# New Theorems for Oscillations to Differential Equations with Mixed Delays

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## Abstract

**:**

## 1. Introduction

- (a)
- $\nu \in C\left(\right[0,\infty ),\mathbb{R})$, $\zeta \in {C}^{2}([0,\infty ),\mathbb{R})$, $\nu \left(\iota \right)<\iota $, $\zeta \left(\iota \right)<\iota $, ${lim}_{\iota \to \infty}\nu \left(\iota \right)=\infty $, ${lim}_{\iota \to \infty}\zeta \left(\iota \right)=\infty $.
- (b)
- $\nu \in C\left(\right[0,\infty ),\mathbb{R})$, $\zeta \in {C}^{2}([0,\infty ),\mathbb{R})$, $\nu \left(\iota \right)>\iota $, $\zeta \left(\iota \right)<\iota $, ${lim}_{\iota \to \infty}\nu \left(\iota \right)=\infty $, ${lim}_{\iota \to \infty}\zeta \left(\iota \right)=\infty $.
- (c)
- $p\in {C}^{1}([0,\infty ),\mathbb{R})$, $r,\tilde{r}\in C([0,\infty ),\mathbb{R})$; $0<p\left(\iota \right)$, $0\le r\left(\iota \right)$, $0\le \tilde{r}\left(\iota \right)$, for all $\iota \ge 0$.
- (d)
- $q\in {C}^{2}([0,\infty ),{\mathbb{R}}_{+})$ with $0\le q\left(\iota \right)\le a<1$.
- (e)
- ${lim}_{\iota \to \infty}P\left(\iota \right)=\infty $ where $P\left(\iota \right)={\int}_{0}^{\iota}{p}^{-1/\alpha}\left(s\right)\phantom{\rule{0.166667em}{0ex}}\mathrm{d}s$.
- (f)
- $\alpha $ and $\beta $ are the quotient of two positive odd integers.

## 2. Preliminary Results

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Remark**

**1.**

## 3. Main Results

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Example**

**1.**

**Example**

**2.**

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Santra, S.S.; Majumder, D.; Bhattacharjee, R.; Bazighifan, O.; Khedher, K.M.; Marin, M.
New Theorems for Oscillations to Differential Equations with Mixed Delays. *Symmetry* **2021**, *13*, 367.
https://doi.org/10.3390/sym13030367

**AMA Style**

Santra SS, Majumder D, Bhattacharjee R, Bazighifan O, Khedher KM, Marin M.
New Theorems for Oscillations to Differential Equations with Mixed Delays. *Symmetry*. 2021; 13(3):367.
https://doi.org/10.3390/sym13030367

**Chicago/Turabian Style**

Santra, Shyam Sundar, Debasish Majumder, Rupak Bhattacharjee, Omar Bazighifan, Khaled Mohamed Khedher, and Marin Marin.
2021. "New Theorems for Oscillations to Differential Equations with Mixed Delays" *Symmetry* 13, no. 3: 367.
https://doi.org/10.3390/sym13030367