局部分数阶拉普拉斯算子的狄利克雷问题弱解的细正则性

In this paper, we study the Holder regularity of weak solutions to the Dirichlet problem associated with the regional fractional Laplacian (-△)^αΩ on a bounded open set Ω ■R(N ≥ 2) with C^(1,1) boundary ■Ω. We prove that when f ∈ L^p(Ω), and g ∈ C(Ω), the following problem (-△)^αΩu = f in Ω, u = g on ■Ω, admits a unique weak solution u ∈ W^(α,2)(Ω) ∩ C(Ω),where p >N/2-2α and 1/2< α < 1. To solve this problem, we consider it into two special cases, i.e.,g ≡ 0 on ■Ω and f ≡ 0 in Ω. Finally, taking into account the preceding two cases, the general conclusion is drawn.     

非线性分数微分方程边值问题正解的存在性(英文)

In this paper,we study a Dirichlet-type boundary value problem（BVP） of nonlinear fractional differential equation with an order α∈(3,4],where the fractional derivative D^α_o⁺is the standard Riemann-Liouville fractional derivative.By constructing the Green function and investigating its properties,we obtain some criteria for the existence of one positive solution and two positive solutions for the above BVP.The Krasnosel’skii fixedpoint theorem in cones is used here.We also give an example to illustrate the applicability of our results.     

一类具时滞变量的类p-Laplacian Liénard微分方程周期解的存在性

本文研究了一类具类p-Laplacian项Liénard微分方程.应用广义Borsuk定理,在系数c(t)为非常值函数的情况下,得到了该方程周期解存在性的一个新的结果.

Abstract：

In this article,we study a kind of p-Laplacian-like Liénard differential equation.By using the generalized Borsuk theorem in coincidence degree theory,a new result on the existence of its periodic solution is obtained in the case that the cofficient c(t)is not a constant function.     

Orbital Stability of Peakons for the Modified Camassa–Holm Equation

In this paper,we investigate the orbital stability of the peaked solitons(peakons)for the modified Camassa–Holm equation with cubic nonlinearity.We consider a minimization problem with an appropriately chosen constraint,from which we establish the orbital stability of the peakons under H¹∩W^1,4norm.     

一类非散度型非线性退化抛物方程解的存在性(英文)

In this paper, we discuss the existence of weak solutions to the initial and boundary value problem of a class of nonlinear degenerate parabolic equations in non-divergence form. Applying the method of parabolic regularization, we prove the existence of weak solutions to the problem. By carefully analyzing the approximate solutions to the problem, we make a series of estimates to the solutions and prove the weak convergence of the approximation solution sequence. Finally we testify the existence of weak solutions to the problem.     

二阶脉冲微方程三点边值问题解的存在性

We study the existence of solutions to the second order three-point boundary value problem: where 0 <η< 1, α∈R, and f : [0, 1]×R×R→R, Ii: R×R→R, Ji : R×R→R(i=1,2,…k) are continuous. Our results is new and different from previous results. In particular, we obtain the Green function of the problem, which makes the problem simpler.     

PROPERTIES OF SOLUTIONS TO A HARMONIC-MAPPING TYPE EQUATION WITH A DIRICHLET BOUNDARY CONDITION

In the present paper, we consider the problem ■ where β₁, β₂ > 0 and β₁ + β₂ < 1, and ? is a convex domain in Rⁿ. The existence, uniqueness,regularity and (2-β₂)/(1-β₁-β₂)-concavity of the positive solutions of the problem(0.1) are proven.     

Large m asymptotics for minimal partitions of the Dirichlet eigenvalue

In this paper,we study large m asymptotics of the l 1 minimal m-partition problem for the Dirichlet eigenvalue.For any smooth domainΩ?Rⁿsuch that|Ω|=1,we prove that the limit lim_m→∞l¹_m(Ω)=c 0 exists,and the constant c 0 is independent of the shape ofΩ.Here,l¹_m(Ω)denotes the minimal value of the normalized sum of the first Laplacian eigenvalues for any m-partition ofΩ.     

The information-based complexity of approximation problem by adaptive Monte Carlo methods

In this paper, we study the complexity of information of approximation problem on the multivariate Sobolev space with bounded mixed derivative MWpr,α(Td), 1 < p < ∞, in the norm of Lq(Td), 1 < q < ∞, by adaptive Monte Carlo methods. Applying the discretization technique and some properties of pseudo-s-scale, we determine the exact asymptotic orders of this problem.     

ORLICZ-LORENTZ SEQUENCE SPACES EQUIPPED WITH THE ORLICZ NORM In memory of Henryk Hudzik,a great man,our teacher and colleague

In this article,we consider Orlicz-Lorentz sequence spaces equipped with the Orlicz norm(λ_φ,ω,‖·‖^O_φ,ω) generated by any Orlicz function and any non-increasing weight sequence.As far as we know,research on such a general case is conducted for the first time.After showing that the Orlicz norm is equal to the Amemiya norm in general and giving some important properties of this norm,we study the problem of existence of order isomorphically isometric copies of l^∞...     

Simultaneous Subgradient Algorithms for the Generalized Split Variational Inclusion Problem in Hilbert Spaces

In this article, we study the generalized split variational inclusion problem. For this purpose, motivated by the projected Landweber algorithm for the split equality problem, we first present a simultaneous subgradient extragradient algorithm and give related convergence theorems for the proposed algorithm. Next, motivated by the alternating CQ-algorithm for the split equality problem, we propose another simultaneous subgradient extragradient algorithm to study the general split variational inclusion problem. As applications, we consider the split equality problem, split feasibility problem, split variational inclusion problem, and variational inclusion problem in Hilbert spaces.     

Multiphysics mixed finite element method with Nitsche's technique for Stokes-poroelasticity problem

In this article, we propose a multiphysics mixed finite element method with Nitsche's technique for Stokes-poroelasticity problem. Firstly, we reformulate the poroelasticity part of the original problem by introducing two pseudo-pressures to into a “fluid–fluid” coupled problem so that we can use the classical stable finite element pairs to deal with this problem conveniently. Then, we prove the existence and uniqueness of weak solution of the reformulated problem. And we use Nitsche's technique to approximate the coupling condition at the interface to propose a loosely-coupled time-stepping method to solve three subproblems at each time step–a Stokes problem, a generalized Stokes problem and a mixed diffusion problem. And the proposed method does not require any restriction on the choice of the discrete approximation spaces on each side of the interface provided that appropriate quadrature methods are adopted. Also, we give the stability analysis and error estimates of the loosely-coupled time-stepping method. Finally, we give the numerical tests to show that the proposed numerical method has a good stability and no “locking” phenomenon.     

An approach for solving nonlinear programming problems

In this paper we use measure theory to solve a wide range of the nonlinear programming problems. First, we transform a nonlinear programming problem to a classical optimal control problem with no restriction on states and controls. The new problem is modified into one consisting of the minimization of a special linear functional over a set of Radon measures; then we obtain an optimal measure corresponding to functional problem which is then approximated by a finite combination of atomic measures and the problem converted approximately to a finite-dimensional linear programming. Then by the solution of the linear programming problem we obtain the approximate optimal control and then, by the solution of the latter problem we obtain an approximate solution for the original problem. Furthermore, we obtain the path from the initial point to the admissible solution.     

Solving some optimal path planning problems using an approach based on measure theory

In this paper we solve a collection of optimal path planning problems using a method based on measure theory. First we consider the problem as an optimization problem and then we convert it to an optimal control problem by defining some artificial control functions. Then we perform a metamorphosis in the space of problem. In fact we define an injection between the set of admissible pairs, containing the control vector function and a collision-free path defined on free space and the space of positive Radon measures. By properties of this kind of measures we obtain a linear programming problem that its solution gives rise to constructing approximate optimal trajectory of the original problem. Some numerical examples are proposed.     

On a singular two dimensional nonlinear evolution equation with nonlocal conditions

This paper deals with a nonclassical initial boundary value problem for a two dimensional parabolic equation with Bessel operator. We prove the existence and uniqueness of the weak solution of the given nonlinear problem. We start by solving the associated linear problem. After writing this latter in its operator form, we establish an a priori bound from which we deduce the uniqueness of the strong solution. For the solvability of the associated linear problem, we prove that the range of the operator generated by the considered problem is dense. On the basis of the obtained results of the linear problem, we apply an iterative process to establish the existence and uniqueness of the nonlinear problem.     

分裂可行问题(SFP)的投影算法

本文探讨了分裂可行问题(SFP)的投影算法．我们先定义了(SFP)的反问题(ISFP), 然后利用正交投影性质,给出了(SFP)与(ISFP)及某些投影不动点问题之间的关系．随后, 给出了求解(SFP)的几种正交投影算法,其中包括精确和不精确投影格式．基于变分不等式中投影算法收敛性的证明思路和分裂可行问题的特有形式,证明了这几种算法的收敛性．最后通过几个算例对讨论的方法进行了初步比较．     

First kind symmetric periodic solutions and their stability for the Kepler problem and anisotropic Kepler problem plus generalized anisotropic perturbation

Motivated by some problems in Celestial Mechanics that combines quasihomogeneous potential in the anisotropic space, we investigate the existence of several families of first kind symmetric periodic solutions for a family of planar perturbed Kepler problem. In addition, we give sufficient conditions for the existence of first kind periodic solutions and also we characterize its type of stability. As an application of this general situation, we discuss the existence of symmetric periodic solutions for the anisotropic Kepler problem plus a generalized anisotropic perturbation, (shortly, p-AKPQ problem) and for the Kepler problem plus a generalized anisotropic perturbation (shortly, p-KPQ problem), as continuation of circular orbits of the two-dimensional Kepler problem. To get this objective, we consider different types of perturbations and then we apply our main result.     

Global Method for Monotone Variational Inequality Problems with Inequality Constraints

We consider optimization methods for monotone variational inequality problems with nonlinear inequality constraints. First, we study the mixed complementarity problem based on the original problem. Then, a merit function for the mixed complementarity problem is proposed, and some desirable properties of the merit function are obtained. Through the merit function, the original variational inequality problem is reformulated as simple bounded minimization. Under certain assumptions, we show that any stationary point of the optimization problem is a solution of the problem considered. Finally, we propose a descent method for the variational inequality problem and prove its global convergence.     

A study of mixed discrete bilevel programs using semidefinite and semi-infinite programming

In this paper, we study the bilevel programming problem with discrete polynomial lower level problem. We start by transforming the problem into a bilevel problem comprising a semidefinite program (SDP for short) in the lower level problem. Then, we are able to deduce some conditions of existence of solutions for the original problem. After that, we again change the bilevel problem with SDP in the lower level problem into a semi-infinite program. With the aid of the exchange technique, for simple bilevel programs, an algorithm for computing a global optimal solution is suggested, the convergence is shown, and a numerical example is given.     

非线性二层规划问题的全局优化方法

对于下层为线性规划问题的一类非线性二层规划问题,利用线性规划的对偶理论,将其转化为一个单层优化问题,同时取下层问题的对偶间隙作为惩罚项,构造了一个相应的罚问题,然后提出了一个求解该类二层规划问题的全局优化方法。最后,数值结果表明,所提出的方法是可行的。