Abstract:

In this paper, the existence and uniqueness of periodic solutions for the second-order evolution equation with weak damping in a Hilbert space $H$

$$ u''(t)+2c u'(t)+Au(t)=f(t, u(t)),\quad t\in \mathbb{R} $$

are discussed, where $ A: D(A)\subset H\to H$ is a positive definite self-adjoint operator with a compact resolvent in $H$, $f: \mathbb{R}\times H\to H$ is continuous, $f(t, x)$ is $\omega$-periodic in $t$, and $c>0$ is the damping coefficient. By applying the semigroup theory of linear operators and fixed-point theorem, we obtain existence and uniqueness results of $\omega$-periodic weak solution

and classical solution of the equations.

Key words: second-order evolution equations with weak damping, periodic solution, weak solution, semigroup of linear operators, existence and uniqueness