The equation \(\bigl (Fu'\bigr )(t)=\bigl (Gu\bigr )(t)+f(t)\), \(t\in {\mathbb {R}}\), where F and G are bounded linear operators, is considered. It is assumed that infinity is a pole of the resolvent of the pencil \(\lambda \mapsto \lambda F-G\) and the spectrum of the pencil is disjoint from the imaginary axis. Under these assumptions, to each free term f bounded on \({\mathbb {R}}\) (in the sense of distributions) there corresponds a unique bounded solution u and \(u(t)=\int _{-\infty }^{\infty }{\mathcal {G}}(s)f(t-s)\,ds\). The kernel \({\mathcal {G}}\) is called Green’s function. In this paper, the representations of Green’s function based on functional calculus in Banach algebras are constructed.

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微分代数方程有界解问题的格林函数表示

方程 \(\bigl (Fu'\bigr )(t)=\bigl (Gu\bigr )(t)+f(t)\), \(t\in {\mathbb {R}}\)，其中F 和 G 是有界线性算子，被考虑。假设无穷大是铅笔 \(\lambda \mapsto \lambda FG\) 的解析度的一个极点，并且铅笔的光谱与虚轴不相交。在这些假设下，对于以 \({\mathbb {R}}\) 为界的每个自由项 f（在分布的意义上），对应一个唯一的有界解 u 和 \(u(t)=\int _{-\ infty }^{\infty }{\mathcal {G}}(s)f(ts)\,ds\)。核\({\mathcal {G}}\) 称为格林函数。本文构建了基于泛函演算的格林函数在 Banach 代数中的表示。