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Periodic solutions of dynamical systems with bounded potential. (English) Zbl 0646.34049
This paper deals with the existence of T-periodic solutions for systems of ordinary differential equations of the form $$-\ddot x=\nabla_ xV(t,x),$$ where $$V\in C^ 2(R\times R^ N,R)$$, $$V(t+T,x)=V(t,x)$$ for any $$(t,x)\in {\mathbb{R}}\times {\mathbb{R}}^ N$$, V is bounded and $$\nabla_ xV(t,x)\to 0$$ as $$| x| \to +\infty$$. The author uses Morse theory to prove the existence of periodic solutions for the given equation. The use of Morse theory is done through defining on a suitable function space E a functional f whose critical points are the T-periodic solutions of the equation. Then he shows that the Palais-Smale condition holds on $$\{x\in E:f(x)\geq c+\epsilon \}$$ and estimates the homology groups $$H_ q(\{x\in E;f(x)\leq c+\epsilon \})$$.
Reviewer: U.D’Ambrosio

##### MSC:
 34C25 Periodic solutions to ordinary differential equations 37-XX Dynamical systems and ergodic theory
##### Keywords:
Morse theory; Palais-Smale condition; estimates; homology groups
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