本项目主要研究Fucík谱意义下的跨共振的Sturm-Liouville边值问题。作为基本的边值条件之一，Sturm-Liouville问题在力学、物理学等领域均具有重要的应用背景，其解的存在性和多重性一直是微分方程定性理论研究的热点。然而对跨共振的一般的Sturm-Liouville问题，人们的研究还很少。对于跨一个共振点情形的Sturm-Liouville边值问题，我们已经得到算子谱集的特征，并给出了最优可解性条件，以及解的具体表达形式。我们将以这些为工作基础，研究跨多个共振点的情况，并给出相应的理论结果。然后，利用这些结果，对Fucík谱意义下的跨共振情形，分析讨论解的存在性和多重性，并将其应用于一般的非线性边值问题。最终，我们旨在建立一种以最优控制理论研究微分方程边值问题解的存在性和多重性问题的框架和方法。

In this project we study the Sturm-Liouville problems across resonance in sence of Fucík spectrum. As one of the basic boundary value conditions, Sturm-Liouville problems possess the important application background in mechanics, physics and other fields. The existence and multiplicity of solutions have always been the hot spot topics in the qualitative theory of ordinary differential equations. However, the research on the Strum-Liouville problems across resonance is still rare. We have proved some optimal results on the existence and uniqueness to the Sturm-Liouville boundary value problems across one resonance point. In addition, we also obtain the spectral expression of the operator under the Sturm-Liouville boundary condition. Based on our previous work, we will consider the case of many resonance points and draw corresponding conclusions from them. Through the application of these results, the Sturm-Liouville problems across resonance in sence of Fucík spectrum can be derived. And we apply these results to the generally nonlinear boundary value problems. Ultimately, our purpose is to find a research method of the solutions to the boundary value problems via the optimal control theory.