Anti-periodic problems for semilinear partial neutral evolution equations

Abstract

We study the anti-periodic problem for the semilinear partial neutral evolution equation in the form \begin{equation*} \begin{array}{l}\displaystyle\frac{d}{dt}[u(t)+h(t,u(t))]+Au(t)=f(t,u(t)),\quad t\in \mathbb{R} \end{array} \end{equation*} in a Banach space $X$, where $h,f$ are given $X$-valued functions, and $-A: D(A)\subseteq X \rightarrow X$ is the infinitesimal generator of a compact analytic semigroup. Some new theorems concerning the existence of anti-periodic mild solutions for the problem are established. The theorems formulated are essential extensions of those given previously for the anti-periodic problems for evolution equations in Banach spaces. The main tools in our study are the analytic semigroup theory of linear operators, fractional powers of closed operators, and the fixed point theorem due to Krasnoselskii. Furthermore, we provide an illustrative example to justify the practical usefulness of the obtained abstract results.

Keywords

• partial neutral evolution equation
• anti-periodicity
• mild solution
• analytic semigroup