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Cím:On the stability of the zero solution of nonlinear second order differential equations.
Szerző:Hatvani, L\'aszl\'o
Forrás:Acta Sci. Math. 57, No.1-4, 367-371 (1993).
Nyelv:English
Absztrakt:The author considers the differential equation (1) $x''+a(t) g(x,x')x'+b(t) f(x) =0$ where the functions $f:\bbfR \to \bbfR$, $g:\bbfR\sp 2 \to \bbfR\sb +=[0,\infty)$, $a:\bbfR\sb +\to \bbfR$, $b:\bbfR\sb +\to(0,\infty)$ are continuous and $xf(x)>0$ for $x \ne 0$. After pointing out an error in a paper of {\it V. M. Starzhinskij} [Priklad. Mat. Mekh. 16, 369-374 (1952; Zbl 0047.178)], he proves the following theorem: Suppose that there exists a $\rho>0$ such that the following conditions are satisfied: (i) for every $v\sb 0 \ne 0$, $\vert v\sb 0 \vert \le \rho$ there is an $\eta=\eta(v\sb 0)>0$ such that $\vert u \vert \le \eta$ implies $g(u,v\sb 0) \ne 0$; (ii) $2a(t) b(t) g(u,v)+b'(t) \ge 0$ for all $t \in \bbfR\sb +$, $\vert u \vert \le \rho$, $\vert v \vert \le \rho$; (iii) there is a $\gamma>0$ such that $a(t) \ge \gamma b(t)$ holds for all $t \in \bbfR\sb +$. Then the equilibrium $x=x'=0$ of (1) is stable.} \RV{W.M\"uller (Berlin) 
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