Fixed point results in $ \mathcal{C}^\star $-algebra-valued bipolar metric spaces with an application

Gunaseelan Mani; Arul Joseph Gnanaprakasam; Hüseyin Işık; Fahd Jarad; Gunaseelan Mani; Arul Joseph Gnanaprakasam; Hüseyin Işık; Fahd Jarad

doi:10.3934/math.2023386

AIMS Mathematics

2023, Volume 8, Issue 4: 7695-7713. doi: 10.3934/math.2023386

Previous Article Next Article

Research article

Fixed point results in $ \mathcal{C}^\star $-algebra-valued bipolar metric spaces with an application

1.
Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602105, Tamil Nadu, India
2.
Department of Mathematics, College of Engineering and Technology, Faculty of Engineering and Technology, SRM Institute of Science and Technology, SRM Nagar, Kattankulathur 603203, Kanchipuram, Chennai, Tamil Nadu, India
3.
Department of Engineering Science, Bandırma Onyedi Eylül University, Bandırma 10200, Balıkesir, Turkey
4.
Department of Mathematics, Cankaya University, 06790 Etimesgut, Ankara, Turkey
5.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan

Received: 07 August 2022 Revised: 05 January 2023 Accepted: 11 January 2023 Published: 18 January 2023
MSC : 46L05, 47H10, 54H25, 54C30

In this work, we prove existence and uniqueness fixed point theorems under Banach and Kannan type contractions on $ \mathcal{C}^{\star} $-algebra-valued bipolar metric spaces. To strengthen our main results, an appropriate example and an effective application are presented.
- $ \mathcal{C}^\star $-algebra,
- $ \mathcal{C}^\star $-algebra-valued bipolar metric space,
- fixed point
Citation: Gunaseelan Mani, Arul Joseph Gnanaprakasam, Hüseyin Işık, Fahd Jarad. Fixed point results in $ \mathcal{C}^\star $-algebra-valued bipolar metric spaces with an application[J]. AIMS Mathematics, 2023, 8(4): 7695-7713. doi: 10.3934/math.2023386

Related Papers:

Abstract

In this work, we prove existence and uniqueness fixed point theorems under Banach and Kannan type contractions on $ \mathcal{C}^{\star} $-algebra-valued bipolar metric spaces. To strengthen our main results, an appropriate example and an effective application are presented.

References

[1]	M. M. Fr$\acute{e}$chet, Sur quelques points du calcul fonctionnel, Rendiconti del Circolo Matematico di Palermo, 22 (1906), 1–72.
[2]	H. Aydi, W. Shatanawi, C. Vetro, On generalized weak G-contraction mapping in G-metric spaces, Comput. Math. Appl., 62 (2011), 4222–4229. https://doi.org/10.1016/j.camwa.2011.10.007 doi: 10.1016/j.camwa.2011.10.007
[3]	Z. Mustafa, B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal., 7 (2006), 289–297.
[4]	T. Rasham, P. Agarwal, L. S. Abbasi, S. Jain, A study of some new multivalued fixed point results in a modular like metric space with graph, J. Anal., 30 (2022), 833–844. https://doi.org/10.1007/s41478-021-00372-z doi: 10.1007/s41478-021-00372-z
[5]	T. Rasham, M. Nazam, H. Aydi, A. Shoaib, C. Park, J. R. Lee, Hybrid pair of multivalued mappings in modular-like metric spaces and applications, AIMS Math., 7 (2022), 10582–10595. https://doi.org/10.3934/math.2022590 doi: 10.3934/math.2022590
[6]	T. Rasham, A. Shoaib, S. Alshoraify, C. Park, J. R. Lee, Study of multivalued fixed point problems for generalized contractions in double controlled dislocated quasi metric type spaces, AIMS Math., 7 (2022), 1058–1073. https://doi.org/10.3934/math.2022063 doi: 10.3934/math.2022063
[7]	M. Gamal, T. Rasham, W. Cholamjiak, F. G. Shi, C. Park, New iterative scheme for fixed point results of weakly compatible maps in multiplicative $G_{M}$-metric space via various contractions with application, AIMS Math., 7 (2022), 13681–13703. https://doi.org/10.3934/math.2022754 doi: 10.3934/math.2022754
[8]	T. Rasham, M. De La Sen, A novel study for hybrid pair of multivalued dominated mappings in b-multiplicative metric space with applications, J. Inequal. Appl., 107 (2022). https://doi.org/10.1186/s13660-022-02845-6 doi: 10.1186/s13660-022-02845-6
[9]	T. Rasham, M. Nazam, H. Aydi, R. P. Agarwal, Existence of common fixed points of generalized $\Delta$-implicit locally contractive mappings on closed ball in multiplicative G-metric spaces with applications, Mathematics, 10 (2022), 3369. https://doi.org/10.3390/math10183369 doi: 10.3390/math10183369
[10]	A. Mutlu, U. G$\ddot{u}$rdal, An infinite dimensional fixed point theorem on function spaces of ordered metric spaces, Kuwait J. Sci., 42 (2015), 36–49. https://doi.org/10.1016/j.langcom.2015.03.001 doi: 10.1016/j.langcom.2015.03.001
[11]	A. Mutlu, U. G$\ddot{u}$rdal, Bipolar metric spaces and some fixed point theorems, J. Nonlinear Sci. Appl., 9 (2016), 5362–5373. http://dx.doi.org/10.22436/jnsa.009.09.05 doi: 10.22436/jnsa.009.09.05
[12]	U. G$\ddot{u}$rdal, A. Mutlu, K. $\ddot{O}$zkan, Fixed point results for $\alpha$-$\psi$-contractive mappings in bipolar metric spaces, J. Inequal. Spec. Funct., 11 (2020), 64–75.
[13]	G. N. V. Kishore, R. P. Agarwal, B. S. Rao, R. V. N. S. Rao, Caristi type cyclic contraction and common fixed point theorems in bipolar metric spaces with applications, Fixed Point Theory A., 2018 (2018), 21. https://doi.org/10.1186/s13663-018-0646-z doi: 10.1186/s13663-018-0646-z
[14]	G. N. V. Kishore, D. R. Prasad, B. S. Rao, V. S. Baghavan, Some applications via common coupled fixed point theorems in bipolar metric spaces, J. Crit. Rev., 7 (2020), 601–607.
[15]	G. N. V. Kishore, K. P. R. Rao, A. Sombabu, R. V. N. S. Rao, Related results to hybrid pair of mappings and applications in bipolar metric spaces, J. Math., 2019 (2019), 8485412. https://doi.org/10.1155/2019/8485412 doi: 10.1155/2019/8485412
[16]	B. S. Rao, G. N. V. Kishore, G. K. Kumar, Geraghty type contraction and common coupled fixed point theorems in bipolar metric spaces with applications to homotopy, Int. J. Math. Trends Technol., 63 (2018), 25–34. http://dx.doi.org/10.14445/22315373/IJMTT-V63P504 doi: 10.14445/22315373/IJMTT-V63P504
[17]	G. N. V. Kishore, K. P. R. Rao, H. Işık, B. S. Rao, A. Sombabu, Covarian mappings and coupled fixed point results in bipolar metric spaces, Int. J. Nonlinear Anal. Appl., 12 (2021), 1–15. http://dx.doi.org/10.22075/IJNAA.2021.4650 doi: 10.22075/IJNAA.2021.4650
[18]	A. Mutlu, K. $\ddot{O}$zkan, U. G$\ddot{u}$rdal, Locally and weakly contractive principle in bipolar metric spaces, TWMS J. Appl. Eng. Math., 10 (2020), 379–388.
[19]	Y. U. Gaba, M. Aphane, H. Aydi, $(\alpha, BK)$-contractions in bipolar metric spaces, J. Math., 2021 (2021), 5562651. https://doi.org/10.1155/2021/5562651 doi: 10.1155/2021/5562651
[20]	K. Roy, M. Saha, R. George, L. Gurand, Z. D. Mitrović, Some covariant and contravariant fixed point theorems over bipolar p-metric spaces and applications, Filomat, 36 (2022), 1755–1767. https://doi.org/10.2298/FIL2205755R doi: 10.2298/FIL2205755R
[21]	Z. H. Ma, L. N. Jiang, H. K. Sun, $C^$-algebras-valued metric spaces and related fixed point theorems, Fixed Point Theory A., 2014* (2014), 206. https://doi.org/10.1186/1687-1812-2014-206 doi: 10.1186/1687-1812-2014-206
[22]	S. Batul, T. Kamran, $C^{\star}$-valued contractive type mappings, Fixed Point Theory A., 2015 (2015), 142. https://doi.org/10.1186/s13663-015-0393-3 doi: 10.1186/s13663-015-0393-3
[23]	M. Gunaseelan, G. Arul Joseph, A. Ul Haq, I. A. Baloch, F. Jarad, Coupled fixed point theorems on $C^{}$-algebra-valued bipolar metric spaces. AIMS Math.*, 7 (2022), 7552–7568. http://dx.doi.org/10.3934/math.2022424 doi: 10.3934/math.2022424
[24]	K. R. Davidson, $C^{\star}$-algebras by example, Fields Institute Monographs, American Mathematical Society, 1996.
[25]	G. J. Murphy, $ C^ $-algebra and operator theory*, London, Academic Press, 1990.
[26]	Q. H. Xu, T. E. D. Bieke, Z. Q. Chen, Introduction to operator algebras and noncommutative Lp spaces, Beijing, Science Press, 2010.

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)