矩阵方程X=Q+A~* (I_m⊕ X-C)~(-1) A的Hermitian正定解

研究了一类来源于插值理论的非线性矩阵方程.利用Kronecker积的性质以及Banach空间单调有界序列收敛原理证明了此类方程正定解的存在唯一性.另外也给出了此方程正定解的范围.     

边界条件含有特征参数的Sturm-Liouville算子的唯一性定理

建立了一类Sturm-Liouville问题的唯一性定理.对于固定的n∈Z,证明了该Sturm-Liouville问题的第n个特征值λ_n（q,a）关于a是严格单调的.对不同系数的a_k,如果能够测得第n个特征值的谱集合{λ_n（q,a_k）}_k=1^+∞,则谱集合{λ_n（q,a_k）}_k=1^+∞能够唯一确定[0,π]上的势函数q（x）.     

一类具时间周期和非局部时滞的非拟单调反应扩散方程的整体解

研究了一类不具有拟单调性的时间周期非局部时滞反应扩散方程的新型整体解.为了克服拟单调性条件缺失带来的困难,借助于两个拟单调辅助方程的周期行波解和相应空间齐次方程的全轨道构造了新型的整体解.     

一类广义欧拉函数方程的可解性

将Lehmer同余式从模素数的平方推广到模任意整数的平方,王容、廖群英定义了一类正整数n广义欧拉函数φ_n (5),并给出了准确计算公式,利用已有的广义欧拉函数计算公式,使用初等的方法和技巧,研究了一类广义欧拉函数方程φ₅(n)=n/d的正整数解.     

奇异二阶连续和离散边值问题正解的存在唯一性

利用一类混合单调算子的一个不动点定理,给出了奇异二阶微分方程边值问题和奇异二阶差分方程边值问题的解的存在及惟一性.     

一类非线性演化方程新的显式行波解

借助Mathematica软件和吴方法,采用双曲函数法,获得了一类非线性演化方程u_tt+au_xx+bu+cu2+du3=0的多组行波解,其中包括周期解与孤子解.这种方法也适用于其他非线性方程或方程组.     

非混合单调二元算子方程(组)解的存在唯一性

利用锥理论及Banach压缩映射原理,在不要求上、下解条件及算子紧性与连续性的条件下,建立了一类满足更一般序关系条件的非混合单调二元算子方程组(?)解的存在唯一性定理,以及非单调二元算子方程T(x,x)=x和非单调一元算子方程Lx=x解的存在唯一性定理,推广了最近相关文献的研究结果.     

一类Curvature方程的两种边值问题解的存在性

利用含有伪单调算子的变分不等式解的存在性定理,证明一类具有Dirichlet边值条件的Curvature方程在W1,p(Ω)空间中存在唯一解.深入研究具有Dirichlet边值条件的Curvature方程和具有Neumann边值条件的Curvature方程之间的关系,利用极大单调算子值域的扰动理论,给出具有Neumann边值条件的Curvature方程在W1,p(Ω)空间中存在解的充分条件.文中采用一些新的证明技巧,推广和补充了以往的相关研究成果.     

一类中立型积分方程的正的概周期解

本文利用混合单调算子中的迭代技巧,研究了一类中立型积分方程正的概周期解的存在性,以前相关结果得以推广.     

具有非局部效应的时滞SEIR系统的周期行波解

该文研究了一类具有非局部效应和非线性发生率的时滞SEIR系统的周期行波解.首先,定义基本再生数R₀并构造适当的上下解,将周期行波解的存在性转化为闭凸集上非单调算子的不动点问题,利用Schauder不动点定理结合极限理论建立该系统周期行波解的存在性.其次,利用反证法结合比较原理,建立当基本再生数R₀<1时该系统周期行波解的不存在性.     

一类非线性不连续集值发展型方程的广义单调迭代法

本文利用广义单调迭代法研究了一类非线性不连续集值发展型方程的数值解法,利用序理论给出其迭代格式,得到了迭代解的收敛性结果.在一种较弱的条件下,给出了离散解集收敛性的若干结论.     

Best Approximation from the Kuhn-Tucker Set of Composite Monotone Inclusions

Kuhn-Tucker points play a fundamental role in the analysis and the numerical solution of monotone inclusion problems, providing in particular both primal and dual solutions. We propose a class of strongly convergent algorithms for constructing the best approximation to a reference point from the set of Kuhn-Tucker points of a general Hilbertian composite monotone inclusion problem. Applications to systems of coupled monotone inclusions are presented. Our framework does not impose additional assumptions on the operators present in the formulation, and it does not require knowledge of the norm of the linear operators involved in the compositions or the inversion of linear operators.     

一类非单调二阶锥互补问题解集的非空性与有界性

本文在二阶锥上引入一类新的映射,称之为笛卡尔P_*(κ)映射,它是单调映射的推广.文中讨论涉及这类映射的二阶锥互补问题的解的存在性和解集的有界性.主要结论为:如果所考虑的互补问题是严格可行的,那么它的解集是非空有界的.     

Use of weak quasi-solutions of the Fredholm first-kind equation in problems with scarce data

This paper addresses the situation where the available data are insufficient to obtain a stable approximate solution. Instead solving the problem directly, it is proposed to convert the linear integral Fredholm first-kind equation to the equation for the anti-derivative of the unknown function. The L_p-compact of monotone bounded functions is the natural set of well-posedness for the modified equation. This transformation does not impose any a priori restriction on the class of solutions. In many applications a weak solution provides sufficient information on the unknown function. A numerical example is presented for application to the important problem in biology of reconstruction of age composition from size distribution data for a population.     

共轭可交换性及其在集合形式单调类定理中的应用

引入了集列的极限运算与二元集运算之间的共轭可交换性的概念,讨论了常见的和重要的集列的极限运算与集代数运算之间所具有的共轭可交换性质,在一般的情形下证明了共轭可交换性是集类的极限运算生成类对集代数运算具有封闭性的一个充分条件,使得集合形式的单调类定理的理论得到了统一的解释,并且证明更简单,思路更清晰,最后还给出了一个新的集合形式的单调类定理.     

Magic Labeling of Disjoint Union Graphs

Let G be a graph with vertex set V (G), edge set E(G) and maximum degree Δ respectively. G is called degree-magic if it admits a labelling of the edges by integers {1, 2, …,|E(G)|} such that for any vertex v the sum of the labels of the edges incident with v is equal to (1+|E(G)|)/2·d(v), where d(v) is the degree of v. Let f be a proper edge coloring of G such that for each vertex v ∈ V (G),|{e:e ∈ E_v, f(e) ≤ Δ/2}|=|{e:e ∈ E_v, f(e) > Δ/2}|, and such an f is called a balanced edge coloring of G. In this paper, we show that if G is a supermagic even graph with a balanced edge coloring and m ≥ 1, then (2m + 1)G is a supermagic graph. If G is a d-magic even graph with a balanced edge coloring and n ≥ 2, then nG is a d-magic graph. Results in this paper generalise some known results.     

Connectedness of Solution Sets of Strong Vector Equilibrium Problems with an Application

In this paper, the connectedness of solution set of a strong vector equilibrium problem in a finite dimensional space, is investigated. Firstly, a nonconvex separation theorem is given, that is, a neither open nor closed set and a compact subset in a finite dimensional space can be strictly separated by a sublinear and strongly monotone function. Secondly, in terms of the nonconvex separation theorem, the union relation between the solution set of the strong vector equilibrium problem and the solution sets of a series of nonlinear scalar problems, is established. Under suitable assumptions, the connectedness and the path connectedness of the solution set of the strong vector equilibrium problem are obtained. In particular, we solve partly the question related to the path connectedness of the solution set of the strong vector equilibrium problem. The question is proposed by Han and Huang in the reference (J Optim Theory Appl, 2016.  https://doi.org/10.1007/s10957-016-1032-9). Finally, as an application, we apply the main results to derive the connectedness of the solution set of a linear vector optimization problem.     

Parallel Algorithms for Solving a Class of Variational Inequalities over the Common Fixed Points Set of a Finite Family of Demicontractive Mappings

Abstract

We propose parallel algorithms for solving a class of variational inequalities over the set of common fixed points for a finite family of demicontractive mappings in real Hilbert spaces. Under some suitable conditions, we prove that the sequence generated by the proposed algorithms converges strongly to a solution of the problem. We apply the proposed algorithms to strongly monotone variational inequality problems with pseudomonotone equilibrium constraints by defining a quasi-nonexpansive and demi-closed mapping whose fixed point set coincides with the solution set of the equilibrium problem.     

Construction algorithms for a class of monotone variational inequalities

This paper is devoted to solve the following monotone variational inequality of finding \(x^*\in \mathrm{Fix}(T)\) such that

$$\begin{aligned} \langle Ax^*,x-x^*\rangle \ge 0,\quad \forall x\in \mathrm{Fix}(T), \end{aligned}$$

where A is a monotone operator and \(\mathrm{Fix}(T)\) is the set of fixed points of nonexpansive operator T. For this purpose, we construct an implicit algorithm and prove its convergence hierarchical to the solution of above monotone variational inequality.

     

具有混合单调非线性项的四阶微分方程两点边值问题

讨论了一类具有混合单调非线性项的四阶微分方程两点边值问题,运用一类混合单调算子的不动点定理及"和型"非线性算子的不动点定理,结合单调迭代技巧和格林函数的性质,获得方程正解存在且唯一的充分条件,并构造两个迭代序列收敛于此唯一解.最后,给出具体的例子验证了定理的正确性.