Abstract
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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The first author is supported by the Russian Foundation for Basic Research under research project No. 19-01-00732 A.
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Communicated by Ti-Jun Xiao.
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Kurbatova, I.V., Pechkurov, A.V. Representations of Green’s function of the bounded solutions problem for a differential-algebraic equation. Banach J. Math. Anal. 14, 707–736 (2020). https://doi.org/10.1007/s43037-019-00036-y
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DOI: https://doi.org/10.1007/s43037-019-00036-y