Multiplicity results for second-order two-point boundary value problems with superlinear or sublinear nonlinearities

We consider boundary value problems for nonlinear second order differential equations of the form

     

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].     

Multiplicity results for second-order m-point boundary value problem

By using fixed-point theorems, some new results for multiplicity of positive solutions for a class of second-order m-point boundary value problem are obtained. The associated Green's function of the problem is also given.     

Multiplicity results for superlinear boundary value problems with impulsive effects

We study the existence and multiplicity of solutions for a nonlinear boundary value problem subject to perturbations of impulsive terms. Under suitable assumptions on the potential of the nonlinearity, the existence of one or two solutions is established. Our approach is based on a local minimum theorem and a two non‐zero critical points for differentiable functionals. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Multiplicity of eigenvalues in certain boundary value problems for the Helmholtz equation

Boundary value problems for two-layer cylindrical waveguides (namely, for a circular two-layer closed waveguide and a dielectric waveguide in an unbounded medium) are considered. It is shown that the same eigenvalues determining the wave numbers of waves in the waveguides can correspond to different solutions of the boundary value problem.     

Error estimates for shooting methods in two-point boundary value problems for second-order equations

This paper is concerned with a procedure for estimating the global discretization error arising when a boundary value problem for a system of second order differential equations is solved by the simple shooting method, without transforming the original problem in an equivalent first order problem. Expressions of the global discretization error are derived for both linear and nonlinear boundary value problems, which reduce the error estimation for a boundary value problem to that for an initial value problem of same dimension. The procedure extends to second order equations a technique for global error estimation given elsewhere for first order equations. As a practical result the accuracy of the estimates for a second order problem is increased compared with the estimates for the equivalent first order problem.     

Multiplicity results for Sturm–Liouville boundary value problems

Multiplicity results for Sturm–Liouville boundary value problems are obtained. Proofs are based on variational methods.     

Multiplicity results for semilinear resonant elliptic problems with discontinuous nonlinearities

In the present paper, some generalized critical points theorems for locally Lipschitz functions are given; as applications, some multiplicity results for zero-boundary-value elliptic resonant problems are obtained. Two classical important theorems are improved.     

Multiplicity results for a class of boundary value problems with impulsive effects

In this paper we study a class of boundary value problems with impulsive effects. Under some natural assumptions, by using Morse theory in combination with the minimax arguments, we establish some new results on the existence of multiple nontrivial solutions for the problems. Recent results from the literature are improved and extended.     

Convergence analysis for least-squares finite element approximations of second-order two-point boundary value problems

In this paper, a C⁰$C^0$ least-squares finite element method for second-order two-point boundary value problems is considered. The problem is recast as a first-order system. Standard and improved optimal error estimates in maximum-norms are established. Superconvergence estimates at interelement, Lobatto, and Gauss points are developed. Numerical experiments are given to illustrate theoretical results.