Existence and asymptotic stability of periodic solutions for a class of neutral delayed evolution equation

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 , $$\frac{\text{d}}{\mathrm{d}t}\left(z\left(t\right)-cz\left(t-\delta \right)\right)+A\left(z\left(t\right)-cz\left(t-\delta \right)\right)=f\left(t, z\left(t\right), z\left(t-\tau \right)\right),\quad t\in \mathbb{R},$$ where ∣ c ∣ < 1, the constants δ , τ > 0 are defined as time lags, A : D ( A ) ⊂ X → X $A : \mathcal{D}\left(A\right)\subset X\to X$ is a closed linear operator, − A generates a strongly continuous semigroup T ( t )( t ⩾ 0), and f : R × X 2 → X $f : \mathbb{R}{\times}{X}^{2}\to X$ 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