Solvability of a singular boundary value problem in a domain of the Lobachevskii plane

Under study is some singular elliptic boundary value problem in a domain of the Lobachevskii plane with an isolated boundary point. We introduce the new function spaces that coincide with the spaces of Sobolev-Nikol’skii-Besov type outside the singular point, and the concept of σ-trace at the singular point. The main result is a proof that the singular boundary value problem is uniquely solvable.     

Resonance and asymptotic solutions of a singular perturbation problem

A singular perturbation problem for a second-order ODE witha pair of singular boundary points and a pair of interior second-orderturning points is studied. Four leading-order uniform approximationsare formally constructed, each is restricted to a region includingone critical point. The neighbouring approximations are formallymatched independently on an overlap domain, yielding an asymptoticapproximation to leading order of the general solution. Twogeneralized formulae for the singular and turning point eigenvaluesthat exhibit the resonance conditions are derived. The resonancecriteria due to the influence of every possible combinationof the critical points are investigated.     

一类四阶奇异边值问题对称正解的最优存在性

本文研究了一类四阶奇异边值问题.通过建立一个特定的锥,利用Leggett-Williams不动点定理,从而在一定的条件下得到一类四阶奇异边值问题对称正解的最优存在性,推广了奇异边值问题对称正解的最优存在性的结果.     

On the Number of Unbounded Solution Branches in a Neighborhood of an Asymptotic Bifurcation Point

We suggest a method for studying asymptotically linear vector fields with a parameter. The method permits one to prove theorems on asymptotic bifurcation points (bifurcation points at infinity) for the case of double degeneration of the principal linear part. We single out a class of fields that have more than two unbounded branches of singular points in a neighborhood of a bifurcation point. Some applications of the general theorems to bifurcations of periodic solutions and subharmonics as well as to the two-point boundary value problem are given.     

奇异二阶三点边值问题的正解

应用锥中的不动点定理研究奇异二阶三点边值问题的正解的存在性。采用一种构造Green函数的方法为出发点，利用分段定义算子的手法讨论更一般的奇异二阶三点边值问题。得到了一个正解的存在性定理。其中的非线性项可以是变号的。     

奇异非线性边值问题的经典Agarwal-O'Regan方法

改进了奇异非线性边值问题的经典Agarwal-O'Regan方法.利用这个改进的方法建立了奇异非线性(p,n-p)共轭边值问题正解的局部存在性与多解性,其中允许非线性项关于时间和空间变元同时奇异.主要工具是锥拉伸与锥压缩型的Guo-Krasnosel'skii不动点定理和精确先验估计技巧.特别的,考察了非自治奇异非线性二阶、三阶、四阶共轭边值问题.     

Singularities of Bernoulli Free Boundaries

In this article we show that, for a large class of Bernoulli problems, a free boundary which is symmetric with respect to a vertical line through an isolated singular point must necessarily have a corner at that point, and we give a formula for the contained angle. The assumptions used admit the possibility of other singular points, even uncountably many, on the free boundary. This result is an extension of the first Stokes conjecture in the theory of hydrodynamic waves. We also show that, even in the presence of singularities, a geometrically simple Bernoulli free boundary is necessarily symmetric.     

B-spline solution of non-linear singular boundary value problems arising in physiology

We use B-spline functions to develop a numerical method for computing approximations to the solution of non-linear singular boundary value problems associated with physiology science. The original differential equation is modified at singular point then the boundary value problem is treated by using B-spline approximation. The numerical method is tested for its efficiency by considering three model problems from physiology.     

一类具$p$-Laplace非线性三点边值问题三个正解的存在性

This paper deals with the existence of three positive solutions for a class of nonlinear singular three-point boundary value problem with p-Laplacian. By means of a fixed point theorem duo to Leggett and Williams, sufficient condition for the existence of at least three positive solutions to the nonlinear singular three-point boundary value problem is established     

Critical points and positive solutions of singular elliptic boundary value problems

Usually we do not think there is variational structure for singular elliptic boundary value problems, so it cannot be considered by using critical points theory. In this paper, we use critical theory on certain convex closed sets to solve positive solutions for singular elliptic boundary value problems, especially use the ordinary differential equation theory of Banach spaces to obtain new results on the existence of multiple positive solutions. The method is useful for other singular problems.     

带可变参数四阶奇异边值问题正解的存在性

本文研究了带有一个可变参数的四阶奇异边值问题,利用Banach空间的不动点理论,在非常弱的条件下,获得了带可变参数四阶奇异边值问题存在正解的结果.     

Projection methods and singular two point boundary value problems

Summary The use of collocation methods and singular splines applied to singular two point boundary value problems is studied. Existence, uniqueness and convergence rates are obtained.     

Cubic spline for a class of singular two-point boundary value problems

In this paper we have presented a method based on cubic splines for solving a class of singular two-point boundary value problems. The original differential equation is modified at the singular point then the boundary value problem is treated by using cubic spline approximation. The tridiagonal system resulting from the spline approximation is efficiently solved by Thomas algorithm. Some model problems are solved, and the numerical results are compared with exact solution.     

EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTION FOR NONLINEAR SINGULAR 2mth-ORDER CONTINUOUS AND DISCRETE LIDSTONE BOUNDARY VALUE PROBLEMS

By fixed point theorem of a mixed monotone operators, we study Lidstone boundary value problems to nonlinear singular 2mth-order differential and difference equations, and provide sufficient conditions for the existence and uniqueness of positive solution to Lidstone boundary value problem for 2mth-order ordinary differential equations and 2mth-order difference equations. The nonlinear term in the differential and difference equation may be singular.     

Local existence of multiple positive solutions to a singular cantilever beam equation

By constructing suitable cone and control functions, we prove some local existence theorems of positive solutions for a singular fourth-order two-point boundary value problem. In mechanics, the problem is called cantilever beam equation. Furthermore, we improve a famous method appeared in the studies of singular boundary value problems. The approximation theorem of completely continuous operators and the Guo-Krasnosel'skii fixed point theorem of cone expansion-compression type play important parts in this work.     

Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces

The existence of multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces is discussed in this paper. Our nonlinearity may be singular in its dependent variable and our analysis relies on a nonlinear alternative of Leray-Schauder type and on a fixed point theorem in cones.     

Fixed-sign solutions for a system of singular focal boundary value problems

We consider a system of focal boundary value problems where the nonlinearities may be singular in the independent variable and may also be singular in the dependent arguments. Using Schauder fixed point theorem, we establish criteria such that the system of boundary value problems has at least one fixed-sign solution.     

负指数的脉冲EMDEN-FOWLER方程奇异边值问题的正解

利用不动点定理得到了脉冲奇异混合边值问题的上下解方法,并且利用此方法得到了负指数的脉冲Emden-Fowler方程奇异混合边值问题正解存在的充分必要条件.     

Boundary value problems of electroelasticity with concentrated singularities

We investigate the solutions of boundary value problems of linear electroelasticity, having growth as a power function in the neighborhood of infinity or in the neighborhood of an isolated singular point. The number of linearly independent solutions of this type is established for homogeneous boundary value problems.     

Banach空间非线性二阶脉冲奇异积分-微分方程积分边值问题的正解

The existence of positive solutions to a boundary value problem of second-order impulsive singular integro-differential equation with integral boundary conditions in a Banach space is obtained by means of fixed point theory.Moreover,an application is also given to illustrate the main result.