论文标题
关于时间尺度的定期强制相对论的摆动方程的解决性
On the Solvability of the Periodically Forced Relativistic Pendulum Equation on Time Scales
论文作者
论文摘要
我们研究了相对论摆动运算符$ \ MATHCAL P $范围的一些属性，也就是说，一组可能的连续$ t $ t $ periodic强迫术语$ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ \ nathcal p x = p $ p $ p $ p $ p $ p $ p $承认$ t $ - periodic解决方案在$ t $ t $ t $ - periodic time caudic time cab $ \ nath $ \ mathb bb t $中。写作$ p（t）= p_0（t）+\上线p $，我们证明存在非空的紧凑型间隔$ \ mathcal i（p_0）$，取决于$ p_0 $，因此问题在\ yny（p_0）$（p_0）$ compline compline时，当问题及时仅在$ p_0 $ p_0 $上，当时只有在$ p_0 $上进行了解决方案。此外，我们为非修复性提供了足够的条件。具体来说，我们证明，如果$ t $很小，则$ \ MATHCAL I（P_0）$是任意$ P_0 $的$ 0 $的社区。本文中的结果改善了连续案例$ \ mathbb t = \ mathbb r $的先前作品中获得的较小条件。
We study some properties of the range of the relativistic pendulum operator $\mathcal P$, that is, the set of possible continuous $T$-periodic forcing terms $p$ for which the equation $\mathcal P x=p$ admits a $T$-periodic solution over a $T$-periodic time scale $\mathbb T$. Writing $p(t)=p_0(t)+\overline p$, we prove the existence of a nonempty compact interval $\mathcal I(p_0)$, depending continuously on $p_0$, such that the problem has a solution if and only if $\overline p\in \mathcal I(p_0)$ and at least two different solutions when $\overline p$ is an interior point. Furthermore, we give sufficient conditions for nondegeneracy; specifically, we prove that if $T$ is small then $\mathcal I(p_0)$ is a neighbourhood of $0$ for arbitrary $p_0$. The results in the present paper improve the smallness condition obtained in previous works for the continuous case $\mathbb T=\mathbb R$.
