# Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients: The Case When the Monodromy Matrix Has Simple Eigenvalues

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## Abstract

**:**

## 1. Introduction

## 2. Notations and Definitions

**Remark**

**1.**

**Proof.**

## 3. Background, Previous Results and the Main Result

**Proposition**

**1.**

**Remark**

**2.**

**i**) The matrices $B,C$ and D in Equation (6) are orthogonal projections, that is

**ii**) In addition, $B,C,$ and D are nonzero matrices.

**Proof.**

**Theorem**

**1.**

**1.**The linear recurrence in $\mathbb{C}$

**2.**The eigenvalues of $A\left(q\right)$ verify the condition

**A1**] ${B}_{n+q}={B}_{n}$ and ${P}_{n+q}={P}_{n}$, for all $n\in {\mathbb{Z}}_{+}$ and some positive integer $q.$

**A2**] ${P}_{n}^{2}={P}_{n}$, for all $n\in {\mathbb{Z}}_{+},$ that is, $\mathcal{P}$ is a family of projections.

**A3**] ${B}_{n}{P}_{n}={P}_{n+1}{B}_{n}$, for all $n\in {\mathbb{Z}}_{+}.$ In particular, this yields that ${B}_{n}x\in ker\left({P}_{n+1}\right)$ for each $x\in ker\left({P}_{n}\right).$

**A4**] For each $n\in {\mathbb{Z}}_{+},$ the map

- (
**i**) - $\parallel {U}_{\mathcal{B}}(n,k){P}_{k}\parallel \le {N}_{1}{e}^{-{\nu}_{1}(n-k)}$ for all $n\ge k\ge 0.$
- (
**ii**) - $\parallel {U}_{\mathcal{B}}(n,k)(I-{P}_{k})\parallel \le {N}_{2}{e}^{{\nu}_{2}(n-k)}$ for all $0\le n<k.$

**Theorem**

**2.**

**A1**]–[

**A4**] above. The following four statements are equivalent:

**1**) The monodromy operator $B\left(q\right):={B}_{q-1}\cdots {B}_{0}$ is hyperbolic (that is, the spectrum of $B\left(q\right)$ does not intersect the unit circle $\mathsf{\Gamma}=\{w\in \mathbb{C}:|w|=1\}$, or equivalently (with the terminology in [9]) it possesses a discrete dichotomy.

**2**) The family $\mathcal{B}$ is $\mathcal{P}$-dichotomic.

**3**) For each bounded sequence ${\left({G}_{n}\right)}_{n\in {\mathbb{Z}}_{+}},{G}_{0}=0$ (of X-valued functions) there exists a unique bounded solution (starting from $ker\left({P}_{0}\right))$ of the difference equation.

**4**) The family $\mathcal{B}$ is Hyers–Ulam stable.

**2**) and (

**3**) still works when X is an infinite dimensional Banach space (see [20]). We use Theorem 2 to prove

**2**⇒

**1**in Theorem 1.

**Lemma**

**1.**

## 4. Proofs

**Proof**

**of**

**Lemma**

**1.**

**Case 1.1.**Let ${b}_{13}\ne 0$. Set

**Case 1.2.**Let ${b}_{23}\ne 0$. Arguing as above we can show that $\left({\phi}_{n+1}\right)$ is unbounded, that is that $\left({\phi}_{n}\right)$ is unbounded as well.

**Case 1.3.**Analogously, we can treat the case ${b}_{33}\ne 0$.

**Case 2.1.**Let ${b}_{13}={b}_{23}={b}_{33}=0$ and ${b}_{12}\ne 0$. Set

**Case 2.2.**Let ${b}_{22}\ne 0$. Similar to the previous case, we can show that $\left({\phi}_{n+1}\right)$ is unbounded, that is that $\left({\phi}_{n}\right)$ is unbounded as well.

**Case 3.**Let ${b}_{12}={b}_{13}=0$ and ${b}_{11}\ne 0$. Then, set

**ii**). □

**Proof**

**of**

**Theorem**

**1.**

**1**⇒

**2.**We argue by contradiction. Suppose that $\sigma \left(A\right(q\left)\right)$ intersects the unit circle. Without loss of generality, assume that x is an eigenvalue of $A\left(q\right)$ and $\left|x\right|=1.$ Let ${Y}_{0}$ and ${X}_{0}$ be as in the Remark 1. From Lemma 1, it follows that the sequence in Equation (14) with $\left({Y}_{0}-{X}_{0}\right)$ instead of ${Z}_{0}$ is unbounded and this contradicts Equation (4).

**2**⇒

**1.**From the assumption and Theorem 2, it follows that the system ${X}_{n+1}={A}_{n}{X}_{n}$ is Hyers–Ulam stable. Thus, for a certain positive constant $L,$ every $\epsilon >0,$ every sequence $\left({f}_{n}\right)$, every ${Y}_{0}$ and some ${X}_{0}$ one has

## 5. An Example

**Example**

**1.**

**Remark**

**3.**

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Buşe, C.; O’Regan, D.; Saierli, O.
Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients: The Case When the Monodromy Matrix Has Simple Eigenvalues. *Symmetry* **2019**, *11*, 339.
https://doi.org/10.3390/sym11030339

**AMA Style**

Buşe C, O’Regan D, Saierli O.
Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients: The Case When the Monodromy Matrix Has Simple Eigenvalues. *Symmetry*. 2019; 11(3):339.
https://doi.org/10.3390/sym11030339

**Chicago/Turabian Style**

Buşe, Constantin, Donal O’Regan, and Olivia Saierli.
2019. "Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients: The Case When the Monodromy Matrix Has Simple Eigenvalues" *Symmetry* 11, no. 3: 339.
https://doi.org/10.3390/sym11030339