The Monotone Iterative Technique for Three-Point Second-Order Integrodifferential Boundary Value Problems with p-Laplacian

A monotone iterative technique is applied to prove the existence of the extremal positive pseudosymmetric solutions for a three-point second-order p-Laplacian integrodifferential boundary value problem.The research of the second author was partially supported by Ministerio de Educacion´ y Ciencia and FEDER, Project MTM2004-06652-C03-01, and by Xunta de Galicia and FEDER, Project PGIDIT05PXIC20702PNS

Adaptive meshless centres and RBF stencils for Poisson equation

We consider adaptive meshless discretisation of the Dirichlet problem for Poisson equation based on numerical differentiation stencils obtained with the help of radial basis functions. New meshless stencil selection and adaptive refinement algorithms are proposed in 2D. Numerical experiments show that the accuracy of the solution is comparable with, and often better than that achieved by the mesh-based adaptive finite element method

Linear, second-order problems with Sturm-Liouville-type multi-point boundary conditions

We consider the linear eigenvalue problem \tag{1} -u" = \lambda u, \quad \text{on $(-1,1)$}, where $\lambda \in \mathbb{R}$, together with the general multi-point boundary conditions \tag{2} \alpha_0^\pm u(\pm 1) + \beta_0^\pm u'(\pm 1) = \sum^{m^\pm}_{i=1} \alpha^\pm_i u(\eta^\pm_i) + \sum_{i=1}^{m^\pm} \beta^\pm_i u'(\eta^\pm_i). We also suppose that: \alpha_0^\pm \ge 0, \quad \alpha_0^\pm + |\beta_0^\pm| > 0, \tag{3} \pm \beta_0^\pm \ge 0, \tag{4} (\frac{\sum_{i=1}^{m^\pm} |\alpha_i^\pm|}{\alpha_0^\pm})^2 + (\frac{\sum_{i=1}^{m^\pm} |\beta_i^\pm|}{\beta_0^\pm})^2 < 1, \tag{5} with the convention that if any denominator in (5) is zero then the corresponding numerator must also be zero, and the corresponding fraction is omitted from (5) (by (3), at least one denominator is nonzero in each condition). In this paper we show that the basic spectral properties of this problem are similar to those of the standard Sturm-Liouville problem with separated boundary conditions. Similar multi-point problems have been considered before under more restrictive hypotheses. For instance, the cases where $\beta_i^\pm = 0$, or $\alpha_i^\pm = 0$, $i = 0,..., m^\pm$ (such conditions have been termed Dirichlet-type or Neumann-type respectively), or the case of a single-point condition at one end point and a Dirichlet-type or Neumann-type multi-point condition at the other end. Different oscillation counting methods have been used in each of these cases, and the results here unify and extend all these previous results to the above general Sturm-Liouville-type boundary conditions

Multiple positive solutions for boundary value problems of second-order differential equations system on the half-line

In this paper, we study the existence of positive solutions for boundary value problems of second-order differential equations system with integral boundary condition on the half-line. By using a three functionals fixed point theorem in a cone and a fixed point theorem in a cone due to Avery-Peterson, we show the existence of at least two and three monotone increasing positive solutions with suitable growth conditions imposed on the nonlinear terms