Spectrum of Discrete Second-Order Difference Operator with Sign-Changing Weight and Its Applications

Abstract

LetT>1be an integer, and let𝕋=1,2,…,T. We discuss the spectrum of discrete linear second-order eigenvalue problemsΔ2ut-1+λmtut=0, t∈𝕋,  u0=uT+1=0, whereλ≠0is a parameter,m:𝕋→ℝchanges sign andmt≠0on𝕋. At last, as an application of this spectrum result, we show the existence of sign-changing solutions of discrete nonlinear second-order problems by using bifurcate technique.