# Solution of Some Impulsive Differential Equations via Coupled Fixed Point

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## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

- (i)
- $d(e,m)=0\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\u27fa\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}e=m,$
- (ii)
- $d(e,m)=d(m,e)$, and
- (iii)
- $d(e,l)\le s\left[d\right(e,m)+d(m,l\left)\right]$

**Definition**

**2.**

**Definition**

**3.**

**Theorem**

**1.**

## 2. Main Result

**Theorem**

**2.**

**Proof.**

- (i)
- for any ${\left\{{l}_{n}\right\}}_{n\in \mathbb{N}}$ in M such that $({l}_{n},{l}_{n+1})\in E\left(G\right)$ and $\underset{n\to \infty}{lim}{l}_{n}=l$, then $({l}_{n},l)\in E\left(G\right)$,and
- (ii)
- for any ${\left\{{l}_{n}\right\}}_{n\in \mathbb{N}}$ in M, such that $({l}_{n+1},{l}_{n})\in E\left(G\right)$ and $\underset{n\to \infty}{lim}{l}_{n}=l$, then $(l,{l}_{n})\in E\left(G\right)$.

**Theorem**

**3.**

**Proof.**

**Remark**

**1.**

**Theorem**

**4.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

## 3. Application

**Assumption**

**1.**

- $f:J\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is continuous.
- ∀$w,z,u,v\in PC\left(\right[0,1\left]\right)$, with $w\le u$ and $z\le y$, we have$$f(t,w(t),z(t\left)\right)\le f(t,u(t),y(t\left)\right)\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}I\left(w\right(t),z(t\left)\right)\le I\left(u\right(t),v(t\left)\right)\phantom{\rule{1.em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}t\in [0,1];$$
- ∃$\alpha ,\beta ,\gamma \in [0,1)$ with $\sum _{i=0}^{\infty}{2}^{i}{\left(\frac{\beta +\gamma}{1-\alpha -\beta}\right)}^{i}<\infty $ such that$$\begin{array}{cc}\hfill |f(t,w\left(t\right),z\left(t\right))-f(t,u\left(t\right),v\left(t\right){|}^{2}& \le \frac{\alpha}{2}\frac{{|w\left(t\right)-f(t,w\left(t\right),z\left(t\right))|}^{2}{[1+|u\left(t\right)-f(t,u\left(t\right),v\left(t\right))|}^{2}]}{{1+|w\left(t\right)-u\left(t\right)|}^{2}}\hfill \\ & +\frac{\beta}{2}\left[{|w\left(t\right)-f(t,w\left(t\right),z\left(t\right))|}^{2}+{|u\left(t\right)-f(t,u\left(t\right),v\left(t\right))|}^{2}\right],\hfill \end{array}$$$${|I(w\left(t\right),z\left(t\right))-I(u\left(t\right),v\left(t\right))|}^{2}\le \frac{\gamma}{2}\left({|w\left(t\right)-u\left(t\right)|}^{2}\right)$$

**Theorem**

**5.**

**Proof.**

## 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**

Boudaoui, A.; Mebarki, K.; Shatanawi, W.; Abodayeh, K.
Solution of Some Impulsive Differential Equations via Coupled Fixed Point. *Symmetry* **2021**, *13*, 501.
https://doi.org/10.3390/sym13030501

**AMA Style**

Boudaoui A, Mebarki K, Shatanawi W, Abodayeh K.
Solution of Some Impulsive Differential Equations via Coupled Fixed Point. *Symmetry*. 2021; 13(3):501.
https://doi.org/10.3390/sym13030501

**Chicago/Turabian Style**

Boudaoui, Ahmed, Khadidja Mebarki, Wasfi Shatanawi, and Kamaleldin Abodayeh.
2021. "Solution of Some Impulsive Differential Equations via Coupled Fixed Point" *Symmetry* 13, no. 3: 501.
https://doi.org/10.3390/sym13030501