Exact number of positive solutions for a class of semipositone problems

This paper is concerned with the boundary value problems y″+λ(y^p−y^q)=0 and y(−1)=y(1)=0, where p>q>−1 and λ>0 is a positive parameter. We discuss the existence of positive solutions and give a complete study.     

Uniqueness of positive solutions for quasilinear boundary value problems

Sufficient conditions for the uniqueness of positive solutions of boundary value problems for quasilinear differential equations of the type

(|u′|^m−2u′)′ + f(t,u,u′)=0, m 2

are established. These problems arise, for example, in the study of the m-Laplace equation in annular regions.     

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.     

Positive solutions of multi-point boundary value problems at resonance

We establish new results on the existence of positive solutions for some multi-point boundary value problems at resonance. Our results are based on a recent Leggett–Williams norm-type theorem due to O’Regan and Zima. We also derive a new result for a three-point problem, previously studied by several authors.     

一类Dirichlet边值问题的正解存在性

本文对Ｄｉｒｉｃｈｌｅｔ边值问题ｙ″－ｆ（ｔ，ｙ）＝０，ｙ（０）＝ｃ＞０，ｙ（１）＝ ０，给出正解的存在性结论，其中函数ｆ（ｔ，ｙ）可以是变号函数，并且可能在ｙ＝０处具有奇异性．     

Multiple positive solutions of nonlinear Neumann problems with time and space singularities

Allowing the nonlinear term to be singular with respect to both the time and space variables, we consider the positive solutions of a nonlinear Neumann boundary value problem. By constructing two height functions and estimating the integrations of these height functions, the existence and multiplicity of positive solutions are established.     

Existence and uniqueness of solutions for third-order nonlinear boundary value problems

In this paper, we present some general results of existence and uniqueness of solutions of nonlinear two-point boundary value problems for third-order nonlinear differential equations by using the Shooting method. As applications we give certain concrete sufficient conditions for the existence and uniqueness.     

二阶边值问题的正解

本文对ｆ没有非负性,同时也没有单调性的情形研究非线性边值问题 正解的存在性．其中ｑ是非负连续函数,并且在端点可以具有奇异性．     

On positive solutions of boundary value problems for second-order functional differential equations on infinite intervals

We study the existence of positive solutions for the following boundary value problem on infinite interval for second-order functional differential equations:

     

Existence of positive solutions of Lidstone boundary value problems

We show that there exists at least one positive solution for Lidstone boundary value problem

     

Positive solutions of second-order boundary value problems with sign-changing nonlinear terms

In this paper the existence results of positive solutions are obtained for second-order boundary value problem

−u″=f(t,u),t∈(0,1),u(0)=u(1)=0,     

Positive solutions for systems of a nonlinear fourth-order singular semipositone boundary value problems

In this paper, we study the existence of positive solutions of a two-point boundary value problem for a system of fourth-order nonlinear singular semipositone differential equations by the fixed point index theorem. Some new existence results are established, and an example is given to demonstrate the application of our main results.     

Uniqueness and parameter dependence of positive solutions to higher order boundary value problems with fractional $q$-derivatives

The authors study a class of nonlinear higher order boundary value problem with fractional $q$-erivativesand dependence on a positive parameter $\lm$.The existence, uniqueness, and dependence of positive solutions on $\lm$ are discussed.Two sequences are constructed so that they converge uniformly to the unique solution of the problems.Two examples are included in the paper. Numerical computations of the examples confirm their theoretical results.     

Positive solutions of superlinear semipositone singular Dirichlet boundary value problems

In this paper, we study a class of superlinear semipositone singular second order Dirichlet boundary value problem. A sufficient condition for the existence of positive solution is obtained under the more simple assumptions.     

Multiple positive solutions for discrete nonlocal boundary value problems

In this paper, we investigate a second-order nonlinear difference equation with sign-changing nonlinearity subject to two different sets of nonlocal boundary conditions. The explicit expressions of the associated Green's functions are presented. By using a recently developed fixed point theorem, we establish sufficient conditions for the existence of multiple positive solutions of the boundary value problem.     

一类两点边值问题的正解个数

本文讨论边值问题y'+λ(yp+μp+yp)=0,y(-1)=y(1)=0,其中λ>0是正参数,μ≥0.对(1-p)(1-q)>0的情形得出了正解的存在唯一性.对(1-p)(1-q)<0的情形,其主要结论是:若p>1>q>-(25+23p)/(23+25p),μ≥0,则存在λ*>0,使得当0<λ<λ*时,此边值问题恰好存在两个正解,当λ=λ*时,存在唯一正解,当λ>λ*时,不存在正解.