Common best proximity point theorems for proximally weak reciprocal continuous mappings

A. Sreelakshmi Unni; V. Pragadeeswarar; Manuel De la Sen; A. Sreelakshmi Unni; V. Pragadeeswarar; Manuel De la Sen

doi:10.3934/math.20231442

AIMS Mathematics

2023, Volume 8, Issue 12: 28176-28187. doi: 10.3934/math.20231442

Previous Article Next Article

Research article

Common best proximity point theorems for proximally weak reciprocal continuous mappings

1.
Department of Mathematics, Amrita School of Physical Sciences, Coimbatore, Amrita Vishwa Vidyapeetham, India
2.
Institute of Research and Development of Processes IIDP, University of the Basque Country, Campus of Leioa, Leioa 48940, Bizkaia, Spain

Received: 14 August 2023 Revised: 21 September 2023 Accepted: 25 September 2023 Published: 13 October 2023
MSC : 47H09, 47H10

The main objective of this paper is to find sufficient conditions for the existence and uniqueness of common best proximity points for discontinuous non-self mappings in the setting of a complete metric space. We introduce and analyze new concepts such as proximally reciprocal continuous, proximally weak reciprocal continuous, R-proximally weak commuting of types $ M_{\Lambda} $ and $ M_{\Gamma} $ for non-self mappings. Furthermore, we obtain a common best proximity point theorem for such mappings. In addition, we provide an example to support our main result.
- compatible mappings,
- weak reciprocal continuity,
- R-weakly commuting of type $ M_{\Lambda} $ and $ M_{\Gamma} $
Citation: A. Sreelakshmi Unni, V. Pragadeeswarar, Manuel De la Sen. Common best proximity point theorems for proximally weak reciprocal continuous mappings[J]. AIMS Mathematics, 2023, 8(12): 28176-28187. doi: 10.3934/math.20231442

Related Papers:

Abstract

The main objective of this paper is to find sufficient conditions for the existence and uniqueness of common best proximity points for discontinuous non-self mappings in the setting of a complete metric space. We introduce and analyze new concepts such as proximally reciprocal continuous, proximally weak reciprocal continuous, R-proximally weak commuting of types $ M_{\Lambda} $ and $ M_{\Gamma} $ for non-self mappings. Furthermore, we obtain a common best proximity point theorem for such mappings. In addition, we provide an example to support our main result.

References

[1]	M. A. Al-Thagafi, N. Shahzad, Convergence and existence results for best proximity points, Nonlinear Anal. Theor., 70 (2009), 3665–3671. https://doi.org/10.1016/j.na.2008.07.022 doi: 10.1016/j.na.2008.07.022
[2]	A. A. Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl., 323 (2006), 1001–1006. https://doi.org/10.1016/j.jmaa.2005.10.081 doi: 10.1016/j.jmaa.2005.10.081
[3]	C. Di Bari, T. Suzuki, C. Vetro, Best proximity points for cyclic Meir-Keeler contractions, Nonlinear Anal. Theor., 69 (2008), 3790–3794. https://doi.org/10.1016/j.na.2007.10.014 doi: 10.1016/j.na.2007.10.014
[4]	V. Pragadeeswarar, M. Marudai, Best proximity points: approximation and optimization in partially ordered metric spaces, Optim. Lett., 7 (2012), 1883–1892. https://doi.org/10.1007/s11590-012-0529-x doi: 10.1007/s11590-012-0529-x
[5]	S. S. Basha, P. Veeramani, Best approximations and best proximity pairs, Acta. Sci. Math., 63 (1997), 289–300.
[6]	S. S. Basha, P. Veeramani, D. V. Pai, Best proximity pair theorems, Indian J. Pure Appl. Math., 32 (2001), 1237–1246. https://doi.org/10.1093/imamat/66.4.411 doi: 10.1093/imamat/66.4.411
[7]	S. S. Basha, Best proximity points: optimal solutions, J. Optim. Theory Appl., 151 (2011), 210–216. https://doi.org/10.1007/s10957-011-9869-4 doi: 10.1007/s10957-011-9869-4
[8]	V. S. Raj, A best proximity point theorem for weakly contractive non-self-mappings, Nonlinear Anal. Theor., 74 (2011), 4804–4808. https://doi.org/10.1016/j.na.2011.04.052 doi: 10.1016/j.na.2011.04.052
[9]	M. R. Haddadi, V. Parvaneh, M. Mursaleen, Global optimal approximate solutions of best proximity points, Filomat, 35 (2021), 1555–1564. https://doi.org/10.2298/FIL2105555H doi: 10.2298/FIL2105555H
[10]	M. Jovanović, Z. Kadelburg, S. Radenović, Common fixed point results in metric-type spaces, Fixed Point Theory Appl., 2010 (2010), 978121. https://doi.org/10.1155/2010/978121 doi: 10.1155/2010/978121
[11]	R. Chugh, S. Kumar, Common fixed points for weakly compatible maps, Proc. Indian Acad. Sci. (Math. Sci.), 111 (2001), 241–247. https://doi.org/10.1007/BF02829594 doi: 10.1007/BF02829594
[12]	P. Debnath, N. Konwar, S. Radenović, Metric fixed point theory, Singapore: Springer, 2021. https://doi.org/10.1007/978-981-16-4896-0
[13]	Z. Mustafa, J. R. Roshan, V. Parvaneh, Z. Kadelburg, Some common fixed point results in ordered partial b-metric spaces, J. Inequal. Appl., 2013 (2013), 562. https://doi.org/10.1186/1029-242X-2013-562 doi: 10.1186/1029-242X-2013-562
[14]	J. R. Roshan, N. Shobkolaei, S. Sedghi, V. Parvaneh, S. Radenović, Common fixed point theorems for three maps in discontinuous Gb metric spaces, Acta Math. Sci., 34 (2014), 1643–1654. https://doi.org/10.1016/S0252-9602(14)60110-7 doi: 10.1016/S0252-9602(14)60110-7
[15]	H. Isik, V. Parvaneh, B. Mohammadi, I. Altun, Common fixed point results for generalized Wardowski type contractive multi-valued mappings, Mathematics, 7 (2019), 1130. https://doi.org/10.3390/math7111130 doi: 10.3390/math7111130
[16]	H. K. Pathak, Y. J. Cho, S. M. Kang, Remarks of R-weakly commuting mappings and common fixed point theorems, Bull. Korean Math. Soc., 34 (1997), 247–257.
[17]	S. S. Basha, N. Shahzad, R. Jeyaraj, Common best proximity points: global optimization of multi-objective functions, Appl. Math. Lett., 24 (2011), 883–886. https://doi.org/10.1016/j.aml.2010.12.043 doi: 10.1016/j.aml.2010.12.043
[18]	N. Shahzad, S. S. Basha, R. Jeyaraj, Common best proximity points: global optimal solutions, J. Optim. Theory Appl., 148 (2011), 69–78. https://doi.org/10.1007/s10957-010-9745-7 doi: 10.1007/s10957-010-9745-7
[19]	S. S. Basha, Common best proximity points: global minimization of multi-objective functions, J. Glob. Optim., 54 (2012), 367–373. https://doi.org/10.1007/s10898-011-9760-8 doi: 10.1007/s10898-011-9760-8
[20]	C. Mongkolkeha, P. Kumam, Some common best proximity points for proximity commuting mappings, Optim. Lett., 7 (2013), 1825–1836. https://doi.org/10.1007/s11590-012-0525-1 doi: 10.1007/s11590-012-0525-1
[21]	M. Gabeleh, J. Markin, Common best proximity pairs via the concept of complete proximal normal structure, Ann. Funct. Anal., 11 (2020), 831–847. https://doi.org/10.1007/s43034-020-00057-x doi: 10.1007/s43034-020-00057-x
[22]	S. Mondal, L. K. Dey, Some common best proximity point theorems in a complete metric space, Afr. Mat., 28 (2017), 85–97. https://doi.org/10.1007/s13370-016-0432-1 doi: 10.1007/s13370-016-0432-1
[23]	P. L$o'$l$o'$, S. M. Vaezpour, J. Esmaily, Common best proximity points theorem for four mappings in metric-type spaces, Fixed Point Theory Appl., 2015 (2015), 47. https://doi.org/10.1186/s13663-015-0298-1 doi: 10.1186/s13663-015-0298-1
[24]	P. L$o'$l$o'$, S. M. Vaezpour, R. Saadati, Common best proximity points results for new proximal C-contraction mappings, Fixed Point Theory Appl., 2016 (2016), 56. https://doi.org/10.1186/s13663-016-0545-0 doi: 10.1186/s13663-016-0545-0
[25]	V. Pragadeeswarar, R. Gopi, M. De la Sen, S. Radenovi'c, Proximally compatible mappings and common best proximity points, Symmetry, 12 (2020), 353. https://doi.org/10.3390/sym12030353 doi: 10.3390/sym12030353
[26]	V. Pragadeeswarar, G. Poonguzali, M. Marudai, S. Radenović, Common best proximity point theorem for multivalued mappings in partially ordered metric spaces, Fixed Point Theory Appl., 2017 (2017), 22. https://doi.org/10.1186/s13663-017-0615-y doi: 10.1186/s13663-017-0615-y
[27]	J. Puangpee, S. Suantai, New hybrid algorithms for global minimization of common best proximity points of some generalized nonexpansive mappings, Filomat, 33 (2019), 2381–2391. https://doi.org/10.2298/FIL1908381P doi: 10.2298/FIL1908381P
[28]	R. P. Pant, A common fixed point theorm under a new condition, Indian J. Pure Appl. Math., 30 (1999), 147–152.
[29]	R. P. Pant, R. K. Bisht, D. Arora, Weak reciprocal continuity and fixed point theorems, Ann. Univ. Ferrara, 57 (2011), 181–190. https://doi.org/10.1007/s11565-011-0119-3 doi: 10.1007/s11565-011-0119-3

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)