The paper deals with the periodic boundary value problem on time scales \[ \begin{aligned} &x^{\Delta \Delta}(t) = f(t, x(\sigma(t))), \quad t \in [a, b]_{\mathbb{T}} \\ &x(a) = x(\sigma^2(b)), \quad x^\Delta (a) = x^\Delta (\sigma(b)),\end{aligned} \] where time scale \(\mathbb{T}\) is closed subset of the interval \([a,b]\), \(\sigma(t) = \inf \{ s \in \mathbb{T}: s > t\},\) \(x^\Delta(t)\) is delta-derivative at point \(t\), i.e., if for any \(\varepsilon > 0\) there is a neighborhood \(U\) of \(t\) in the time-scale topology such that \(| x(\sigma(t)) - x(s) - x^\Delta (t)[\sigma(t) - s]| \leq \varepsilon | \sigma(t) - s| \) for all \(s \in U\). Using the Schauder fixed point theorem and concept of lower and upper solutions, existence of solutions is proved. By some restriction on graininess function \(\sigma (t) - t\) and Lipschitz constant for \(f\), a monotone iterative method is investigated.