与耦合不同边值条件的Hill方程相关的Green函数之间的关系
Relationship of the Green's functions related to the Hill's equation coupled to different boundary value conditions
AI总结：
本文研究Hill方程在不同边值条件下Green函数之间的关系，通过线性组合表示不同问题的Green函数，并利用其常符号性质及不动点理论证明非线性问题解的存在性。
摘要：
本文推导与耦合各种边值条件的Hill方程相关的Green函数的若干性质。特别地，其思路是研究二阶微分算子耦合Neumann、Dirichlet、周期和混合边值条件时的Green函数，方法是将给定问题的Green函数表示为其他问题Green函数的线性组合。这将使我们能够在这些Green函数具有常符号时对其进行比较。最后，线性问题Green函数的这些性质将成为推导非线性问题解的存在性的基础。这些结果来源于应用于Banach空间中适当锥上相关算子的不动点理论。
英文摘要：
In this paper we will deduce several properties of the Green's functions related to the Hill's equation coupled to various boundary value conditions. In particular, the idea is to study the Green's functions of the second order differential operator coupled to Neumann, Dirichlet, Periodic and Mixed boundary conditions, by expressing the Green's function of a given problem as a linear combination of the Green's function of the other ones. This will allow us to compare different Green's functions when they have constant sign. Finally, such properties of the Green's function of the linear problem will be fundamental to deduce the existence of solutions to the nonlinear problem. The results are derived from the fixed point theory applied to related operators defined on suitable cones in Banach spaces.
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