#
Common Fixed-Point and Fixed-Circle Results for a Class of Discontinuous F-Contractive Mappings †

^{†}

## Abstract

**:**

## 1. Introduction and Preliminaries

**Definition**

**1.**

**(F1)**- F is strictly increasing;
**(F2)**- For each sequence ${\left\{{u}_{n}\right\}}_{n\in \mathbb{N}}\subset \left(0,+\infty \right)$, ${lim}_{n\to +\infty}{u}_{n}=0$ if and only if ${lim}_{n\to +\infty}F\left({u}_{n}\right)=-\infty ;$
**(F3)**- There is $t\in \left(0,1\right)$ such that ${lim}_{u\to {0}^{+}}{u}^{t}F\left(u\right)=0.$

## 2. Common Fixed Point with Discontinuity of the Contraction

**Theorem**

**1.**

- (i)
- $\delta +F(\eta (\mathsf{\Phi}\theta ,\mathsf{\Psi}\xi ))\le F(\mathsf{\Gamma}({\mathsf{\Lambda}}_{0}(\theta ,\xi )))$ for all $\theta ,\xi \in \mathcal{W}$, where $\mathsf{\Gamma}:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ has the property $\mathsf{\Gamma}(s)<s$ for each $s>0$;
- (ii)
- For a given $\u03f5>0$, there exist $\kappa >0$ such that $\u03f5<{\mathsf{\Lambda}}_{0}(\theta ,\xi )<\u03f5+\kappa $ implies that $\eta (\mathsf{\Phi}\theta ,\mathsf{\Psi}\xi )\le \u03f5$.

**Proof.**

**Remark**

**1.**

**Example**

**1.**

## 3. A Fixed-Circle Result

**Theorem**

**2.**

- (i)
- For all $\theta \in {C}_{{\theta}_{0},\rho}$, there exists $\tau >0$ such that$$\rho \le \Delta (\theta ,{\theta}_{0})<\rho +\tau \u27f9\eta (\mathsf{\Phi}\theta ,{\theta}_{0})\le \rho ;$$
- (ii)
- For all $\theta \in \mathcal{W}$,$$\eta (\mathsf{\Phi}\theta ,\theta )>0\u27f9\eta (\mathsf{\Phi}\theta ,\theta )\le \mathsf{\Gamma}(\Delta (\theta ,{\theta}_{0})),$$

**Proof.**

**Example**

**2.**

## 4. Conclusions and Future Work

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Debnath, P.
Common Fixed-Point and Fixed-Circle Results for a Class of Discontinuous *F*-Contractive Mappings *Mathematics* **2022**, *10*, 1605.
https://doi.org/10.3390/math10091605

**AMA Style**

Debnath P.
Common Fixed-Point and Fixed-Circle Results for a Class of Discontinuous *F*-Contractive Mappings *Mathematics*. 2022; 10(9):1605.
https://doi.org/10.3390/math10091605

**Chicago/Turabian Style**

Debnath, Pradip.
2022. "Common Fixed-Point and Fixed-Circle Results for a Class of Discontinuous *F*-Contractive Mappings *Mathematics* 10, no. 9: 1605.
https://doi.org/10.3390/math10091605