Novel results on fixed-point methodologies for hybrid contraction mappings in $ M_{b} $-metric spaces with an application

Mustafa Mudhesh; Hasanen A. Hammad; Eskandar Ameer; Muhammad Arshad; Fahd Jarad; Mustafa Mudhesh; Hasanen A. Hammad; Eskandar Ameer; Muhammad Arshad; Fahd Jarad

doi:10.3934/math.2023077

AIMS Mathematics

2023, Volume 8, Issue 1: 1530-1549. doi: 10.3934/math.2023077

Previous Article Next Article

Research article

Novel results on fixed-point methodologies for hybrid contraction mappings in $ M_{b} $-metric spaces with an application

1.
Department of Mathematics, International Islamic University, H-10, Islamabad - 44000, Pakistan
2.
Department of Mathematics, Unaizah College of Sciences and Arts, Qassim University, Buraydah 52571, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt
4.
Department of Mathematics, Taiz University, Taiz, 6803, Yemen
5.
Department of Mathematics, Čankaya University, Etimesgut 06790, Ankara, Turkey
6.
Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
7.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan

Received: 05 May 2022 Revised: 17 October 2022 Accepted: 19 October 2022 Published: 21 October 2022
MSC : 46S40, 47H10, 54H25

By combining the results of Wardowski's cyclic contraction operators and admissible multi-valued mappings, the motif of $ \eta $-cyclic $ \left(\alpha _{\ast }, \beta _{\ast }\right) $-admissible type $ \digamma $-contraction multivalued mappings are presented. Moreover, some novel fixed point theorems for such mappings are proved in the context of $ M_{b} $-metric spaces. Also, two examples are given to clarify and strengthen our theoretical study. Finally, the existence of a solution of a pair of ordinary differential equations is discussed as an application.
Citation: Mustafa Mudhesh, Hasanen A. Hammad, Eskandar Ameer, Muhammad Arshad, Fahd Jarad. Novel results on fixed-point methodologies for hybrid contraction mappings in $ M_{b} $-metric spaces with an application[J]. AIMS Mathematics, 2023, 8(1): 1530-1549. doi: 10.3934/math.2023077

Related Papers:

Abstract

By combining the results of Wardowski's cyclic contraction operators and admissible multi-valued mappings, the motif of $ \eta $-cyclic $ \left(\alpha _{\ast }, \beta _{\ast }\right) $-admissible type $ \digamma $-contraction multivalued mappings are presented. Moreover, some novel fixed point theorems for such mappings are proved in the context of $ M_{b} $-metric spaces. Also, two examples are given to clarify and strengthen our theoretical study. Finally, the existence of a solution of a pair of ordinary differential equations is discussed as an application.

References

[1]	S. Czerwik, Contraction mappings in $b$-metric spaces, Acta Math. Inform. Univ. Ostrav., 1 (1993), 5–11.
[2]	S. G. Matthews, Partial metric topology, Ann. N. Y. Acad. Sci., 728 (1994), 183–197. https://doi.org/10.1111/j.1749-6632.1994.tb44144.x
[3]	Z. Ma, L. Jiang, H. Sun, $C^$-algebra-valued metric spaces and related fixed point theorems, Fixed Point Theory A., 206* (2014), 1–11. https://doi.org/10.1186/s13663-015-0471-6 doi: 10.1186/s13663-015-0471-6
[4]	M. Asadi, E. Karapınar, P. Salimi, New extension of $p$-metric spaces with some fixed-point results on $M$-metric spaces, J. Ineq. Appl., 2014 (2014), 1–9. https://doi.org/10.1186/1029-242X-2014-18 doi: 10.1186/1029-242X-2014-18
[5]	I. Altun, H. Sahin, D. Turkoglu, Fixed point results for multivalued mappings of Feng-Liu type on $M$-metric spaces, J. Nonlin. Funct. Anal., 2018 (2018), 1–8. https://doi.org/10.22436/jnsa.009.06.36 doi: 10.22436/jnsa.009.06.36
[6]	H. Sahin, I. Altun, D. Turkoglu, Two fixed point results for multivalued $F$-contractions on $M$-metric spaces, RACSAM, 113 (2019), 1839–1849. https://doi.org/10.1007/s13398-018-0585-x doi: 10.1007/s13398-018-0585-x
[7]	P. R. Patle, D. K. Patel, H. Aydi, D. Gopal, N. Mlaiki, Nadler and Kannan type set valued mappings in $M$-metric spaces and an application, Mathematics, 7 (2019), 1–14. https://doi.org/10.3390/math7040373 doi: 10.3390/math7040373
[8]	H. Monfared, M. Azhini, M. Asadi, Fixed point results on $M$-metric spaces, J. Math. Anal., 7 (2016), 85–101.
[9]	H. Monfared, M. Azhini, M. Asadi, $C$-class and $F\left(\psi, \varphi \right) $-contractions on $M$-metric spaces, J. Nonlin. Anal. Appl., 8 (2017), 209–224.
[10]	N. Mlaiki, $F_{m}$-contractive and $F_{m}$-expanding mappings in $M$-metric spaces, J. Math. Comput. Sci., 18 (2018), 262–271. https://doi.org/10.22436/jmcs.018.03.02 doi: 10.22436/jmcs.018.03.02
[11]	N. Mlaiki, A. Zarrad, N. Souayah, A. Mukheimer, T. Abdeljawed, Fixed point theorem in $M_{b}$-metric spaces, J. Math. Anal., 7 (2016), 1–9.
[12]	P. Hu, F. Gu, Some fixed point theorems of $\lambda$-contractive mappings in Menger $PSM$-spaces, J. Nonlin. Funct. Anal., 33 (2020), 1–12. https://doi.org/10.23952/jnfa.2020.33 doi: 10.23952/jnfa.2020.33
[13]	M. A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc., 40 (1973), 604–608. https://doi.org/10.1090/S0002-9939-1973-0334176-5 doi: 10.1090/S0002-9939-1973-0334176-5
[14]	N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl., 141 (1989), 177–188. https://doi.org/10.1016/0022-247X(89)90214-X doi: 10.1016/0022-247X(89)90214-X
[15]	O. Popescu, Some new fixed point theorems for $\alpha$-Geraghty contraction type maps in metric spaces, Fixed Point Theory A., 190 (2014), 1–12. https://doi.org/10.1186/1687-1812-2014-190 doi: 10.1186/1687-1812-2014-190
[16]	M. Arshad, M. Mudhesh, A. Hussain, E. Ameer, Recent thought of $\alpha_{\ast}$-geraghty $F$-contraction with application, J. Math. Ext., 16 (2021), 1–28.
[17]	S. Alizadeh, F. Moradlou, P. Salimi, Some fixed point results for $\left(\alpha, \beta \right) $-$\left(\psi, \phi \right) $-contractive mappings, Filomat, 28 (2014), 635–647. https://doi.org/10.1186/1687-1812-2014-190 doi: 10.1186/1687-1812-2014-190
[18]	E. Ameer, H. Huang, M. Nazam, M. Arshad, Fixed point theorems for multivalued $\gamma $-$FG$-contractions with $\left(\alpha_{\ast }, \beta _{\ast }\right) $-admissible mappings in partial $b$-metric spaces and application, U.P.B. Sci. Bull., S. A, 81 (2019), 97–108.
[19]	S. K. Padhan, GVV. J. Rao, A. Al-Rawashdeh, H. K. Nashine, R. P. Agarwal, Existence of fixed point for $\gamma $-$FG$-contractive condition via cyclic $\left(\alpha, \beta \right) $-admissible mappings in $b$-metric spaces, J. Nonlinear Sci. Appl., 10 (2017), 5495–5508. https://doi.org/10.22436/jnsa.010.10.31 doi: 10.22436/jnsa.010.10.31
[20]	H. Isik, B. Samet, C. Vetro, Cyclic admissible contraction and applications to functional equations in dynamic programming, Fixed Point Theory A., 2015 (2015), 1–19. https://doi.org/10.1186/s13663-015-0410-6 doi: 10.1186/s13663-015-0410-6
[21]	M. S. Sezen, Cyclic $\left(\alpha, \beta \right) $-admissible mappings in modular spaces and applications to integral equations, Universal J. Math. Appl., 2 (2019), 85–93.
[22]	H. A. Hammad, P. Agarwal, L. G. J. Guirao, Applications to boundary value problems and homotopy theory via tripled fixed point techniques in partially metric spaces, Mathematics, 9 (2021), 2012. https://doi.org/10.3390/math9162012 doi: 10.3390/math9162012
[23]	H. A. Hammad, H. Aydi, M. D. la Sen, Analytical solution for differential and nonlinear integral equations via $F_{\varpi _{e}}$-Suzuki contractions in modified $\varpi _{e}$-metric-like spaces, J. Func. Space., 2021 (2021), 6128586.
[24]	H. A. Hammad, H. Aydi, M. D. la Sen, Solutions of fractional differential type equations by fixed point techniques for multivalued contractions, Complexity, 2021 (2021), 5730853. https://doi.org/10.1155/2021/5730853 doi: 10.1155/2021/5730853
[25]	H. A. Hammad, M. D. la Sen, Tripled fixed point techniques for solving system of tripled-fractional differential equations, AIMS Math., 6 (2021), 2330–2343. https://doi.org/10.3934/math.2021141 doi: 10.3934/math.2021141
[26]	R. A. Rashwan, H. A. Hammad, M. G. Mahmoud, Common fixed point results for weakly compatible mappings under implicit relations in complex valued g-metric spaces, Inform. Sci. Lett., 8 (2019), 111–119. https://doi.org/10.18576/isl/080305 doi: 10.18576/isl/080305
[27]	A. Hammad, M. D. la Sen, Fixed-point results for a generalized almost $(s, q)$-Jaggi $F$-contraction-type on $b$-metric-like spaces, Mathematics, 8 (2020), 63. https://doi.org/10.3390/math8010063 doi: 10.3390/math8010063
[28]	S. Anwar, M. Nazam, H. H. Al Sulami, A. Hussain, K. Javed, M. Arshad, Existence fixed-point theorems in the partial $b$-metric spaces and an application to the boundary value problem, AIMS Math., 7 (2022), 8188–8205. https://doi.org/10.3934/math.2022456 doi: 10.3934/math.2022456
[29]	B. Rodjanadid, J. Tanthanuch, Some fixed point results on $M_{b}$-metric space via simulation functions, Thai J. Math., 18 (2020), 113–125.
[30]	D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory A., 2012 (2012), 1–6. https://doi.org/10.1186/1687-1812-2012-94 doi: 10.1186/1687-1812-2012-94
[31]	A. Felhi, Some fixed point results for multi-valued contractive mappings in partial b-metric spaces, J. Adv. Math. Stud., 9 (2016), 208–225.
[32]	I. Altun, G. Minak, H. Daǧ, Multivalued $F$-contractions on complete metric spaces, J. Nonlin. Convex A., 16 (2015), 659–666. https://doi.org/10.2298/FIL1602441A doi: 10.2298/FIL1602441A
[33]	M. Delfani, A. Farajzadeh, C. F. Wen, Some fixed point theorems of generalized $F_{t}$-contraction mappings in $b$-metric spaces, J. Nonlin. Var. Anal., 5 (2021), 615–625.

Reader Comments

Your name:*

Email:*
© 2023 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)