| Title:
|
Common best proximity point theorems for certain types of mappings (English) |
| Author:
|
Murali, Arunachalam |
| Author:
|
Muthunagai, Krishnan |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0011-4642 |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
151 |
| Issue:
|
2 |
| Year:
|
2026 |
| Pages:
|
305-325 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $S$ and $T$ be two single-valued non-self-mappings from a nonempty set $\mathcal {P}$ to another nonempty set $\mathcal {Q}$. As they are non-self-mappings, the equations $Sx=x$ and $Tx=x$ do not have a common solution. In other words, they do not have a common fixed point. So one intends to find an element $x,$ close to $Sx$ and $Tx,$ which is called the common best proximity point. The common best proximity theorem guarantees the existence of such a best proximity point of the mappings $S$ and $T.$ In this article, we prove the existence and uniqueness of the common best proximity point for a pair of non-self-mappings for rational type contractive conditions on complex valued metric spaces. In addition, by transforming non-self-mappings into self-mappings in complex valued metric spaces, we prove the existence and uniqueness of a common best proximity point for Kannan type rational expression mappings and Chatterjea type rational expression contractive mappings. Moreover, we introduce contraction conditions involving a control function of some kind and prove the existence and uniqueness of a common best proximity point for such conditions. Our key findings extend and integrate some previously published results. (English) |
| Keyword:
|
best proximity point |
| Keyword:
|
fixed point |
| Keyword:
|
rational type contractive condition |
| Keyword:
|
complex valued metric space |
| MSC:
|
39B32 |
| MSC:
|
41A52 |
| MSC:
|
47H10 |
| MSC:
|
54H25 |
| MSC:
|
55M20 |
| DOI:
|
10.21136/MB.2025.0098-24 |
| . |
| Date available:
|
2026-05-19T08:24:22Z |
| Last updated:
|
2026-05-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153626 |
| . |
| Reference:
|
[1] Aghayan, S. M., Zireh, A., Ebadian, A.: Common best proximity points in complex valued metric spaces.Bull. Int. Math. Virtual Inst. 7 (2017), 549-560. Zbl 1412.54045, MR 3649799 |
| Reference:
|
[2] Al-Thagafi, M. A., Shahzad, N.: Convergence and existence results for best proximity points.Nonlinear Anal., Theory Methods Appl., Ser. A 70 (2009), 3665-3671. Zbl 1197.47067, MR 2504453, 10.1016/j.na.2008.07.022 |
| Reference:
|
[3] Azam, A., Fisher, B., Khan, M.: Common fixed point theorems in complex valued metric spaces.Numer. Funct. Anal. Optim. 32 (2011), 243-253. Zbl 1245.54036, MR 2748327, 10.1080/01630563.2011.533046 |
| Reference:
|
[4] Chatterjea, S. K.: Fixed-point theorems.C. R. Acad. Bulg. Sci. 25 (1972), 727-730. Zbl 0274.54033, MR 0324493 |
| Reference:
|
[5] Choudhury, B. S., Metiya, N., Maity, P.: Best proximity point results in complex valued metric spaces.Int. J. Anal. 2014 (2014), Article ID 827862, 6 pages. Zbl 1390.54029, MR 3261246, 10.1155/2014/827862 |
| Reference:
|
[6] Eldred, A. A., Veeramani, P.: Existence and convergence of best proximity points.J. Math. Anal. Appl. 323 (2006), 1001-1006. Zbl 1105.54021, MR 2260159, 10.1016/j.jmaa.2005.10.081 |
| Reference:
|
[7] Fan, K.: Extensions of two fixed point theorems of F. E. Browder.Math. Z. 112 (1969), 234-240. Zbl 0185.39503, MR 0251603, 10.1007/BF01110225 |
| Reference:
|
[8] Gabeleh, M.: Global optimal solutions of non-self mappings.Sci. Bull., Ser. A, Appl. Math. Phys., Politeh. Univ. Buchar. 75 (2013), 67-74. Zbl 1299.47109, MR 3130208 |
| Reference:
|
[9] Gabeleh, M.: Best proximity points for weak proximal contractions.Bull. Malays. Math. Sci. Soc. (2) 38 (2015), 143-154. Zbl 1310.41028, MR 3394044, 10.1007/s40840-014-0009-9 |
| Reference:
|
[10] Hussain, N., Azam, A., Ahmad, J., Arshad, M.: Common fixed point results in complex valued metric spaces with application to integral equations.Filomat 28 (2014), 1363-1380. Zbl 1462.54066, MR 3360044, 10.2298/FIL1407363H |
| Reference:
|
[11] Jungck, G.: Commuting mappings and fixed points.Am. Math. Mon. 83 (1976), 261-263. Zbl 0321.54025, MR 0400196, 10.1080/00029890.1976.11994093 |
| Reference:
|
[12] Kannan, R.: Some results on fixed points.Bull. Calcutta Math. Soc. 60 (1968), 71-76. Zbl 0209.27104, MR 0257837 |
| Reference:
|
[13] Mondal, S., Dey, L. K.: Some common best proximity point theorems in a complete metric space.Afr. Mat. 28 (2017), 85-97. Zbl 1489.54177, MR 3613622, 10.1007/s13370-016-0432-1 |
| Reference:
|
[14] Murali, A., Muthunagai, K.: Best proximity point theorems for generalized rational type contraction conditions involving control functions on complex valued metric spaces.Adv. Fixed Point Theory 13 (2023), Article ID 13, 11 pages. 10.28919/afpt/8299 |
| Reference:
|
[15] Murali, A., Muthunagai, K.: Fixed points for interpolative and rational type mappings on complex valued metric spaces.AIP Conf. Proc. 3180 (2024), Article ID 020001. 10.1063/5.0224350 |
| Reference:
|
[16] Prolla, J. B.: Fixed-point theorems for set-valued mappings and existence of best approximations.Numer. Funct. Anal. Optimization 5 (1983), 449-455. Zbl 0513.41015, MR 0703107, 10.1080/01630568308816149 |
| Reference:
|
[17] Raj, V. S.: A best proximity point theorem for weakly contractive non-self-mappings.Nonlinear Anal., Theory Methods Appl., Ser. A 74 (2011), 4804-4808. Zbl 1228.54046, MR 2810719, 10.1016/j.na.2011.04.052 |
| Reference:
|
[18] Reich, S.: Approximate selections, best approximations, fixed points, and invariant sets.J. Math. Anal. Appl. 62 (1978), 104-113. Zbl 0375.47031, MR 0514991, 10.1016/0022-247X(78)90222-6 |
| Reference:
|
[19] Rouzkard, F., Imdad, M.: Some common fixed point theorems on complex valued metric spaces.Comput. Math. Appl. 64 (2012), 1866-1874. Zbl 1268.54033, MR 2960809, 10.1016/j.camwa.2012.02.063 |
| Reference:
|
[20] Basha, S. Sadiq: Common best proximity points: Global minimization of multi-objective functions.J. Glob. Optim. 54 (2012), 367-373. Zbl 1281.90058, MR 2979635, 10.1007/s10898-011-9760-8 |
| Reference:
|
[21] Basha, S. Sadiq, Veeramani, P.: Best proximity pairs and best approximations.Acta. Sci. Math. 63 (1997), 289-300. Zbl 0909.47042, MR 1459793 |
| Reference:
|
[22] Basha, S. Sadiq, Veeramani, P.: Best proximity pair theorems for multifunctions with open fibres.J. Approximation Theory 103 (2000), 119-129. Zbl 0965.41020, MR 1744381, 10.1006/jath.1999.3415 |
| Reference:
|
[23] Sehgal, V. M., Singh, S. P.: A theorem on best approximations.Numer. Funct. Anal. Optimization 10 (1989), 181-184. Zbl 0635.41022, MR 0978810, 10.1080/01630568908816298 |
| Reference:
|
[24] Sintunavarat, W., Kumam, P.: Generalized common fixed point theorems in complex valued metric spaces and applications.J. Inequal. Appl. 2012 (2012), Article ID 84, 12 pages. Zbl 1295.54090, MR 2922731, 10.1186/1029-242X-2012-84 |
| Reference:
|
[25] Vetrivel, V., Veeramani, P., Bhattacharyya, P.: Some extensions of Fan's best approximation theorem.Numer. Funct. Anal. Optimization 13 (1992), 397-402. Zbl 0763.41026, MR 1179367, 10.1080/01630569208816486 |
| . |
