Eigenvalues of discrete Sturm-Liouville problems with nonlinear eigenparameter dependent boundary conditions

Abstract

We consider the discrete right denite Sturm-Liouville problems with nonlinear eigenparameter dependent boundary conditions

-∇(p(t)Δy(t)) + q(t)y(t) = λr(t)y(t), t ∈ {1, 2, ... , T},

b₀y(0) = b₁Δy(0), (c₀+c₁λ+c₂λ²)y(T +1) = (d₀+d₁λ+d₂λ²)(-p(T)∇y(T +1)),

where T > 1 is an integer and λ is the spectrum parameter. We obtain the existence of the eigenvalues, the oscillation properties of the eigenfunctions and the interlacing results of the eigenvalues of the above problem with the eigenvalues of the Dirichlet problem and the Neumann problem.

Mathematics Subject Classification (2010): 39A06, 39A12, 39A21, 39A70.

Keywords: Discrete Sturm-Liouville problems, nonlinear eigenparameter dependent boundary condition, spectrum, oscillation properties, interlacing