## Solving a collection of free coexistence-like problems in stability.(English)Zbl 0683.34033

The aim of the paper is to find all the $$C^ 1$$ real maps f, defined in a neighbourhood of $$0\in {\mathbb{R}}$$, such that the origin is Lyapunov stable for the system $\frac{d^ 2}{dt^ 2}x=g(x),\quad \frac{d^ 2}{dt^ 2}y=y(g'(x)+\frac{3\alpha g(x)}{1+\alpha x}),\quad x,y\in {\mathbb{R}},$ where $$g(x)=-xf(x)$$ and $$\alpha\in {\mathbb{R}}$$.
Reviewer: G.Zampieri

### MSC:

 34D20 Stability of solutions to ordinary differential equations

Hill’s equations
Full Text:

### References:

 [1] W. Magnus - S. WINKLER, Hill’s Equation , pp. 1 - 127 , Interscience , New York ( 1966 ). MR 197830 | Zbl 0158.09604 · Zbl 0158.09604 [2] G. Zampieri , Liapunov stability for some central forces , J. Differential Equations , 74 , n. 2, pp. 254-265 ( 1988 ). MR 952898 | Zbl 0668.34051 · Zbl 0668.34051 [3] G. Zampifri , Some problems in stability of the equilibrium related to coexistence of solutions to Hill’s equations, in the Proceedings of the Equadiff. 87 , Lecture Notes in Pure and Applied Mathematics , Dekker , Zbl 0685.34060 · Zbl 0685.34060
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