关于多复变数微分从属的一个注记

我们知道，微分从属是非常有用的，在单叶函数，微分方程和微分不等式等方面都有一定的作用．在这个方面S S Miller，P T Maeanu和龚升等人都作了大量的工作(见[1-3])．本人在这个也做了一些探索，丰富了这个领域的内容(见[4-6])．本文将给出一个很有意思的结果．它将为研究多复变的微分不等式起到一定的作用．为了方便先给出一些记号如下：     

关于一般微分单项式的值分布

本文对一般的微分单项式的值分布进行了讨论，把已有的定性描述改进为定量的描述．     

关于Briot—Bouquet微分从属

本文主要研究如下的Ｂｒｉｏｔ－Ｂｏｕｑｕｅｔ微分从属Ｐ（Ｚ）＋ＺＰ＇（Ｚ）／βＰ（Ｚ）＋γ＜１＋ａｚ／１＋ｂｚ，ｐ（０）＝１其中ｐ在｜ｚ｜Ｄ＜１内解析，β、γ、ａ、ｂ为适当的实数，并由此解决了Ｓ．Ｓ．Ｍｉｌｌｅｒ和Ｐ．Ｔ．Ｍｏｃａｎｕ提出的一。     

带参数2n 阶非对称微分算子的一些线性同构定理

本文给出了一类带参数２ｎ阶非对称微分算子Ａ_λ的一些正则性定理，籍此可刻画一类飞行器在其运行过程中的双向平稳行为．     

多复变数微分从属的最优控制

讨论了多复变函数中的微分从属的最优控制的问题，取得了一些结果，得到了一些应用．它是单复变函数论中的相应结果[1，2]的推广．     

关于微分多项式的一些结果

设ｆ是平面内超越亚纯函数，满足Ｎ（ｒ，ｆ）＝Ｓ（ｒ，ｆ）．Ｐ［ｆ］是ｆ的多项式，Ｑ［ｆ］是ｆ的微分多项式。本文考虑了形如Ｐ［ｆ］ｆ＇＋Ｑ［ｆ］（特殊情形为ｆ￣ｎｆ＇＋Ｑ［ｆ］及Ｐ［ｆ］ｆ＇－Ｃ，Ｃ为常数）的微分多项式的值分布，推广并改进了Ｗ．Ｋ．Ｈａｙｍａｎ，Ｊ．Ｃｌｕｎｉｅ，Ｅ．Ｍｕｅｓ等人关于整函数的一些结果。     

微分模式

１．论述ｄｘ与△ｘ的关系；ｄｘ是从△ｘ→０转化而来．比△ｘ更为抽象；２对多元函数引出几．种可微性概念，更加细致地刻划其可微性。     

微分单项式的唯一性定理

本文主要讨论亚纯函数的微分单项式的唯一性问题，推广和改进了仪洪勋和Ｃ．Ｃ．Ｙａｎｇ等人的某些结果．     

带转向点的非线性系统的奇摄动

本文考虑如下非线性系统εＹ″＝Ｆ（ｔ，Ｙ，Ｙ′，ε），－１＜ｔ＜１，Ｙ（－１，ε），Ｙ（１，ε）＝设Ｆ的部分或所有的分量ｆ．满足，称Ｆ在ｔ＝０处具有一般的转点．本文讨论系统在Ｆ具有一般转点时解存在的充分条件且研究解在转向点ｔ＝０处的渐近性态．对于内部层、边界层现象分别进行了讨论．     

二元函数全微分求积的一个简单方法

利用分离积分的方法，给出了二元函数全微分P（x，y）出＋Q（x，y）dy=du（x,y）求积的一个简便公式．其特点是只需计算函数P和函数Q的不定积分，从而避免原方法在积分路径选取时的麻烦以及由积分路径选取不当所导致的错误．     

Rational Solutions to the KdV Equation in Terms of Particular Polynomials

Here, we construct rational solutions to the KdV equation by particular polynomials. We get the solutions in terms of determinants of the order $n$ for any positive integer $n$, and we call these solutions, solutions of the order $n$. Therefore, we obtain a very efficient method to get rational solutions to the KdV equation, and we can construct explicit solutions very easily. In the following, we present some solutions until order $10$.     

三维空间中半线性波动方程组的整体解和自相似解

本文证明了一类半线性波动方程组Cauchy问题整体解的存在唯一性．特别地,证明了自相似解的存在唯一性．同时还得到了渐近自相似解．     

Asymptotic forms of positive solutions of quasilinear ordinary differential equations with singular nonlinearities

In this paper we consider positive solutions of second order quasilinear ordinary differential equations with singular nonlinearities. We obtain asymptotic equivalence theorems for asymptotically superlinear solutions and decaying solutions. By using these theorems, exact asymptotic forms of such solutions are determined. Furthermore, we can establish the uniqueness of decaying solutions as an application of our results.     

Solutions of the generalized Lennard-Jones system

In this paper, we study solution structures of the following generalized Lennard-Jones system in R~n,x=(-α/|x|~(α+2)+β/|x|~(β+2))x,with 0 α β. Considering periodic solutions with zero angular momentum, we prove that the corresponding problem degenerates to 1-dimensional and possesses infinitely many periodic solutions which must be oscillating line solutions or constant solutions. Considering solutions with non-zero angular momentum, we compute Morse indices of the circular solutions first, and then apply the mountain pass theorem to show the existence of non-circular solutions with non-zero topological degrees. We further prove that besides circular solutions the system possesses in fact countably many periodic solutions with arbitrarily large topological degree, infinitely many quasi-periodic solutions, and infinitely many asymptotic solutions.     

Bifurcations and exact solutions of nonlinear Schrodinger equation with an anti-cubic nonlinearity

In this paper, we consider the nonlinear Schr\"{o}dinger equation with an anti-cubic nonlinearity. By using the method of dynamical systems, we obtain bifurcations of the phase portraits of the corresponding planar dynamical system under different parameter conditions. Corresponding to different level curves defined by the Hamiltonian, we derive all exact explicit parametric representations of the bounded solutions (including periodic peakon solutions, periodic solutions, homoclinic solutions, heteroclinic solutions and compacton solutions).     

具有无穷时滞的Volterra反应扩散方程组正解与有界正解的存在唯一性

本文利用Ｃ-ｈ空间理论,ＢＣ_ｈ空间理论和上下解方法研究具有无穷时滞的 Ｖｏｌｔｅｒｒａ反应扩散方程组正解和有界正解的存在唯一性,并给出有关应用的例子．     

Existence of solutions of the nonautonomous abstract Cauchy problem of second order

In this paper the existence of solutions of a nonautonomous abstract Cauchy problem of second order is considered. Assuming appropriate conditions on the operator of the equation, we establish the existence of mild solutions and, in some cases, we construct an evolution operator associated to the homogeneous equation. Using this evolution operator we obtain existence of solutions for the inhomogeneous equation. Finally, we apply our results to study the existence of solutions of the nonautonomous wave equation.     

向量优化问题的近似解研究

向量优化是数学规划领域中十分重要的研究方向之一,其相关基础理论与基本方法的研究具有非常重要的理论意义与应用价值.近年来,关于近似解的定义及其性质研究已成为向量优化理论与方法研究的热点.现主要介绍国内学者,特别是我们团队在向量优化问题的各类近似解和统一解概念及其发展和各类近似解与统一解的性质研究方面取得的一些重要进展.最后,提出了与向量优化问题的近似解与统一解相关的一些公开问题.     

慢增长系数微分方程的解及其导数的不动点

运用值分布理论研究了高阶慢增长系数线性微分方程的解及其导数的不动点问题.当存在某个系数对方程的解的性质起主要支配作用时,得到了方程解及其导数的不动点收敛指数的精确估计,推广了有关文献中的结论.     

Existence and multiplicity of positive solutions for singular quasilinear problems

In this work we combine perturbation arguments and variational methods to study the existence and multiplicity of positive solutions for a class of singular p-Laplacian problems. In the first two theorems we prove the existence of solutions in the sense of distributions. By strengthening the hypotheses, in the third and last result, we establish the existence of two ordered positive weak solutions.