Existence and regularity of periodic solutions for a class of neutral evolution equation with delay

The purpose of this paper is to investigate the existence and $C^1$-regularity of $\omega$-periodic mild solutions for a class of neutral evolution equation with two-constant delays in Banach space $X$

$$\frac{d}{dt}(z(t)-cz(t-\delta ))+A(z(t)-cz(t-\delta ))=f(t,z(t),z(t-\tau )),t\in \mathbb{R},$$

where $\mid c\mid <1$, the constants $\tau ,\delta >0$ are defined as time lags, $A:\mathcal{D}(A)\subset X\to X$ is a sectorial operator and has compact resolvent, that is, $-A$ generates exponentially stable, compact analytic operator semigroup $T(t)(t⩾0)$, and $f:\mathbb{R}\times X\times X\to X$ is nonlinear mapping which is $\omega$-periodic in $t$. By using the theory of analytic operator semigroups, fixed point theorems, and the fractional powers of the sectorial operator, we establish the existence and $C^1$-regularity results of $\omega$-periodic mild solutions for the equation for the first time when $f$ satisfies the appropriate growth conditions. In the end, we present an example to demonstrate the applications of our main results.