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