# Some New Observations for F-Contractions in Vector-Valued Metric Spaces of Perov’s Type

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## Abstract

**:**

## 1. Introduction and Preliminaries

**Definition 1**

**.**Let $\left(\right)$ be a vector-valued metric space and $T:X\to X$ be a map. If there exist $F\in {\mathcal{F}}^{m}$ and $\tau ={\left(\right)}_{{\tau}_{i}}^{}i=1m$ such that:

**(F1)**F is strictly increasing in each variable, i.e., for all ${\left(\right)}^{-}m,{\left(\right)}^{-}m$ such that $u\prec v,$ and, then, $F\left(u\right)\prec F\left(v\right),$

**(F2)**For each sequence, $\left(\right)open="\{"\; close="\}">{u}_{n}$ of ${\mathbb{R}}_{+}^{m}={\left(\right)}^{0}m$

**(F3)**There exists $k\in \left(\right)open="("\; close=")">0,1$ such that ${lim}_{{u}_{i}\to {0}^{+}}{u}_{i}^{k}{v}_{i}=0$ for each $i\in \left(\right)open="\{"\; close="\}">1,2,\dots ,m$ where:

**Theorem 1.**

**.**Let $\left(\right)$ be a complete vector-valued metric space and $T:X\to X$ be a Perov type F-contraction. Then, T has a unique fixed point.

**Remark 1.**

**(F1)**–

**(F3)**are equal to those introduced by Wardowski in his work of 2012 [3]. Note also that it is sufficient to consider only the case $m=2$, i.e., Banach space $\left(\right)$ where the norm ${\u2225.\u2225}_{e}$ is Euclidean, and it is ordered by the cone $P=\left\{\right(a,b):a\ge 0$ and $b\ge 0\}$. Cases $m=3,4,\dots $ add only the burden of writing but do not change the essence of the results. Thus, in all announced articles for vector-valued metric spaces of Perov’s type, $d(u,v)$ can be written as an ordered pair $d\left(\right)open="("\; close=")">u,v)$ of non-negative real numbers ${d}_{1}\left(\right)open="("\; close=")">u,v$ and ${d}_{2}\left(\right)open="("\; close=")">u,v$. The functions ${d}_{i}:{X}^{2}\to [0,+\infty )$ are in fact pseudometrics defined on the non-empty set X (for more details, see [16], Proposition 2.1).

## 2. Main Results

**(F1)**–

**(F3)**from [3] for all $i=1,2,\dots ,m.$

**(F1)**. Indeed, this follows from ([18], Corollary 2) or from ([22], Theorem 2.1).

**Definition 2.**

**Theorem 2.**

**Proof.**

**Remark 2.**

**(F1)**in Theorem 2, we get a second direction of generalization of main results from [2]. In our approach, the method of the system of inequalities (11) shows that many (and maybe all) well-known results in the setting of ordinary metric spaces with known contractive conditions (see [31,32,33]) are equivalent to the corresponding ones in generalized metric spaces of Perov’s type.

**Theorem 3.**

**1**. f has a unique fixed point ${u}^{*}\in X;$

**2.**The Picard iterative sequence ${u}_{n}={f}^{n}\left(u\right),n\in \mathbb{N}$ converges to ${u}^{*}$ for all $u\in X;$

**3.**$d\left(\right)open="("\; close=")">{u}_{n},{u}^{*}\left(\right)open="("\; close=")">d\left(\right)open="("\; close=")">{u}_{0},{u}_{1}$

**4.**if $g:X\to X$ satisfies the condition $d\left(\right)open="("\; close=")">f\left(u\right),g\left(u\right)$ for all $u\in X$ and some $c\in {\mathbb{R}}^{m},$ then, for the sequence ${v}_{n}={g}^{n}\left(\right)open="("\; close=")">{u}_{0}$ the following inequality

**Proof.**

**Corollary 1.**

**Proof.**

**Corollary 2.**

**Proof.**

**Example 1.**

**Case 1.**$u=0,v={u}_{n}.$ Then, (31) became:

**Case 2.**$u={u}_{n},v={u}_{m},n<m.$ In this case, (31) became:

**Remark 3.**

- generalized partial metric space;
- generalized metric like space;
- generalized b-metric space;
- generalized partial b-metric space;
- generalized b-metric like space.

- partial metric space;
- metric like space;
- b-metric space;
- partial b-metric space;
- b-metric like space.

**Example 2.**Let $\left(\right)open="("\; close=")">\mathcal{X},{\overline{d}}^{ml}$ where $\mathcal{X}=\left(\right)open="("\; close=")">C\left(\right)open="["\; close="]">0,1$ is the set of real continuous functions on $\left(\right)$ and ${\overline{d}}^{ml}\left(\right)open="("\; close=")">u,v\left(\right)open="("\; close=")">\left(\right)open="|"\; close="|">u\left(t\right)$ for all $u,v\in \left(\right)open="("\; close=")">C\left(\right)open="["\; close="]">0,1.$ This is an example of metric-like space that is not a partial metric space.

**Definition 3.**

**Theorem 4.**

## 3. Applications in Nonlinear Fractional Differential Equations

**Theorem 5.**

**(i)**There exists ${\nu}_{0}\in \left(\right)open="("\; close=")">C\left(\right)open="["\; close="]">0,1$ such that $\xi \left(\right)open="("\; close=")">{\nu}_{0}\left(t\right),\underset{0}{\overset{t}{\int}}\mathcal{T}{\nu}_{0}\left(t\right)$ for all $t\in \left(\right)open="["\; close="]">0,1$ where the map $\mathcal{T}:C\left(\right)open="("\; close=")">\left(\right)open="["\; close="]">0,1\to C\left(\right)open="("\; close=")">\left(\right)open="["\; close="]">0,1$ is defined as:

**(ii)**There exists $\tau 0$ such that for all $\mu ,\nu \in \mathcal{X}$

**(iii)**For each $t\in \left(\right)open="["\; close="]">0,1$ and $\mu ,\nu \in C\left(\right)open="("\; close=")">\left(\right)open="["\; close="]">0,1,\xi \left(\right)open="("\; close=")">\nu \left(t\right),\mu \left(t\right)$ yields $\xi \left(\right)open="("\; close=")">\mathcal{T}\mu \left(t\right),\mathcal{T}\left(\nu \right)$

**(iv)**For each $t\in \left(\right)open="["\; close="]">0,1$ if $\left(\right)$ is a sequence in $C\left(\right)open="("\; close=")">\left(\right)open="["\; close="]">0,1$ such that ${\nu}_{n}\to \nu $ in $C\left(\right)open="("\; close=")">\left(\right)open="["\; close="]">0,1$ and $\xi \left(\right)open="("\; close=")">{\nu}_{n}\left(t\right),{\nu}_{n-1}\left(t\right)$ for all $n\in \mathbb{N},$ then $\xi \left(\right)open="("\; close=")">{\nu}_{n}\left(t\right),\nu \left(t\right)$ for all $n\in \mathbb{N}.$

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

- Banach, S. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math.
**1922**, 3, 133–181. [Google Scholar] [CrossRef] - Altun, I.; Olgun, M. Fixed point results for Perov type F-contractions and an application. J. Fixed Point Theory Appl.
**2020**, 22, 46. [Google Scholar] - Wardowski, D. Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl.
**2012**, 2012, 94. [Google Scholar] [CrossRef] [Green Version] - Kurepa, Đ.R. Tableaux ramifies d’ensambles. C. R. Acad. Sci. Paris
**1934**, 198, 1563–1565. [Google Scholar] - Perov, A.I. On Cauchy problem for a system of ordinary differential equations. Priblizhen. Met. Reshen. Diff. Uravn.
**1964**, 2, 115–134. [Google Scholar] - Abbas, M.; Jungck, G. Common fixed point results for noncommuting mappings without continuity in cone metric space. J. Math. Anal. Appl.
**2008**, 341, 416–420. [Google Scholar] [CrossRef] [Green Version] - Abbas, M.; Nazir, T.; Rakočević, V. Common fixed points results of multivalued Perov type contractions on cone metric spaces with a directed graph. Bull. Belg. Math. Soc. Simon Stevin
**2018**, 25, 331–354. [Google Scholar] - Abbas, M.; Rakočević, V.; Hussain, A. Best proximity point of Zamfirescu contractions of Perov type on regular cone metric spaces. Fixed Point Theory
**2020**, 21, 3–18. [Google Scholar] - Altun, I.; Hussain, N.; Qasim, M.; Alsulami, H.H. A new fixed point result of Perov type and its application to a semilinear operator system. Mathematics
**2019**, 7, 1019. [Google Scholar] [CrossRef] [Green Version] - Aleksić, S.; Paunović, L.; Radenović, S.; Vetro, F. Some critical remarks on the paper ‘A note on the metrizability of TVS-cone metric spaces’. Vojn. Teh. Glas. Tech. Cour.
**2017**, 65, 1–8. [Google Scholar] - Karapinar, E. Couple fixed point theorems for nonlinear contractions in cone metric spaces. Comput. Math. Appl.
**2010**, 59, 3656–3668. [Google Scholar] - Aleksić, S.; Kadelburg, Z.; Mitrović, Z.D.; Radenović, S. A new survey: Cone metric spaces. J. Int. Math. Virtual Inst.
**2019**, 9, 93–121. [Google Scholar] - Cvetković, M. On the equivalence between Perov fixed point theorems and Banach contraction principle. Filomat
**2017**, 31, 3137–3146. [Google Scholar] - Hardy, G.E.; Rogers, T.D. A generalization of a fixed point theorem of Reich. Can. Math. Bull.
**1973**, 16, 201–206. [Google Scholar] - Huang, L.G.; Zhang, X. Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl.
**2007**, 332, 1468–1476. [Google Scholar] - Jachymski, J.; Klima, J. Around Perov’s fixed point theorem for mappings on generalized metric spaces. Fixed Point Theory
**2016**, 17, 367–380. [Google Scholar] - Janković, S.; Kadelburg, Z.; Radenović, S. On cone metric spaces: A survey. Nonlinear Anal.
**2011**, 74, 2591–2601. [Google Scholar] - Popescu, O.; Stan, G. Two fixed point theorems concerning F-contraction in complete metric spaces. Symmetry
**2020**, 12, 58. [Google Scholar] [CrossRef] [Green Version] - Radenović, S.; Vetro, F.; Xu, S. Some results of Perov type mappings. J. Adv. Math. Stud.
**2017**, 10, 396–409. [Google Scholar] - Radenović, S.; Vetro, F. Some remarks on Perov type mappings in cone metric spaces. Mediterr. J. Math.
**2017**, 14, 240. [Google Scholar] - Vetro, F.; Radenović, S. Some results of Perov type on rectangular cone metric spaces. J. Fixed Point Theory Appl.
**2018**, 20, 1–16. [Google Scholar] [CrossRef] - Vujaković, J.; Radenović, S. On some F-contraction of Piri-Kumam-Dung-type mappings in metric spaces. Vojno Tehnički Glasnik
**2020**, 68, 697–714. [Google Scholar] - Wardowski, D.; Van Dung, N. Fixed points of F-weak contractions on complete metric spaces. Demonstr. Math.
**2014**, 47, 146–155. [Google Scholar] [CrossRef] - Wardowski, D. Solving existence problems via F-contractions. Proc. Am. Math. Soc.
**2018**, 146, 1585–1598. [Google Scholar] - Xu, S.; Dolićanin, Ć.; Radenović, S. Some remarks on results of Perov type. J. Adv. Math. Stud.
**2016**, 9, 361–369. [Google Scholar] - Zabrejko, P.P. K-metric and K-normed linear spaces; survey. Collect. Math.
**1997**, 48, 825–859. [Google Scholar] - Karapinar, E.; Fulga, A.; Agarwal, R.P. A survey: F-contractions with related fixed point results. J. Fixed Point Theory Appl.
**2020**, 22, 69. [Google Scholar] [CrossRef] - Agarwal, R.P.; Aksoy, U.; Karapinar, E.; Erhan, I.M. F-contraction mappings on metric-like spaces in connection with integral equations on time scales. RACSAM
**2020**, 114, 147. [Google Scholar] [CrossRef] - Aydi, H.; Karapinar, E.; Yazidi, H. Modified F-Contractions via α-Admissible Mappings and Application to Integral Equations. Filomat
**2017**, 31, 1141–1148. [Google Scholar] [CrossRef] - Alsulami, H.H.; Karapinar, E.; Piri, H. Fixed Points of Modified F-Contractive Mappings in Complete Metric-Like Space Cone metric spaces. J. Funct. Spaces
**2015**, 2015, 270971. [Google Scholar] [CrossRef] - Ćirić, L.B. Some Recent Results in Metrical Fixed Point Theory; University of Belgrade: Belgrade, Serbia, 2003. [Google Scholar]
- Collaco, P.; Silva, J.C. A complete comparison of 23 contraction conditions. Nonlinear Anal. TMA
**1997**, 30, 471–476. [Google Scholar] - Rhoades, B.E. A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc.
**1997**, 226, 257–290. [Google Scholar] - Caputo, M. Linear model of dissipation whose Q is almost frequency independent—II. Geophys. J. Int.
**1967**, 13, 529–539. [Google Scholar] - Kilbas, A.A.; Srivastava, H.H.; Trujillo, J.J. Theory and Appolications of Fractional Differential Equations; North-Holland Mathematics Studies; Elsevier: Amsterdam, The Netherlands, 2006; Volume 204. [Google Scholar]
- Samko, S.G.; Kilbas, A.A.; Marichey, O.I. Fractional Integral and Derivative; Gordon and Breach: London, UK, 1998. [Google Scholar]
- Tuan, N.H.; Mohammadi, H.; Rezapour, S. A mathematical model for COVID-19 transmission by using the Caputo fractional derivative. Chaos Solitons Fractals
**2020**, 140, 110107. [Google Scholar] [CrossRef] - Rezapour, S.; Mohammadi, H.; Jajarmi, A. A new mathematical model for Zika virus transmission. Adv. Differ. Equ.
**2020**, 2020, 589. [Google Scholar] [CrossRef] - Podlubny, I. Fractional Differential Equations; Academic Press: San Diego, CA, USA, 1999. [Google Scholar]

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Mirkov, N.; Radenović, S.; Radojević, S.
Some New Observations for F-Contractions in Vector-Valued Metric Spaces of Perov’s Type. *Axioms* **2021**, *10*, 127.
https://doi.org/10.3390/axioms10020127

**AMA Style**

Mirkov N, Radenović S, Radojević S.
Some New Observations for F-Contractions in Vector-Valued Metric Spaces of Perov’s Type. *Axioms*. 2021; 10(2):127.
https://doi.org/10.3390/axioms10020127

**Chicago/Turabian Style**

Mirkov, Nikola, Stojan Radenović, and Slobodan Radojević.
2021. "Some New Observations for F-Contractions in Vector-Valued Metric Spaces of Perov’s Type" *Axioms* 10, no. 2: 127.
https://doi.org/10.3390/axioms10020127