The $n$ th order eigenvalue problem: $$\Delta^n x(t)=(-1)^{n-k}\lambda f(t,x(t)),t\in[0,T],x(0)=x(1)=\cdots=x(k-1)=x(T+k+1)=\cdots=x(T+n)=0,$ is considered, where $n\ge 2$ and $k\in\{1,2,\ldots,n-1\}$ are given. Eigenvalues $\lambda$ are determined for $f$ continuous and the case where the limits $f_0(t)=\lim\limits_{n\to 0^+}\frac{f(t,u)}{u}$ and $f_\infty(t)=\lim\limits_{n\to\infty}\frac{f(t,u)}{u}$ exist for all $t\in[0,T]$ . Guo′s fixed point theorem
is applied to operators defined on annular regions in a cone.