Some coupled fixed point theorems on multiplicative metric spaces with an application

Baqir Hussain; Saif Ur Rehman; Mohammed M.M. Jaradat; Sami Ullah Khan; Muhammad Arshad; Baqir Hussain; Saif Ur Rehman; Mohammed M.M. Jaradat; Sami Ullah Khan; Muhammad Arshad

doi:10.3934/math.2022805

AIMS Mathematics

2022, Volume 7, Issue 8: 14631-14651. doi: 10.3934/math.2022805

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Some coupled fixed point theorems on multiplicative metric spaces with an application

1.
Institute of Numerical Sciences, Department of Mathematics, Gomal University, Dera Ismail Khan 29050, Pakistan
2.
Mathematics Program, Department of Mathematics, Statistics and Physics, College of Arts and Sciences, Qatar University, Doha 2713, Qatar
3.
Department of Math & State, International Islamic University, Islamabad, Pakistan

Received: 04 January 2022 Revised: 28 February 2022 Accepted: 09 March 2022 Published: 08 June 2022
MSC : 47H10, 54H25

This article aims to prove some coupled fixed point (FP) theorems for nonlinear contractive type mapping in multiplicative metric space (MM-space). Our presented work consists of the maximum type and some other expressions in the framework of MM-space. We also provide illustrative examples and an application in support of our generalized results. Our offered results expand and develop a variety of the latest outcomes in the existing literature. Moreover, we present an application of the two Urysohn integral equations to support our work.
- coupled fixed point,
- multiplicative metric space,
- contraction conditions, Urysohn integral equations
Citation: Baqir Hussain, Saif Ur Rehman, Mohammed M.M. Jaradat, Sami Ullah Khan, Muhammad Arshad. Some coupled fixed point theorems on multiplicative metric spaces with an application[J]. AIMS Mathematics, 2022, 7(8): 14631-14651. doi: 10.3934/math.2022805

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Abstract

This article aims to prove some coupled fixed point (FP) theorems for nonlinear contractive type mapping in multiplicative metric space (MM-space). Our presented work consists of the maximum type and some other expressions in the framework of MM-space. We also provide illustrative examples and an application in support of our generalized results. Our offered results expand and develop a variety of the latest outcomes in the existing literature. Moreover, we present an application of the two Urysohn integral equations to support our work.

References

[1]	S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133–181. https://doi.org/10.4064/fm-3-1-133-181 doi: 10.4064/fm-3-1-133-181
[2]	D. Chatterjea, Generalized contraction principal, Int. J. Math. Math. Sci., 6 (1983), 89–94.
[3]	R. Kannan, Some results on fixed points, Bulletin Calcutta Math. Society, 60 (1968), 71–76.
[4]	S. U. Rehman, S. Jabeen, Muhammad, H. Ullah, Hanifullah, Some multi-valued contraction theorems on $H-$cone metric, J. Adv. Studies Topol., 10 (2019), 11–24.
[5]	R. Sharma, V. Gupta, M. Kushwaha, New results for compatible mappings of type A and subsequential continuous mappings, Appl. Appl. Math.: Int. J., 15 (2020), 282–295. https://doi.org/10.2298/AADM210120017S doi: 10.2298/AADM210120017S
[6]	M. Grossman, R. Katz, Non-Newtonian Calculus, Lee Press, Pigeon Cove, MA (1972).
[7]	A. E. Bashirov, E. M. Kurpnar, A. Ozyapic, Multiplicative calculus and its applications, J. Math. Anal. Appl., 337 (2008), 36–48. https://doi.org/10.1016/j.jmaa.2007.03.081 doi: 10.1016/j.jmaa.2007.03.081
[8]	B. Rome, M. Sarwar, Characterization of multiplicative metric completeness, Int. J. Anal. Appl., 10 (2016), 90–94.
[9]	B. Zada, U. Riaz, Some fixed point results on multiplicative-metric-like spaces, Turkish J. Anal. Number Theory, 4 (2016), 118–131.
[10]	S. Shukla, Some critical remarks on the multiplicative metric spaces and fixed point results, J. Adv. Math. Studies, 9 (2016), 454–458.
[11]	M. U. Ali, Caristi mapping in multiplicative metric spaces, Sci. Int. (Lahore), 27 (2015), 3917–3919.
[12]	M. Ozavsar, A. C. Cervikel, Fixed points of multiplicative contraction mappings on multiplicative metric spaces, arXiv:1205.5131v1 [math.GM], 2 (2012), 1205–1531.
[13]	P. Kumar, S. Kumar, S. M. Kang, Common fixed points for weakly compatible mappings in multiplicative metric spaces, Int. J. Math. Anal., 9 (2015), 2087–2097. https://doi.org/10.12988/ijma.2015.56162 doi: 10.12988/ijma.2015.56162
[14]	X. He, M. Song, D. Chen, Common fixed points for weak commutative mappings on a multiplicative metric space, Fixed Point Theory A., 48 (2014), 9 pages. https://doi.org/10.1186/1687-1812-2014-48 doi: 10.1186/1687-1812-2014-48
[15]	S. M. Kang, P. Nagpal, S. K. Garg, S. Kumar, Fixed points for multiplicative expansive mappings in multiplicative metric spaces, Int. J. Math. Anal., 9 (2015), 1939–1946. https://doi.org/10.12988/ijma.2015.54130 doi: 10.12988/ijma.2015.54130
[16]	S. M. Kang, P. Kumar, P. Nagpal, S. K. Garg, Common fixed points for compatible mappings and its variants in multiplicative metric spaces, Int. J. Pure Appl. Math., 102 (2015), 383–406. https://doi.org/10.12732/ijpam.v102i2.14 doi: 10.12732/ijpam.v102i2.14
[17]	F. Gu, Y. J. Cho, Common fixed point results for four maps satisfying phi-contractive condition in multiplicative metric spaces, Fixed Point Theory A., 165 (2015), 2189. https://doi.org/10.1186/s13663-015-0412-4 doi: 10.1186/s13663-015-0412-4
[18]	C. Mongkolkeha, W. Sintunavarat, Best proximity points for multiplicative proximal contraction mapping on multiplicative metric spaces, J. Nonlinear Sci. Appl., 8 (2015), 1134–1140. https://doi.org/10.22436/jnsa.008.06.22 doi: 10.22436/jnsa.008.06.22
[19]	V. Lakshikantham, L. Ciric, Coupled fixed point theorems for nonlinear contractions in partiallyordered metric spaces, Nonlinear Anal. Theor., 12 (2009), 4341–4349. https://doi.org/10.1016/j.na.2008.09.020 doi: 10.1016/j.na.2008.09.020
[20]	Gordji, M. Eshaghi, Y. J. Cho, Coupled fixed-point theorems for contractions in partial ordered metric spaces and applications, Math. Probl. Eng., (2012) Article ID: 150363.
[21]	Y. J. Cho, B. E. Rhoades, R. Saadati, B. Samet, W. Shantawi, Nonlinear coupled fixed point theorems in ordered generalized metric spaces with integral type, Fixed Point Theory A., 1 (2012), 14 pages. https://doi.org/10.1186/1687-1812-2012-8 doi: 10.1186/1687-1812-2012-8
[22]	N. J. Huang, Y. P. Fang, Y. J. Cho, Fixed point and coupled fixed point theorems for multi-valued increasing operators in ordered metric spaces, Fixed Point Theory A., 3 (2002), 91–98.
[23]	W. Sintunavarat, Y. J. Cho, P. Kumam, Coupled coincidence point theorems for contractions without commutative condition in intuitionistic fuzzy normed spaces, Fixed Point Theory Appl., 81 (2011), 1897–1906. https://doi.org/10.1186/1687-1812-2011-81 doi: 10.1186/1687-1812-2011-81
[24]	X. Li, S. U. Rehman, S. U. Khan, H. Aydi, J. Ahmad, N. Hussain, Strong coupled fixed point results and applications to Urysohn integral equations, Dynam. Syst. Appl., 30 (2021), 197–218. https://doi.org/10.46719/dsa20213023 doi: 10.46719/dsa20213023
[25]	W. Sintunavart, P. Kumam, Coupled coincidence and coupled common fixed point theorems in partially ordered metric spaces, Thai J. Math., 10 (2012), 551–563. https://doi.org/10.1186/1687-1812-2012-128 doi: 10.1186/1687-1812-2012-128
[26]	F. Sabetghadam, H. P. Masiha, Some coupled fixed point theorems in cone metric space, Fixed Point Theory A., (2009), Article ID: 125426, 8 pages. https://doi.org/10.1155/2009/125426
[27]	S. U. Rehman, S. U. Khan, A. Ghaffar, S. W. Yao, M. Inc, Some novel generalized strong coupled fixed point findings in cone metric spaces with application to integral equation, J. Function Spaces, (2021), Article ID: 5541981, 9 pages. https://doi.org/10.1155/2021/5541981
[28]	Y. Jiang, F. Gu, Common coupled fixed point results in multiplicative metric spaces and applications, J. Nonlinear Sci. Appl., 10 (2017), 1881–1895. https://doi.org/10.22436/jnsa.010.04.48 doi: 10.22436/jnsa.010.04.48
[29]	L. Shanjit, Y. Rohen, T. C. Singh, P. P. Murthy, Coupled fixed point theorems in partially ordered multiplicative metric space and its applications, Int. J. Pure Appl. Math., 108 (2016), 1314–3395. https://doi.org/10.12732/ijpam.v108i1.7 doi: 10.12732/ijpam.v108i1.7
[30]	T. Rugumisa, S. Kumar, A fixed point theorem for nonself mappings in multiplicative metric spaces, Konuralp J. Math., 8 (2020), 1–6. https://doi.org/10.18576/jant/060103 doi: 10.18576/jant/060103
[31]	T. G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal., 65 (2006), 1379–1393. https://doi.org/10.1016/j.na.2005.10.017 doi: 10.1016/j.na.2005.10.017
[32]	G. X. Chen, S. Jabeen, S. U. Rehman, A. M. Khalil, F. Abbas, A. Kanwal, et al., Coupled fixed point analysis in fuzzy cone metric spaces with application to nonlinear integral equations, Adv. Differ. Equ-Ny., (2020), 25 pages. https: //doi.org/10.1186/s13662-020-03132-8
[33]	V. Gupta, W. Shatanawi, N. Mani, Fixed point theorems for $(\psi, \beta)-$Geraghty contraction type maps in ordered metric spaces and some applications to integral and ordinary differential equations, J. Fixed Point Theory A., 19 (2017), 1251–1267. https://doi.org/10.1007/s11784-016-0303-2 doi: 10.1007/s11784-016-0303-2
[34]	I. Shamas, S. U. Rehman, H. Aydi, T. Mahmood, E. Ameer, Unique fixed-point results in fuzzy metric spaces with an application to Fredholm integral equations, J. Function Spaces, (2021), Article ID 4429173, 12 pages. https://doi.org/10.1155/2021/4429173
[35]	I. Shamas, S. U. Rehman, N. Jan, A. Gumaei, M. Al-Rakhami, A new approach to Fuzzy differential equations using weakly-compatible self-mappings in fuzzy metric spaces, J. Function Spaces, (2021), Article ID 6123154, 13 pages. https://doi.org/10.1155/2021/6123154
[36]	M. T. Waheed, S. U. Rehman, N. Jan, A. Gumaei, M. Al-Rakhami, An approach of Lebesgue integral in fuzzy cone metric spaces via unique coupled fixed point theorems, J. Function Spaces, (2021), Article ID: 8766367, 14 pages. https://doi.org/10.1155/2021/8766367

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