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Please, is there any reference for proposition below or does it perhaps follow from a standard fact? I've got it for some other problem but I actually do not know how to comment it in my article.

Consider a nonlinear system $$ \dot x_k=\sum_{j=1}^ma_{kj}(t)x_j+\psi_k(t,x),\qquad k=1,\ldots, m. \quad (1)$$ Here the functions are as follows $$a_{ij}\in C[0,\infty),\quad\psi_k\in C(A),\quad A=\{(t,x)\in\mathbb{R}^{m+1}\mid t\ge 0,\quad \|x\|_\infty\le r\},$$ and $r>0$ is a constant. The functions $\psi$ are also locally Lipschitz in the second argument and for some constants $\lambda>1,\quad c\ge 0$ it follows that $$|\psi_k(t,x)|\le c\|x\|_\infty^\lambda,\quad (t,x)\in A.$$

Introduce a notation $$p_k(t)=a_{kk}(t)+\sum_{j\in J_k}|a_{kj}(t)|,\quad J_k=\{1,\ldots,m\}\backslash\{k\}.$$

Proposition. Assume that $$\sup_{t\ge 0}\Big\{\int_0^tp_k(s)ds\Big\}<\infty$$and $$c\cdot\sup_{t\ge 0}\Big\{ \int_0^te^{(\lambda-1)\int_0^\xi p_k(s)ds}d\xi\Big\}<\infty,\quad k=1,\ldots, m.$$ Then the zero solution to system (1) is stable in the sense of Lyapunov.

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  • $\begingroup$ If this question is different from this recent question, can you explain how? mathoverflow.net/q/182478/55893 It looks quite a lot like a duplicate. $\endgroup$ Oct 4, 2014 at 15:01
  • $\begingroup$ @JoonasIlmavirta "If this question is different from this recent question, can you explain how?" It's the difference between a linear and nonlinear system. $\endgroup$
    – user59037
    Oct 4, 2014 at 15:06
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    $\begingroup$ Ah, indeed, the difference is in linearity. It might be a good idea to mention in your question that this is a nonlinear version of an earlier one. Or, if you are the author of the earlier question, you could edit the question to include the nonlinear case. $\endgroup$ Oct 4, 2014 at 15:13

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