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Periodic solutions of Lagrangian systems of relativistic oscillators. (English) Zbl 1235.34130
The authors study the existence of periodic solutions for a system of the type $(\phi(u'))'=\nabla_uF(x,u)+h(x),$ where $$\phi=\nabla\Phi$$, with $$\Phi$$ strictly convex, is a homeomorphism of the ball $$B_a\subset{\mathbb R}^n$$ onto $${\mathbb R}^n$$. An example is given by a “relativistic” differential equation. Different situations are considered, with $$F$$ being either coercive, or convex, or periodic. The approach is mostly variational, but requires the use of results on an auxiliary system based upon fixed point theory and Leray–Schauder degree.

##### MSC:
 34C25 Periodic solutions to ordinary differential equations 47H11 Degree theory for nonlinear operators 49J40 Variational inequalities 58E30 Variational principles in infinite-dimensional spaces 58E35 Variational inequalities (global problems) in infinite-dimensional spaces 78A35 Motion of charged particles