Differential Equation has a unique solution periodic

Question

Let $A(t)$ continuous and periodic of period $S$ in $\mathbb{R}$. Suppose $x' = Ax$ has $\varphi \equiv 0$ as the only periodic solution of period $S$. Show that there exists $\delta> 0$ such that for every continuous function $f:\mathbb{R} \times E \longrightarrow E $, $S$ period of periodic with variable the first $|D_2f(t,x)|<\delta$ to all $(t,x)$ so $x' = A(t)x + f(t,x)$ has a unique periodic solution $\varphi_f$ of period $S$ also prove that if $f\rightarrow0$ uniformly then $\varphi \rightarrow 0$ uniformly.