Based on a Morse-Smale structure, we study planar global attractors Af of the scalar reaction-advection-diffusion equation ut = uxx + f(x, u, ux) in one space dimension. We assume Neumann boundary conditions on the unit interval, dissipativeness of f , and hyperbolicity of equilibria. We call Af Sturm attractor because our results strongly rely on nonlinear nodal properties of Sturm type. Plana...

In this paper we consider eigenvalue problems on time scales involving linear Hamiltonian dynamic systems. We give conditions that ensure that the eigenvalues of the problem are isolated and bounded below. The presented results are applicable also to Sturm-Liouville dynamic equations of higher order, and further special cases of our systems are linear Hamiltonian differential systems as well as...

The main purpose of this paper is to describe the extension of the successful modified integral series methods for Schrödinger problems to more general Sturm-Liouville eigenvalue problems. We present a robust and reliable modified Neumann method which can handle a wide variety of problems. This modified Neumann method is closely related to the second-order Pruess method, but provides for higher...

By using the Leggett-Williams fixed theorem, we establish the existence of multiple positive solutions for second-order nonhomogeneous Sturm-Liouville boundary value problems with linear functional boundary conditions. One explicit example with singularity is presented to demonstrate the application of our main results.