# On Suzuki Mappings in Modular Spaces

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries on Modular Vector Spaces

**Definition**

**1.**

- (1)
- $\rho \left(x\right)=0\iff x=0;$
- (2)
- $\rho (-x)=\rho \left(x\right),\phantom{\rule{1.em}{0ex}}\forall x\in X$;
- (3)
- $\rho (\alpha x+(1-\alpha y\left)\right)\le \rho \left(x\right)+\rho \left(y\right),\phantom{\rule{1.em}{0ex}}\forall \alpha \in [0,1]$.

**Definition**

**2.**

**Definition**

**3.**

- (a)
- $\left\{{x}_{\ell}\right\}$ is said to be modular-convergent (or ρ-convergent) to $x\in {X}_{\rho}$ if $\underset{\ell \to \infty}{lim}\rho ({x}_{\ell}-x)=0$. It is worth mentioning the uniqueness of the ρ-limit, whenever it exists.
- (b)
- $\left\{{x}_{\ell}\right\}\in {X}_{\rho}$ is called modular or ρ-Cauchy if $\underset{l,k\to \infty}{lim}\rho ({x}_{\ell}-{x}_{k})=0$.
- (c)
- If all the modular-Cauchy sequences are also modular-convergent, then the modular space ${X}_{\rho}$ is called ρ-complete.
- (d)
- A subset $S\subset {X}_{\rho}$ which contains the ρ-limits of all its modular-convergent sequences $\left\{{x}_{\ell}\right\}\subset S$ is called ρ-closed.
- (e)
- A subset $S\subset {X}_{\rho}$ satisfying ${\mathrm{diam}}_{\rho}\left(S\right)=sup\{\rho (x-y)\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}x,y\in S\}<\infty$ is referred to as ρ-bounded.
- (f)
- ρ satisfies the Fatou property if, for any sequence $\left\{{x}_{\ell}\right\}$ in ${X}_{\rho}$ which is ρ-convergent to $x\in {X}_{\rho}$, the inequality $\rho \left(x\right)\le \underset{\ell \to \infty}{lim\; inf}\rho \left({x}_{\ell}\right)$ holds.
- (g)
- ρ is said to satisfy condition ${\Delta}_{2}$ if $\rho \left(2x\right)\le K\rho \left(x\right),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\forall x\in {X}_{\rho}$, for a constant element $K\ge 0$. The minimal possible value of K is usually denoted by $\omega \left(2\right)$.

**Remark**

**1.**

**Definition**

**4.**

**Lemma**

**1.**

**Definition**

**5.**

- (1)
- the ρ-type function defined as$$\tau :S\to [0,\infty ],\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\tau \left(x\right)=\underset{\ell \to \infty}{lim\; sup}\rho (x-{x}_{\ell});$$
- (2)
- the asymptotic radius of $\left\{{x}_{\ell}\right\}$ with respect to S, meaning the value $r\left(S\right)=inf\left\{\tau \right(x)\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}x\in S\}$;
- (3)
- the asymptotic center of $\left\{{x}_{\ell}\right\}$ with respect to S defined as the set $A\left(S\right)=\{x\in S\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.166667em}{0ex}}\tau (x)=r(S\left)\right\}$;
- (4)
- the minimizing sequences of the ρ-type function, namely sequences $\left\{{c}_{\ell}\right\}$ in S satisfying $\underset{n\to \infty}{lim}\tau \left({c}_{\ell}\right)=r\left(S\right)$.

**Lemma**

**2.**

## 3. Condition (C) of Suzuki in Modular Spaces

**Definition**

**6.**

**Lemma**

**3.**

**Definition**

**7.**

**Lemma**

**4.**

- (i)
- for each $x\in {X}_{\rho}$, one has $\rho (\mathcal{M}x-{\mathcal{M}}^{2}x)\le \rho (x-\mathcal{M}x)$;
- (ii)
- for any $x,y\in {X}_{\rho}$ either $\frac{1}{\omega \left(2\right)}\rho (x-\mathcal{M}x)\le \rho (x-y)$ or $\frac{1}{\omega \left(2\right)}\rho (\mathcal{M}x-{\mathcal{M}}^{2}x)\le \rho (\mathcal{M}x-y)$.

**Proof.**

**Lemma**

**5.**

- (i)
- $\underset{\ell \to \infty}{lim\; sup}max\{{\mu}_{\ell},{\nu}_{\ell}\}=max\{\underset{\ell \to \infty}{lim\; sup}{\mu}_{\ell},\underset{\ell \to \infty}{lim\; sup}{\nu}_{\ell}\}$;
- (ii)
- let ${\eta}_{\ell}={\alpha}_{\ell}{\mu}_{\ell}+(1-{\alpha}_{\ell}){\nu}_{\ell}$, with ${\alpha}_{\ell}\in [0,1]$ convergent to a real number $\alpha \in [0,1]$. Then, $\underset{\ell \to \infty}{lim\; sup}{\eta}_{\ell}\le \alpha \underset{\ell \to \infty}{lim\; sup}{\mu}_{\ell}+(1-\alpha )\underset{\ell \to \infty}{lim\; sup}{\nu}_{\ell}$.

**Theorem**

**1.**

**Proof.**

**Lemma**

**6.**

**Corollary**

**1.**

**Proof.**

- $\rho \left(x\right)=\rho \left(\right|x\left|\right),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\forall x\in \mathbb{R}$;
- $\rho $ is a nondecreasing function on ${\mathbb{R}}_{+}$;
- $\rho $ is a continuous function.

**Remark**

**2.**

## 4. Kirk’s Lemma in Modular Spaces

**Lemma**

**7.**

- (i)
- ${x}_{\ell +1}$ is the point of $\sigma [{x}_{\ell},{y}_{\ell}]$ for which $d({x}_{\ell},{x}_{\ell +1})=\alpha d({x}_{\ell},{y}_{\ell})$;
- (ii)
- $d({y}_{\ell},{y}_{\ell +1})\le d({x}_{\ell},{x}_{\ell +1})$.

**Lemma**

**8.**

- (i)
- ${x}_{\ell +1}=(1-\alpha ){x}_{\ell}+\omega \left(\alpha \right){y}_{\ell};$
- (ii)
- $\rho ({y}_{\ell +1}-{y}_{\ell})\le \rho ({x}_{\ell +1}-{x}_{\ell}).$

**Proof.**

**Corollary**

**2.**

**Proof.**

**Example**

**1.**

- (i)
- ${x}_{\ell +1}=(1-\alpha ){x}_{\ell}+\alpha {y}_{\ell};$
- (ii)
- $\rho ({y}_{\ell +1}-{y}_{\ell})\le \rho ({x}_{\ell +1}-{x}_{\ell})$.

**Example**

**2.**

- (i)
- ${x}_{\ell +1}=(1-\alpha ){x}_{\ell}+{\alpha}^{2}{y}_{\ell};$
- (ii)
- $\rho ({y}_{\ell +1}-{y}_{\ell})\le \rho ({x}_{\ell +1}-{x}_{\ell}).$

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Bejenaru, A.; Postolache, M.
On Suzuki Mappings in Modular Spaces. *Symmetry* **2019**, *11*, 319.
https://doi.org/10.3390/sym11030319

**AMA Style**

Bejenaru A, Postolache M.
On Suzuki Mappings in Modular Spaces. *Symmetry*. 2019; 11(3):319.
https://doi.org/10.3390/sym11030319

**Chicago/Turabian Style**

Bejenaru, Andreea, and Mihai Postolache.
2019. "On Suzuki Mappings in Modular Spaces" *Symmetry* 11, no. 3: 319.
https://doi.org/10.3390/sym11030319