AI 中文总结
本文推导具有线性泛函扰动的微分方程的格林函数与伴随问题，建立不同泛函方程的联系，刻画格林函数常号区域并以示例验证。
AI 中文摘要
本文研究一类受两点边值条件约束的泛函微分方程，其泛函依赖通过形如\\(\sum_{k=1}^{l}\gamma_{k}(t)\mathcal{C}_{k}(u)\\)的算子引入，其中\\(\mathcal{C}_{k}:C(I) \rightarrow \mathbb{R}\\)（\\(k=1,\ldots,l\\)，\\(I:=[a,b]\\)）为线性连续算子，且对所有\\(k=1,\ldots,l\\)，\\(\gamma_{k} \in \mathcal{L}^{1}(I)\\)。该形式涵盖分段常自变量方程及积分型依赖方程等多种类型。我们通过推导并刻画其格林函数、计算相关伴随问题来分析这类方程，该方法可建立不同类型泛函方程间的联系，将分段常自变量方程与脉冲微分方程、非局部边值问题关联起来。此外，我们推导了一系列比较原理与结果，用于刻画原问题及其伴随问题的格林函数保持常号的区域。最后，通过代表性示例说明上述理论发现。
英文摘要
In this paper we study a functional differential equation subject to two-point boundary value conditions, where the functional dependence is introduced through an operator of the form \begin{equation*} \sum_{k=1}^{l}γ_{k}(t)\mathcal{C}_{k}(u), \end{equation*} where $\mathcal{C}_{k}:C(I) \rightarrow \mathbb{R}$, $k=1, \ldots l$, ($I:=[a,b]$) are linear continuous operators and $γ_{k} \in \mathcal{L}^{1}(I)$ for all $k=1, \ldots, l$. This formulation encompasses, among others, equations with piecewise constant arguments and those with integral-type dependence. We analyze this class of equations by deducing and characterizing their Green's function, as well as by computing the related adjoint problem. This approach enables us to establish connections among different types of functional equations and to relate equations with piecewise constant arguments to impulsive differential equations and non local boundary value problems. Next, we develop a series of comparison principles and results that allow us to characterize the regions where the Green's function of the original problem and of its adjoint maintain a constant sign. Finally, we illustrate the theoretical finding with representative examples.
Comments45 pages, 7 figures
