### An instability theorem for a certain vector differential equation of fourth order.

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Sufficient conditions are established for the asymptotic stability of the zero solution of the equation (1.1) with $p\equiv 0$ and the boundedness of all solutions of the equation (1.1) with $p\ne 0$. Our result includes and improves several results in the literature ([4], [5], [8]).

We study the asymptotic behavior of solutions to a nonlinear differential equation of the second order whose coefficient of nonlinearity is a bounded function for arbitrarily large values of $x$ in $R$. We obtain certain sufficient conditions which guarantee boundedness of solutions, their convergence to zero as $x\to \infty $ and their unboundedness.

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