The nth order eigenvalue problem: Δnx(t)=(1)nkλf(t,x(t)),t[0,T],x(0)=x(1)==x(k1)=x(T+k+1)==x(T+n)=0, is considered, where n2 and k{1,2,,n1} are given. Eigenvalues λ are determined for f continuous and the case where the limits f0(t)=limn0+f(t,u)u and f(t)=limnf(t,u)u exist for all t[0,T]. Guo's fixed point theorem is applied to operators defined on annular regions in a cone.