Eigenvalues of discrete linear second-order periodic and antiperiodic eigenvalue problems with sign-changing weight

Eigenvalue problems with sign-changing coefficients

We consider a class of eigenvalue problems involving coefficients changing sign on the domain of interest. We describe the main spectral properties of these problems according to the features of the coefficients. Then, under some assumptions on the mesh, we explain how one can use classical finite element methods to approximate the spectrum as well as the eigenfunctions while avoiding spurious ...

Sign-Changing Solutions for Discrete Second-Order Three-Point Boundary Value Problems

Singular Eigenvalue Problems for Second Order Linear Ordinary Differential Equations

We consider linear differential equations of the form (p(t)x′)′ + λq(t)x = 0 (p(t) > 0, q(t) > 0) (A) on an infinite interval [a,∞) and study the problem of finding those values of λ for which (A) has principal solutions x0(t;λ) vanishing at t = a. This problem may well be called a singular eigenvalue problem, since requiring x0(t;λ) to be a principal solution can be considered as a boundary co...

Subharmonic solutions and homoclinic orbits of second order discrete Hamiltonian systems with potential changing sign

Periodic and antiperiodic eigenvalues for half-linear version of Hill’s equation

The nonlinear eigenvalue problem of the differential equation ( |x′|p−2 x′ )′ + (λ+ c(t)) |x|p−2 x = 0, p > 1, with respect to the periodic boundary conditions: x(0) = x(T ), x′(0) = x′(T ), or to the antiperiodic boundary conditions: x(0) = −x(T ), x′(0) = −x′(T ) are considered. Various results on the set of eigenvalues concerning both problems are presented. Some estimates are given for the ...