# Fixed Points Theorems for Unsaturated and Saturated Classes of Contractive Mappings in Banach Spaces

^{1}

^{2}

^{3}

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^{†}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1**

**Theorem**

**2**

**Definition**

**1**

## 2. Unsaturated Classes of Contractive Mappings

**Definition**

**2.**

**Remark**

**1.**

**Definition**

**3.**

#### 2.1. Banach Contractions

**Example**

**1.**

**Proposition**

**1.**

**Theorem**

**3**

#### 2.2. Kannan Contractions

**Example**

**2.**

**Proposition**

**2.**

**Theorem**

**4**

#### 2.3. Chatterjea Contractions

**Example**

**3.**

**Proposition**

**3.**

**Theorem**

**5**

#### 2.4. Almost Contractions

**Example**

**4**

**Proposition**

**4.**

**Theorem**

**6**

**Remark**

**2.**

#### 2.5. Nonexpansive Mappings

**Example**

**5**

**Proposition**

**5.**

**Theorem**

**7**

**Remark**

**3.**

## 3. Saturated Classes of Contractive Mappings

#### 3.1. Strictly Pseudo-Contractive Mappings

**Example**

**6.**

**Theorem**

**8.**

**Proof.**

**Remark**

**4.**

#### 3.2. Demicontractive Mappings

**Example**

**7**

**Theorem**

**9.**

**Proof.**

**Remark**

**5.**

**Open**

**problem**

**1.**

**Open**

**problem**

**2.**

## 4. Conclusions and Further Study

- 1.
- Based on the technique of enriching contractive type mappings T by means of the averaged operator ${T}_{\lambda}$, we introduced the concept of saturated class of contractive mappings;
- 2.
- We have shown that, from this perspective, the contractive type mappings in metric fixed-point theory could be separated into two distinct classes, unsaturated contractive mappings and saturated contractive mappings;
- 3.
- We identified some important unsaturated mapping classes by surveying some significant fixed-point results reported recently for five remarkable classes of contractive mappings. We have shown that the technique of enriching contractive type mappings provided genuine new fixed-point results for all mappings;
- 4.
- In the second part of the paper, we also identified two important saturated classes of contractive mappings: the class of strictly pseudocontractive mappings and the class of demicontractive mappings. The essential feature of these saturated classes of mappings is that they cannot be expanded by means of the technique of enriching contractive mappings;
- 5.
- We also formulated two open problems, which are intended to establish whether Ćirić-Reich-Rus contractions and Ćirić quasi contractions are saturated or unsaturated contraction classes;
- 6.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Berinde, V.; Păcurar, M.
Fixed Points Theorems for Unsaturated and Saturated Classes of Contractive Mappings in Banach Spaces. *Symmetry* **2021**, *13*, 713.
https://doi.org/10.3390/sym13040713

**AMA Style**

Berinde V, Păcurar M.
Fixed Points Theorems for Unsaturated and Saturated Classes of Contractive Mappings in Banach Spaces. *Symmetry*. 2021; 13(4):713.
https://doi.org/10.3390/sym13040713

**Chicago/Turabian Style**

Berinde, Vasile, and Mădălina Păcurar.
2021. "Fixed Points Theorems for Unsaturated and Saturated Classes of Contractive Mappings in Banach Spaces" *Symmetry* 13, no. 4: 713.
https://doi.org/10.3390/sym13040713