Differential and Integral Equations

Periodic solutions of the forced relativistic pendulum

Haöim Brezis and Jean Mawhin

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The existence of at least one classical T-periodic solution is proved for differential equations of the form \begin{eqnarray*} (\phi(u'))' - g(x,u) = h(x) \end{eqnarray*} when $\phi : (-a,a) \to {\mathbb R}$ 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 $h \in L^1_{loc}(R)$ 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 show, using variational inequalities techniques, that such a minimum solves the differential equation. A special case is the ``relativistic forced pendulum equation'' \begin{eqnarray*} \Big (\frac{u'}{\sqrt{1 - u'^2}} \Big )' + A \sin u = h(x). \end{eqnarray*}

Article information

Differential Integral Equations, Volume 23, Number 9/10 (2010), 801-810.

First available in Project Euclid: 20 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34C25: Periodic solutions 49J40: Variational methods including variational inequalities [See also 47J20] 58E30: Variational principles 58E35: Variational inequalities (global problems)


Brezis, Haöim; Mawhin, Jean. Periodic solutions of the forced relativistic pendulum. Differential Integral Equations 23 (2010), no. 9/10, 801--810. https://projecteuclid.org/euclid.die/1356019113

Export citation