Problem of instability in the first approximation

Let $E$ be a Banach space, $A$ be a continuous linear operator such that $\sigma(A)\cap\{\lambda: \mathrm{Re}\,\lambda>0\}\ne\varnothing$, and $F(t, x)$ be a continuous function on $[0,\infty)\times E$ satisfying the condition $||F(t, x)||\leqslant q||x||$ ($q=\mathrm{const}$). An example of a system ${dx}/{dt}=Ax+F(t, x)$ is given which has an exponentially stable zero solution for certain $F(t, x)$ with arbitrarily small $q$.