Spectral and oscillation theory for general second order Sturm-Liouville difference equations

Roman Šimon Hilscher

doi:10.1186/1687-1847-2012-82

Abstract

Abstract In this article we establish an oscillation theorem for second order Sturm-Liouville difference equations with general nonlinear dependence on the spectral parameter λ. This nonlinear dependence on λ is allowed both in the leading coefficient and in the potential. We extend the traditional notions of eigenvalues and eigenfunctions to this more general setting. Our main result generalizes the recently obtained oscillation theorem for second order Sturm-Liouville difference equations, in which the leading coefficient is constant in λ. Problems with Dirichlet boundary conditions as well as with variable endpoints are considered. Mathematics Subject Classification 2010: 39A21; 39A12.

Highlights

In this article we consider the second order Sturm-Liouville difference equation (rk(λ) xk) + qk(λ)xk+1 = 0, k ∈ [0, N − 1]Z,(SLλ) where rk : R ® R for k Î [0, N]Z and qk : R ® R for k Î [0, N - 1]Z are given differentiable functions of the spectral parameter l such that rk(λ) = 0 and rk(λ) ≤ 0, k ∈ [0, N]Z, qk(λ) ≥ 0, k ∈ [0, N − 1]Z. (1:1)Here N Î N is a fixed number with N ≥ 2 and [a, b]Z := [a, b]∩Z, and the dot denotes the differentiation with respect to l
With equation (SLl) we consider the Dirichlet boundary conditions, that is, we study the eigenvalue problem (SLλ), λ ∈ R, x0 = 0 = xN+1
By a standard argument from linear algebra it follows that the eigenvalues of (E0) with l Î C are real and that the eigenfunctions corresponding to different eigenvalues are orthogonal with respect to the inner product x, y w :=

Summary

Introduction

The following result shows that under the monotonicity assumption (1.1) the oscillation behavior in l is not allowed for the above type of solutions near any finite value of l, compare with [9, Theorem 4.3]. Since in this case we have mk(l0) = 1 (by the definition of a generalized zero at k + 1) and hk(λ−0 ) = hk(λ0) = 1 , hk+1(λ−0 ) = 1 , and hk+1(l0) = 0, it follows that the equations in (2.8) and (2.9) hold as the identity 1 = 1.

Objectives

Results

Conclusion