Existence of sign-changing periodic solutions of second order difference equations

We are concerned with determining values of r, for which there exist sign changing solutions of nonlinear periodic boundary value problemsΔ2u(t-1)+rf(u(t))=0,t∈T,u(0)=u(T),u(1)=u(T+1)The proof of our main result is based upon global bifurcation techniques.