On the relativistic pendulum-type equation. (English) Zbl 1474.34135

In this paper, the authors consider the equation \[\Bigl(\frac{u'}{\sqrt{1-{u'}^2}}\Bigr)'+F'(u)=0\tag{1.1}\] and firstly look for solutions of the relativistic Dirichlet boundary value problem. They assume:
(\(F_1\)) \(F\in C^1(\mathbb{R})\) with \(F(0)=F'(0)=0\),
(\(F_2\)) \(F\) is \(C^2\) at \(u=0\) and \(F''(0)=\alpha^2>0\)
and prove the following theorem.
Theorem 3.1. Let \((F_1)\), \((F_2)\) hold. If \(T>\frac{\pi}{\alpha}\), then (1.1) with \(u(0)=u(T)=0\) has at least two nontrivial solutions: \(u_1(t)>0\) on \((0,T)\) and \(u_2(t)<0\) on \((0,T)\).
This result is used to find non-trivial periodic solutions to (1.1).
Theorem 4.1. (i) Let \((F_1)\), \((F_2)\) hold and let \(F\) be even. If \(T>\pi /\alpha\) equation (1.1) has a pair of periodic solutions \(\pm z_1\) with minimal period \(2T\).
(ii) If \(F\in C^2(\mathbb{R})\) and \(F''(u)\le\alpha^2\forall u\in\mathbb{R}\) and \(T<\pi /\alpha\) the only \(2T\) periodic solution to (1.1) is \(u\equiv 0\).
In Section 5, the authors consider the equation \[\Bigl(\frac{u'}{\sqrt{1-{u'}^2}}\Bigr)'+F'(u)-h^2G'(u)=0,\] where \(h\) is the constant angular momentum and prove the existence of periodic solutions provided \(T\) is larger than a suitable value.
Finally, in Section 6, the authors prove the existence of multiple periodic solutions to the periodically forced spherical pendulum equation \[\Bigl(\frac{u'}{\sqrt{1-{u'}^2}}\Bigr)'+F'(u)-h^2G'(u)=f(t),\] where \(F\) and \(G\) are periodic.
The proofs are based on the critical point theory carried out by A. Szulkin [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 3, 77–109 (1986; Zbl 0612.58011)] to handle functionals \(I\) which are not smooth.


34B15 Nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences


Zbl 0612.58011