Existence of positive ground state solutions for a critical Kirchhoff type problem with sign-changing potential

In this work, we are interested in studying the following Kirchhoff type problem

where $\Omega ?{\mathbb{R}}^N(N\ge 3)$ is a smooth bounded domain, $2^1=\frac{2N}{N?2}$ is the critical Sobolev exponent, $0<q<1,\lambda >0$, and $f\in L^{\infty }\left(\Omega \right)$ with the set $\{x\in \Omega :f\left(x\right)>0\}$ of positive measures, and $g\in L^{\infty }\left(\Omega \right)$ with $g\left(x\right)\ge 0,g?0.$ By the Nehari method and variational method, the existence of positive ground state solutions is obtained.     

On a backward problem for the Kirchhoff’s model of parabolic type

We study for the first time the backward problem for nonlocal nonlinear boundary value problem of Kirchhoff’s model of parabolic type. First, we show that the problem is severely ill-posed in the sense of Hadamard. We propose two methods: the Fourier truncation method for stabilizing the problem with homogeneous source and the quasi-reversibility method for regularizing the problem with nonlinear source. Under some priori assumptions on the exact solution, we establish some stability estimates in the $H_0^1$ norm.     

克希霍夫积分偏移在探地雷达中的应用及实现

详细分析和论述了克希霍夫积分偏移原理及其在探地雷达成像中的应用,针对单目标和多目标做了仿真实验.实验结果证明:将克希霍夫积分偏移原理应用在探地雷达成像中具有精度和效率较高、算法简单等优势.     

Bilinear optimal control of a Kirchhoff plate

We consider the problem of optimal control of a Kirchhoff plate. A bilinear control is used as a force to make the plate close to a desired profile taking into the account, a quadratic cost of control. We prove the existence of an optimal control and characterize it uniquely through the solution of an optimality system.     

Existence of multiple solutions for nonhomogeneous Schrödinger–Kirchhoff system involving the fractional p-Laplacian with sign-changing potential

In this paper, we prove the existence of multiple solutions for the following Schrödinger–Kirchhoff system involving the fractional $p$-Laplacian

$\begin{array}{c}\hfill \left\{\begin{array}{c}M\left({?}_{{\mathbb{R}}^{2N}}\frac{{|u\left(x\right)?u\left(y\right)|}^p}{{|x?y|}^{N+ps}}dxdy\right){(?\Delta )}_p^{s}u+V\left(x\right){\left|u\right|}^{p?2}u=F_u(x,u,v)+\lambda g\left(x\right),x\in {\mathbb{R}}^N,\hfill \\ M\left({?}_{{\mathbb{R}}^{2N}}\frac{{|v\left(x\right)?v\left(y\right)|}^p}{{|x?y|}^{N+ps}}dxdy\right){(?\Delta )}_p^{s}u+V\left(x\right){\left|v\right|}^{p?2}v=F_v(x,u,v)+\lambda h\left(x\right),x\in {\mathbb{R}}^N,\hfill \\ u\left(x\right)\to 0,v\left(x\right)\to 0,\text{as}\left|x\right|\to +\infty ,\hfill \end{array}\right.\end{array}$

where ${(?\Delta )}_p^s$ denotes the fractional $p$-Laplacian of order $s\in (0,1)$, $2\le p<\infty$, $ps<N$, $F_u=\frac{?F}{?u}$, $F_v=\frac{?F}{?v}$, $V\left(x\right)$ is allowed to be sign-changing, $\lambda >0$ and $g,h:{\mathbb{R}}^N\to \mathbb{R}$ is a perturbation. Under some certain assumptions on $f$, we obtain the existence of multiple solutions for this problem via Ekeland’s variational principle and mountain pass theorem.     

The backward problem for a time-fractional diffusion-wave equation in a bounded domain

This paper is devoted to solve the backward problem for a time-fractional diffusion-wave equation in a bounded domain. Based on the series expression of the solution for the direct problem, the backward problem for searching the initial data is converted into solving the Fredholm integral equation of the first kind. The existence, uniqueness and conditional stability for the backward problem are investigated. We use the Tikhonov regularization method to deal with the integral equation and obtain the series expression of the regularized solution for the backward problem. Furthermore, the convergence rate for the regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Numerical results for five examples in one-dimensional case and two-dimensional case show that the proposed method is efficient and stable.     

Solvability of a non-linear Cauchy problem for an elliptic equation

ABSTRACT

We study a non-linear operator equation originating from a Cauchy problem for an elliptic equation. The problem appears in applications where surface measurements are used to calculate the temperature below the earth surface. The Cauchy problem is ill-posed and small perturbations to the used data can result in large changes in the solution. Since the problem is non-linear certain assumptions on the coefficients are needed. We reformulate the problem as an non-linear operator equation and show that under suitable assumptions the operator is well-defined. The proof is based on making a change of variables and removing the non-linearity from the leading term of the equation. As a part of the proof we obtain an iterative procedure that is convergent and can be used for evaluating the operator. Numerical results show that the proposed procedure converges faster than a simple fixed point iteration for the equation in the the original variables.     

A mollification method for a Cauchy problem for the Helmholtz equation

The Cauchy problem for the Helmholtz equation is considered. This problem is severely ill-posed, that is, the solution does not depend continuously on the data. To solve the problem numerically a mollification method is proposed. Convergence on error estimates between the exact solution and its approximation are obtained. Some numerical examples are given to show that the method works effectively.     

Existence and multiplicity of solutions for a nonlocal problem with critical Sobolev exponent

In this work, we are interested in considering the following nonlocal problem

where $\Omega ?{\mathbb{R}}^N(N\ge 4)$ is a smooth bounded domain, $a\ge 0,b>0,1<q<2,\mu ,\lambda >0$ and $2^1=\frac{2N}{N?2}$ is the critical Sobolev exponent. By using the variational method and the critical point theorem, some existence and multiplicity results are obtained.     

Existence of solutions for generalized equilibrium problem in G-convex space

In this paper, we introduce and study a kind of generalized equilibrium problem in a G-convex space. By means of the fixed-point theorems, we obtain some existence theorems of solutions for the generalized equilibrium problems.     

Modified quasi-boundary value method for a Cauchy problem of semi-linear elliptic equation

In this paper, we investigate a Cauchy problem for the semi-linear elliptic equation. This problem is well known to be severely ill-posed and regularization methods are required. We use a modified quasi-boundary value method to deal with it, and a convergence estimate for the regularized solution is obtained under an a priori bound assumption for the exact solution. Finally, some numerical results show that our given method works well.     

Existence of multiple positive solutions for functional differential equations

In this paper, the existence of at least three positive solutions for the boundary value problem (BVP) of second-order functional differential equation with the form y″(t) + f(t, y^t) = 0, for t ε [0,1], y(t) -βy′(t) =η(t), for t ε [−τ,0], −γy(t) + Δy′(t) = ζ(t), for t ε [1, 1 + a], is studied. Moreover, we investigate the existence of at least three partially symmetric positive solutions for the above BVP with Δ = βγ.     

Existence results for fractional integrodifferential equations with nonlocal condition via resolvent operators

In this paper, we prove the existence of solutions of fractional integrodifferential equations by using the resolvent operators and fixed point theorem. An example is given to illustrate the abstract results.     

用无网格方法求解声波散射问题

在均匀介质中,对软表面障碍、时间调和声波散射问题归结为Helmholtz方程的Dirichlet外问题.应用无网格方法求解Helmholtz方程的Dirichlet外问题,并给出了一个数值例子,与Nystrom方法进行了对比,表明该方法是较精确的.     

A finite difference method for a boundary value problem related to the Kolmogorov equation

We propose an explicit and an implicit difference scheme for a boundary value problem related to the Kolmogorov equation. Stability conditions and convergence results are given.     

A regularization method for a Cauchy problem of Laplace's equation in an annular domain

In this paper, we propose a new regularization method based on a finite-dimensional subspace generated from fundamental solutions for solving a Cauchy problem of Laplace's equation in an annular domain. Based on a conditional stability for the Cauchy problem of Laplace's equation, we obtain a convergence estimate under the suitable choice of a regularization parameter and an a-priori bound assumption on the solution. A numerical example is provided to show the effectiveness of the proposed method from both accuracy and stability.     

A solution procedure for type E simple assembly line balancing problem

This paper presents a type E simple assembly line balancing problem (SALBP-E) that combines models SALBP-1 and SALBP-2. Furthermore, this study develops a solution procedure for the proposed model. The proposed model provides a better understanding of management practice that optimizes assembly line efficiency while simultaneously minimizing total idle time. Computational results indicated that, under the given upper bound of cycle time (ct_max), the proposed model can solve problems optimally with minimal variables, constraints, and computing time.     

A quasi-reversibility regularization method for the Cauchy problem of the Helmholtz equation

In this paper, a Cauchy problem for the Helmholtz equation is considered. It is known that such a problem is severely ill-posed, i.e. the solution does not depend continuously on the given Cauchy data. We propose a quasi-reversibility regularization method to solve it. Convergence estimates are established under two different a priori assumptions for an exact solution. Numerical results obtained by two different schemes show that our proposed methods work well.