## Differential and Integral Equations

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

### Periodic solutions of the forced relativistic pendulum

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356019113

**Mathematical Reviews number (MathSciNet)**

MR2675583

**Zentralblatt MATH identifier**

1240.34207

**Subjects**

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

#### Citation

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