In this paper, we first deal with a class of neutral evolution equation with delay in ordered Banach space X \( \frac{\text {d}}{\text {d}t}(z(t)-cz(t-\delta ))+A(z(t)-cz(t-\delta ))=f(t,\ z(t),\ z(t-\tau )),\quad t\in \mathbb {R}, \) where \(\mid c\mid <1\) , the constants \(\delta>0, \tau >0\) are defined as time lags, \(A:\mathcal {D}(A)\subset X\rightarrow X\) is a closed linear operator and \(-A\) generates a positive \(C_{0}\) -semigroup \(T(t)(t\geqslant 0)\) , and \(f:\mathbb {R}\times X^{2}\rightarrow X\) is continuous function which is \(\omega \) -periodic in t. Under the assumption that the \(T(t)(t\geqslant 0)\) is compact or non-compact, by applying the characteristics of positive operator semigroups, the monotone iterative technique, and the non-compact measurement, we provide important ordered conditions on the nonlinear term f to guarantee that the above equation has \(\omega \) -periodic solutions and positive periodic solutions. Finally, two example of our main results are presented.
