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

Abstract EN

Let T∈ℕ with T>5.

Let ?:={1,…,T}.

We study the Fučik spectrum Σ of the discrete problem Δ2u(t-1)+λu+(t)-μu-(t)=0, t∈?, u(0)=u(T+1)=0, where u+(t)=max{u(t),0}, u-(t)=max{-u(t),0}.

We give an expression of Σ via the matching-extension method.

We also use such discrete spectrum theory to study nonlinear boundary value problems of difference equations at resonance.