# An Operator Method for the Stability of Inhomogeneous Wave Equations

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Theorem**

**1.**

- (i)
- ${\int}_{{a}_{0}}^{t}p(s)ds$ exists for each $t\in I$;
- (ii)
- ${\int}_{{a}_{0}}^{t}q(y)exp\left\{{\int}_{{a}_{0}}^{y}p(s)ds\right\}dy$ exists for any $t\in I$.Moreover, assume that $\phi :I\to [0,\infty )$ is a function such that:
- (iii)
- ${\int}_{{a}_{0}}^{b}\phi (y)exp\left\{{\int}_{{a}_{0}}^{y}p(s)ds\right\}dy$ exists.

## 3. Main Results

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Remark**

**1.**

## 4. Discussions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Theorem 1 | X | t | a | b | ${\mathit{a}}_{0}$ | $\mathit{v}(\mathit{t})$ | $\mathit{p}(\mathit{t})$ | $\mathit{q}(\mathit{t})$ | $\mathit{\phi}(\mathit{t})$ |
---|---|---|---|---|---|---|---|---|---|

(17) | $\mathbb{R}$ | r | 0 | c | ${r}_{0}$ | $w(r)$ | $\frac{(n-1){c}^{2}}{r({c}^{2}-{r}^{2})}-\frac{1}{c+r}$ | $\frac{g(r)}{c-r}$ | $\frac{k\phi (r)}{c-r}$ |

Theorem 1 | X | t | a | b | ${\mathit{a}}_{0}$ | $\mathit{v}(\mathit{t})$ | $\mathit{p}(\mathit{t})$ | $\mathit{q}(\mathit{t})$ | $\mathit{\phi}(\mathit{t})$ |
---|---|---|---|---|---|---|---|---|---|

(20) | $\mathbb{R}$ | r | 0 | c | ${r}_{0}$ | $v(r)$ | 0 | $-\frac{{w}_{0}(r)}{c+r}$ | $\text{therightsideof}$ (20) |

Theorem 1 | X | t | a | b | ${\mathit{a}}_{0}$ | $\mathit{v}(\mathit{t})$ | $\mathit{p}(\mathit{t})$ | $\mathit{q}(\mathit{t})$ | $\mathit{\phi}(\mathit{t})$ |
---|---|---|---|---|---|---|---|---|---|

(33) | $\mathbb{R}$ | r | c | ∞ | ${r}_{0}$ | $w(r)$ | $\frac{(n-1){c}^{2}}{r({c}^{2}-{r}^{2})}-\frac{1}{c+r}$ | $\frac{g(r)}{c-r}$ | $\frac{k\phi (r)}{r-c}$ |

Theorem 1 | X | t | a | b | ${\mathit{a}}_{0}$ | $\mathit{v}(\mathit{t})$ | $\mathit{p}(\mathit{t})$ | $\mathit{q}(\mathit{t})$ | $\mathit{\phi}(\mathit{t})$ |
---|---|---|---|---|---|---|---|---|---|

(35) | $\mathbb{R}$ | r | c | ∞ | ${r}_{0}$ | $v(r)$ | 0 | $-\frac{{w}_{0}(r)}{c+r}$ | $\text{therightsideof}$ (35) |

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**MDPI and ACS Style**

Choi, G.; Jung, S.-M.; Roh, J.
An Operator Method for the Stability of Inhomogeneous Wave Equations. *Symmetry* **2019**, *11*, 324.
https://doi.org/10.3390/sym11030324

**AMA Style**

Choi G, Jung S-M, Roh J.
An Operator Method for the Stability of Inhomogeneous Wave Equations. *Symmetry*. 2019; 11(3):324.
https://doi.org/10.3390/sym11030324

**Chicago/Turabian Style**

Choi, Ginkyu, Soon-Mo Jung, and Jaiok Roh.
2019. "An Operator Method for the Stability of Inhomogeneous Wave Equations" *Symmetry* 11, no. 3: 324.
https://doi.org/10.3390/sym11030324