Positive solutions of higher-order nonlinear fractional differential systems with nonlocal boundary conditions

The paper deals with the existence and multiplicity of positive solutions for a system of higher-order nonlinear fractional differential equations with nonlocal boundary conditions. The main tool used in the proof is fixed point index theory in cone. Some limit type conditions for ensuring the existence of positive solutions are given.     

The Bolzano mean-value theorem and partial differential equations

We study the existence of solutions to abstract equations of the form $0=Au+F(u)$, $u\in K?E$, where A is an abstract differential operator acting in a Banach space E, K is a closed convex set of constraints being invariant with respect to resolvents of A and a perturbation F satisfies a certain tangency condition. Such problems are closely related to the so-called Poincaré–Miranda theorem, being the multi-dimensional counterpart of the celebrated Bolzano intermediate value theorem. In fact our main results should be regarded as infinite-dimensional variants of Bolzano and Miranda–Poincaré theorems. Along with single-valued problems we deal with set-valued ones, yielding the existence of the so-called constrained equilibria of set-valued maps. The abstract results are applied to show existence of (strong) steady state solutions to some weakly coupled systems of drift reaction–diffusion equations or differential inclusions of this type. In particular we get the existence of strong solutions to the Dirichlet, Neumann and periodic boundary problems for elliptic partial differential inclusions under the presence of state constraints of different type. Certain aspects of the Bernstein theory for bvp for second order ODE are studied, too. No assumptions concerning structural coupling (monotonicity, cooperativity) are undertaken.     

赋值Banach代数的偏序D^-度量空间和公共不动点定理

基于完备的D^*-度量空间,提出具有Banach代数的偏序D^*-度量空间的定义,通过迭代序列的构造,证明了新定义下不动点定理的存在性及唯一性理论.随后引入了关于两个连续自映射的公共不动点定理,并对其存在性和唯一性进行了研究.所得结果推广了现有文献.     

Existence of positive solutions to a boundary value problem for a delayed singular high order fractional differential equation with sign-changing nonlinearity

In this paper, we discuss the existence of positive solutions to the boundary value problem for a high order fractional differential equation with delay and singularities including changing sign nonlinearity. By using the properties of the Green function, Guo-krasnosel"skii fixed point theorem, Leray-Schauder"s nonlinear alternative theorem, some existence results of positive solutions are obtained, respectively.     

Learning the optimal kernel for Fisher discriminant analysis via second order cone programming

Kernel Fisher discriminant analysis (KFDA) is a popular classification technique which requires the user to predefine an appropriate kernel. Since the performance of KFDA depends on the choice of the kernel, the problem of kernel selection becomes very important. In this paper we treat the kernel selection problem as an optimization problem over the convex set of finitely many basic kernels, and formulate it as a second order cone programming (SOCP) problem. This formulation seems to be promising because the resulting SOCP can be efficiently solved by employing interior point methods. The efficacy of the optimal kernel, selected from a given convex set of basic kernels, is demonstrated on UCI machine learning benchmark datasets.     

凹泛函型锥拉伸与压缩不动点定理

本文通过凹泛函构造了Banach空间中的一个收缩核,并利用收缩核给出了凹泛函型锥拉伸与压缩不动点定理,扩展和完善了已有文献中的一些结果.     

年龄分布下种群资源开发动态博弈的Nash均衡

研究具有年龄结构的种群资源开发中的动态博弈问题.应用~Kakutani~多值映射不动点定理证明了Nash均衡的存在性,借助切锥－法锥和共轭系统技巧刻画了均衡策略.结果表明,在一定条件下,均衡策略具有Bang-Bang结构.     

奇异三阶微分方程边值问题的正解

讨论了奇异三阶微分方程边值问题的正解存在性.通过与一个线性算子相关的第一特征值的讨论,运用不动点指数定理,得到了正解存在的结果.其中允许h(t)在t=0和t=1处奇异.     

贴膜条件下飞秒激光诱导硅基表面锥状微结构

利用波长为800 nm的飞秒脉冲激光,对表面贴有聚对苯二甲酸乙二醇酯透明膜的单晶硅片进行扫描,研究了不同激光制备参数对微结构形成的影响.结果表明,锥状微结构是否形成取决于激光能量密度,能量密度太小时不能形成锥状结构,能量密度太大时易破坏锥状突起;而激光扫描速度可直接影响锥状微结构的质量,扫描速度太小时也易破坏锥状结构突起,扫描速度太大时,由于作用深度太浅使得锥状结构轮廓不分明.在此基础上对实验参数进行优化,得到了较理想的锥状微结构.最后通过分析指出,贴膜条件下锥状微结构的形成是由激光烧蚀作用和氧化作用共同引起的,且激光烧蚀作用占主导.