Asymptotic behavior of a third-order nonlinear differential equation. (English) Zbl 1054.34078
Summary: Consider the third-order nonlinear differential equation
\[
x'''+ \psi(x, x')x''+ f(x, x')= p(t),
\]
where \(\psi\), \(f\), \(f_x\in C(\mathbb{R}\times \mathbb{R},\mathbb{R})\) and \(p\in C([0, \infty),\mathbb{R})\). We obtain sufficient conditions for every solution to the equation to be bounded; we also establish criteria for every solution to the equation to converge to zero.
MSC:
| 34D05 | Asymptotic properties of solutions to ordinary differential equations |
| 34C11 | Growth and boundedness of solutions to ordinary differential equations |
References:
| [1] | Barbashin, E. A., Lyapunov Functions (1970), Nauka: Nauka Moscow · Zbl 0241.34057 |
| [2] | Ezeilo, J. O.C., Stability results for the solutions of some third and fourth order differential equations, Ann. Mat. Pura Appl., 66, 233-249 (1964) · Zbl 0126.30403 |
| [3] | Gera, M.; Graef, J. R.; Gregus, M., On oscillatory and asymptotic properties of solutions of certain nonlinear third order differential equations, Nonlinear Anal., 32, 417-425 (1998) · Zbl 0945.34021 |
| [4] | Qian, C., On global stability of third-order nonlinear differential equations, Nonlinear Anal., 42, 651-661 (2000) · Zbl 0969.34048 |
| [5] | Reissig, R.; Sansone, G.; Conti, R., Nonlinear Differential Equations of Higher Order (1974), Noordhoff: Noordhoff Leyden · Zbl 0275.34001 |
| [6] | Tunc, C., On the ultimate boundedness result of the solutions of certain fourth order differential equation, Ann. Differential Equations, 14, 475-485 (1998) · Zbl 0967.34055 |
| [7] | Yoshizawa, T., Asymptotic behavior of solutions of a system of differential equations, Contrib. Differential Equations, I, 371-387 (1963) · Zbl 0127.30802 |
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