Geodesic
Problem of instability in the first approximation
Matematičeskie zametki, Tome 23 (1978) no. 5, pp. 721-723

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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 $q$. 
@article{MZM_1978_23_5_a7,
     author = {V. E. Slyusarchuk},
     title = {Problem of instability in the first approximation},
     journal = {Matemati\v{c}eskie zametki},
     pages = {721--723},
     publisher = {mathdoc},
     volume = {23},
     number = {5},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1978_23_5_a7/}
}
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V. E. Slyusarchuk. Problem of instability in the first approximation. Matematičeskie zametki, Tome 23 (1978) no. 5, pp. 721-723. http://geodesic.mathdoc.fr/item/MZM_1978_23_5_a7/