. Singular integral eqution (1.1) is solvable if and only if the Noether's conditions \[det(p(t) \pm q(t)) \ne 0\] are satisfied. 2. Index of sigular integral eqution (1.1) is calculated by the formula \[Ind{\kern 1pt} {\kern 1pt} {\kern 1pt} \mathcal{K} = \frac{1}{{8\pi }}{\{ arg\frac{{\det (p(t) - q(t))}}{{\det (p(t) + q(t))}}\} _\Gamma }\] where p(t) and q(t) are matrices of coeffioents of so-called corresponding system of equtions. All these results have been generalized for systems of singular integral equtions with two Carleman's shifts and complex conjugate of unknown functions.     

An application of Schauder's fixed point theorem with respect to higher order BVPs

We shall provide conditions on the function . The higher order boundary value problem

has at least one solution.

     

具有p-Laplace算子的一类高阶奇异边值问题解的存在性

本文研究下面问题的正解其中Φp(s)=|s|p-2s,p>1．f在点x(i)=0,i=0,．．．,n-2可能是奇异的．证明建立在Leray-Schauder拓扑度和Vitali收敛定理的基础上．     

四阶奇异边值问题的正解

研究了一类四阶奇异边值问题正解的存在性,在f和g满足比超线性和次线性条件更广泛的极限条件下,利用锥压缩和拉伸不动点定理获得了正解的存在性结果,推广和包含了一些已知结果.     

依赖于一阶导数的二阶脉冲微分方程边值问题的正解(英)

本文研究一类二阶脉冲微分方程：■的正解存在性．其中,0＜η＜1,0＜α＜1,f：[0,1]×[0,∞)×R→[0,∞),I_i：[0,∞)×R→R,J_i：[0,∞)×R→R,(i=1,2,…,k)均为连续函数．本文所用方法是文献[5]推广的Krasnoselskii不动点定理,此定理为解决依赖于一阶导数的边值问题提供了理论依据．基于此定理,获得了问题正解存在性定理．特别地,我们获得此类问题的Green函数,使问题的解决更直观和简单．     

Eigenvalues of Fourth-order Singular Sturm-Liouville Boundary Value Problems

In this paper, by using Krasnoselskii''s fixed-point theorem, some sufficient conditions of existence of positive solutions for the following fourth-order nonlinear Sturm-Liouville eigenvalue problem:\begin{equation*}\left\{\begin{array}{lll} \frac{1}{p(t)}(p(t)u'')''(t)+ \lambda f(t,u)=0, t\in(0,1), \\ u(0)=u(1)=0, \\ \alpha u''(0)- \beta \lim_{t \rightarrow 0^{+}} p(t)u''(t)=0, \\ \gamma u''(1)+\delta\lim_{t \rightarrow 1^{-}} p(t)u''(t)=0, \end{array}\right.\end{equation*} are established, where $\alpha,\beta,\gamma,\delta \geq 0,$ and $~\beta\gamma+\alpha\gamma+\alpha\delta >0$. The function $p$ may be singular at $t=0$ or $1$, and $f$ satisfies Carath\''{e}odory condition.     

Existence Theory for an Arbitrary Order Fractional Differential Equation with Deviating Argument

In this paper, we are concerned with the existence criteria for positive solutions of the following nonlinear arbitrary order fractional differential equations with deviating argument

Existence of solution for a class of nth-order multi-point boundary value problem

In this paper, we are concerned with the following nth-order ordinary differential equation $$x^{(n)}(t)+f(t,x(t),x'(t),\ldots,x^{(n-1)}(t))=0,\quad t\in (0,1),$$ with the nonlinear boundary conditions $$\begin{array}{l}x^{(i)}(0)=0,\quad i=0,1,\ldots,n-3,\\[3pt]g(x^{(n-2)}(0),x^{(n-1)}(0),x(\xi_1),\ldots,x(\xi_{m-2}))=A,\\[3pt]h(x^{(n-2)}(1),x^{(n-1)}(1),x(\eta_1),\ldots,x(\eta_{l-2}))=B,\end{array}$$ here A,B∈R, f:[0,1]×R ⁿ →R is continuous, g:[0,1]×R ^m →R is continuous, h:[0,1]×R ^l →R is continuous, ξ _i ∈(0,1), i=1,…,m?2, and η _j ∈(0,1), j=1,…,l?2. The existence result is given by using a priori estimate, Nagumo condition, the method of upper and lower solutions and Leray-Schauder degree. We also give an example to demonstrate our result.     

Solvability of a higher-order multi-point boundary value problem at resonance

Based on the coincidence degree theory of Mawhin, we get a new general existence result for the following higher-order multi-point boundary value problem at resonance

一类n-阶m-点奇异边值问题的正解

研究n-阶m-点奇异边值问题其中h(t)允许在t=0,t=1处奇异,f(t,v_0,v_1,…,v_(n-2))允许在v_i=0(i=0,1,…,n-2)处奇异.利用锥拉伸与压缩不动点定理得到了上述奇异边值问题正解的存在性.