四阶非线性差分方程周期解的存在性

本文研究一类四阶非线性差分方程周期解的存在性.首先建立四阶差分方程的变分泛函,然后将四阶差分方程周期解的存在性问题转化成相应泛函临界点的存在性问题加以求解.利用临界点理论和一些空间分解技巧,获得了该方程至少存在一个周期解的充分条件.通过一个实例,说明所得定理在应用中的有效性.所得结果丰富了对四阶差分方程周期解问题的研究.     

时标上脉冲动力方程周期边值问题的拟线性化方法

本文研究了一类时标上脉冲动力方程周期边值问题解的收敛性问题.利用时标上一阶脉冲动力不等式、上下解和单调迭代技巧证明了该问题解的一致收敛性结果,并进一步采用拟线性化方法和分析技巧获得了该方程在周期边值条件下两个逼近解序列高阶收敛的充分性判据.本文所得结果发展了时标上动力方程定性理论的结果.     

超线性二阶常微分方程Dirichlet边值问题多重解的存在性研究

本文对于一个超线性二阶常微分方程的边值问题,利用变分方法,将微分方程解的存在性转化为求解某个泛函临界点的存在性,获得Sobolev空间中新的解的存在性定理,得到了一类超线性二阶常微分方程边值问题无穷多解的存在性定理.     

一类二阶非线性差分方程的多重周期解

本文研究了一类二阶非自治非线性差分方程多重周期解的存在性问题.将这类方程的周期解转化为定义在一个适当空间上泛函的临界点,利用变分原理和Clark定理,得到了此类方程周期解个数的下界估计.     

离散广义Emden-Fowler方程边值问题的多重解

本文研究了离散广义Emden-Fowler方程边值问题多重解的存在性。通过将这类边值问题的解转化为定义在一个适当空间上泛函的临界点,并利用Morse理论中的三临界点定理,文中得到了该问题存在3个解的充分条件,并举例说明了所获得的主要结果是有效的。     

二阶周期边值问题正解的存在性

周期边值问题已成为方程研究领域的一个重要分支,它在许多实际问题中有着更为广泛的应用,本文主要研究了二阶周期边值问题正解的存在性.利用锥上不动点指数理论研究了二阶周期边值问题方程组的正解的存在性,通过相应的线性问题的第一特征值和拓扑度乘积定理,建立了正解的存在性定理.最后,我们给出具体的例子说明了该正解存在性定理的结论.     

一类四阶非线性椭圆方程的无穷多个变号解

在工程实际中,含有双调和算子的四阶椭圆问题?~2u+c?u=f(x,u),x∈?,可用来描述悬索桥的非线性振动.当悬索桥处于平衡位置且不受外力的理想情形下,相应的边界条件为u|_(??)=?u|_(??)=0.本文研究了一类四阶椭圆边值问题,其中非线性项f在0处渐近线性、在∞处超二次.证明方法为下降流不变集方法,主要结果是证明了这类四阶椭圆边值问题存在一个变号解以及存在无穷多个变号解的两个定理.所得结果及其证明方法均不同于现有文献中的结果.     

非线性三阶两点边值问题变号解的逐次逼近方法

本文研究了一个非线性三阶两点边值问题变号解的存在性与逐次逼近,其中非线性项关于空间变元单调增并且关于时间变元奇异.利用Green函数,将该问题转化为一个等价积分方程,其中相伴积分算子是全连续并且增的.在适当的条件下借助于全连续增算子构造了两个逐次迭代序列.这些序列从常值函数开始并且一致收敛于此问题的变号解.结论说明这种变号解的存在性仅仅依赖于非线性项在某个有界集合上的增长,而与非线性项在这个集合以外的状态无关.最后,数值算例证实新的逼近方法对于数值计算是有效的.     

Klein-Gordon-Maxwell系统解的存在性和多解性

Klein-Gordon-Maxwell系统具有很强的物理背景,它提供了带电粒子物质和它所产生的电磁场之间作用的“二元模型”描述．根据这个模型,粒子物质是一个非线性场方程的孤波解,且电磁场的作用是由场方程与麦克斯韦方程耦合的衡量电位描述的．本文利用变分方法和临界点理论研究一类Klein-Gordon-Maxwell系统解的存在性和多重性．首先,利用山路引理,我们证明了系统非平凡解的存在性,其中一个解是非负的,一个解是非正的．其次,运用喷泉定理,文中证明系统在非线性项满足一定条件下无穷多高能量解的存在性．本文所得结果推广了以前的结论．     

Ekeland变分原理导出的向量均衡系统解的存在性

本文研究了定义在度量空间上的向量均衡系统解的存在性问题。利用非线性标量化方法,将向量优化问题转化为数量优化问题,得到了Ekeland变分原理的一个向量形式推广。用向量形式Ekeland变分原理,证明了向量均衡系统解的存在性定理。结果表明,如果函数满足向量形式Ekeland变分原理和上半连续性条件,那么向量均衡系统的解集非空。     

二阶常微分方程周期边值问题的正解

非线性二阶周期边值问题可描述天体力学、工程和生物中出现的许多周期现象,其广泛的应用引起了许多学者的关注.本文主要研究二阶周期边值问题正解的存在性,其中非线性项包含一阶导数项.设非线性项满足Caratheodory条件,利用零点指数理论和分析技巧,本文建立了二阶周期边值问题正解的存在性定理,推广并改进了一些已知结果.最后给出一个例子说明主要结果.     

时间标度上二阶周期边值问题解的存在性

运用Mawhin的重合度定理,讨论了一类时间标度上非线性动态方程周期边值问题的解的存在性,得到判别方法并举例作以说明.     

A numerical method for unilateral problems

Variational inequalities connected with Signorini's problem have appeared as a natural generalization of the minimum potential-energy theorem for bodies with unilateral constraints. In this paper, we describe numerical experience on the use of variational inequalities and Pade approximants to obtain approximate solutions to a class of unilateral boundary value problems of elasticity, like those describing the equilibrium configuration of an elastic membrane stretched over an elastic obstacle. These problems have the peculiar feature of being alternatively formulated as nonlinear boundary value problems without constraints for which the technique of Pade approximants can be successfully employed. The variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution.     

Cubic splines collocation methods for unilateral problems

Variational inequalities theory not only provides us a general unified frame work for study many unrelated moving and free boundary vary problems, but also gives more efficient numerical methods for solving them. In this paper, we describe numerical experience on the use of variational inequalities and cubic splines collocation technique to obtain approximate solution to a class of unilateral boundary value problems of elasticity, like those describing the equilbrium configuration of an elastic string stretched over an elastic obstacle. The variational inequality formulation is used to discuss the problem of uniqueness and existence of the solution of the unilateral problems.     

Banach空间中二阶周期边值问题的解

本文研究Banach空间中弱拓扑下二阶周期边值问题的解。在上解小于等于下解的假设条件下证明了二阶周期边值问题解的存在性。     

Periodic and slightly periodic boundary value problems in elastostatics on bodies unbounded in several directions

General properties of solutions to elastostatic boundary value problems in which some or all of the functions involved are periodic are studied with particular attention given to problems on bodies unbounded in a direction other than the direction “of periodicity”. It is shown that, even though the displacement corresponding to a periodic strain may, in a very nontrivial sense, be nonperiodic, it does satisfy a “semiperiodicity” condition. Conditions which assure the periodicity of the displacement corresponding to a periodic strain are developed as are conditions which assure that the solution to a periodic boundary value problem has periodic strain. This leads to a discussion of the uniqueness of the solutions to various boundary value problems which, in themselves, are not necessarily periodic but whose corresponding null boundary value problem is periodic. As a special example a uniqueness theorem for the displacement problem and a uniqueness theorem for the traction problem on a homogeneous (but not necessarily isotropic) half-plane are proven using the arbitrariness of the periodicity. Throughout the paper, counterexamples demonstrate the necessity of many of the conditions assumed.     

非线性Sturm-Liouville奇异边值问题正解的存在唯一性

本文用正则锥上的非紧减算子不动点定理,讨论了一类非线性Sturm-Liouville奇异边值问题正解的存在性和唯一性。对此问题的讨论,我们构造了一个新的正则锥。这样的方法完全可以应用到其它奇异边值上去,用以讨论正解的存在性。     

二阶积分微分方程的周期边值问题

本文研究了二阶积分微分方程的周期边值问题,在反向上下解的条件下,利用Fredholm定理和比较原则得到其极解的存在性。     

The diffuse Nitsche method: Dirichlet constraints on phase‐field boundaries

We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase‐field approximations of sharp domains. Leveraging the properties of the phase‐field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase‐field solutions of the Allen‐Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase‐field in the diffuse boundary region and a uniform mesh for the representation of the physics‐based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty‐type methods. In the context of imaging‐based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, ie, the interface width of the phase‐field, the voxel spacing, and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body.