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New results on the asymptotic behavior of a third-order nonlinear differential equation. (English) Zbl 1198.34085
The author studies the equation \[ x'''+g(x,x')x''+f(x,x') = p(t) , \] where \(g,f,g_{x},f_{x}\in C({\mathbb R}\times {\mathbb R},{\mathbb R})\) and \(p\in C([0,\infty ),{\mathbb R})\). It is assumed that the solutions \(x(t)\) of the considered equation exist and are unique. Sufficient conditions are established such that for all solutions there holds: (a) There exists a constant \(D>0\) such that \(|x^{(i)}(t)|\leq D\), for sufficiently large \(t\). (b) \(x^{(i)}(t)\rightarrow 0\), as \(t\rightarrow \infty \); \(i=0,1,2\).

MSC:
34D05 Asymptotic properties of solutions to ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
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