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.
