Spectrum of Linear Difference Operators and the Solvability of Nonlinear Discrete Problems

Abstract

LetT∈ℕwithT>5. Let𝕋:={1,…,T}. We study the Fučik spectrumΣof the discrete problemΔ2u(t-1)+λu+(t)-μu-(t)=0,t∈𝕋,u(0)=u(T+1)=0, whereu+(t)=max⁡{u(t),0},u-(t)=max⁡{-u(t),0}. We give an expression ofΣvia the matching-extension method. We also use such discrete spectrum theory to study nonlinear boundary value problems of difference equations at resonance.