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

Abstract

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 $ \begin{document}$ \frac{d}{dt}(z(t)-cz(t-\delta))+A(z(t)-cz(t-\delta)) = f(t, \ z(t), \ z(t-\tau)), \quad t\in\mathbb{R}, $\end{document} where $ \mid c\mid 0 $ are defined as time lags, $ A:\mathcal{D}(A)\subset X\rightarrow X $ is a sectorial operator and has compact resolvent, that is, $ -A $ generates exponentially stable, compact analytic operator semigroup $ T(t)(t\geqslant0) $, and $ f:\mathbb{R}\times X\times X\rightarrow 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.

Keywords

• neutral evolution equation with delay
• periodic mild solution
• analytic operator semigroup
• existence and $ c^{1} $-regularity
• fixed point theorem