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.


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