#
Fixed Points of Kannan Maps in the Variable Exponent Sequence Spaces ℓ_{p(·)}

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Basic Notation and Terminology

**Definition**

**1**

**Proposition**

**1**

- (i)
- $\upsilon \left(x\right)=0$ if and only if $x=0$;
- (ii)
- $\upsilon \left(\gamma x\right)=\upsilon \left(x\right)$, if $\left|\gamma \right|=1$;
- (iii)
- $\upsilon (s\phantom{\rule{4pt}{0ex}}x+(1-s\left)\phantom{\rule{4pt}{0ex}}y\right)\le s\phantom{\rule{4pt}{0ex}}\upsilon \left(x\right)+(1-s)\phantom{\rule{4pt}{0ex}}\upsilon \left(y\right)$, for any $s\in [0,1]$,

**Definition**

**2**

- 1.
- A sequence $\left\{{x}_{n}\right\}\subset {\ell}_{p(\xb7)}$ is υ-convergent to $x\in {\ell}_{p(\xb7)}$ if and only if $\upsilon ({x}_{n}-x)\to 0$. Note that the υ-limit is unique if it exists.
- 2.
- A sequence $\left\{{x}_{n}\right\}\subset {\ell}_{p(\xb7)}$ is υ-Cauchy if $\upsilon ({x}_{n}-{x}_{m})\to 0$ as $n,m\to \infty $.
- 3.
- A set $C\subset {\ell}_{p(\xb7)}$ is υ-closed if for any υ-converging sequence $\left\{{x}_{n}\right\}\subset C$ to x one has $x\in C$.
- 4.
- A set $C\subset {\ell}_{p(\xb7)}$ is υ-bounded if ${\delta}_{\upsilon}\left(C\right)=sup\{\upsilon (x-y);x,y\in C\}<\infty $.
- 5.
- The x-centered υ-ball of radius r is defined as$${B}_{\upsilon}(x,r)=\{y\in {\ell}_{p(\xb7)};\phantom{\rule{0.277778em}{0ex}}\upsilon (x-y)\le r\},$$

**Definition**

**3.**

## 3. Modular Kannan Mappings in ${\ell}_{p(\xb7)}$

**Definition**

**4.**

- (1)
- Kannan υ-contraction if $L<1/2$;
- (2)
- Kannan υ-nonexpansive if $L=1/2$.

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Definition**

**5.**

- (i)
- ${\ell}_{p(\xb7)}$ satisfies the property $\left(R\right)$ if any decreasing sequence of nonempty υ-bounded and υ-closed convex subsets have a nonempty intersection;
- (ii)
- ${\ell}_{p(\xb7)}$ satisfies the υ-quasi-normal property if for any nonempty υ-bounded and υ-closed convex subset K with more than one point, there exists $x\in K$ such that$$\upsilon (x-y)<{\delta}_{\upsilon}\left(K\right)=sup\{\upsilon (a-b);\phantom{\rule{4pt}{0ex}}a,b\in K\},$$

**Remark**

**1.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Remark**

**2.**

**Corollary**

**2.**

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Abdou, A.A.N.; Khamsi, M.A. Fixed Points of Kannan Maps in the Variable Exponent Sequence Spaces *ℓ*_{p(·)}. *Mathematics* **2020**, *8*, 76.
https://doi.org/10.3390/math8010076

**AMA Style**

Abdou AAN, Khamsi MA. Fixed Points of Kannan Maps in the Variable Exponent Sequence Spaces *ℓ*_{p(·)}. *Mathematics*. 2020; 8(1):76.
https://doi.org/10.3390/math8010076

**Chicago/Turabian Style**

Abdou, Afrah A. N., and Mohamed Amine Khamsi. 2020. "Fixed Points of Kannan Maps in the Variable Exponent Sequence Spaces *ℓ*_{p(·)}" *Mathematics* 8, no. 1: 76.
https://doi.org/10.3390/math8010076