# The Gelfand--Shilov type estimate for Green's function of the bounded solutions problem

## TL;DR

This paper establishes a Gelfand-Shilov type estimate for Green's function associated with the bounded solutions problem in ordinary differential equations, extending classical exponential estimates to a broader context.

## Contribution

The paper introduces a novel Gelfand-Shilov type estimate for Green's function, generalizing the matrix exponential estimate to bounded solutions of ODEs.

## Key findings

- Derived a new estimate for Green's function similar to Gelfand-Shilov bounds.
- Extended classical exponential estimates to bounded solutions framework.
- Provided theoretical foundation for analyzing stability and boundedness in ODEs.

## Abstract

An analog of the Gelfand--Shilov estimate of the matrix exponential is proved for Green's function of the problem of bounded solutions of the ordinary differential equation $x'(t)-Ax(t)=f(t)$.