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Locally lipschitzian guiding function method for ODEs. (English) Zbl 0935.34007
The author treats the existence of periodic solutions to the problem $x'(t) = f(t,x(t)), \quad x(0)=x(T), \tag{*}$ where $$f:[0,T] \times \mathbb{R}^n \to \mathbb{R}^n$$ is a Carathéodory function with integrably bound growth. Under the assumption that $$f$$ has a locally Lipschitzian guiding function $$V$$ with Ind$$(V) \neq 0$$ it is proved that problem (*) admits at least one solution. Moreover, it is shown that every coercive direct potential has Ind$$(V)=1$$, and so it can be used as a guiding function in the above theorem.

##### MSC:
 34A60 Ordinary differential inclusions 34C25 Periodic solutions to ordinary differential equations
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##### References:
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