Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 20 (2025), 149 -- 172

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This work is licensed under a Creative Commons Attribution 4.0 International License.

EXISTENCE OF COUPLED COINCIDENCE BEST PROXIMITY POINT

Pulak Konar and Sumit Chandok

Abstract. This work of optimality is engaged with couple coincidence best proximity point. We have developed some optimal results where control functions and non-linear weak inequalities are evident. Further using our results, we establish some fixed point results. Our work is supported by an illustration.

2020 Mathematics Subject Classification: 47H10; 54H10; 54H25.
Keywords: Control function; coupled best proximity point; coupled coincidence best proximity point; Partially ordered set.

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References

  1. A. Abkar, M. Gabeleh, Best proximity points of non-self mappings, Top, 21 (2013), 287--295. MR3068488 Zbl1310.47073

  2. A. Abkar, M. Gabeleh, Global optimal solutions of noncyclic mappings in metric spaces, J. Optim. Theory Appl. 153(2012), 298--305. MR2908645 Zbl1262.90134

  3. Ya. I. Alber, S. Guerre-Delabriere, Principles of weakly contractive maps in Hilbert spaces, new results in operator theory, Advances and Appl. ( ed. by I. Goldberg and Yu. Lyubich ), Birkhäuser Verlag, 98 (1997), 7--22. MR1478463 Zbl0897.47044

  4. S. Banach, Sur lés oprations dans lés ensembles abstraits et leurs applications aux quations intgrales, Fund Math. 3(1922), 133--181. Zbl 0005.20901

  5. S. S. Basha, Best proximity points: optimal solutions, JOTA, 151(2011), 210--216.MR2836473 Zbl 1226.90135

  6. S. S. Basha, Discrete optimization in partially ordered sets, J. Glob Optim. 54(2012), 511--517. MR2988196 Zbl 1261.90039

  7. V. Berinde, Approximating fixed points of weak contractions using the Picard iteration, Nonlin. Anal. Forum. 9(2004), 43--53. MR211366 Zbl 1078.47042

  8. S. K. Bhandari, D. Gopal, P. Konar, Probabilistic α - min Ciric Type Contraction Results using a Control Function, AIMS Math. 5(2), 2020: 1186–1198. MR4141907 Zbl 1485.54033

  9. J Caballero, J Harjani, K Sadarangani, A best proximity point theorem for Geraghty-contractions, Fixed Point Theory and Appl. 2012(2012): 231. MR3017227 Zbl 1281.54021

  10. S. Chandok, B. S. Choudhury, N. Metiya, Fixed point results in ordered metric spaces for rational type expressions with auxiliary functions, Journal of the Egyptian Mathematical Society 23(2015), 95--101. MR3317306 Zbl1317.54016

  11. C. E. Chidume, H. Zegeye, S. J. Aneke, Approximation of fixed points of weakly contractive nonself maps in Banach spaces, J. Math. Anal. Appl., 270(1)(2002), 189-199. MR1911760 Zbl 1005.47053

  12. B. S. Choudhury, P. Konar, B. E. Rhoades, N. Metiya, Fixed point theorems for generalized weakly contractive mappings, Nonlinear Anal. TMA 74(2011), 2116--2126.MR2781742 Zbl 1218.54036

  13. B. S. Choudhury, A. Kundu, A couple coincidence point result in partially ordered metric spaces for compatible mappings, Nonlin. Anal. 73(2010), 2524--2531. MR2674088 Zbl1229.54051

  14. B. S. Choudhury, P. Maity, P. Konar, A global optimality result using nonself mappings, Opsearch, 51(2014), 312--320. MR3216707 Zbl1397.54050

  15. B. S. Choudhury, P. Maity, P. Konar, A global optimality result using geraghty type contraction, IJOCTA, 4(2014), 99--104. MR3232677 Zbl1339.49006

  16. B. S. Choudhury, N. Metiya, G. Maniu, P. Konar, Best proximity results: optimization by approximate solutions, Fixed Point Theory Appl. (2016) 2016: 79. MR3522360 Zbl1342.54027

  17. B. S. Choudhury, N. Metiya, P. Konar, Existence and Uniqueness of coupled best proximity point in patially ordered metric spaces, J. Nonlinear Anal. Opt., 7(2016), 145 - 153. MR3601239 Zbl1413.54108

  18. D. Doric, Common fixed point for generalized (ψ, φ)-weak contractions, Appl. Math. Lett. 22(2009), 1896--1900. MR2558564 Zbl1203.54040

  19. P. N. Dutta, B. S. Choudhury, A generalisation of contraction principle in metric spaces, Fixed Point Theory Appl. 2008(2008), Article ID 406368. MR2470177 Zbl1177.54024

  20. A. A. Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323(2006), 1001-1006. MR2260159 Zbl1105.54021

  21. K. Fan, Extensions of two fixed point theorems of F.E. Browder, Math Z. 122(1969), 234 -- 240. Zbl0204.23201

  22. T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65(2006), 1379--1393. MR2260159 Zbl1106.47047

  23. D. Guo, V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications, Nonlinear Anal. 11(1987), 623-632. MR 886653 Zbl0635.47045

  24. J. Harjani, K. Sadarangani, Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal. 71(2009), 3403-3410. MR2532760 Zbl1221.54058

  25. E. Karapinar, On best proximity point of ψ- Geraghty contractions, Fixed Point Theory and Appl. 2013(2013) : 200. MR3095321 Zbl1295.41037

  26. S. Karpagam, S. Agrawal, Best proximity points for cyclic Meir-Keeler contraction maps, Nonlin. Anal. 74(2011), 1040--1046. MR2746787

  27. W. A. Kirk, B. Sims, Handbook of metric fixed point theory, Springer Netherlands 2001, XIII, 703p. MR1904271 Zbl0970.54001

  28. P. Konar, S. Chandok, S. Dutta, M. De la Sen, Coupled Optimal Results with an Application to Nonlinear Integral Equations, Axioms, 10(2021), 73. 10.3390/axioms10020073

  29. P. Kumam, V. Pragadeeswarar, M. Marudai, K. Sitthithakerngkiet, Coupled best proximity points in ordered metric spaces, Fixed Point Theory Appl. 2014(2014) : 107. MR3258670 Zbl1312.54026

  30. V. Lakshmikantham, L. Ciric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlin. Anal., 70(2009), 4341--4349. MR2514765 Zbl1176.54032

  31. E. Prajisha, P. Shaini, Coupled coincidence point theorems of mappings in partially ordered metric spaces, J. Nonlin. Anal. Optim. Vol. 11(1)(2020), 29--40. MR4086415 Zbl1474.54222

  32. B. E. Rhoades, Some theorems on weakly contractive maps, Nonlin. Anal. 47(4)(200l), 2683--2693. MR1972392 Zbl1042.47521

  33. V. Sankar Raj, A best proximity point theorem for weakly contractive non-self-mappings, Nonlin. Anal. 74(2011), 4804--4808. MR2810719 Zbl1228.54046

  34. W. Sintunavarat, P. Kumam, Coupled best proximity point theorem in metric spaces, Fixed Point Theory and Appl. 2012(2012) : 93. MR2956826 Zbl1308.41036

  35. Q. Zhang, Y. Song, Fixed point theory for generalized φ - weak contractions, Appl. Math. Lett. 22(2009), 75--78. MR2484285 Zbl1163.47304



Pulak Konar -Corresponding author
School of Advanced Sciences, Department of Mathematics,
VIT University - Chennai Campus
Vandalur Kelambakkam Road, Chennai, 600127, India.
e-mail: pulakkonar@gmail.com

Sumit Chandok
Department of Mathematics, Thapar Institute of Engineering & Technology.
Patiala - 147-004, Punjab, India.
e-mail: sumit.chandok@thapar.edu

https://www.utgjiu.ro/math/sma