Abstract:

In this paper, we deal with the existence of periodic solutions for the semilinear evolution equation in a Banach space $X$,

$ u'(t)+Au(t)=f(t,\,u(t)),\quad t\in{\Bbb R}, $

where $A: D(A)\subset X\to X$ is a closed linear operator and $ -A$ generates a $C_{0}$-semigroup $X$, $f:{\Bbb R}\times X\to X$ is a continuous mapping and $f(t,\,x)$ is $\omega$-periodic in $t$. Existence results of $\omega$-periodic mild solutions are obtained by using operator semigroup theory, estimation technique of noncompact measure and fixed point theorem. Examples of applications in parabolic partial differential equations and weakly damped wave equations are present.

Key words: Semilinear evolution equation, Semigroup of linear operators, Measure of noncompactness, Periodic mild solutions, Existence