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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 $F:[0,T]\times \bbfR^N\rightarrow \bbfR$ is measurable with respect to $t$ for each $x\in \bbfR^N$ and continuously differentiable in $x$ for a.e. $t\in [0,T]$ and there exist $a\in C(\bbfR^+,\bbfR^+)$ and $b\in L^1(0,T;\bbfR^+)$ such that $|F(t,x)|+|\nabla F(t,x)|\leq a(|x|)b(t)$ for all $x\in \bbfR^N$ 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.

34C25Periodic solutions of ODE
37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
58E05Abstract critical point theory
Full Text: DOI
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