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

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:

 3.4e+06 Asymptotic expansions of solutions to ordinary differential equations

boundedness

Zbl 0212.114
Full Text:

### References:

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