## Stability and boundedness of solutions of nonlinear vector differential equations of third order.(English)Zbl 1340.34189

The author investigates stability properties of solutions of the system $\dot x=y,\quad \dot y=z,\quad \dot z=-\Psi(y)z-\Phi(x)y-cx+P(t),\quad t\geq 0,\tag{1}$ where $$x,y,z\in {\mathbb R}^n$$, $$\Psi,\Phi:{\mathbb R}^n\to {\mathbb R}^{n\times n}$$ are symmetric, positive definite matrix-valued functions, $$P:[0,\infty)\to {\mathbb R}^n$$, and $$c>0$$ is a real constant. The main result of the paper reads as follows.
Theorem. Suppose that there exist positive constants $$a_0,b_0$$ such that $$a_0b_0>c$$, $$b_0\leq \lambda_i(\Phi(x))\leq b_0+\mu,$$ and $$\lambda_i(\Psi(y))\geq a_0$$, $$i=1,\dots,n$$, where $$\mu=4cb_0^{-2}(a_0b_0-c)>0$$, and $$\lambda_i(.)$$ are eigenvalues of the matrix indicated. Then the zero solution of (1) is globally asymptotically stable.

### MSC:

 34D20 Stability of solutions to ordinary differential equations 34C11 Growth and boundedness of solutions to ordinary differential equations 34D05 Asymptotic properties of solutions to ordinary differential equations
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