Periodic solutions of the forced relativistic pendulum. (English) Zbl 1240.34207

Brézis, Haïm; Mawhin, Jean

Differ. Integral Equ. 23, No. 9-10, 801-810 (2010).

Summary: The existence of at least one classical \(T\)-periodic solution is proved for differential equations of the form \(\phi (u'))'-g(x,u)=h(x)\), when \(\phi:(-a,a)\rightarrow \mathbb {R}\) is an increasing homeomorphism; \(g\) is a Carathéodory function \(T\)-periodic with respect to \(x\), \(2\pi \)-periodic with respect to \(u\) of mean value zero with respect to \(u\); and \(h\in L^1_{loc}(\mathbb {R})\) is \(T\)-periodic and has mean value zero. The problem is reduced to finding a minimum for the corresponding action integral over a closed convex subset of the space of \(T\)-periodic Lipschitz functions, and then to showing, using variational inequalities techniques, that such a minimum solves the differential equation. A special case is the “relativistic forced pendulum equation” \[ \frac {u'}{\sqrt {1-{u'}^2}}+A\sin u=h(x). \]

MSC:

34C25	Periodic solutions to ordinary differential equations
49J40	Variational inequalities
58E30	Variational principles in infinite-dimensional spaces
58E35	Variational inequalities (global problems) in infinite-dimensional spaces

Keywords:

periodic solution; relativistic pendulum; variational inequalities technique