# New Extensions of Kannan’s and Reich’s Fixed Point Theorems for Multivalued Maps Using Wardowski’s Technique with Application to Integral Equations

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction and Preliminaries

**Theorem**

**1.**

**Theorem**

**2.**

**Definition**

**1.**

**(F1)**F is strictly increasing, i.e., for all $u,v\in \left(0,+\infty \right),u<v$ implies $F\left(u\right)<F\left(v\right);$

**(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 exists $t\in \left(0,1\right)$ such that ${lim}_{u\to {0}^{+}}{u}^{t}F\left(u\right)=0.$

**Definition**

**2.**

**Remark**

**1.**

- 1.
- In the MS $\left(\mathcal{CB}\right(\Im ),\mathcal{H})$, $\theta \in \Im $ is a fixed point of Υ if and only if $\Delta (\theta ,\mathrm{{\rm Y}}\theta )=0$.
- 2.
- The metric function $\zeta :\Im \times \Im \to [0,\infty )$ is continuous in the sense that if $\left\{{\theta}_{n}\right\},\left\{{\vartheta}_{n}\right\}$ are two sequences in ℑ with $({\theta}_{n},{\vartheta}_{n})\to (\theta ,\vartheta )$ for some $\theta ,\vartheta \in \Im $, as $n\to \infty $, then $\zeta ({\theta}_{n},{\vartheta}_{n})\to \zeta (\theta ,\vartheta )$ as $n\to \infty $. Similarly, the function Δ is continuous because if ${\theta}_{n}\to \theta $ as $n\to \infty $, then $\Delta ({\theta}_{n},\mathcal{A})\to \Delta (\theta ,\mathcal{A})$ as $n\to \infty $ for any $\mathcal{A}\subseteq \Im $.

**Lemma**

**1.**

- 1.
- $\Delta (\mu ,\mathcal{B})\le \zeta (\mu ,\gamma )$ if $\gamma \in \mathcal{B}$ and $\mu \in \Im $;
- 2.
- $\Delta (\mu ,\mathcal{B})\le \mathcal{H}(\mathcal{A},\mathcal{B})$ if $\mu \in \mathcal{A}$.

**Lemma**

**2.**

**Definition**

**3.**

**Remark**

**2.**

**Definition**

**4.**

**Remark**

**3.**

## 2. Multivalued Kannan Type F-contraction

**Definition**

**5.**

**Theorem**

**3.**

**Proof.**

**Case I:**There exists a subsequence $\left\{{\theta}_{{n}_{k}}\right\}$ of $\left\{{\theta}_{n}\right\}$ such that $\Gamma {\theta}_{{n}_{k}}=\Gamma \theta $ for all $k\in \mathbb{N}$.

**Case II:**There exists ${n}_{1}\in \mathbb{N}$ such that $\Gamma {\theta}_{n}\ne \Gamma \theta $ for all $n\ge {n}_{1}$. Then

**Remark**

**4.**

**Example**

**1.**

**Example**

**2.**

## 3. Multivalued Reich Type F-Contraction

**Definition**

**6.**

**Theorem**

**4.**

**Proof.**

**Remark**

**5.**

- 1.
- In Theorem 4, if we take $c=0$, then Theorem 3 is obtained. Thus GMKFC introduced in this paper is a particular case of GMRFC when $c=0$.
- 2.
- Theorem 4 is more general than Theorem 2.7 in [21] in terms of relaxation of degrees of freedom of the constants involved.

## 4. An Application to Integral Equations

**Theorem**

**5.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Kannan, R. Some results on fixed points. Bull. Calcutta Math. Soc.
**1968**, 60, 71–77. [Google Scholar] - Subrahmanyam, P.V. Completeness and fixed points. Monatsh. Math.
**1975**, 80, 325–330. [Google Scholar] [CrossRef] - Reich, S. Some remarks concerning contraction mappings. Can. Math. Bull.
**1971**, 14, 121–124. [Google Scholar] [CrossRef] - Wardowski, D. Fixed points of a new type of contractive mappings in complete metric space. Fixed Point Theory Appl.
**2012**, 2012, 94. [Google Scholar] [CrossRef] [Green Version] - Nadler, S.B. Multi-valued contraction mappings. Pac. J. Math.
**1969**, 30, 475–488. [Google Scholar] [CrossRef] - Boriceanu, M.; Petrusel, A.; Rus, I. Fixed point theorems for some multivalued generalized contraction in b-metric spaces. Int. J. Math. Stat.
**2010**, 6, 65–76. [Google Scholar] - Czerwik, S. Nonlinear set-valued contraction mappings in b-metric spaces. Atti Sem. Mat. Univ. Modena
**1998**, 46, 263–276. [Google Scholar] - Reich, S. Fixed points of contractive functions. Boll. Unione Mat. Ital.
**1972**, 4, 26–42. [Google Scholar] - Altun, I.; Minak, G.; Dag, H. Multivalued F-contractions on complete metric spaces. J. Nonlinear Convex Anal.
**2015**, 16, 659–666. [Google Scholar] [CrossRef] [Green Version] - Aydi, H.; Chen, C.M.; Karapinar, E. Interpolative Ćirić-Reich-Rus type contractions via the Branciari distance. Mathematics
**2019**, 7, 84. [Google Scholar] [CrossRef] [Green Version] - Aydi, H.; Karapinar, E.; Hierro, A.F.R. ω-Interpolative Ćirić-Reich-Rus-type contractions. Mathematics
**2019**, 7, 57. [Google Scholar] [CrossRef] [Green Version] - Bojor, F. Fixed point theorems for Reich type contractions on metric spaces with a graph. Nonlinear Anal. Theory Methods Appl.
**2012**, 75, 3895–3901. [Google Scholar] [CrossRef] - Bojor, F. Fixed points of Kannan mappings in metric spaces endowed with a graph. An. Univ. Ovidius Constanta Ser. Mat.
**2012**, 20, 31–40. [Google Scholar] [CrossRef] - Choudhury, B.S.; Kundu, A. A Kannan-like contraction in partially ordered spaces. Demonstr. Math.
**2013**, XLVI. [Google Scholar] [CrossRef] [Green Version] - 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. [Google Scholar] [CrossRef] [Green Version] - Debnath, P.; de La Sen, M. Fixed points of interpolative Ćirić-Reich-Rus-type contractions in b-metric spaces. Symmetry
**2020**, 12, 12. [Google Scholar] [CrossRef] [Green Version] - 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**. in Press. [Google Scholar] - Debnath, P.; Neog, M.; Radenović, S. Set valued Reich type G-contractions in a complete metric space with graph. Rend. Circ. Mat. Palermo II Ser.
**2019**. in Press. [Google Scholar] - Gornicki, J. Fixed point theorems for Kannan type mappings. J. Fixed Point Theory Appl.
**2017**, 19, 2145–2152. [Google Scholar] [CrossRef] [Green Version] - Karapinar, E.; Agarwal, R.P.; Aydi, H. Interpolative Reich-Rus-Ćirić type contractions on partial metric spaces. Mathematics
**2018**, 6, 256. [Google Scholar] [CrossRef] [Green Version] - Mohammadi, B.; Parvaneh, V.; Aydi, H. On extended interpolative Ćirić-Reich-Rus type F-contractions and an application. J. Inequal. Appl.
**2019**, 2019, 290. [Google Scholar] [CrossRef] - Sgroi, M.; Vetro, C. Multi-valued F-contractions and the solution of certain functional and integral equations. Filomat
**2013**, 27, 1259–1268. [Google Scholar] [CrossRef] [Green Version] - Ali, M.U.; Kamran, T. Multi-valued F-contractions and related fixed point theorems with an application. Filomat
**2016**, 30, 3779–3793. [Google Scholar] [CrossRef] [Green Version]

© 2020 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.; Srivastava, H.M.
New Extensions of Kannan’s and Reich’s Fixed Point Theorems for Multivalued Maps Using Wardowski’s Technique with Application to Integral Equations. *Symmetry* **2020**, *12*, 1090.
https://doi.org/10.3390/sym12071090

**AMA Style**

Debnath P, Srivastava HM.
New Extensions of Kannan’s and Reich’s Fixed Point Theorems for Multivalued Maps Using Wardowski’s Technique with Application to Integral Equations. *Symmetry*. 2020; 12(7):1090.
https://doi.org/10.3390/sym12071090

**Chicago/Turabian Style**

Debnath, Pradip, and Hari Mohan Srivastava.
2020. "New Extensions of Kannan’s and Reich’s Fixed Point Theorems for Multivalued Maps Using Wardowski’s Technique with Application to Integral Equations" *Symmetry* 12, no. 7: 1090.
https://doi.org/10.3390/sym12071090