Green's function of the problem of bounded solutions in the case of a block triangular coefficient

TL;DR

This paper investigates the Green's function for bounded solutions of linear differential equations where the coefficient matrix has a block triangular structure, expressing the kernel as convolutions of Green's functions of diagonal blocks.

Contribution

It extends the theory of Green's functions to block triangular operator matrices, providing a new representation involving convolutions of diagonal block Green's functions.

Findings

Green's function blocks are sums of convolutions of diagonal block Green's functions.

Provides a new representation for Green's function in block triangular cases.

Applicable to bounded solutions when the spectrum avoids the imaginary axis.

Abstract

It is known that the equation $x'(t)=Ax(t)+f(t)$, where $A$ is a bounded linear operator, has a unique bounded solution $x$ for any bounded continuous free term~$f$ if and only if the spectrum of the coefficient $A$ does not intersect the imaginary axis. The solution can be represented in the form \begin{equation*} x(t)=\int_{-\infty}^{\infty}\mathcal G(s)f(t-s)\,ds. \end{equation*} The kernel $\mathcal G$ is called Green's function. In this paper, the case when $A$ admits a representation by a block triangular operator matrix is considered. It is shown that the blocks of $\mathcal G$ are sums of special convolutions of Green's functions of diagonal blocks of $A$.