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

We consider a higher order two-point boundary value problem, and establish some upper and lower type estimates for positive solutions of the problem. Sufficient conditions for the existence and nonexistence of positive solutions for the problem are obtained. An example is included to illustrate the results.     

Solvability of a third-order two-point boundary value problem

The upper and lower solutions method and a new maximum principle are employed to establish several existence results for a third-order two-point boundary value problem.     

Multiple solutions for a two-point boundary value problem

In this paper, using a recent result of Ricceri, we prove two multiplicity theorems for the problem −u^″=λf(u)+μg(x,u), u(0)=u(1)=0, extending a previous result that G. Bonanno obtained for μ=0.     

Fundamental solution for linear two-point boundary value problem

The quadratic functional minimization with differential restrictions represented by the command linear systems is considered. The optimal solution determination implies the solving of a linear problem with two points boundary values. The proposed method implies the construction of a fundamental solution S(t)—a n×n matrix—and of a vector h(t) defining an adjoint variable λ(t) depending of the state variable x(t). From the extremum necessary conditions it is obtained the Riccati matrix differential equation having the S(t) as unknown fundamental solution is obtained. The paper analyzes the existence of the Riccati equation solution S(t) and establishes as the optimal solution of the proposed optimum problem. Also a superior limit of the minimum for the considered quadratic functionals class are evaluated.     

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

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

     

Orderh 2 method for a singular two-point boundary value problem

A method is described based on auniform mesh for the singular two-point boundary value problem:y+(/x)y+f(x, y)=0, 0<x1,y(0)=0,y(1)=A, and it is shown to be orderh ² convergent forall 1.     

Multiplicity results for a two-point boundary value double eigenvalue problem

The purpose of this note is to prove the existence of three solutions for a two-point boundary value double eigenvalue problem. The approach is fully based on a recent three critical points theorem of B. Ricceri [A three critical points theorem revisited, Nonlinear Anal., 70/9 (2009) 3084–3089].     

Solving singular two-point boundary value problem in reproducing kernel space

In this paper, we present a new method for solving singular two-point boundary value problem for certain ordinary differential equation having singular coefficients. Its exact solution is represented in the form of series in reproducing kernel space. In the mean time, the n  -term approximation _un(x)$u_n(x)$ to the exact solution u(x)$u(x)$ is obtained and is proved to converge to the exact solution. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other.     

Positive solutions of nonlinear singular third-order two-point boundary value problem

In this paper, we are concerned with the existence of single and multiple positive solutions to the nonlinear singular third-order two-point boundary value problem

     

Stability of Ritz procedure for nonlinear two-point boundary value problem

In this paper we establish the stability of Ritz procedure in sense of Michlin and Tucker for a class of nonlinear two-point boundary value problems which has been considered by Varga, Schultz and Ciarlet in [1].     

A priori estimates and solvability of the third two-point boundary value problem

We study a priori estimates and solvability of a nonlinear two-point boundary value problem for systems of second-order ordinary differential equations with leading positively homogeneous nonlinearity of order > 1 vanishing on a single surface. Assuming that an a priori estimate holds, we prove the invariance of the solvability of the problem under a continuous change of the leading nonlinear homogeneous terms and under arbitrary perturbations that do not affect the behavior of the leading nonlinear homogeneous terms at infinity.     

Quintic nonpolynomial spline solutions for fourth order two-point boundary value problem

In this paper, we develop quintic nonpolynomial spline methods for the numerical solution of fourth order two-point boundary value problems. Using this spline function a few consistency relations are derived for computing approximations to the solution of the problem. The present approach gives better approximations and generalizes all the existing polynomial spline methods up to order four. This approach has less computational cost. Convergence analysis of these methods is discussed. Two numerical examples are included to illustrate the practical usefulness of our methods.     

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.     

Existence of three solutions for a perturbed two-point boundary value problem

The aim of this note is to establish the existence of three solutions for a perturbed two-point boundary value problem depending of two real parameters. The approach is based on variational methods.