Positive solutions of a second-order integral boundary value problem

In this paper we study the existence of positive solutions of a second-order integral boundary value problems for ordinary differential equations. Our results presented here unify, generalize and substantially improve the existing results in the literature. Moreover, it is worthwhile to point out that our method will dispense with constructing a new Green function.     

Sign-changing solutions on a kind of fourth-order Neumann boundary value problem

In this paper, we consider the fourth-order Neumann boundary value problem u(4)(t)−2u^″(t)+u(t)=f(t,u(t)) for all t∈[0,1] and subject to u^′(0)=u^′(1)=u^?(0)=u^?(1)=0. Using the fixed point index and the critical group, we establish the existence theorem of solutions that guarantees the problem has at least one positive solution and two sign-changing solutions under certain conditions.     

Multiple solutions for fourth-order boundary value problem

In this paper, we study the existence and multiplicity of nontrivial solutions for the fourth-order two point boundary value problems. Making use of the theory of fixed point index in cone and Leray-Schauder degree, under general conditions on nonlinearity, we prove that there exist at least six different nontrivial solutions for the fourth-order two point boundary value problems. Furthermore, if the nonlinearity is odd, we obtain that there exist at least eight different nontrivial solutions.     

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,     

Existence of multiple solutions to second-order discrete Neumann boundary value problems

By using the invariant set of descending flow and variational method, we establish the existence of multiple solutions to a class of second-order discrete Neumann boundary value problems. The solutions include sign-changing solutions, positive solutions, and negative solutions. An example is given to illustrate our results.     

一类含有p-拉普拉斯算子的二阶边值问题的单调迭代正解（英文）

研究了一类含有p-拉普拉斯算子的微分方程积分边值问题.运用迭代技巧,给出了这一类边值问题的单调正解,值得感兴趣的是微分方程中的非线性项含有一阶导数.     

Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations

This paper is devoted to study the existence of multiple positive solutions for the second order Dirichlet boundary value problem with impulse effects. The main results here is the generalization of Liu and Li [L. Liu, F.Y. Li, Multiple positive solution of nonlinear two-point boundary value problems, J. Math. Anal. Appl. 203 (1996) 610-625] for ordinary differential equations. Existence is established via the theory of fixed point index in cones.     

Multiple positive solutions of periodic boundary value problems for second order impulsive differential equations

This paper is devoted to study the existence of multiple positive solutions for the second order periodic boundary value problem with impulse effects. The main results here are the generalization of Jiang [Daqing Jiang, On the existence of positive solutions to second order periodic BVPs, Acta Math. Sci. 18 (1998) 31–35] for ordinary differential equations. Existence is established via the theory of fixed point index in cones.     

Multiple positive solutions for a system of impulsive integral boundary value problems with sign-changing nonlinearities

In this paper we investigate a system of impulsive integral boundary value problems with sign-changing nonlinearities. Using the fixed point theorem in double cones, we prove the existence of multiple positive solutions.     

Positive solutions to a multi-point higher order boundary value problem

The authors consider a higher order multi-point boundary value problem. Some existence and nonexistence results for positive solutions of the problem are obtained by using Krasnosel'skii's fixed point theorem. Examples are included to illustrate the results.     

Positive solutions for nonlinear discrete second-order boundary value problems with parameter dependence

In this paper, we consider the nonlinear discrete boundary value problem

     

Triple positive periodic solutions of nonlinear singular second-order boundary value problems

This paper studies the positive solutions of the nonlinear second-order periodic boundary value problem u″(t) + λ(t)u(t) = f(t,u(t)),a.e.t ∈ [0,2π],u(0) = u(2π),u′(0) = u′(2π),where f(t,u) is a local Carath′eodory function.This shows that the problem is singular with respect to both the time variable t and space variable u.By applying the Leggett–Williams and Krasnosel'skii fixed point theorems on cones,an existence theorem of triple positive solutions is established.In order to use these theorems,the exact a priori estimations for the bound of solution are given,and some proper height functions are introduced by the estimations.     

Existence of solutions for anti-periodic boundary value problems

By using Schauder’s fixed point theorem and the lower and upper solutions method, we discuss the anti-periodic boundary value problem for a class of second-order differential equations. Some sufficient conditions for the existence of solutions are obtained.     

Existence and approximation of solutions for nonlinear second-order four-point boundary value problems

Applying the method of upper and lower solutions, Leray–Schauder degree theory and one-sided Nagumo condition, we obtain the existence and uniqueness results for a class of nonlinear second-order four-point boundary value problems. By the generalized approximation method, a monotone iteration sequence which converges uniformly to the unique solution of the nonlinear problem and converges quadratically to the unique solution of the linear problem is also obtained.     

Positive solutions for higher order multi-point boundary value problems with nonhomogeneous boundary conditions

We study nth order boundary value problems with a nonlinear term f(t,x) subject to nonhomogeneous multi-point boundary conditions. Criteria for the existence of positive solutions of such problems are established. Conditions are determined by the relationship between the behavior of f(t,x)/x near 0 and ∞ when compared with the smallest positive characteristic value of an associated linear integral operator. This work improves and extends some recent results in the literature for the second order problems. The results are illustrated with examples.     

A note on singular nonlinear boundary value problems

In this paper we study the existence and multiplicity of solutions of the following operator equation in Banach space E:

u=λAu,0<λ<+∞,u∈P?{θ},     

Positive solutions of m-point boundary value problems

The second-order m-point boundary value problem

     

Multiple positive solutions for second-order m-point boundary value problems

In this paper, the second-order m-point boundary value problem