| Popis: |
Abstract The aim of this paper is to study the periodic problem for neutral evolution equation (u(t)−G(t,u(t−ξ)))′+Au(t)=F(t,u(t),u(t−τ)),t∈R, $$ \bigl(u(t)-G\bigl(t,u(t-\xi )\bigr)\bigr)'+Au(t)=F \bigl(t,u(t),u(t-\tau )\bigr), \quad t\in \mathbb{R}, $$ in a Banach space X, where A:D(A)⊂X→X $A:D(A)\subset X\rightarrow X$ is a closed linear operator, and −A generates a compact analytic operator semigroup T(t) $T(t)$ ( t≥0 $t\geq 0$). With the aid of the analytic operator semigroup theories and some fixed point theorems, we obtain the existence and uniqueness of periodic mild solution for the neutral evolution equation. The regularity of periodic mild solutions for the evolution equation with delay is studied, and some existence results of the classical and strong solutions are obtained. In the end, we give an example to illustrate the applicability of abstract results. Our works greatly improve and generalize the relevant results of existing literature. |
