Existence of solutions for some discrete boundary value problems with a parameter

In this paper, the existence and multiplicity results of solutions are obtained for the discrete nonlinear two point boundary value problem (BVP) ; u(0)=0=Δu(T), where T is a positive integer, Z(1,T)={1,2,…,T}, Δ is the forward difference operator defined by Δu(k)=u(k+1)-u(k) and f:Z(1,T)×R→R is continuous, λ∈R⁺ is a parameter. By using the critical point theory and Morse theory, we obtain that the above (BVP) has solutions for λ being in some different intervals.     

Multiple solutions of some boundary value problems with parameters

In this paper, we study the existence and multiplicity of nontrivial solutions for the following second-order Dirichlet nonlinear boundary value problem with odd order derivative: −u^″(t)+au^′(t)+bu(t)=f(t,u(t)) for all t∈[0,1] with u(0)=u(1)=0, where a,b∈R¹, f∈C¹([0,1]×R¹,R¹). By using the Morse theory, we impose certain conditions on f which are able to guarantee that the problem has at least one nontrivial solution, two nontrivial solutions and infinitely many solutions, separately.     

Existence of infinitely many mountain pass solutions for some fourth-order boundary value problems with a parameter

In this paper, for the fourth-order boundary value problem (BVP) ,0<t<1,u(0)=u(1)=u^″(0)=u^″(1)=0, where f:[0,1]×R→R is continuous, η≤0 is a parameter, the existence of infinitely many mountain pass solutions are obtained with the variational methods and critical point theory. We prove the conclusion by combining sub-sup solution method, Mountain pass theorem in order intervals, Leray-Schauder degree theory and Morse theory.     

Existence of nontrivial solutions for discrete elliptic boundary value problems

In this article, the existence of nontrivial solutions for the discrete elliptic boundary value problems is considered by using the extremum principle. Such system admits at least 2ⁿ nontrivial solutions when the nonlinear term is superlinear or sublinear. An explanation example is also given. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Multiple solutions of some fourth-order boundary value problems

For some fourth-order boundary value problems, several new existence theorems on multiple positive, negative and sign-changing solutions are obtained. The critical point theory and the supersolution and subsolution method are employed to discuss this problem.     

Multiple solutions for boundary value problems of second-order difference equations with resonance

In this paper, we study the existence of multiple solutions for boundary value problems of second-order difference equations with resonance at both infinity and zero by using Morse theory, critical point theory, minimax methods and bifurcation theory.     

Existence of solutions for some higher order boundary value problems

In this paper, we are concerned with the existence of solutions for the higher order boundary value problem in the form

     

一类四阶差分边值问题解的存在性与临界点方法

利用临界点方法研究了一类四阶差分边值问题,分别得到了存在一个解和两个解的若干充分条件,展示了临界点理论在研究差分边值问题中的重要作用.     

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.     

Infinitely many mountain pass solutions on a kind of fourth-order Neumann boundary value problem

In this paper, the existence of infinitely many mountain pass solutions are obtained for the fourth-order boundary value problem (BVP) u⁽⁴⁾(t)-2u^″(t)+u(t)=f(u(t)),0<t<1, u^′(0)=u^′(1)=u^?(0)=u^?(1)=0, where f:R→R is continuous. The study of the problem is based on the variational methods and critical point theory. We prove the conclusion by using sub-sup solution method, Mountain Pass Theorem in Order Intervals, Leray-Schauder degree theory and Morse theory.     

Existence and multiple solutions for a nonlinear system with a parameter

In this paper, the critical point theory and the Morse theory are employed to discuss the system of nonlinear equations of the form Au=λf(u)$Au=\lambda f\left(u\right)$, where the matrix A$A$ is not necessarily positive definite. Some existence and multiple solutions are obtained.     

Existence of solutions for three-point boundary value problems for second order equations

Shooting methods are employed to obtain solutions of the three-point boundary value problem for the second order equation, where is continuous, and and conditions are imposed implying that solutions of such problems are unique, when they exist.

     

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.     

Positive solutions for some second-order four-point boundary value problems

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

     

Existence of three positive solutions for some second-order m-point boundary value problems

By using fixed-point theorems, some new results for multiplicity of positive solutions for some second order m-point boundary value problems are obtained.The associated Green＇s function of these problems are also given.     

Positive solutions of some nonlocal fourth-order boundary value problem

By the use of the Krasnosel’skii’s fixed point theorem, the existence of one or two positive solutions for the nonlocal fourth-order boundary value problem

     

Existence of positive solutions for three-point boundary value problems at resonance

With the help of the theorem of a fixed point index for A-proper semilinear operators established by Cremins, we get a existence theorem concerning the existence of positive solution for the second order ordinary differential equation of three-point boundary value problems at resonance.