受迫摆方程周期解的存在性

Existence of Periodic Solutionsfor the Fourth-Order ForcedPendulum Equation

摘要: 本文运用 Mawhin 延拓定理确保以下具有周期边界条件的四阶受迫摆方程x⁽⁴⁾+kx"+a(t)sin x=e(t)至少具有一个非平凡正解, 其中 k 是负常数，a(t) 是一个连续的 T −周期函数且在 [0, T ] 上不变号， e(t) 是一个连续的 T −周期函数且 e(t) 不恒为 0。 作为应用，我们给出一些例子来说明这些定理的适用性。

Abstract: In this paper, we apply the Mawhin's continuation theorem to ensure that a fourth- order forced pendulum equation of the form x⁽⁴⁾+kx"+a(t)sin x=e(t) with periodic boundary conditions possesses at least one nontrivial positive solution, where k is a constant, a(t) is a continuous T−periodic function and does not change the sign on [0; T], e(t) is a continuous T−periodic function and e(t) is not equal to 0. As applications, we will give some examples to illustrate the application of these theorems.