In this paper, we use the Leray–Schauder degree theory to study the following singular periodic problems: $x^{″}+f(x^{\prime })+g(t,x)=s$, $x(0)-x(T)=0=x^{\prime }(0)-x^{\prime }(T)$, where $f:ℝ\to ℝ$ is a continuous function with $f(0)=0$, function $g:ℝ/T\mathbb{Z}\times {ℝ}_{+}\to ℝ$ is continuous with an attractive singularity at the origin, and $s$ is a constant. We consider the case where the friction term $f$ satisfies a local superlinear growth condition but not necessarily of the Nagumo type, and function $g$ does not need to satisfy coercivity conditions. An Ambrosetti–Prodi type result is obtained.

中文翻译：

---

没有矫顽力条件的 Rayleigh 方程的奇异周期 Ambrosetti-Prodi 问题

在本文中，我们使用 Leray-Schauder 度数理论来研究以下奇异周期问题：$X^{”}+F(X^{\text{'}})+G(吨,X)=s$,$X(0)-X(吨)=0=X^{\text{'}}(0)-X^{\text{'}}(吨)$， 在哪里$F：ℝ\to ℝ$是一个连续函数$F(0)=0$， 功能$G：ℝ/吨\mathbb{Z}\times {ℝ}_{+}\to ℝ$是连续的，在原点有一个有吸引力的奇点，并且$s$是一个常数。我们考虑摩擦项的情况$F$满足局部超线性生长条件，但不一定是 Nagumo 类型，并且函数$G$不需要满足矫顽力条件。获得了Ambrosetti-Prodi类型的结果。