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.     

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

We present some results for positive solutions of a system of nonlinear second-order difference equations, subject to multi-point boundary conditions.     

On some boundary value problems for a system of two second-order equations

For a two-point homogeneous boundary value problem for a system of two nonlinear second-order differential equations, we suggest sufficient solvability conditions (in particular, stated, like Bernstein conditions, in terms of the growth of the absolute values of the right-hand sides of the system with respect to the derivatives of the unknown functions). We obtain a priori estimates for solutions.     

On a multi-point discrete boundary value problem

We study the existence and non-existence of positive solutions for a system of nonlinear second-order difference equations subject to multi-point boundary conditions. The proof of the existence of positive solutions is based upon the Schauder fixed point theorem.     

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.     

Positive solutions to a system of second-order nonlocal boundary value problems

In this paper we study the existence of positive solutions to a system of second-order nonlocal boundary value problems by using fixed point index theory in a cone. Our hypotheses imposed on nonlinearities are those which characterize systems of nonlocal boundary value problems, and our boundary value conditions are expressed in terms of possibly nonlinear functions of Riemann–Stieltjes integrals, thus generalizing and unifying the boundary value conditions in the literature. Therefore our results cannot be routinely deduced from the ones for single nonlocal problem in the literature.     

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.     

Positive solution of second-order multi-point boundary value problems for finite difference equation with a p-Laplacian

In this paper, we consider discrete second-order multi-point boundary value problem with a p-Laplacian. By giving condition on f and applying Krasnosel’skii fixed point theorem, we ensure the existence of at least one positive solution and show the existence of eigenvalue intervals.     

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.     

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.     

Nonlinear second-order multivalued boundary value problems

In this paper we study nonlinear second-order differential inclusions involving the ordinary vectorp-Laplacian, a multivalued maximal monotone operator and nonlinear multivalued boundary conditions. Our framework is general and unifying and incorporates gradient systems, evolutionary variational inequalities and the classical boundary value problems, namely the Dirichlet, the Neumann and the periodic problems. Using notions and techniques from the nonlinear operator theory and from multivalued analysis, we obtain solutions for both the ‘convex’ and ‘nonconvex’ problems. Finally, we present the cases of special interest, which fit into our framework, illustrating the generality of our results.     

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

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

     

On the solvability of two-point,second-order boundary value problems

We gain solvability of a system of nonlinear, second-order ordinary differential equations subject to a range of boundary conditions. The ideas involve differential inequalities and fixed point methods. In particular, maximum principles are not employed.     

On solvability of second-order Sturm-Liouville boundary value problems at resonance

In this paper, based on of the concept , which is a generalized form of the first resonant point to the Picard problem , , we study the solvability of second-order Sturm-Liouville boundary value problems at resonance , , , and improve the previous results about problems derived by Chaitan P. Gupta, R.Iannacci and M. N. Nkashama, and Ma Ruyun, respectively.

     

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.     

Existence and multiplicity for positive solutions of a system of higher-order multi-point boundary value problems

We investigate the existence and multiplicity of positive solutions of multi-point boundary value problems for systems of nonlinear higher-order ordinary differential equations.     

Existence and multiplicity for positive solutions of a second-order multi-point discrete boundary value problem

We investigate the existence and multiplicity of positive solutions for a system of nonlinear second-order difference equations with multi-point boundary conditions.