# Existence of the Solutions of Nonlinear Fractional Differential Equations Using the Fixed Point Technique in Extended b-Metric Spaces

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction and Preliminaries

**Definition**

**1**

- 1.
- $d(x,y)=0$ if and only if $x=y$;
- 2.
- $d(x,y)=d(y,x)$;
- 3.
- $d(x,z)\le s\left[d\right(x,y)+d(y,z\left)\right]$.

**Example**

**1**

**.**The space ${L}_{p}(0<p<1)$ of all real functions $x\left(t\right)$, $t\in [0,1]$ such that

**Definition**

**2**

**.**Let $(X,d)$ be a b-metric space. Let p be a positive integer; $p\ge 2$, ${A}_{1},{A}_{2},\dots ,{A}_{p}$ be nonempty and closed subsets of $X,Y={\displaystyle \bigcup _{i=1}^{p}}{A}_{i}$ and $T:Y\to Y$. Then, T is called a cyclic operator if

- 1.
- ${A}_{i},i\in \{1,2,\dots p\}$ are nonempty subsets;
- 2.
- $T\left({A}_{1}\right)\subseteq {A}_{2},\dots ,T\left({A}_{p-1}\right)\subseteq {A}_{p},T\left({A}_{p}\right)\subseteq {A}_{1}$.

**Definition**

**3.**

- 1.
- ${d}_{\theta}(x,y)=0$ if and only if $x=y$;
- 2.
- ${d}_{\theta}(x,y)=d(y,x)$;
- 3.
- ${d}_{\theta}(x,z)\le \theta (x,z)[d(x,y)+d(y,z)]$.

**Definition**

**4.**

- (i)
- Convergent if and only if $x\in X$ exists such that ${d}_{\theta}({x}_{n},x)\to 0$ as $n\to \infty $ we write $\underset{n\to \infty}{lim}{x}_{n}=x.$
- (ii)
- Cauchy if and only if ${d}_{\theta}({x}_{n},{x}_{m})\to 0$ as $n,m\to \infty .$

**Lemma**

**1.**

## 2. Fixed Point Results

**Theorem**

**1.**

- (i)
- $T\left({A}_{i}\right)\subseteq {A}_{i+1}$, for all $i\in \{1,2,\dots ,p\}$;
- (ii)
- $d(Tx,Ty)\le \lambda d(x,y)$ for all $x\in {A}_{i},y\in {A}_{i+1}$ where $\lambda \in [0,1)$ be such that for each $x\in X,\underset{n,m\to \infty}{lim}\theta ({x}_{n},{x}_{m})<\frac{1}{\lambda}$ where ${x}_{n}={T}^{n}\left(x\right)$, $n=1,2,\dots $.

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Example**

**2.**

## 3. Applications to Nonlinear Fractional Differential Equations

**Theorem**

**4.**

**Theorem**

**5.**

**Proof.**

**Remark**

**1.**

**Theorem**

**6.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Guran, L.; Bota, M.-F.
Existence of the Solutions of Nonlinear Fractional Differential Equations Using the Fixed Point Technique in Extended *b*-Metric Spaces. *Symmetry* **2021**, *13*, 158.
https://doi.org/10.3390/sym13020158

**AMA Style**

Guran L, Bota M-F.
Existence of the Solutions of Nonlinear Fractional Differential Equations Using the Fixed Point Technique in Extended *b*-Metric Spaces. *Symmetry*. 2021; 13(2):158.
https://doi.org/10.3390/sym13020158

**Chicago/Turabian Style**

Guran, Liliana, and Monica-Felicia Bota.
2021. "Existence of the Solutions of Nonlinear Fractional Differential Equations Using the Fixed Point Technique in Extended *b*-Metric Spaces" *Symmetry* 13, no. 2: 158.
https://doi.org/10.3390/sym13020158