Values of $\lambda_1,\lambda_2,\ldots,\lambda_n$ are determined for which there exist positive solutions of the iterative system of dynamic equations, \[ u_i^{\Delta\Delta}(t)+\lambda_ia_i(t)f_i(u_{i+1}(\sigma(t)))=0,\quad1\leq i\leq n,\quad u_{n+1}(t)=u_1(t). \] for $t\in[a,b]_\mathbb T$, are determined for which there exist positive solutions of the iterative system of dynamic equations, $u_i(a)=0=u_i(\sigma^2(b))$, $1\leq i\leq n$, where $\mathbb T$ is a time scale. A Guo-Krasnosel'skii fixed-point theorem is applied.