## Periodic solutions of a class of non-autonomous second-order systems.(English)Zbl 0971.34027

The authors consider the problem $u''(t)=\nabla F(t,u(t)), \quad \text{for a.e. }t\in [0,T], \quad u(0)=u(T),\;u'(0)=u'(T),$ with $$T>0$$ and $$F:[0,T]\times \mathbb{R}^N\rightarrow \mathbb{R}$$ is measurable with respect to $$t$$ for each $$x\in \mathbb{R}^N$$ and continuously differentiable in $$x$$ for a.e. $$t\in [0,T]$$ and there exist $$a\in C(\mathbb{R}^+,\mathbb{R}^+)$$ and $$b\in L^1(0,T;\mathbb{R}^+)$$ such that $$|F(t,x)|+|\nabla F(t,x)|\leq a(|x|)b(t)$$ for all $$x\in \mathbb{R}^N$$ and a.e. $$t\in [0,T]$$. Under these assumptions it follows that the corresponding action functional on $$H^1_T$$ is continuously differentiable and weakly lower semicontinuous. It has been proved that if the function $$F$$ is convex or $$\gamma$$ subadditive then at least one solution exists.
Here, the authors prove an existence result assuming that the potential is the sum of a subconvex and a subquadratic function by the least action principle.

### MSC:

 34C25 Periodic solutions to ordinary differential equations 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

### Keywords:

periodic solutions; variational methods
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### References:

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