Nodal solutions of multi-point boundary value problems

We study the nonlinear boundary value problem consisting of the equation y^″+w(t)f(y)=0 on [a,b] and a multi-point boundary condition. By relating it to the eigenvalues of a linear Sturm-Liouville problem with a two-point separated boundary condition, we obtain results on the existence and nonexistence of nodal solutions of this problem. We also discuss the changes in the existence question for different types of nodal solutions as the problem changes.     

A note on multi-point boundary value problems

Multi-point boundary value problems for a second-order ordinary differential equation are considered in this note. An existence result is obtained with the help of coincidence degree theory.     

Asymptotic solutions of singularly perturbed second-order differential equations and application to multi-point boundary value problems

In this work, a singularly perturbed second-order ordinary differential equation is solved by applying a new Liouville–Green transform and the asymptotic solutions are obtained. As an application, we employ our results in discussing a second-order multi-point boundary value problem.     

Positive solutions of nonlinear multi-point boundary value problems

This paper deals with the existence of positive solutions of nonlinear differential equation

$$\begin{aligned} u^{\prime \prime }(t)+ a(t) f(u(t) )=0,\quad 0<t <1, \end{aligned}$$

subject to the boundary conditions

$$\begin{aligned} u(0)=\sum _{i=1}^{m-2} a_i u (\xi _i) ,\quad u^{\prime } (1) = \sum _{i=1}^{m-2} b_i u^{\prime } (\xi _i), \end{aligned}$$

where \( \xi _i \in (0,1) \) with \( 0< \xi _1<\xi _2< \cdots<\xi _{m-2} < 1,\) and \(a_i,b_i \) satisfy   \(a_i,b_i\in [0,\infty ),~~ 0< \sum _{i=1}^{m-2} a_i <1,\) and \( \sum _{i=1}^{m-2} b_i <1. \) By using Schauder’s fixed point theorem, we show that it has at least one positive solution if f is nonnegative and continuous. Positive solutions of the above boundary value problem satisfy the Harnack inequality

$$\begin{aligned} \displaystyle \inf _{0 \le t \le 1} u(t) \ge \gamma \Vert u\Vert _\infty . \end{aligned}$$

     

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.     

On a system of second-order multi-point boundary value problems

We study the existence of positive solutions for a system of nonlinear second-order ordinary differential equations subject to some multi-point boundary conditions. The nonexistence of positive solutions is also investigated.     

A numerical algorithm for nonlinear multi-point boundary value problems

In this paper, an algorithm is presented for solving second-order nonlinear multi-point boundary value problems (BVPs). The method is based on an iterative technique and the reproducing kernel method (RKM). Two numerical examples are provided to show the reliability and efficiency of the present method.     

Positive solutions of multi-point boundary value problems at resonance

We establish new results on the existence of positive solutions for some multi-point boundary value problems at resonance. Our results are based on a recent Leggett–Williams norm-type theorem due to O’Regan and Zima. We also derive a new result for a three-point problem, previously studied by several authors.     

Twin positive solutions for quasi-linear multi-point boundary value problems

In this paper, we investigate two classes of quasi-linear multi-point boundary value problems with sign-changing nonlinearity. By applications of fixed point index theory, sufficient conditions for the existence of twin positive solutions are established.     

The existence of approximate solutions to mixed boundary value problems

Approximate solutions, similar to the type used in the Complex Variable Boundary Element Method, are shown to exist for two dimensional mixed boundary value potential problems on multiply connected domains. These approximate solutions can be used numerically to obtain least squares solutions or solutions which interpolate given boundary conditions. Areas of application include fluid flow around obstacles and heat flow in a domain with insulated boundary segments. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 191–199, 1999     

On existence and multiplicity of positive solutions to singular multi-point boundary value problems

The existence and multiplicity of positive solutions are established for the multi-point boundary value problem

     

Positive solutions of multi-point boundary value problems with multivalued operators at resonance

We establish new results on the existence of positive solutions for a kind multi-point boundary value problem with multivalued operator. Our results are based on a recent Leggett-Williams theorem for coincidences of multivalued operators due to O’Regan and Zima. The most interesting point is the acquisition of positive solutions for the resonance case. And an example is constructed to show that our result here is valid.     

Positive solutions for third order multi-point singular boundary value problems

We study a third order singular boundary value problem with multi-point boundary conditions. Sufficient conditions are obtained for the existence of positive solutions of the problem. Recent results in the literature are significantly extended and improved. Our analysis is mainly based on a nonlinear alternative of Leray-Schauder.     

An adaptive algorithm for solving stochastic multi-point boundary value problems

In this paper, we present an adaptive multiple-shooting method to solve stochastic multi-point boundary value problems. We first analyze the strong order of convergence of the underlying multiple shooting method. We then proceed to describe the proposed strategy to adaptively choose the location of shooting points. We analyze the effect of method parameters on the performance of the overall scheme using a benchmark linear two-point stochastic boundary value problem. We illustrate the effectiveness of this approach on several (one and two dimensional) test problems by comparing our results with other non-adaptive alternative techniques proposed in the literature.     

A compactness result for differentiable functions with an application to boundary value problems

We characterize the relative compactness of subsets of the space ${\mathcal{BC}^m([0,+\infty [;E)}$ of bounded and m-differentiable functions defined on [0, +∞[ with values in a Banach space E. Moreover, we apply this characterization to prove the existence of solutions of a boundary value problem in Banach spaces.     

Positive solutions for a system of second-order multi-point boundary value problems

In this paper, we present some results for positive solutions of a system of nonlinear second-order ordinary differential equations subject to multi-point boundary conditions.     

On shifted Jacobi spectral method for high-order multi-point boundary value problems

This paper reports a spectral tau method for numerically solving multi-point boundary value problems (BVPs) of linear high-order ordinary differential equations. The construction of the shifted Jacobi tau approximation is based on conventional differentiation. This use of differentiation allows the imposition of the governing equation at the whole set of grid points and the straight forward implementation of multiple boundary conditions. Extension of the tau method for high-order multi-point BVPs with variable coefficients is treated using the shifted Jacobi Gauss–Lobatto quadrature. Shifted Jacobi collocation method is developed for solving nonlinear high-order multi-point BVPs. The performance of the proposed methods is investigated by considering several examples. Accurate results and high convergence rates are achieved.     

The use of Sinc-collocation method for solving multi-point boundary value problems

Multi-point boundary value problems have received considerable interest in the mathematical applications in different areas of science and engineering. In this work, our goal is to obtain numerically the approximate solution of these problems by using the Sinc-collocation method. Some properties of the Sinc-collocation method required for our subsequent development are given and are utilized to reduce the computation of solution of multi-point boundary value problems to some algebraic equations. It is well known that the Sinc procedure converges to the solution at an exponential rate. Numerical examples are included to demonstrate the validity and applicability of the new technique.