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.     

Nontrivial solutions of boundary value problems of second-order difference equations

We employ the critical point theory to establish the existence of nontrivial solutions for some boundary value problems of second-order difference equations.     

Multiplicity of solutions for nonlinear second order impulsive differential equations with linear derivative dependence via variational methods

This paper uses critical point theory and variational methods to investigate the multiple solutions of boundary value problems for second order impulsive differential equations. The conditions for the existence of multiple solutions are established. An example is constructed to illustrate the proposed result.     

Nontrivial solutions for discrete boundary value problems with multiple resonance via computations of the critical groups

In this paper, we study the existence of nontrivial solutions for a class of second-order difference equations with multiple resonance at both infinity and the origin by applying the critical point theory and Morse 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

     

Nontrivial solutions of semilinear equations at resonance

In this paper we prove new existence results concerning nontrivial solutions to semilinear elliptic problem at resonance. The methods used here are based on combining the minimax methods and the Morse theory.     

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.     

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

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

     

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.

     

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.     

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.     

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.     

Fast homoclinic solutions for some second order non-autonomous systems

This paper is concerned with the existence of homoclinic solutions for the following second order non-autonomous system

(FHS)     

Multiple positive solutions for superlinear second order singular boundary value problems with derivative dependence

The existence of at least two positive solutions is presented for the singular second-order boundary value problem
{1/p（t）（ p（t）x′（t））′＋Φ（t）f（t,x（t）,p（t）x′（t））=0,0〈t〈1,
limt→0 p（t）x′（t）=0,x（1）=0
by using the fixed point index, where f may be singular at x = 0 and px ′= 0.     

一类四阶边值问题的正解

§ 1　IntroductionThe deformations of an elastic beam are described by a fourth-order two-pointbound-ary value problem[1 ] .The boundary conditions are given according to the controls at theends of the beam. For example,the nonlinear fourth order problemu(4) (x) =λa(x) f(u(x) ) ,u(0 ) =u′(0 ) =u′(1 ) =u (1 ) =0 (1 .1 ) λdescribes the deformations of an elastic beam whose one end fixed and the other slidingclamped.The existence of solutions of (1 .1 ) λhas been studied by Gupta[1 ] . But …     

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 positive solutions for second order nonlinear dynamic BVPs by variational methods

The aim of this paper is to employ variational techniques and critical point theory to prove some sufficient conditions for the existence of multiple positive solutions to a nonlinear second order dynamic equation with homogeneous Dirichlet boundary conditions.