In this paper, we study the existence of nontrivial solutions to the following nonlinear differential equation with derivative term:$$\begin{aligned} {\left\{ \begin{array}{l}u''(t)+a(t)u(t)=f\big (t,u(t),u'(t)\big ),\quad t\in [0,\omega ],\\ u(0)=u(\omega ),\quad u'(0)=u'(\omega ),\end{array}\right. } \end{aligned}$$where a: \([0,\omega ]\rightarrow \mathbb {R}^{+}\big (\mathbb {R}^{+}=[0,+\infty )\big )\) is a continuous function with \(a(t)\not \equiv 0\), f: \([0,\omega ]\times \mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}\) is continuous and may be sign-changing and unbounded from below. Without making any nonnegative assumption on nonlinearity, using the first eigenvalue corresponding to the relevant linear operator and the topological degree, the existence of nontrivial solutions to the above periodic boundary value problem is established in \(C^1[0,\omega ]\). Finally, an example is given to demonstrate the validity of our main result.

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具有导数项的非线性二阶周期边值问题的非平凡解的存在性

在本文中，我们研究了以下带有导数项的非线性微分方程的非平凡解的存在：$$ \ begin {aligned} {\ left \ {\ begin {array} {l} u''（t）+ a（ t）u（t）= f \ big（t，u（t），u'（t）\ big），\ quat t \ in [0，\ omega]，\\ u（0）= u（\ omega ），\ quad u'（0）= u'（\ omega），\ end {array} \ right。} \ {端对齐} $$其中一个：\（[0，\ω-\ RIGHTARROW \ mathbb {R} ^ {+} \大（\ mathbb {R} ^ {+} = [0，+ \ infty） \ big）\）是具有\（a（t）\ not \ equiv 0 \），f的连续函数：\（[0，\ omega] \ times \ mathbb {R} \ times \ mathbb {R} \ rightarrow \ mathbb {R} \）是连续的，并且可能会发生变化且从下方不受限制。在不对非线性进行任何非负假设的情况下，使用与相关线性算子和拓扑度相对应的第一特征值，在\（C ^ 1 [0，\ omega] \中建立了上述周期边值问题的非平凡解的存在性）。最后，通过一个例子来说明我们主要结果的有效性。