# Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients

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## Abstract

**:**

## 1. Introduction

## 2. Definitions and Notations

**Definition**

**1.**

_{j}) is called an ε-approximative solution of the linear recurrence

**Definition**

**2.**

## 3. Background and the Main Result

**Proposition**

**1.**

**Proposition**

**2.**

**Remark**

**1.**

**Theorem**

**1.**

**Corollary**

**1.**

**Remark**

**2.**

**Lemma**

**1.**

## 4. Proofs

**Proof.**

**Case I.**Let $x=y=z$ and $\left|x\right|=1$. We use the notation of the previous sections.

**I.1.**When $B={\left({b}_{rs}\right)}_{r,s\in \{1,2,3\}}\ne {0}_{3}$, there exists a pair $(i,j)$ with $i,j\in \{1,2,3\}$ such that ${b}_{ij}\ne 0$. We analyze three cases:

**I.1.1.**Let $j=3$ and ${b}_{13}\ne 0$. Set

**I.1.2.**Let $j=2$ and ${b}_{12}\ne 0$. Set

**I.1.3.**Let $j=1$ and ${b}_{11}\ne 0$. Set

**I.2.**Let $B={0}_{3}$ and $C\ne {0}_{3}$ be of the form

**I.2.1.**Let $j=3$ and ${c}_{13}\ne 0$. Set ${F}_{k}={F}_{k}^{1}$. As above, we have

**I.2.2.**Let $j=2$ and ${c}_{12}\ne 0$. Set ${F}_{k}={F}_{k}^{2}$. Then, we obtain

**I.2.3.**let $j=1$ and ${c}_{11}\ne 0$. Set ${F}_{k}={F}_{k}^{3}$. As in the previous cases, we obtain

**I.3.**When $B={0}_{3}$ and $C={0}_{3}$, then $A{\left(q\right)}^{n}={x}^{n}{I}_{3}$ for all $n\in {\mathbb{Z}}_{+}$. Set ${F}_{k}={F}_{k}^{1}$. Then,

**Case II.**When the characteristic polynomial ${p}_{A\left(q\right)}\left(\lambda \right)$ is given by

**II.1.**When $\left|x\right|=1$ and $\left|y\right|<1.$

**II.1.1.**When $B\ne {0}_{3}$, let us first assume that ${b}_{13}\ne 0$ and set ${F}_{k}={F}_{k}^{1}$. Then,

**II.1.2.**When ${b}_{12}\ne 0$, set ${F}_{k}={F}_{k}^{2}$. Then,

**I.1.3.**, so we omit the details.

**II.1.2.**Let $B={0}_{3}$. As $C\ne {0}_{3}$ (see Remark 1), we can proceed in a similar manner as in Case

**I.2.**.

**II.2.**When $\left|x\right|=\left|y\right|=1.$

**II.2.1.**Let $\left|x\right|=\left|y\right|=1$ and $B={0}_{3}$. Let ${F}_{k}={F}_{k}^{1}$ and ${u}_{0}$ be as defined above. An easy calculation yields

**II.2.2.**When $\left|x\right|=\left|y\right|=1$ and $B\ne {0}_{3}$, it can be treated like in Case

**II.1.1.**.

**II. 3.**When $\left|x\right|=1$ and $\left|y\right|>1$. Taking into account that D is not the zero matrix (of order 3), we can choose a sequence $\left({F}_{k}\right)$ and a pair $(i,j)$ such that the sequence ${\left({\left[{P}_{y}\mathsf{\Phi}\left(nq\right)\right]}_{i,j}\right)}_{n}$ is unbounded (we omit the details).

**Proof.**

## 5. Examples

**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. *Symmetry* **2019**, *11*, 512.
https://doi.org/10.3390/sym11040512

**AMA Style**

Buşe C, O’Regan D, Saierli O.
Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients. *Symmetry*. 2019; 11(4):512.
https://doi.org/10.3390/sym11040512

**Chicago/Turabian Style**

Buşe, Constantin, Donal O’Regan, and Olivia Saierli.
2019. "Hyers-Ulam Stability for Linear Differences with Time Dependent and Periodic Coefficients" *Symmetry* 11, no. 4: 512.
https://doi.org/10.3390/sym11040512