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On the stability of the trivial solution of a non-autonomous quasilinear equation of second order. (Russian) Zbl 0636.34044

A second order differential equation (1) \(\ddot y+\lambda (t)y=F(t,y,\dot y)\) where i) \(\lambda \in C^ 2([a,b))\), \(\lambda (t)\to \lambda_ 0\) as \(t\to b\) \((b=\infty\) is included) ii) \(| F| \leq Q(t)(| y| +| \dot y|)^{1+\alpha},\) Q(t)\(\geq 0\), \(\alpha\geq 0\), is considered. Sufficient conditions for the stability, asymptotic stability and instability of the zero solution of equation (1) are presented. For example, the following theorem is proved. Theorem. If \(\lambda_ 0\neq 0\), \({\dot \lambda}=o(1)\), \({\ddot \lambda}\), \(Q\in L([a,b))\), then the zero solution of (1) is stable in the sense of Lyapunov.
Reviewer: Yu.N.Bibikov

MSC:

34D20 Stability of solutions to ordinary differential equations
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