离散广义Emden-Fowler方程边值问题的多解存在性

应用临界点理论,获得了一类离散广义Emden-Fowler方程边值问题存在多个解的条件.     

一类分数阶微分方程边值问题的三个正解

利用锥上Avery-Peterson不动点定理,研究了一类分数阶微分方程积分边值问题正解的存在性,给出了该边值问题至少存在三个正解的充分条件.     

Positive solutions of discrete third-order three-point right focal boundary value problems

We consider linear and nonlinear boundary value problems for third-order difference equations with three-point right focal boundary conditions. In the linear case, the existence of positive solutions corresponding to the first eigenvalue of the problem is established and an interval estimate for the first eigenvalue is obtained. In the nonlinear case, sufficient conditions for the existence and nonexistence of positive solutions are obtained. An example is included at the end of the paper to illustrate the sharpness of our results.     

On the existence of solutions for discrete elliptic boundary value problems

Using critical point theory and some monotonicity results we consider the existence of solutions to a boundary value problem connected with the discrete elliptic equation with a positive parameter.     

On boundary value problems for a discrete generalized Emden-Fowler equation

In this paper, some sufficient conditions for the existence of solutions to the boundary value problems of a class of second order difference equation are obtained by using the critical point theory.     

一类二阶离散Neumann边值问题正解的存在性

研究一类二阶离散Neumann边值问题正解的存在性,运用不动点指数理论获得了方程存在正解的最优条件,并给出一个具体例子说明这一结果。     

The generalized Thomas–Fermi singular boundary value problems for neutral atoms

This paper presents an upper and lower solution theory for singular boundary value problems modelling the Thomas–Fermi equation, subject to a boundary condition corresponding to the neutral atom with Bohr radius equal to its existence interval. Furthermore, we derive sufficient conditions for the existence–construction of the above‐mentioned upper–lower solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

奇异P-Lapace差分方程边值问题正解的存在唯一性

利用混合单调算子不动点定理研究了一维非线性奇异P-Lapace差分方程边值问题,得到P-Lapace差分方程边值问题的存在唯一正解的充要条件.     

An initial boundary value problem for a pseudoparabolic equation with a nonlinear boundary condition

An initial boundary value problem for a quasilinear equation of pseudoparabolic type with a nonlinear boundary condition of the Neumann–Dirichlet type is investigated in this work. From a physical point of view, the initial boundary value problem considered here is a mathematical model of quasistationary processes in semiconductors and magnets, which takes into account a wide variety of physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions in problems where the boundary conditions are linear with respect to the desired function and its derivatives. Among these methods, the Galerkin method leads to the simplest calculations. On the basis of a priori estimates, we prove a local existence theorem and uniqueness for a weak generalized solution of the initial boundary value problem for the quasilinear pseudoparabolic equation. A special place in the theory of nonlinear equations is occupied by the study of unbounded solutions, or, as they are called in another way, blow-up regimes. Nonlinear evolutionary problems admitting unbounded solutions are globally unsolvable. In the article, sufficient conditions for the blow-up of a solution in a finite time in a limited area with a nonlinear Neumann–Dirichlet boundary condition are obtained.     

Necessary conditions for the existence of positive solutions to fractional boundary value problems at resonance

In this article, we obtain some properties of the positive solutions to a class of fractional differential equation multipoint boundary value problems at resonance. Necessary conditions for the existence of positive solutions are established by means of the spectral theory of linear operator. As application, necessary and sufficient conditions for the existence of positive solutions are given for the case that the nonlinearity has a fixed sign. Some examples are presented to illustrate our results.     

Existence of solutions for discrete p-Laplacian with potential boundary conditions

We are concerned with the existence of solutions for some discrete p-Laplacian equations subjected to a potential type boundary condition. Our approach is a variational one and relies on Szulkin's critical point theory. We obtain the existence of solutions in a coercive case as well as the existence of non-trivial solutions when the corresponding energy functional has a ‘mountain pass’ geometry.     

一个非局部方程解的存在性

本文旨在讨论一个线性非局部方程和相应非线性方程解存在的条件，对非局部积分核作一定限制后，得到在积分核变号条件下解的存在唯一性结果。     

Optimal control problems for the reaction–diffusion–convection equation with variable coefficients

The solvability of optimal control problems is proved on both weak and strong solutions of a boundary value problem for the nonlinear reaction–diffusion–convection equation with variable coefficients. In the second case, the requirements for smoothness of the multiplicative control are reduced. The study of extremal problems is based on the proof of the solvability of the corresponding boundary value problems and on the qualitative analysis of their solutions properties. The large data existence results for weak solutions, the maximum principle as well as the local existence and uniqueness of a strong solution are established. Moreover, an optimal feedback control problem is considered. Using methods of the theory of topological degree for set-valued perturbations (with aspheric image sets) of generalized monotone operators, we obtain sufficient conditions for the solvability of this problem in the class of weak solutions.     

On Positive Solutions for Semilinear Elliptic Equations with an Indefinite Nonlinearity Via Bifurcation

Existence of a positive solution was establsihed via bifurcation theory for sernilinear elliptic boundary value problems. With the aid of maximum principles and a priori estimates, global behavior of the bifurcation curve was obtained. In particular, a general sufficient condition was given for existence of multiple positive solutions when the nonlinearity is sub-critical and indefinite.     

Eigenvalue problems for anisotropic discrete boundary value problems

In this paper, we prove the existence of a continuous spectrum for a family of discrete boundary value problems. The main existence results are obtained by using critical point theory. The equations studied in the paper represent a discrete variant of some recent anisotropic variable exponent problems, which deserve as models in different fields of mathematical physics.     

一类奇异次线性边值问题正解存在的充分必要条件

本文研究一类奇异次线性边值问题正解的存在性，得到Ｃ［０，１］正解和Ｃ１［０，１］正解存在的充分必要条件，也得到正解的唯一性．     

Nontrivial solutions for a nonlinear discrete elliptic equation with periodic boundary conditions

In this paper, existence of nontrivial solutions for a nonlinear discrete elliptic equation with periodic boundary conditions is considered by using the monotone operator principle, Mountain pass lemma, Linking theorem and some results in Morse theory. The obtained results are also valid and new for the corresponding difference periodic boundary value problem.     

一类含一阶导数项两点边值问题单调正解的存在性

研究一类含有一阶导数项的二阶微分方程在非齐次边界条件下的两点边值问题单调解的存在性.利用锥拉伸与锥压缩不动点定理,给出了边值问题单调递增正解和单调递减负解存在的充分条件,并且对解的凸性进行了分析.     

A parametrix for the linear elastic equation with diffractive boundary and its application

In this paper, we construct parametrices near diffractive points for the boundary value problems for the linear elastic equation with free boundary condition or Dirichlet boundary condition. Naturally, our construction is similar to that for the wave equation case. However, since the linear elastic equation is a second order system, our method is more complicated. As an application to the existence of the parametrices, we prove the theorem on propagation of singularities for solutions of the boundary value problem.     

A necessary and sufficient condition for the existence of positive solutions of fourth-order singular boundary value problems

This paper studies positive solutions to a class of superlinear (sublinear) fourth-order singular boundary value problems by means of operator approximation theory and fixed point theorems. A necessary and sufficient condition for the existence of C³[0, 1] positive solutions is obtained, which extends and includes some known results.