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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
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