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On global stability of third-order nonlinear differential equations. (English) Zbl 0969.34048
It is considered the third-order nonlinear differential equation \[ x'''+ \psi(x,x')x''+ f(x,x')= 0, \tag{1} \] with \(\psi,f,\psi_{x}\in C(\mathbb{R}\times \mathbb{R},\mathbb{R}).\) The main result is the following theorem: Assume that
(1) \(xf(x,0)>0\) for \(x\neq 0, \)
(2) \(\int_{0}^{y}f(0,v) dv>0\) for \(y\neq 0, \)
(3) \(\lim_{|x|\to\infty}\int_{0}^{x}f(u,0) du=\infty\)
and there is a positiv number \(B\) such that
(4) \(\psi(x,y)\geq B,\)
(5) \(B[f(x,y)- f(x,0)-\int_{0}^{y}\psi_{x}(x,v)v dv]y\geq y\int_{0}^{y}f_{x}(x,v) dv,\)
(6) \(B[f(x,y)- f(x,0)-\int_{0}^{y}\psi_{x}(x,v)v dv]y +\psi(x,y)> \int_{0}^{y}f_{x}(x,v) dv + B\) for \(y\neq 0,\)
(7) \(4B\int_{0}^{x}f(u,0) du \{\int_{0}^{y}[f(x,v)-f(x,0)] dv+ B\int_{0}^{y}[\psi(x,v)-B]v dv\}>y^{2}f^{2}(x,0)\) for \(xy\neq 0.\)
Then the trivial solution to (1) is globally asymptotically stable.

34D23 Global stability of solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
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