Fixed point approach to solve nonlinear fractional differential equations in orthogonal $ \mathcal{F} $-metric spaces

Abdullah Eqal Al-Mazrooei; Jamshaid Ahmad; Abdullah Eqal Al-Mazrooei; Jamshaid Ahmad

doi:10.3934/math.2023255

AIMS Mathematics

2023, Volume 8, Issue 3: 5080-5098. doi: 10.3934/math.2023255

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Research article

Fixed point approach to solve nonlinear fractional differential equations in orthogonal $ \mathcal{F} $-metric spaces

Abdullah Eqal Al-Mazrooei ¹,
Jamshaid Ahmad ^{2
,
,}

1.
Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
2.
Department of Mathematics, University of Jeddah, P. O. Box 80327, Jeddah 21589, Saudi Arabia

Received: 27 September 2022 Revised: 18 November 2022 Accepted: 05 December 2022 Published: 13 December 2022
MSC : 46S40, 47H10, 54H25

In this paper, we introduce the notion of a generalized ($ \alpha $, $ \Theta _{\mathcal{F}}) $-contraction in the context of an orthogonal $ \mathcal{F} $-complete metric space and obtain some new fixed point results for this newly introduced contraction. A nontrivial example is also provided to satisfy the validity of the established results. As consequences of our obtained results, we derive the leading results in [Fixed Point Theory Appl., 2015,185, 2015] and [Symmetry, 2020, 12,832]. As an application, we investigate the existence and uniqueness of the solution for a nonlinear fractional differential equation.
Citation: Abdullah Eqal Al-Mazrooei, Jamshaid Ahmad. Fixed point approach to solve nonlinear fractional differential equations in orthogonal $ \mathcal{F} $-metric spaces[J]. AIMS Mathematics, 2023, 8(3): 5080-5098. doi: 10.3934/math.2023255

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Abstract

In this paper, we introduce the notion of a generalized ($ \alpha $, $ \Theta _{\mathcal{F}}) $-contraction in the context of an orthogonal $ \mathcal{F} $-complete metric space and obtain some new fixed point results for this newly introduced contraction. A nontrivial example is also provided to satisfy the validity of the established results. As consequences of our obtained results, we derive the leading results in [Fixed Point Theory Appl., 2015,185, 2015] and [Symmetry, 2020, 12,832]. As an application, we investigate the existence and uniqueness of the solution for a nonlinear fractional differential equation.

References

[1]	M. Bestvina, Real trees in topology, geometry and group theory, arXiv: math/9712210.
[2]	W. Kirk, Some recent results in metric fixed point theory, J. Fixed Point Theory Appl., 2 (2007), 195–207. http://dx.doi.org/10.1007/s11784-007-0031-8 doi: 10.1007/s11784-007-0031-8
[3]	C. Semple, M. Steel, Phylogenetics, Oxford: Oxford University Press, 2003.
[4]	M. Frechet, Sur quelques points du calcul fonctionnel, Rend. Circ. Matem. Palermo, 22 (1906), 1–72. http://dx.doi.org/10.1007/BF03018603 doi: 10.1007/BF03018603
[5]	S. Czerwik, Contraction mappings in $b$-metric spaces, Acta Mathematica et Informatica Universitatis Ostraviensis, 1 (1993), 5–11.
[6]	A. Branciari, A fixed point theorem of Banach-Caccioppoli type on a class of generalizedmetric spaces, Publ. Math. Debrecen, 57 (2000), 31–37. http://dx.doi.org/10.5486/PMD.2000.2133 doi: 10.5486/PMD.2000.2133
[7]	M. Jleli, B. Samet, On a new generalization of metric spaces, J. Fixed Point Theory Appl., 20 (2018), 128. http://dx.doi.org/10.1007/s11784-018-0606-6 doi: 10.1007/s11784-018-0606-6
[8]	M. Gordji, D. Rameani, M. De La Sen, Y. Cho, On orthogonal sets and Banach fixed point theorem, Fixed Point Theory, 18 (2017), 569–578. http://dx.doi.org/10.24193/fpt-ro.2017.2.45 doi: 10.24193/fpt-ro.2017.2.45
[9]	T. Kanwal, A. Hussain, H. Baghani, M. De La Sen, New fixed point theorems in orthogonal $\mathcal{F}$-metric spaces with application to fractional differential equation, Symmetry, 12 (2020), 832. http://dx.doi.org/10.3390/sym12050832 doi: 10.3390/sym12050832
[10]	I. Bakhtin, The contraction mapping principle in almost metric spaces, Funct. Anal, 30 (1989), 26–37.
[11]	M. Khamsi, N. Hussain, KKM mappings in metric type spaces, Nonlinear Anal.-Theor., 73 (2010), 3123–3129. http://dx.doi.org/10.1016/j.na.2010.06.084 doi: 10.1016/j.na.2010.06.084
[12]	J. Ahmad, A. Al-Rawashdeh, A. Al-Mazrooei, Fixed point results for ($\alpha, \bot _{\mathcal{F}}$)-contractions in orthogonal $F$-metric spaces with applications, J. Funct. Space., 2022 (2022), 8532797. http://dx.doi.org/10.1155/2022/8532797 doi: 10.1155/2022/8532797
[13]	L. Alnaser, D. Lateef, H. Fouad, J. Ahmad, Relation theoretic contraction results in $F$-metric spaces, J. Nonlinear Sci. Appl., 12 (2019), 337–344. http://dx.doi.org/10.22436/jnsa.012.05.06 doi: 10.22436/jnsa.012.05.06
[14]	S. Al-Mezel, J. Ahmad, G. Marino, Fixed point theorems for generalized ($\alpha \beta $-$\psi $)-contractions in $F$-metric spaces with applications, Mathematics, 8 (2020), 584. http://dx.doi.org/10.3390/math8040584 doi: 10.3390/math8040584
[15]	M. Alansari, S. Mohammed, A. Azam, Fuzzy fixed point results in $F$-metric spaces with applications, J. Funct. Space., 2020 (2020), 5142815. http://dx.doi.org/10.1155/2020/5142815 doi: 10.1155/2020/5142815
[16]	D. Lateef, J. Ahmad, Dass and Gupta's Fixed point theorem in $F $-metric spaces, J. Nonlinear Sci. Appl., 12 (2019), 405–411. http://dx.doi.org/10.22436/jnsa.012.06.06 doi: 10.22436/jnsa.012.06.06
[17]	A. Hussain, T. Kanwal, Existence and uniqueness for a neutral differential problem with unbounded delay via fixed point results, T. A. Razmadze Math. In., 172 (2018), 481–490. http://dx.doi.org/10.1016/j.trmi.2018.08.006 doi: 10.1016/j.trmi.2018.08.006
[18]	Z. Ahmadi, R. Lashkaripour, H. Baghani, A fixed point problem with constraint inequalities via a contraction in incomplete metric spaces, Filomat, 32 (2018), 3365–3379. http://dx.doi.org/10.2298/FIL1809365A doi: 10.2298/FIL1809365A
[19]	H. Baghani, M. Ramezani, Coincidence and fixed points for multivalued mappings in incomplete metric spaces with applications, Filomat, 33 (2019), 13–26. http://dx.doi.org/10.2298/FIL1901013B doi: 10.2298/FIL1901013B
[20]	A. Ran, M. Reuring, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Am. Math. Soc., 132 (2004), 1435–1443. http://dx.doi.org/10.1090/S0002-9939-03-07220-4 doi: 10.1090/S0002-9939-03-07220-4
[21]	K. Javed, H. Aydi, F. Uddin, M. Arshad, On orthogonal partial $b$-metric spaces with an application, J. Math., 2021 (2021), 6692063. http://dx.doi.org/10.1155/2021/6692063 doi: 10.1155/2021/6692063
[22]	B. Samet, C. Vetro, P. Vetro, Fixed point theorems for $ \alpha $-$\psi $-contractive type mappings, Nonlinear Anal.-Theor., 75 (2012), 2154–2165. http://dx.doi.org/10.1016/j.na.2011.10.014 doi: 10.1016/j.na.2011.10.014
[23]	M. Ramezani, Orthogonal metric space and convex contractions, Int. J. Nonlinear Anal., 6 (2015), 127–132. http://dx.doi.org/ 10.22075/IJNAA.2015.261 doi: 10.22075/IJNAA.2015.261
[24]	A. Asif, M. Nazam, M. Arshad, S. Kim, $F$-metric, $F$ -contraction and common fixed point theorems with applications, Mathematics, 7 (2019), 586. http://dx.doi.org/10.3390/math7070586 doi: 10.3390/math7070586
[25]	G. Mani, A. Gnanaprakasam, N. Kausar, M. Munir, Salahuddin. Orthogonal $F$-contraction mapping on $O$-complete metric space with applications, Int. J. Fuzzy Log. Inte., 21 (2021), 243–250. http://dx.doi.org/10.5391/IJFIS.2021.21.3.243 doi: 10.5391/IJFIS.2021.21.3.243
[26]	S. Banach, Sur les operations dans les ensembles abstraits et leur applications aux equations integrales, Fund. Math., 3 (1922), 133–181. http://dx.doi.org/10.4064/fm-3-1-133-181 doi: 10.4064/fm-3-1-133-181
[27]	J. Ahmad, A. Al-Rawashdeh, A. Azam, Fixed point results for $ \{ \alpha, \xi \}$-expansive locally contractive mappings, J. Inequal. Appl., 2014 (2014), 364. http://dx.doi.org/10.1186/1029-242X-2014-364 doi: 10.1186/1029-242X-2014-364
[28]	J. Ahmad, A. Al-Rawashdeh, A. Azam, New fixed point theorems for generalized $F$-contractions in complete metric spaces, Fixed Point Theory Appl., 2015 (2015), 80. http://dx.doi.org/10.1186/s13663-015-0333-2 doi: 10.1186/s13663-015-0333-2
[29]	M. Jleli, B. Samet, A new generalization of the Banach contraction principle, J. Inequal. Appl., 2014 (2014), 38. http://dx.doi.org/10.1186/1029-242X-2014-38 doi: 10.1186/1029-242X-2014-38
[30]	J. Ahmad, A. Al-Mazrooei, Y. Cho, Y. Yang, Fixed point results for generalized $\Theta $-contractions, J. Nonlinear Sci. Appl., 10 (2017), 2350–2358. http://dx.doi.org/10.22436/jnsa.010.05.07 doi: 10.22436/jnsa.010.05.07
[31]	N. Hussain, V. Parvaneh, B. Samet, C. Vetro, Some fixed point theorems for generalized contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2015 (2015), 185. http://dx.doi.org/10.1186/s13663-015-0433-z doi: 10.1186/s13663-015-0433-z
[32]	N. Hussain, J. Ahmad, New Suzuki-Berinde type fixed point results, Carpathian J. Math., 33 (2017), 59–72. http://dx.doi.org/10.37193/CJM.2017.01.07 doi: 10.37193/CJM.2017.01.07
[33]	Z. Li, S. Jiang, Fixed point theorems of JS-quasi-contractions, Fixed Point Theory Appl., 2016 (2016), 40. http://dx.doi.org/10.1186/s13663-016-0526-3 doi: 10.1186/s13663-016-0526-3
[34]	F. Vetro, A generalization of Nadler fixed point theorem, Carpathian J. Math, 31 (2015), 403–410.
[35]	H. Ali, H. Isik, H. Aydi, E. Ameer, J. Lee, M. Arshad, On multivalued Suzuki-type $\Theta $-contractions and related applications, Open Math., 18 (2020), 386–399. http://dx.doi.org/10.1515/math-2020-0139 doi: 10.1515/math-2020-0139
[36]	E. Ameer, H. Aydi, M. Arshad, A. Hussain, A. Khan, Ćirić type multi-valued $\alpha _{\ast }$-$\eta _{\ast }$-$\Theta $ -contractions on $b$-meric spaces with applications, Int. J. Nonlinear Anal., 12 (2021), 597–614. http://dx.doi.org/10.22075/IJNAA.2021.4865 doi: 10.22075/IJNAA.2021.4865
[37]	L. Ćirić, Generalized contractions and fixed-point theorems, Publ. Inst. Math., 12 (1971), 9–26.
[38]	R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc., 60 (1968), 71–76.
[39]	S. Chatterjea, Fixed point theorem, C. R. Acad. Bulg. Sci., 25 (1972), 727–730.
[40]	S. Reich, Kannan's fixed point theorem, Bull. Univ. Mat. Italiana, 4 (1971), 1–11.
[41]	H. Mohammadi, S. Kumar, S. Rezapour, S. Etemad, A theoretical study of the Caputo-Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control, Chaos Soliton. Fract., 144 (2021), 110668. http://dx.doi.org/10.1016/j.chaos.2021.110668 doi: 10.1016/j.chaos.2021.110668
[42]	S. Kumar, P. Shaw, A. Abdel-Aty, E. Mahmoud, A numerical study on fractional differential equation with population growth model, Numer. Meth. Part. D. E., in press. http://dx.doi.org/10.1002/num.22684
[43]	P. Shaw, S. Kumar, S. Momani, S. Hadid, Dynamical analysis of fractional plant disease model with curative and preventive treatments, Chaos, Soliton. Fract., 164 (2022), 112705. http://dx.doi.org/10.1016/j.chaos.2022.112705 doi: 10.1016/j.chaos.2022.112705
[44]	D. Gopal, M. Abbas, D. Patel, C. Vetro, Fixed point of $ \alpha $-type $F$-contractive mappings with an application to nonlinear fractional differential equation, Acta Math. Sci., 36 (2016), 957–970. http://dx.doi.org/10.1016/S0252-9602(16)30052-2 doi: 10.1016/S0252-9602(16)30052-2
[45]	D. Baleanu, S. Rezapour, H. Mohammadi, Some existence results on nonlinear fractional differential equations, Philos. Trans. A Math. Phys. Eng. Sci., 371 (2013), 20120144. http://dx.doi.org/10.1098/rsta.2012.0144 doi: 10.1098/rsta.2012.0144

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