# Set-Valued Interpolative Hardy–Rogers and Set-Valued Reich–Rus–Ćirić-Type Contractions in b-Metric Spaces

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- (1)
- $\delta (u,v)=0$ if and only if $u=v$;
- (2)
- $\delta (u,v)=\delta (v,u)$ for all $u,v\in \mathsf{\Omega}$;
- (3)
- there exists a real number $s\ge 1$ such that $\delta (u,z)\le s\left[\delta \right(u,v)+\delta (v,z\left)\right]$ for all $u,v,z\in \mathsf{\Omega}$.

**Definition**

**2.**

**Theorem**

**1.**

**Lemma**

**1.**

**Remark**

**1.**

**Lemma**

**2.**

**Remark**

**2.**

## 3. Main Results

**Definition**

**3.**

**Theorem**

**2.**

**Proof.**

**Example**

**1.**

**Definition**

**4.**

**Theorem**

**3.**

**Proof.**

**Example**

**2.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Banach, S. Sur les opérations dans les ensembles abstraits et leur applications aux tequations integrales. Fund. Math.
**1922**, 3, 133–181. [Google Scholar] [CrossRef] - Kannan, R. Some results on fixed points. Bull. Calc. Math. Soc.
**1968**, 60, 71–77. [Google Scholar] - Chatterjea, S.K. Fixed point theorems. C. R. Acad. Bulgare Sci.
**1972**, 25, 727–730. [Google Scholar] [CrossRef] - Ćirić, L. Fixed point theorems for multivalued contractions in complete metric spaces. J. Math. Anal. Appl.
**2008**, 348, 499–507. [Google Scholar] [CrossRef] - Meir, A.; Keeler, E. A theorem on contraction mappings. J. Math. Anal. Appl.
**1969**, 28, 326–329. [Google Scholar] [CrossRef] [Green Version] - Boyd, D.W.; Wong, J.S. On nonlinear contractions. Proc. Am. Math. Soc.
**1969**, 20, 458–464. [Google Scholar] [CrossRef] - Park, S. Some general fixed point theorems on topological vector spaces. Appl. Set-Valued Anal. Optim.
**2019**, 1, 19–28. [Google Scholar] - Petrusel, A. Local fixed point results for graphic contractions. J. Nonlinear Var. Anal.
**2019**, 3, 141–1448. [Google Scholar] - Zaslavski, A.J. Two fixed point results for a class of mappings of contractive type. J. Nonlinear Var. Anal.
**2018**, 2, 113–119. [Google Scholar] - Bakhtin, I.A. The contraction mapping principle in almost metric spaces. Funct. Anal.
**1989**, 30, 26–37. [Google Scholar] - Czerwik, S. Contraction mappings in b-metric spaces. Acta Math. Univ. Ostrav.
**1993**, 1, 5–11. [Google Scholar] - Alharbi, N.; Aydi, H.; Felhi, A.; Ozel, C.; Sahmim, S. α-contractive mappings on rectangular b-metric spaces and an application to integral equations. J. Math. Anal.
**2018**, 9, 47–60. [Google Scholar] - Ali, M.U.; Kamran, T.; Postolache, M. Solution of Volterra integgral inclusion in b-metric spaces via new fixed point theorem. Nonlinear Anal. Modell. Control
**2017**, 22, 17–30. [Google Scholar] [CrossRef] - Aydi, H.; Chen, C.M.; Karapinar, E. Interpolative Ćirić-Reich-Rus type contractions via the Branciari distance. Mathematics
**2019**, 7, 84. [Google Scholar] [CrossRef] - Aydi, H.; Karapinar, E.; Hierro, A.F.R. ω-Interpolative Ćirić-Reich-Rus-type contractions. Mathematics
**2019**, 7, 57. [Google Scholar] [CrossRef] - Collaco, P.; Silva, J.C. A complete comparison of 25 contraction conditions. Nonlinear Anal. Theory Methods Appl.
**1997**, 30, 471–476. [Google Scholar] [CrossRef] - Ege, O. Complex valued rectangular b-metric spaces and an application to linear equations. J. Nonlinear Sci. Appl.
**2015**, 8, 1014–1021. [Google Scholar] [CrossRef] - Jeong, G.S.; Rhoades, B.E. Maps for which F(T) = F(T
^{n}). Fixed Point Theory Appl.**2005**, 6, 71–105. [Google Scholar] - Kamran, T.; Postolache, M.; Ali, M.U.; Kiran, Q. Feng and Liu type F-contraction in b-metric spaces with application to integral equations. J. Math. Anal.
**2016**, 7, 18–27. [Google Scholar] - Rhoades, B.E. A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc.
**1977**, 226, 257–290. [Google Scholar] [CrossRef] - Shatanawi, W.; Pitea, A.; Lazović, R. Contraction conditions using comparison functions on b-metric spaces. Fixed Point Theory Appl.
**2014**, 2014, 135. [Google Scholar] [CrossRef] - Debnath, P.; Mitrović, Z.; Radenović, S. Interpolative Hardy-Rogers and Reich-Rus-Ćirić type contractions in b-metric spaces and rectangular b-metric spaces. Math. Vesnik.
**2019**. to appear. [Google Scholar] - Karapinar, E.; Agarwal, R.P.; Aydi, H. Interpolative Reich-Rus-Ćirić type contractions on partial metric spaces. Mathematics
**2018**, 6, 256. [Google Scholar] [CrossRef] - Karapinar, E.; Alahtani, O.; Aydi, H. On interpolative Hardy-Rogers type contractions. Symmetry
**2018**, 11, 8. [Google Scholar] [CrossRef] - Markin, J.T. A fixed point theorem for set- valued mappings. Bull. Am. Math. Soc.
**1968**, 74, 639–640. [Google Scholar] [CrossRef] - Nadler, S.B. Multi-valued contraction mappings. Pac. J. Math.
**1969**, 30, 475–488. [Google Scholar] [CrossRef] - Kamran, T. Mizoguchi-Takahashi’s type fixed point theorem. Comput. Math. Appl.
**2009**, 57, 507–511. [Google Scholar] [CrossRef] - Berinde, M.; Berinde, V. On a general class of multivalued weakly picard mappings. J. Math. Anal. Appl.
**2007**, 326, 772–782. [Google Scholar] [CrossRef] - Miculescu, R.; Mihail, A. New fixed point theorems for set-valued contractions in b-metric spaces. J. Fixed Point Theory Appl.
**2017**, 19, 2153–2163. [Google Scholar] [CrossRef]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Debnath, P.; de La Sen, M.
Set-Valued Interpolative Hardy–Rogers and Set-Valued Reich–Rus–Ćirić-Type Contractions in *b*-Metric Spaces. *Mathematics* **2019**, *7*, 849.
https://doi.org/10.3390/math7090849

**AMA Style**

Debnath P, de La Sen M.
Set-Valued Interpolative Hardy–Rogers and Set-Valued Reich–Rus–Ćirić-Type Contractions in *b*-Metric Spaces. *Mathematics*. 2019; 7(9):849.
https://doi.org/10.3390/math7090849

**Chicago/Turabian Style**

Debnath, Pradip, and Manuel de La Sen.
2019. "Set-Valued Interpolative Hardy–Rogers and Set-Valued Reich–Rus–Ćirić-Type Contractions in *b*-Metric Spaces" *Mathematics* 7, no. 9: 849.
https://doi.org/10.3390/math7090849