Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Solution of a fuzzy global optimization problem by fixed point methodology using a weak coupled contraction

  • 92 Accesses

Abstract

In the present work, we consider the global optimization problem of obtaining distance between two subsets of a fuzzy metric space and solve it by fixed point methodology through the determination of two different pairs of points each of which determines the fuzzy distance. We use fuzzy weak coupled contractions for that purpose. The problem is well studied in metric spaces where it is known as a proximity point problem. We use geometric notions in fuzzy metric spaces. Our result is valid for arbitrary continuous t-norms associated with the fuzzy metric space. The problem is solved by reducing it to that of finding optimal approximate solution of a fuzzy coupled fixed point equation. We also obtain a coupled fixed point result as a consequence of our main theorem. The main result is illustrated with an example.

This is a preview of subscription content, log in to check access.

References

  1. Alber YI, Guerre-Delabriere S (1997) Principle of weakly contractive maps in Hilbert spaces. Operator theory: advances and applications, vol 98. Birkhauser Verlag, Basel, pp 7–22

  2. Bari CD, Suzuki T, Vetro C (2008) Best proximity points for cyclic Meir–Keeler contractions. Nonlinear Anal 69(11):3790–3794

  3. Cho YJ, Kadelburg Z, Saadati R, Shatanawi W (2012) Coupled fixed point theorems under weak contractions. Discrete Dyn Nat Soc. https://doi.org/10.1155/2012/184534

  4. Choudhury BS, Das P (2014) Coupled coincidence point results in partially ordered probabilistic metric spaces. Asian Eur J Math 7(2)

  5. Choudhury BS, Kundu K (2014) Two coupled weak contraction theorems in partially ordered metric spaces. RACSAM 108(2):335–351

  6. Choudhury BS, Maity P (2011) Coupled fixed point results in generalized metric spaces. Math Comput Model 54:73–79

  7. Choudhury BS, Maity P (2016) Best proximity point results in generalized metric spaces. Vietnam J Math 44(2):339–349

  8. Choudhury BS, Metiya N (2010) Fixed points of weak contractions in cone metric spaces. Nonlinear Anal 72:1589–1593

  9. Choudhury BS, Das K, Das P (2013a) Coupled coincidence point results for compatible mappings in partially ordered fuzzy metric spaces. Fuzzy Sets Syst 222(1):84–97

  10. Choudhury BS, Metiya N, Postolache M (2013b) A generalized weak contraction principle with applications to coupled coincidence point problems. Fixed Point Theory Appl 2013:152

  11. Choudhury BS, Das K, Das P (2014) Coupled coincidence point results in partially ordered fuzzy metric spaces. Ann Fuzzy Math Inf 7:619–628

  12. Ćirić L (2009) Some new results for Banach contractions and Edelstein contractive mappings on fuzzy metric spaces. Chaos Solitons Fractals 42(1):146–154

  13. Dutta PN, Choudhury BS (2008) A generalisation of contraction principle in metric spaces. Fixed Point Theory Appl. https://doi.org/10.1155/2008/406368

  14. Eldred AA, Veeramani P (2006) Existence and convergence of best proximity points. J Math Anal Appl 323(2):1001–1006

  15. George A, Veeramani P (1994) On some result in fuzzy metric space. Fuzzy Sets Syst 64(3):395–399

  16. Grabice M (1988) Fixed points in fuzzy metric spaces. Fuzzy Sets Syst 27(3):385–389

  17. Guo D, Lakshmikantham V (1987) Coupled fixed points of nonlinear operators with applications. Nonlinear Anal 11(5):623–632

  18. Hu XQ (2011) Common coupled fixed point theorems for contractive mappings in fuzzy metric spaces. Fixed Point Theory Appl. https://doi.org/10.1155/2011/363716

  19. Ilchev A, Zlatanov B (2016) Error estimates for approximation of coupled best proximity points for cyclic contractive maps. Appl Math Comput 290:412–425

  20. Jain M, Tas K, Kumar S, Gupta N (2012) Coupled fixed point theorems for a pair of weakly compatible maps along with CLRg property in fuzzy metric spaces. J Appl Math. https://doi.org/10.1155/2012/961210

  21. Jleli M, Samet B (2013) Best proximity points for \(\alpha -\psi \)- proximal contractive type mappings and applications. Bull Sci Math 137(8):977–995

  22. Jleli M, Karapinar K, Samet B (2012) Best proximity point result for MK-proximal contractions. Abstr Appl Anal. https://doi.org/10.1155/2012/193085

  23. Karapinar E (2010) Coupled fixed point theorems for nonlinear contractions in cone metric spaces. Comput Math Appl 59(12):3656–3668

  24. Karapinar E (2012) Best proximity points of cyclic mappings. Appl Math Lett 25(11):1761–1766

  25. Kramosil I, Michalek J (1975) Fuzzy metric and statistical metric spaces. Kybernetica 11:336–344

  26. Lakshmikantham V, Ćirić L (2009) Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal 70(12):4341–4349

  27. López JR, Ramaguera S (2004) The Hausdorff fuzzy metric on compact sets. Fuzzy Sets Syst 147(2):273–283

  28. Luong NV, Thuan NX (2011) Coupled fixed points in partially ordered metric spaces and application. Nonlinear Anal 74(3):983–992

  29. Mihet D (2007) On fuzzy contractive mappings in fuzzy metric spaces. Fuzzy Sets Syst 158:915–921

  30. Raj VS (2011) A best proximity point theorem for weakly contractive non-self-mappings. Nonlinear Anal 74(14):4804–4808

  31. Raj VS (2013) Best proximity point theorems for non-self mappings. Fixed Point Theory 14:447–454

  32. Saha P, Choudhury BS, Das P (2016) A new contractive mapping principle in fuzzy metric spaces. Bull dell’Uni Math Ital 8(4):287–296

  33. Saha P, Choudhury BS, Das P (2016) Weak coupled coincidence point results having a partially ordering in fuzzy metric spaces. Fuzzy Inf Eng 8:199–216

  34. Saha P, Guria S, Choudhury BS (2019) Determining fuzzy distance through non-self fuzzy contractions. Yugosl J Oper Res. https://doi.org/10.2298/YJOR180515002S

  35. Saha P, Choudhury BS, Das P (to appear) A weak contraction in a fuzzy metric spaces. J Uncertain Syst

  36. Schweizer B, Sklar A (1960) Statistical metric spaces. Pac J Math 10:313–334

  37. Shayanpour H, Nematizadeh A (2017) Some results on common best proximity point in fuzzy metric spaces. Bol Soc Paran Mat 35:177–194

  38. Sintunavarat W, Kumam P (2012) Coupled best proximity point theorem in metric spaces. Fixed Point Theory Appl. https://doi.org/10.1186/1687-1812-2012-93

  39. Vetro C, Salimi P (2013) Best proximity point results in non-Archimedean fuzzy metric spaces. Fuzzy Inf Eng 5(4):417–429

  40. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

  41. Zhang Q, Song Y (2009) Fixed point theory for generalized \(\phi \)-weak contractions. Appl Math Lett 22(1):75–78

  42. Zhu XH, Xiao J (2011) Note on coupled fixed point theorems for contractions in fuzzy metric spaces. Nonlinear Anal 74(16):5475–5479

Download references

Author information

Correspondence to S. Guria.

Ethics declarations

Conflicts of interest

The corresponding author is a senior research fellow sponsored by UGC, India, and therefore, he is grateful to UGC for the support. At the same time, all the authors are thankful to IIEST, Shibpur, India, for their support. There is no conflict of interest among the authors.

Human and animal rights

Further, we would like to mention that this article does not contain any studies with animals and does not involve any studies over human being.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by V. Loia.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Saha, P., Guria, S., Choudhury, B.S. et al. Solution of a fuzzy global optimization problem by fixed point methodology using a weak coupled contraction. Soft Comput 24, 4121–4129 (2020). https://doi.org/10.1007/s00500-019-04179-w

Download citation

Keywords

  • Fuzzy metric space
  • Coupled contraction
  • Coupled proximity point
  • Fuzzy P-property
  • Global optimality