带分段常数自变量的反射方程
Reflection equation with piecewise constant argument
AI总结：
本研究聚焦带反射与分段常数自变量的非局部微分方程，通过构造格林函数推导线性问题解的显式表达式，提出常号参数刻画公式，用单调方法证明非线性解的存在性，是首次结合两类方程开展的研究，可应用于生物医学等领域。
摘要：
在本研究中，我们针对非局部微分方程展开研究，重点关注自变量带反射且依赖关系为分段常数的一类方程。我们的研究思路包括：通过构建对应的格林函数推导线性问题解的显式表达式，同时提出一个新的公式，用以刻画所分析方程中涉及的、能让格林函数保持常号的参数集合。此外，我们利用单调方法证明了非线性问题解的存在性。\n上述方法被专门应用于带周期条件的线性问题$v'(t) + mv(-t) + Mv([t]) = h(t)$（其中$t \in [-T,T]$），我们证明了相关非线性问题的若干存在性结果，并精确划定了格林函数$H_{m,M}$在其定义域上保持常号的区域。\n所研究的这类方程有望应用于生物医学、量子力学等领域。此外，本研究取得了重要进展——据作者所知，这是首次将带对合的方程与带分段常数自变量的方程结合起来开展研究。
英文摘要：
In this work, we study nonlocal differential equations with particular focus on those with reflection in their argument and piecewise constant dependence. The approach entails deriving the explicit expression of the solution to the linear problem by constructing the corresponding Green's function, as well as developing a novel formula to delineate the set of parameters involved in the analyzed equations for which the Green's function exhibits a constant sign. Furthermore, we demonstrate the existence of solutions for nonlinear problems through the utilisation of the monotone method. The aforementioned methodology is specifically applied to the linear problem with periodic conditions $v'(t) + mv(-t) + Mv([t]) = h(t)$ for $t \in [-T,T]$, proving several existence results for the associated nonlinear problem and precisely delimiting the region where the Green's function $H_{m,M}$ has a constant sign on its domain of definition. The equations studied have the potential to be applied in fields such as biomedicine or quantum mechanics. Furthermore, this work represents a significant advance, as it is, as far the authors know, the first time that equations with involution and piecewise constant arguments have been studied together.
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