A new approach to integral equations via contraction results in multiplicative metric spaces

Hizbullah; Saif Ur Rehman; Sami Ullah Khan; Gauhar Rahman; Kamsing Nonlaopon; Hizbullah; Saif Ur Rehman; Sami Ullah Khan; Gauhar Rahman; Kamsing Nonlaopon

doi:10.3934/math.20221089

AIMS Mathematics

2022, Volume 7, Issue 11: 19891-19901. doi: 10.3934/math.20221089

Previous Article Next Article

Research article

A new approach to integral equations via contraction results in multiplicative metric spaces

1.
Institute of Numerical Sciences, Department of Mathematics, Gomal University, Dera Ismail Khan 29050, Pakistan
2.
Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan
3.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Received: 24 June 2022 Revised: 27 August 2022 Accepted: 29 August 2022 Published: 08 September 2022
MSC : 47H10, 54H25

In this research study, some striking features of single-valued fixed point theorems on multiplicative metric spaces have been established. Our displayed work consists of some unique fixed point theorems under generalized contraction with maximum and minimum conditions. In support of our work, we demonstrate some illustrative examples to justify all the conditions of our main theorems. In addition, a nonlinear integral equation is presented as an application to express the validity of our work. The offered outcomes in this study extend and improve many of the results proved in recent decades.
- fixed point,
- multiplicative metric space,
- contraction conditions,
- integral equations
Citation: Hizbullah, Saif Ur Rehman, Sami Ullah Khan, Gauhar Rahman, Kamsing Nonlaopon. A new approach to integral equations via contraction results in multiplicative metric spaces[J]. AIMS Mathematics, 2022, 7(11): 19891-19901. doi: 10.3934/math.20221089

Related Papers:

Abstract

In this research study, some striking features of single-valued fixed point theorems on multiplicative metric spaces have been established. Our displayed work consists of some unique fixed point theorems under generalized contraction with maximum and minimum conditions. In support of our work, we demonstrate some illustrative examples to justify all the conditions of our main theorems. In addition, a nonlinear integral equation is presented as an application to express the validity of our work. The offered outcomes in this study extend and improve many of the results proved in recent decades.

References

[1]	E. Picard, Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives, J. Math. Pures Appl., 6 (1890), 145–210.
[2]	S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133–181.
[3]	S. K. Chatterjea, Fixed point theorems, Dokl. Bolgarskata Akad. Nauk., 25 (1972), 727–730.
[4]	R. Kannan, Some results on fixed points, Bull. Cal. Math. Soc., 60 (1968), 71–76.
[5]	G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007), 1468–1476. https://doi.org/10.1016/j.jmaa.2005.03.087 doi: 10.1016/j.jmaa.2005.03.087
[6]	S. Jabeen, S. U. Rehman, Z. Zheng, W. Wei, Weakly compatible and Quasi-contraction results in fuzzy cone metric spaces with application to the Urysohn type integral equations, Adv. Differ. Equ., 2020 (2020), 280. https://doi.org/10.1186/s13662-020-02743-5 doi: 10.1186/s13662-020-02743-5
[7]	S. Mehmood, S. U. Rehman, I. Ullah, R. A. R. Bantan, M. Elgarhy, Integral equations approach in complex-valued generalized $b$-metric spaces, J. Math., 2022 (2022), 7454498. https://doi.org/10.1155/2022/7454498 doi: 10.1155/2022/7454498
[8]	S. U. Rehmana, S. Jabeenb, Muhammad, H. Ullah, Hanifullah, Some multi-valued contraction theorems on $H$-cone metric, J. Adv. Stud. Topol., 10 (2019), 11–24.
[9]	S. U. Rehman, H. Aydi, G. X. Chen, S. Jabeen, S. U. Khan, Some set-valued and multi-valued contraction results in fuzzy cone metric spaces, J. Ineq. Appl., 2021 (2021), 110. https://doi.org/10.1186/s13660-021-02646-3 doi: 10.1186/s13660-021-02646-3
[10]	I. Shamas, S. U. Rehman, H. Aydi, T. Mahmood, E. Ameer, Unique fixed-point results in fuzzy metric spaces with an application Fredholm integral equations, J. Funct. Spaces, 2021 (2021), 4429173. https://doi.org/10.1155/2021/4429173 doi: 10.1155/2021/4429173
[11]	M. Grossman, R. Katz, Non-newtonian calculus, Lee Press, 1972.
[12]	A. E. Bashirov, E. M. Kurpnar, A. Özyapic, 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
[13]	B. Rome, M. Sarwar, Characterization of multiplicative metric completeness, Int. J. Anal. Appl., 10 (2016), 90–94.
[14]	X. He, M. Song, D. Chen, Common fixed points for weak commutative function on a multiplicative metric space, Fixed point Theory Appl., 2014 (2014), 48. https://doi.org/10.1186/1687-1812-2014-48 doi: 10.1186/1687-1812-2014-48
[15]	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
[16]	M. U. Ali, Caristi mapping in multiplicative metric spaces, Sci. Int. (Lahore), 27 (2015), 3917–3919.
[17]	T. Došenović, S. Radenović, Some critical remarks on the paper: An essential remark on fixed point results on multiplicative metric spaces, J. Adv. Math. Stud., 10 (2017), 20–24.
[18]	T. Došenović, M. Postolache, S. Radenović, On the multiplicative metric spaces: Survey, Fixed Point Theory Appl., 2016 (2016), 92. https://doi.org/10.1186/s13663-016-0584-6 doi: 10.1186/s13663-016-0584-6
[19]	F. Gu, Y. J. Cho, Common fixed point results for four maps satisfying $\phi$-contractive condition in multiplicative metric spaces, Fixed Point Theory Appl., 2015 (2015), 165. https://doi.org/10.1186/s13663-015-0412-4 doi: 10.1186/s13663-015-0412-4
[20]	S. M. Kang, P. Nagpal, S. K. Garg, S. Kumar, Fixed points for multiplicative expansive functions in multiplicative metric spaces, Int. J. Math. Anal., 9 (2015), 1939–1946. http://dx.doi.org/10.12988/ijma.2015.54130 doi: 10.12988/ijma.2015.54130
[21]	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. http://dx.doi.org/10.12732/ijpam.v102i2.14 doi: 10.12732/ijpam.v102i2.14
[22]	P. Kumar, S. Kumar, Common fixed points for weakly compatible mappings in multiplicative metric spaces, Int. J. Math. Anal., 9 (2015), 2087–2097. http://dx.doi.org/10.12988/ijma.2015.56162 doi: 10.12988/ijma.2015.56162
[23]	C. Mongkolkeha, W. Sintunavarat, Best proximity points for multiplicative proximal contraction function on multiplicative metric spaces, J. Nonlinear Sci. Appl., 8 (2015), 1134–1140.
[24]	M. Özavsar, A. C. Çevikel, Fixed points of multiplicative contraction functions on multiplicative metric spaces, J. Eng. Technol. Appl. Sci., 2 (2017), 65–79. https://doi.org/10.30931/jetas.338608 doi: 10.30931/jetas.338608
[25]	M. Sarwar, B. Rome, Some unique fixed point theorems in multiplicative metric space, arXiv, 2014. https://doi.org/10.48550/arXiv.1410.3384
[26]	S. Shukla, Some critical remarks on the multiplicative metric spaces and fixed point results, J. Adv. Math. Stud., 9 (2016), 454–458.

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)