带有p-Laplace算子的分数阶边值问题的特征区间

本文研究了一类带有p-Laplace算子的分数阶微分方程两点边值问题.利用锥上的不动点定理,得到了这类边值问题的特征区间,推广了整数阶边值问题情形的结论.     

含有广义p-Laplace算子的非线性边值问题解的存在性的研究

该文研究了两类含有广义p-Laplace算子的非线性边值问题. 首先, 利用变分不等式解的存在性的结果, 证明了含有广义p-Laplace算子的非线性Dirichlet边值问题解的存在性. 然后, 提出了一类含有广义p-Laplace算子的非线性Neumann边值问题. 通过深入挖掘这两类非线性边值问题间的关系, 借助于极大单调算子值域的一个扰动结果, 证明了含有广义p-Laplace算子的非线性Neumann边值问题解的存在性. 文中采用了一些新的证明技巧,推广和补充了作者以往的一些研究工作.     

一类非线性三阶微分方程边值问题解的存在唯一性

研究了一类非线性三阶微分方程边值问题解的存在唯一性.首先分析了近年来国内外三阶微分方程边值问题的研究成果,提出了边值条件中含非线性函数的非线性三阶微分方程边值问题.然后寻找相关线性问题的解决途径,利用Banach不动点定理,证明了提出的边值问题存在唯一解.最后,举例阐述了主要结果的应用.     

Cauchy-Riemann方程的Riemann-Hilbert边值问题

本文考虑多柱域上非齐次的Cauchy-Riemann方程的Riemann-Hilbert边值问题.讨论了上述边值问题可解的充分必要条件,并给出了边值问题解的积分表达式.     

二阶差分边值问题的正解

本文研究了非线性项具有半正定和混合单调性的二阶差分边值问题正解的存在性.利用Krasnosel'skii不动点定理和锥拉伸与锥压缩不动点定理,讨论了二阶半正定非线性差分边值问题以及非线性项具有半正定和混合单调性的特征值问题正解的存在性,给出了这几类差分边值问题的正解存在性定理,改进和推广了具有正定非线性项的二阶差分边值问题的一些结果,并将所得结果应用于一个具体的二阶半正定非线性差分边值问题中.     

二阶Sturm-Liouville特征值问题解的存在与非存在性

讨论了二阶Sturm-Liouville特征值边值问题解的存在性与非存在性,得到了边值问题至少有一个正解的特征值λ的存在区间的结论.进一步,给出了边值问题没有正解的特征值存在区间.     

带有p-Laplace算子的分数阶边值问题的特征区间（英文）

本文研究了一类带有p-Laplace算子的分数阶微分方程两点边值问题.利用锥上的不动点定理,得到了这类边值问题的特征区间,推广了整数阶边值问题情形的结论.     

含有一阶导数的四阶边值问题正解的存在性

研究了含有一阶导数的四阶边值问题,利用度理论构造出的一个新的不动点定理,由Green函数的性质,给出了一类含有导数的四阶边值问题正解存在的充分条件,得到了边值问题正解存在的一些新结果.     

Clifford分析中双曲调和函数的一个带位移的非线性边值问题

讨论了Clifford分析中双曲调和函数的一个带位移的非线性边值问题,先讨论了解析函数的一个边值问题的解的存在性,然后利用Clifford分析中双曲调和函数与解析函数的关系讨论了此边值问题的解,并给出了解的积分表达式.     

含p-Laplace算子的Sturm-Liouville边值问题正解的性质

研究了含p-Laplace算子的Sturm-Liouville边值问题正解的性质.利用p-Laplace算子的性质,使用L’Hpital(洛必达)法则和闭区间上连续函数的最值性定理,研究了含p-Laplace算子的Sturm-Liouville边值问题,得到了其正解存在的两个必要条件.最后给出了主要结论的应用.结论丰富了边值问题研究领域的内容,为利用计算机使用迭代技术求这类边值问题的近似解提供了新的渠道,推广了一些文献的结论.     

Adjoint and self-adjoint boundary value problems on a geometric graph

We consider boundary value problems of arbitrary order for linear differential equations on a geometric graph. Solutions of boundary value problems are coordinated at the interior vertices of the graph and satisfy given conditions at the boundary vertices. For considered boundary value problems, we construct adjoint boundary value problems and obtain a self-adjointness criterion. We describe the structure of the solution set of homogeneous self-adjoint boundary value problems with alternating coefficients of a differential equation and obtain nondegeneracy conditions for these boundary value problems.     

The Basic Mixed Problem for an Anisotropic Elastic Body

The mixed boundary value problem is considered for an anisotropic elastic body under the condition that a boundary value of the displacement vector is given on some part of the boundary and a boundary value of the generalized stress vector on the remainder. Using the potential method and the theory of singular integral equations with discontinuous coefficients, the existence of a solution of the mixed boundary value problem is proved.     

New fixed point theorems for mixed monotone operators and local existence-uniqueness of positive solutions for nonlinear boundary value problems

In this article we present a new fixed point theorem for a class of general mixed monotone operators, which extends the existing corresponding results. Moreover, we establish some pleasant properties of nonlinear eigenvalue problems for mixed monotone operators. Based on them the local existence-uniqueness of positive solutions for nonlinear boundary value problems which include Neumann boundary value problems, three-point boundary value problems and elliptic boundary value problems for Lane-Emden-Fowler equations is proved. The theorems for nonlinear boundary value problems obtained here are very general.     

Universal boundary value problem for equations of mathematical physics

A new statement of a boundary value problem for partial differential equations is discussed. An arbitrary solution to a linear elliptic, hyperbolic, or parabolic second-order differential equation is considered in a given domain of Euclidean space without any constraints imposed on the boundary values of the solution or its derivatives. The following question is studied: What conditions should hold for the boundary values of a function and its normal derivative if this function is a solution to the linear differential equation under consideration? A linear integral equation is defined for the boundary values of a solution and its normal derivative; this equation is called a universal boundary value equation. A universal boundary value problem is a linear differential equation together with a universal boundary value equation. In this paper, the universal boundary value problem is studied for equations of mathematical physics such as the Laplace equation, wave equation, and heat equation. Applications of the analysis of the universal boundary value problem to problems of cosmology and quantum mechanics are pointed out.     

二阶非线性椭圆型方程带位移的边值问题

In this paper, we consider the boundary value problem with the shift for nonlinear uniformly elliptic equations of second order in a multiply connected domain. For this sake, we propose a modified boundary value problem for nonlinear elliptic systems of first order equations, and give a priori estimates of solutions for the modified boundary value problem. Afterwards we prove by using the Schauder fixedpoint theorem that this boundary value problem with some conditions has a solution. The result obtained is the generlization of the corresponding theorem on the Poincare boundary value problem.     

Boundary Value Problems for the Stationary Schrodinger Equation on Riemannian Manifolds

We suggest a new approach to the statement of boundary value problems for elliptic partial differential equations on arbitrary Riemannian manifolds which is based on the consideration of equivalence classes of functions on a manifold. Using this approach, we establish some interrelation between the solvability of boundary value problems and solvability of exterior boundary problems for the stationary Schrodinger equation. Also we prove the comparison and uniqueness theorems for solutions to boundary value problems in this statement and obtain sufficient conditions for solvability of boundary value problems when the coefficient in the Schrodinger equation is changed.     

Elliptic boundary problems on manifolds with polycylindrical ends

We investigate general Shapiro-Lopatinsky elliptic boundary value problems on manifolds with polycylindrical ends. This is accomplished by compactifying such a manifold to a manifold with corners of in general higher codimension, and we then deal with boundary value problems for cusp differential operators. We introduce an adapted Boutet de Monvel's calculus of pseudodifferential boundary value problems, and construct parametrices for elliptic cusp operators within this calculus. Fredholm solvability and elliptic regularity up to the boundary and up to infinity for boundary value problems on manifolds with polycylindrical ends follows.     

一类Dirichlet边值逆问题

给出解析函数的一类Dirichlet边值逆问题的数学提法.依据解析函数Dirichlet边值问题和广义Dirichlet边值问题的理论,讨论了此边值逆问题的可解性.利用解析函数Dirichlet边值问题的Schwarz公式,给出了该边值逆问题的可解条件和解的表示式.     

STABILITY OF A KIND OF COMPOUND BOUNDARY VALUE PROBLEM WITH RESPECT TO THE PERTURBATION OF BOUNDARY CURVE

In this paper, we discuss the stability of general compound boundary value prob-lems combining Riemann boundary value problem for an open arc and Hilbert bound-ary value problem for a unit circle with respect to the perturbation of boundary curve.     

Application of reproducing kernel method to third order three-point boundary value problems

This paper investigates the analytical approximate solutions of third order three-point boundary value problems using reproducing kernel method. The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel method can not be used directly to solve third order three-point boundary value problems, since there is no method of obtaining reproducing kernel satisfying three-point boundary conditions. This paper presents a method for solving reproducing kernel satisfying three-point boundary conditions so that reproducing kernel method can be used to solve third order three-point boundary value problems. Results of numerical examples demonstrate that the method is quite accurate and efficient for singular second order three-point boundary value problems.