Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem

We study the existence of positive solutions of the nonlinear boundary value problem

     

Existence and uniqueness of positive solutions for a nonlinear fourth-order boundary value problem

This work presents sufficient conditions for the existence and uniqueness of a positive solution for a nonlinear fourth-order differential equation under Lidstone boundary conditions. Our analysis relies on a fixed point theorem in partially ordered sets.     

一类四阶边值问题正解的存在性

讨论了四阶常微分方程边值问题u^(4)=βu″-au=ψ(t)f(u),u(0)=u(1)=u″（0)=u″(1)=0的正解的存在性,利用锥拉伸与锥压缩不动点定理证明了,当f(u)在u=0及u=∞超线性或次线性增长时,该问题至少存在一个正解。     

一类四阶两点边值问题正解的存在性

运用上下解方法及拓扑度理论讨论了非齐次边界条件下四阶两点边值问题u″″(t)=f(u(t)),t∈(0,1),u(0)=u″(0)=u″(1)=0,u(1)=λ,其中λ＞0为参数,f∈C([0,+∞),[0,+∞)).在非线性项满足一定的增长条件下,获得了上述问题存在正解时λ的取值范围.     

On positive solutions for fourth-order boundary value problem with impulse

In this paper, we consider a fourth-order boundary value problem with impulse. First, we establish criteria for the existence of one or more than one positive solution of a non-eigenvalue problem. Second, we are concerned with determining values of λ$\lambda$, for which there exist positive solutions for an eigenvalue problem. In both problems, we shall use the Krasnoselskii fixed point theorem.     

Three positive solutions to a discrete focal boundary value problem

We are concerned with the discrete focal boundary value problem Δ³x(t − k) = f(x(t)), x(a) = Δx(t₂) = Δ²x(b + 1) = 0. Under various assumptions on f and the integers a, t₂, and b we prove the existence of three positive solutions of this boundary value problem. To prove our results we use fixed point theorems concerning cones in a Banach space.     

Existence of solutions for a fully nonlinear fourth-order two-point boundary value problem

In this paper, we investigate the existence of solutions of a fully nonlinear fourth-order differential equation $$x^{(4)}=f(t,x,x',x'',x'''),\quad t\in [0,1]$$ with one of the following sets of boundary value conditions; $$x'(0)=x(1)=a_{0}x''(0)-b_{0}x'''(0)=a_{1}x''(1)+b_{1}x'''(1)=0,$$ $$x(0)=x'(1)=a_{0}x''(0)-b_{0}x'''(0)=a_{1}x''(1)+b_{1}x'''(1)=0.$$ By using the Leray-Schauder degree theory, the existence of solutions for the above boundary value problems are obtained. Meanwhile, as an application of our results, an example is given.     

Nonexistence and existence of solutions for a fourth-order discrete mixed boundary value problem

In this paper, a fourth-order nonlinear difference equation is considered. By using the critical point theory, we establish various sets of sufficient conditions of the nonexistence and existence of solutions for mixed boundary value problem and give some new results. Results obtained generalize and complement the existing ones.     

Multiple symmetric positive solutions of fourth-order two point boundary value problem

In this paper, by using the Krasnoselskii fixed point theorem, we study the existence of 2m or 2m + 1 symmetric positive solutions of fourth-order two point boundary value problem $y^{(4)} (t) - f(t,y(t),y''(t)) = 0,y(0) = y(1) = y'(0) = y''(1) = 0$ .     

Three positive solutions of a nonlinear three-point boundary value problem

In this paper, existence criteria for three positive solutions of the nonlinear three-point boundary value problem

     

Positive solutions and eigenvalue intervals of a nonlinear singular fourth-order boundary value problem

We consider the classical nonlinear fourth-order two-point boundary value problem . In this problem, the nonlinear term h(t)f(t, u(t), u′(t), u″(t)) contains the first and second derivatives of the unknown function, and the function h(t)f(t, x, y, z) may be singular at t = 0, t = 1 and at x = 0, y = 0, z = 0. By introducing suitable height functions and applying the fixed point theorem on the cone, we establish several local existence theorems on positive solutions and obtain the corresponding eigenvalue intervals.     

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.     

On positive solutions for a nonlinear boundary value problem with impulse

In this paper we study nonlinear second order differential equations subject to separated linear boundary conditions and to linear impulse conditions. Sign properties of an associated Green’s function are investigated and existence results for positive solutions of the nonlinear boundary value problem with impulse are established. Upper and lower bounds for positive solutions are also given.     

Positive solutions for systems of a nonlinear fourth-order singular semipositone Sturm-Liouville boundary value problem

This paper concerns the existence of positive solutions for systems of a fourth-order singular semipositone Sturm-Liouville boundary value problem. By applying the fixed point index theorem, some sufficient conditions for positive solutions are established. An example is given to demonstrate the application of our main results.     

Nodal solutions for a fourth-order two-point boundary value problem

We consider boundary value problems of fourth-order differential equations of the form