On generalized vector equilibrium problems

In this paper, we study the generalized vector equilibrium problems in real Hausdorff topological vector space settings. The concepts of weak solutions and strong solutions are introduced. Several new results of existence for the weak solutions and strong solutions of generalized vector equilibrium problems are derived. The new results extend and modify various existence theorems for similar problems.     

Asymptotic expansions of solutions to inverse problems for a hyperbolic equation with a small parameter multiplying the highest derivative

Two inverse problems for a hyperbolic equation with a small parameter multiplying the highest derivative are considered. The existence and uniqueness of their solutions are proved. As the small parameter tends to zero, the solutions of the inverse problems are proved to converge to solutions of inverse problems for a parabolic equation.     

Initial-irregular oblique derivative problems for nonlinear parabolic systems of second order equations with measurable coefficients

The initial-irregular oblique derivative boundary value problems for nonlinear and nondivergence parabolic systems of second order equations in multiply connected domains are dealt with where coefficients of systems of equations are meaurable. The uniqueness theorem of solutions for the above problems and somea priori estimates of solutions for the problems are given. And by using the above estimates of solutions and the Leray-Schauder theorem, the existence of solutions of the initial-boundary value problems is proved. The results are generalizations of corresponding theorems in literature. Project supported by the National Natural Science Foundation of China (Grant No. 19671006).     

轴对称断裂动力学问题的自相似解

通过复变函数论的方法,对复合材料的圆盘状裂纹的轴对称性扩展问题进行研究.采用自相似方法,并根据任意的自相似指数的断裂动力学问题,进行自相似求解,导出解析解的一般表示.应用该法可以迅速地将所论问题转化为Riemann-Hilbert问题,并可以相当简单地得到问题的闭合解.利用这些解并采用叠加原理,就可以求得任意复杂问题的解.     

INITIAL-OBLIQUE DERIVATIVE PROBLEMS FOR NONLINEAR PARABOLIC SYSTEMS WITH MEASURABLE COEFFICIENTS

This paper deals with some initial-oblique derivative boundary value problems for nonlinear nondivergent parabolic systems of several second order equations with measurable coefficients in multiply connected domains. Firstly, a priori estimates of solutions for the initial-boundary value problems are given, and then by using the above estimates of solutions and the Leray-Schauder theorem, the existence and uniqueness of solutions for the problems are proved.     

高阶微分积分方程的单调迭代法及其应用

首先利用上下解方法以及微分不等式理论给出了n阶微分积分方程的初值问题解的存在性及其单调迭代法,然后将所得结果应用到n阶微分方程的两点边值问题,得到了n阶非线性两点边值问题解的存在性及其单调迭代法,所得结果推广了已有的结果．     

Method of Characteristics for Optimal Control Problems and Conservation Laws

In this paper, notions of global generalized solutions of Cauchy problems for the Hamilton–Jacobi–Bellman equation and for a quasilinear equation (a conservation law) are introduced in terms of characteristics of the Hamilton–Jacobi equation. Theorems on the existence and uniqueness of generalized solutions are proved. Representative formulas for generalized solutions are obtained and a relation between generalized solutions of the mentioned problems is justified. These results tie nonlinear scalar optimal control problems and one-dimensional stationary conservation laws.     

反平面集中力作用下不同材料界面共线裂纹问题

本文研究在反平面集中力作用下,不同弹性材料界面上的共线裂纹问题.运用复变函数的解析延拓方法并结合对奇性主部的分析,获得了一般解;求出了几种典型情况的封闭解;算出了应力强度因子.本文解答的若干特殊情形,与前人成果吻合.通过比较,我们发现,在局部对称加载下,本文结果与同种材料的相应解答完全相同.     

OBLIQUE DERIVATIVE PROBLEMS FOR SECOND ORDER NONLINEAR MIXED EQUATIONS WITH DEGENERATE LINE

The present article deals with oblique derivative problems for some nonlinear mixed equations with parabolic degeneracy, which include the Tricomi problem as a special case. First, the formulation of the problems for the equations is given; next, the representation and estimates of solutions for the above problems are obtained; finally, the existence of solutions for the problems is proved by the successive iteration and the compactness principle of solutions of the problems. In this article, the author uses the complex method, namely, the complex functions in the elliptic domain and the hyperbolic complex functions in hyperbolic domain are used.     

A method of convolution of Fourier expansions as applied to solving boundary value problems with intersecting interface lines

An efficient method of construction of solutions to a set of boundary value problems with additional interface conditions, more complicated boundary conditions, and so on on the basis of known solutions to classical boundary value problems is proposed. The method is based on the representation of solutions to classical and more complicated problems in the form of expansions into Fourier series with subsequent reduction of one series to the other. As a result, formulas directly expressing solutions to more complicated problems in terms of solutions to classical problems are obtained. On the basis of the well-known solution to the Dirichlet problem on a half plane, solutions to boundary value problems with interface conditions (including generalized conditions of the type of a crack and a screen) on intersecting straight lines for boundary conditions of the first and the third kind are obtained.     

一类混合似变分不等式问题解的存在唯一性

本文中,我们首先给出了一类混合似变分不等式问题．接着,在Banach空间中研究了它的解的存在性和唯一性．最后,讨论了混合似变分不等式问题的扰动问题,并证明了扰动问题的解的存在唯一性定理．     

一类具非线性边界条件的高阶方程的双参数奇摄动问题

研究了一类含双参数的非线性高阶微分方程的奇摄动问题.运用合成展开法构造了问题的形式渐近解,并运用微分不等式理论证明了原问题解的存在性及所得形式渐近解的一致有效性.     

Scalarization in vector optimization

In this paper some scalar optimization problems are presented whose optimal solutions are also solutions of a general vector optimization problem. This will be done for weakly minimal and minimal solutions, respectively. Finally the results will be applied to a certain class of approximation problems.     

向量映射的鞍点和Lagrange对偶问题

本文研究拓扑向量空间广义锥-次类凸映射向量优化问题的鞍点最优性条件和Lagrange对偶问题,建立向量优化问题的Fritz John鞍点和Kuhn-Tucker鞍点的最优性条件及其与向量优化问题的有效解和弱有效解之间的联系。通过对偶问题和向量优化问题的标量化刻画各解之间的关系,给出目标映射是广义锥-次类凸的向量优化问题在其约束映射满足广义Slater约束规格的条件下的对偶定理。     

Estimates of solutions of initial-irregular oblique derivative problems for linear parabolic equations of second order with measurable coefficients

The initial-irregular oblique derivative boundary value problems for linear and nondivergence parabolic complex equations of second order in multiply connected domains are dealt with, where the coefficients of equations are measurable. Firstly the uniqueness of solutions for the above problems is introduced, and then somea priori estimates of solutions for the problems are given. By using the above estimates and the Leray-Schauder theorem, the existence of solutions of the initial-boundary value problems can be proved. The results are generalizations of corresponding theorems in literature. Project supported by the National Natural Science Foundation of China (Grant No. 19671006).     

On quasivariational inclusion problems of type I and related problems

The quasivariational inclusion problems are formulated and sufficient conditions on the existence of solutions are shown. As special cases, we obtain several results on the existence of solutions of a general vector ideal (proper, Pareto, weak) quasi-optimization problems, of quasivariational inequalities, and of vector quasi-equilibrium problems. Further, we prove theorems on the existence for solutions of the sum of these inclusions. As corollaries, we shall show several results on the existence of solutions to another problems in the vector optimization problems concerning multivalued mappings. This work was supported by the National Science Council of the Republic of China and the Academy of Sciences and Technologies of Vietnam. The authors wish to express their gratitude to the referees for their valuable suggestions.     

对称集值向量拟均衡问题解集的稳定性

本文研究了对称集值向量拟均衡问题解集的稳定性.证明了在约束映射满足一定连续性与目标映射是锥-恰当拟凸的集值映射条件下,对称集值向量拟均衡问题的解集是稳定的,还证明了每个对称集值向量拟均衡问题的解集至少存在一个本质连通区.     

Optimal Path Planning Based on Visibility

Various problems associated with optimal path planning for mobile observers such as mobile robots equipped with cameras to obtain maximum visual coverage of a surface in the three-dimensional Euclidean space are considered. The existence of solutions to these problems is discussed first. Then, optimality conditions are derived by considering local path perturbations. Numerical algorithms for solving the corresponding approximate problems are proposed. Detailed solutions to the optimal path planning problems for a few examples are given.     

The closed-form particular solutions for Laplace and biharmonic operators using a Gaussian function

Particular solutions play a critical role in solving inhomogeneous problems using boundary methods such as boundary element methods or boundary meshless methods. In this short article, we derive the closed-form particular solutions for the Laplace and biharmonic operators using the Gaussian radial basis function. The derived particular solutions are implemented numerically to solve boundary value problems using the method of particular solutions and the localized method of approximate particular solutions. Two examples in 2D and 3D are given to show the effectiveness of the derived particular solutions.     

Properties of solutions of a mixed problem for a nonlinear ultraparabolic equation

Mixed problems for a nonlinear ultraparabolic equation are considered in domains bounded and unbounded with respect to the space variables. Conditions for the existence and uniqueness of solutions of these problems are established and some estimates for these solutions are obtained.