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On the new variational principles and duality for periodic solutions of Lagrange equations with superlinear nonlinearities. (English) Zbl 0998.34033

The authors consider the Euler-Lagrange equation \[ {d\over dt} L_{x'}(t,x'(t))+V_x(t,x(t))=0\quad \text{a.e. in} \mathbb{R}, \tag{1} \] where \(T>0\) is arbitrary, \(L, V: \mathbb{R}\times \mathbb{R}^n \rightarrow \mathbb{R}\) are convex, Gateaux differentiable in the second variable, \(T-\)periodic and measurable functions it \(t\).
Here, the existence of periodic solutions to (1) is proved, where a periodic solution to (1) is understood as a pair \((x,p)\) of \(T-\)periodic absolutely continuous functions such that \[ {d\over dt} p(t)+ V_x(t,x(t))=0, \quad p(t)=L_{x'}(t,x'(t)). \]
The authors look for critical points of the “minmax” type of the functional \[ J(x) = \int^T_0 (-V(t,x(t))+L(t,x'(t))) dt \] which is superlinear and defined on the space of absolutely continuous \(T-\)periodic functions \(x: \mathbb{R}\rightarrow \mathbb{R}^n\).
The study of (1) is based on duality methods analogous to the methods developed for (1) in sublinear cases. The duality results proved here yield a measure of a duality gap between corresponding functionals for approximate solutions to (1), which has been proved before for the sublinear case, only.

MSC:

34C25 Periodic solutions to ordinary differential equations
49J35 Existence of solutions for minimax problems
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