# A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay

## Abstract

**:**

## 1. Introduction and Preliminary Results

**Theorem**

**1**

- (i)
- If$$\underset{t\to \infty}{lim\; inf}{\int}_{t-\tau \left(t\right)}^{t}p\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds>\frac{1}{e},$$
- (ii)
- If$$\underset{t\to \infty}{lim\; sup}{\int}_{t-\tau \left(t\right)}^{t}p\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds<\frac{1}{e},$$$${\int}_{t-\tau \left(t\right)}^{t}p\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds\le \frac{1}{e}\phantom{\rule{1.em}{0ex}}for\phantom{\rule{4.pt}{0ex}}all\phantom{\rule{4.pt}{0ex}}large\phantom{\rule{4.pt}{0ex}}t,$$

**Theorem**

**2**

**.**Let the function τ in Equation (1) be constant, and function p be nonnegative, bounded and uniformly continuous. Assume further that the function $t\mapsto {\int}_{t-\tau}^{t}p\left(s\right)\phantom{\rule{0.166667em}{0ex}}ds$ is slowly varying at infinity. Then,

## 2. Results

**Theorem**

**3.**

**Lemma**

**1**

**Proof**

**of**

**Theorem**

**3.**

**Lemma**

**2.**

**Lemma**

**3.**

**Proof**

**of**

**Lemma**

**2.**

**Theorem**

**4.**

**Proof.**

## 3. Example

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Garab, Á.
A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay. *Symmetry* **2019**, *11*, 1332.
https://doi.org/10.3390/sym11111332

**AMA Style**

Garab Á.
A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay. *Symmetry*. 2019; 11(11):1332.
https://doi.org/10.3390/sym11111332

**Chicago/Turabian Style**

Garab, Ábel.
2019. "A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay" *Symmetry* 11, no. 11: 1332.
https://doi.org/10.3390/sym11111332