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Periodic solutions of dynamical systems by a saddle point theorem of Rabinowitz. (English) Zbl 0729.58044
The author considers the problem of finding periodic solutions of differential equations of the type $(1)\quad \ddot x+\nabla_ xV(t,x)=0\quad (x\in {\mathbb{R}}^ n,\quad V\in C^ 1({\mathbb{R}}\times {\mathbb{R}}^ n,{\mathbb{R}}),\quad V(t+T,x)=V(t,x)).$ Using some special cases of linking techniques [see, for instance, V. Benci and P. H. Rabinowitz, Invent. Math. 52, 241-273 (1979; Zbl 0465.49006) and P. H. Rabinowitz, Lect. Notes Math. 648, 97-115 (1978; Zbl 0377.35020)] he establishes some sufficient conditions for periodic solutions to exist:
Theorem 1. If V(t,x)$$\leq 0\forall (t,x)\in {\mathbb{R}}\times {\mathbb{R}}^ n$$, $$V\not\equiv 0$$, V(t,x)$$\to 0$$ and $$\nabla_ xV(t,x)\to 0$$ as $$\| x\| \to \infty$$ uniformly in t, then equation (1) has at least one T- periodic solution x(t) such that $$V(t,x(t))<0$$ for some t and there exists a sequence of distinct subharmonic solutions.
Theorem 2. If $$V(t,x)\to +\infty$$ as $$\| x\| \to \infty$$ uniformly in t and $$\nabla_ xV(t,x)$$ is bounded, then for every $$k\in N$$ there exists a kT-periodic solution $$x_ k(t)$$ such that $$\| x_ k\|_{L\infty}\to +\infty$$ as $$k\to +\infty.$$
A generalization of Theorem 1, in which V(t,x) is bounded and may change its sign is considered, too.

##### MSC:
 37G99 Local and nonlocal bifurcation theory for dynamical systems 35B10 Periodic solutions to PDEs
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##### References:
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