Asymptotic properties of solutions of a certain third-order differential equation with an oscillatory restoring term. (English) Zbl 0701.34070

Andres, Jan

Acta Univ. Palacki. Olomuc., Fac. Rerum Nat. 91, Math. 27, 201-210 (1988).

The author improves results of K. E. Swick [SIAM J. Appl. Math. 19, 96-102 (1970; Zbl 0212.114)] concerning the boundedness of all solutions of the equation: \[ (1)\quad x'''+ax''+g(x)x'+h(x)=p(t), \] where \(a>0\) is a constant, \(g,h\in C^ 1(-\infty,+\infty)\), h is an oscillatory function of x, and p(t)\(\in C(0,\infty)\), \(\int^{\infty}_{0}| p(\tau)d\tau | <\infty.\) Swick has shown that if there exists positive constants b and c such that \(1)\quad (1/x)\int^{x}_{0}g(\xi)d\xi \geq b,\) 2) \(h'(x)\leq c\), \(c<ab\), \(3)\quad h(x)sign x\geq 0,\) then all solutions of equation (1) are bounded and \(\lim_{t\to \infty}(x(t))\) exists, while lim x\({}'=\lim x''=0\). The author shows that these conditions may be replaced by conditions at a single point \(x=0\), and that condition (3) can be considerably weakened.

Reviewer: V.Komkov

MSC:

34E05

Asymptotic expansions of solutions to ordinary differential equations

Keywords:

boundedness

Citations:

Zbl 0212.114

References:

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