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On the relativistic pendulum-type equation. (English) Zbl 07217165
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.
##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations 49J40 Variational inequalities 35Q75 PDEs in connection with relativity and gravitational theory