# Contractive Inequalities for Some Asymptotically Regular Set-Valued Mappings and Their Fixed Points

^{1}

^{2}

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## Abstract

**:**

## 1. Preliminaries

**Definition**

**1.**

**Lemma**

**1.**

- 1.
- $\mathsf{\Delta}(\mu ,V)\le \delta (\mu ,\gamma )$for any$\gamma \in V$and$\mu \in X$;
- 2.
- $\mathsf{\Delta}(\mu ,V)\le \mathcal{PH}(U,V)$for any$\mu \in U$.

**Lemma**

**2.**

**Lemma**

**3.**

**Definition**

**2.**

**Definition**

**3.**

**Remark**

**1.**

- 1.
- R is$\mathcal{PH}$-continuous on a subset S of X if it is continuous on every point of S.
- 2.
- If R is a set-valued contraction, then it is$\mathcal{PH}$-continuous.

**Definition**

**4.**

## 2. Geraghty-Type Contractive Inequality

**Definition**

**5.**

**Definition**

**6.**

**Theorem**

**1.**

**Proof.**

**Example**

**1.**

## 3. Boyd and Wong-Type Contractive Inequality

- $\psi \left(t\right)<t$ for all $t>0$,
- $\psi $ is upper semi-continuous from right (i.e., ${t}_{n}\to t>0$ implies that ${lim\; sup}_{n\to \infty}\psi \left({t}_{n}\right)\le \psi \left(t\right)$).

**Theorem**

**2.**

**Proof.**

**Example**

**2.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Debnath, P.; de La Sen, M.
Contractive Inequalities for Some Asymptotically Regular Set-Valued Mappings and Their Fixed Points. *Symmetry* **2020**, *12*, 411.
https://doi.org/10.3390/sym12030411

**AMA Style**

Debnath P, de La Sen M.
Contractive Inequalities for Some Asymptotically Regular Set-Valued Mappings and Their Fixed Points. *Symmetry*. 2020; 12(3):411.
https://doi.org/10.3390/sym12030411

**Chicago/Turabian Style**

Debnath, Pradip, and Manuel de La Sen.
2020. "Contractive Inequalities for Some Asymptotically Regular Set-Valued Mappings and Their Fixed Points" *Symmetry* 12, no. 3: 411.
https://doi.org/10.3390/sym12030411