Abstract The purpose of this work is to study the existence and asymptotic stability of ω -periodic mild solutions to a class of neutral delayed evolution equation in Banach space X d d t (z (t) − c z (t − δ) ) + A (z (t) − c z (t − δ) ) = f (t, z (t), z (t − τ) ), t ∈ R, ddt (z (t) -cz (t-) ) +A (z (t) -cz (t-) ) =f (t, z (t), z (t-) ), t R, where ∣ c ∣ 0 are defined as time lags, A: D (A) ⊂ X → X A: D (A) X X is a closed linear operator, − A generates a strongly continuous semigroup T (t) (t ⩾ 0), and f: R × X 2 → X f: RX^2 X is continuous function which is ω -periodic in t. Firstly, by using theory of operator semigroup and fixed point theorems, the existence results of ω -periodic mild solutions to the above equation are obtained under the assumption that the T (t) (t ⩾ 0) is compact. Next, in the case that T (t) (t ⩾ 0) is non-compact, we investigate the asymptotic stability of periodic solution for the equation based on the contract
Yang et al. (Thu,) studied this question.