On generalized $ \Im_b $-contractions and related applications

Muhammad Rashid; Muhammad Sarwar; Muhammad Fawad; Saber Mansour; Hassen Aydi; Muhammad Rashid; Muhammad Sarwar; Muhammad Fawad; Saber Mansour; Hassen Aydi

doi:10.3934/math.20231064

AIMS Mathematics

2023, Volume 8, Issue 9: 20892-20913. doi: 10.3934/math.20231064

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Research article

On generalized $ \Im_b $-contractions and related applications

1.
Department of Mathematics, University of Malakand, Chakdara Dir(L), Khyber Pakhtunkhwa, Pakistan, Email: mrashidghani26@gmail.com, muhmmadfawad123@gmail.com
2.
Department of Mathematics, Umm Al-Qura University, Faculty of Applied sciences, P.O. Box 14035, Holly Makkah 21955, Saudi Arabia, Email: samansour@uqu.edu.sa
3.
Université de Sousse, Institut Supérieur d'Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia
4.
China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
5.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa

Received: 02 January 2023 Revised: 27 April 2023 Accepted: 03 May 2023 Published: 30 June 2023
MSC : 34A08, 54H25, 47H10

In this manuscript, using the idea of $ b $-simulation functions, certain common fixed point results via generalized $ \Im_b $-contractions are investigated in the context of b-metric spaces. These findings generalize and supplement various numbers of established results from the existing literature. Examples and applications are also provided for the authenticity of the presented work. Besides, we ensure the existence of a unique common solution for systems of Volterra-Hammerstein integral and Urysohn integral equations, respectively, by applying the established results.
- fixed point,
- $ b $-metric space,
- $ b $-simulation function,
- compatible mapping,
- $ \Im $-contraction,
- generalized $ \Im_b $-contraction
Citation: Muhammad Rashid, Muhammad Sarwar, Muhammad Fawad, Saber Mansour, Hassen Aydi. On generalized $ \Im_b $-contractions and related applications[J]. AIMS Mathematics, 2023, 8(9): 20892-20913. doi: 10.3934/math.20231064

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Abstract

In this manuscript, using the idea of $ b $-simulation functions, certain common fixed point results via generalized $ \Im_b $-contractions are investigated in the context of b-metric spaces. These findings generalize and supplement various numbers of established results from the existing literature. Examples and applications are also provided for the authenticity of the presented work. Besides, we ensure the existence of a unique common solution for systems of Volterra-Hammerstein integral and Urysohn integral equations, respectively, by applying the established results.

References

[1]	S. Radenović, T. Dosenović, T. A. Lampert, Z. Golubović, A note on some recent fixed point results for cyclic contractions in $b$-metric spaces and an application to interal equations, Appl. Math. Comput., 273 (2016), 155–164. https://doi.org/10.1016/j.amc.2015.09.089 doi: 10.1016/j.amc.2015.09.089
[2]	M. Shoaib, M. Sarwar, K. Shah, P. Kumum, Fixed point results and its applications to the systems of non-linear integral and differential equations of arbitrary order, J. Nonlinear Sci. Appl., 9 (2016), 4949–4962. https://doi.org/10.22436/jnsa.009.06.128 doi: 10.22436/jnsa.009.06.128
[3]	A. Azam, B. Fisher, M. S. Khan, Common fixed point theorems in complex valued metric spaces, Number. Funct. Anal. optim., 32 (2011), 243–253. https://doi.org/10.1080/01630563.2011.533046 doi: 10.1080/01630563.2011.533046
[4]	L. Gholizadeh, A fixed point theorem in generalized ordered metric spaces with applications, J. Nonlinear Sci. Appl., 6 (2013), 244–251. https://doi.org/10.22436/jnsa.006.04.02 doi: 10.22436/jnsa.006.04.02
[5]	M. B. Zada, M. Sarwar, C. Tunc, Fixed point theorems in b-metric spaces and their applications to non-linear fractional differential and integral equations, J. Fixed Point Theory Appl., 20 (2018), 25. https://doi.org/10.1007/s11784-018-0510-0 doi: 10.1007/s11784-018-0510-0
[6]	P. Salimi, N. Hussain, S. Shukla, S. Fathollahi, Radenović, fixed point results for cyclic $\alpha$-$\psi \varphi\varphi$-$contractions$ with applications to integral equations, J. Comput. Appl. Math., 290 (2015), 445–458. https://doi.org/10.1016/j.cam.2015.05.017 doi: 10.1016/j.cam.2015.05.017
[7]	W. Sintunavarat, M. B Zada, M. Sarwar, Common solution of Urysohn integral equations with the help of common fixed point results in complex valued metric spaces, RACSAM, 11 (2017), 531–545. https://doi.org/10.1007/s13398-016-0309-z doi: 10.1007/s13398-016-0309-z
[8]	R. A. Rashwan, S. M. Saleh, Solution of nonlinear equations via fixed point in G-metric spaces, Int. J. Appl. Math. Res., 3 (2014), 561–571.
[9]	H. K. Pathak, M. S. Khan, R. Tiwari, A common fixed point theorem and its applications to nonlinear integral equations, Comput. Math. Appl., 53 (2007), 961–971. https://doi.org/10.1016/j.camwa.2006.08.046 doi: 10.1016/j.camwa.2006.08.046
[10]	A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Am. Math. Soc., 132 (2004), 1435–1443. https://doi.org/10.1090/S0002-9939-03-07220-4 doi: 10.1090/S0002-9939-03-07220-4
[11]	H. Q. Wang, X. H. Song, L. L. Chen, W. Xie, A secret sharing scheme for quantum gray and color images based on encryption, Int. J. Theor. Phys., 58 (2019), 1626–1650. https://doi.org/10.1007/s10773-019-04057-z doi: 10.1007/s10773-019-04057-z
[12]	S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundam. Math., 3 (1922), 133–181. https://doi.org/10.4064/fm-3-1-133-181 doi: 10.4064/fm-3-1-133-181
[13]	L. Chen, L. Gao, D. Chen, Fixed point theorems of mean nonexpansive set-valued mappings in Banach spaces, J. Fixed point theory Appl., 19 (2017), 2019–2143. https://doi.org/10.1007/s11784-017-0401-9 doi: 10.1007/s11784-017-0401-9
[14]	S. Reich, Kannan's fixed point theorem, Boll. Unione Mat. Ital., 4 (1971), 1–11.
[15]	A. Baklouti, L. Mifdal, S. Dellagi, A. Chelbi, An optimal preventive maintenance policy for a solar photovoltaic system, Sustainability, 12 (2020), 4266. https://doi.org/10.3390/su12104266 doi: 10.3390/su12104266
[16]	A. Baklouti, Quadratic Hom-Lie triple systems, J. Geom. Phys., 121 (2017), 166–175.
[17]	A. Baklouti, S. Hidri, Tools to specify semi-simple Jordan triple systems, Differ. Geom. Appl., 83 (2022), 101900. https://doi.org/10.1016/j.difgeo.2022.101900 doi: 10.1016/j.difgeo.2022.101900
[18]	C. Mongkolkeh, Y. J. Cho, P. Kumam, Fixed point theorem for simulation functions in $b$-metric spaces via the wt-distance, Appl. Gen. Topol., 18 (2017), 91–105. https://doi.org/10.4995/agt.2017.6322 doi: 10.4995/agt.2017.6322
[19]	S. Reich, A. J. Zaslavski, Genericity in nonlinear analysis, Springer: London, UK, 2014.
[20]	I. A. Bakhtin, The contraction mapping principle in quasimetric spaces, Funct. Anal, Unianowsk Gos. Ped. Inst., 30 (1989), 26–37.
[21]	S. M. Abusalim, M. S. Noorani, Fixed point and common fixed point theorem on ordered cone $b$-metric spaces, Abstr. Appl. Anal., 2013, 81528. https://doi.org/10.1155/2013/815289
[22]	A. Roldan, E. Karapinar, C. Roldan, J. Martinez-Moreno, Coincidence point theorems on metric spaces via simulation functions, J. Comput. Appl. Math., 275 (2015), 345–355. https://doi.org/10.1016/j.cam.2014.07.011 doi: 10.1016/j.cam.2014.07.011
[23]	B. Rodjanadid, P. Cherachapridi, J. Tanthaunch, Some fixed point theorems in $b$-metric spaces with $b$-simulation functions, Thai J. Math., 19 (2021), 1625–1636.
[24]	A. A. Jawaher, S. Sedghi, I. Altun, Fixed point results on $b$-metric spaces via $b$-simulation functions, Thai J. Math., 19 (2021), 1575–1584.
[25]	M. Fréchet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palerm., 22 (1906), 1–72. https://doi.org/10.1007/BF03018603 doi: 10.1007/BF03018603
[26]	S. Czerwik, Nonlinear set-valued contraction mappings in $b$-metric spaces, Atti Sem. Mat. Fis. Univ. Modena., 46 (1998), 263–276.
[27]	S. Czerwik, Contraction mappings in $b$-metric spaces, Acta Math. inform. Univ. Ostraviensis, 1 (1993), 5–11.
[28]	F. Khojasteh, S. Shukla, S. Radenović, A new approach to the study of fixed point theory for simulation functions, Filomat, 29 (2015), 1189–1194. https://doi.org/10.2298/FIL1506189K doi: 10.2298/FIL1506189K
[29]	M. Olgun, O. Bicer, T. Alyildiz, A new aspect to Picard operators with simulation functions, Turk. J. Math., 40 (2016), 832–837. https://doi.org/10.3906/mat-1505-26 doi: 10.3906/mat-1505-26
[30]	M. Demma, R. Saadati, P. Vetro, Fixed point results on $b$-metric space via Picard sequences and $b$-simulation functions, Iran. J. Math. Sci. Info., 11 (2016), 123–136. https://doi.org/10.3917/top.136.0123 doi: 10.3917/top.136.0123
[31]	G. Jungck, Compatible mappings and common fixed points, Int. J. Math. Math. Sci., 9 (1986), 771–779. https://doi.org/10.1155/S0161171286000935 doi: 10.1155/S0161171286000935
[32]	V. Ozturk, D. Turkoglu, Common fixed point theorems for mappings satisfying (E.A) property in $b$-metric spaces, J. Nonlinear Sci. Appl., 8 (2015), 1127–1133. https://doi.org/10.22436/jnsa.008.06.21 doi: 10.22436/jnsa.008.06.21
[33]	M. Demma, R. Saadati, P. Vetro, Fixed point results on $b$-metric space via Picard sequences and $b$-simulation functions, Iran. J. Math. Sci. Info., 11 (2016), 123–136.
[34]	F. Khojasteh, S. Shukla, S. Radenović, A new approach to the study of fixed point theory for simulation functions, Filomat, 29 (2015), 1189–1194. https://doi.org/10.2298/FIL1506189K doi: 10.2298/FIL1506189K

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