# An estimate of Green's function of the problem of bounded solutions in the case of a triangular coefficient

## TL;DR

This paper provides an estimate for Green's function in bounded solutions problems for differential equations with triangular matrix coefficients, extending Van Loan's matrix exponential estimate.

## Contribution

It introduces a generalized estimate of Green's function for differential equations with triangular matrix coefficients, building upon Van Loan's matrix exponential estimate.

## Key findings

- Derived a new estimate for Green's function in bounded solutions problems.
- Extended Van Loan's matrix exponential estimate to Green's function.
- Applicable to differential equations with triangular matrix coefficients.

## Abstract

An estimate of Green's function of the bounded solutions problem for the ordinary differential equation $x'(t)-Bx(t)=f(t)$ is proposed. It is assumed that the matrix coefficient $B$ is triangular. This estimate is a generalization of the estimate of the matrix exponential proved by Ch. F. Van Loan.