# On the Exponents of Exponential Dichotomies

^{1}

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## Abstract

**:**

## 1. Introduction

**Theorem**

**1**

**.**Let $\dot{x}=A\left(t\right)x$ have an exponential dichotomy on ${\mathbb{R}}_{+}$ with exponents α and β. Given $0<\tilde{\alpha}<\alpha $ and $0<\tilde{\beta}<\beta $ there exists $\epsilon >0$ such that if $B\left(t\right)$ is a piecewise continuous matrix such that ${sup}_{t\in {\mathbb{R}}_{+}}\left|B\left(t\right)\right|<\epsilon $ then the linear system $\dot{x}=[A\left(t\right)+B\left(t\right)]x$ has an exponential dichotomy on ${\mathbb{R}}_{+}$ with exponents $\tilde{\alpha},\tilde{\beta}$ (but the constant may be larger).

**Theorem**

**2.**

## 2. Properties of Exponential Dichotomies

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Corollary**

**1.**

**Proof.**

## 3. The Main Result

**Theorem**

**3.**

**Proof.**

**Proposition 3.**

## 4. Asymptotically Constant Matrices

- (A1)
- $\underset{t\to \infty}{lim}A\left(t\right)=A$ and ${\int}_{0}^{\infty}|A\left(t\right)-A|dt<\infty$;
- (A2)
- A has two semi-simple eigenvalues $-\alpha <0$ and $\beta >0$;
- (A3)
- there exists $\mu >0$ such that all others eigenvalues $\lambda $ of A satisfy either $\Re \lambda \le -(\alpha +\mu )$ or $\Re \lambda \ge \beta +\mu $.

**Proposition**

**4.**

**Theorem**

**4.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Battelli, F.; Fečkan, M.
On the Exponents of Exponential Dichotomies. *Mathematics* **2020**, *8*, 651.
https://doi.org/10.3390/math8040651

**AMA Style**

Battelli F, Fečkan M.
On the Exponents of Exponential Dichotomies. *Mathematics*. 2020; 8(4):651.
https://doi.org/10.3390/math8040651

**Chicago/Turabian Style**

Battelli, Flaviano, and Michal Fečkan.
2020. "On the Exponents of Exponential Dichotomies" *Mathematics* 8, no. 4: 651.
https://doi.org/10.3390/math8040651