Kybernetika 58 no. 3, 335-353, 2022

Fixed point result in controlled fuzzy metric spaces with application to dynamic market equilibrium

Rakesh Tiwari, Vladimir Rakočević and Shraddha RajputDOI: 10.14736/kyb-2022-3-0335


In this paper, we introduce $\Theta_f$-type controlled fuzzy metric spaces and establish some fixed point results in this spaces. We provide suitable examples to validate our result. We also employ an application to substantiate the utility of our established result for finding the unique solution of an integral equation emerging in the dynamic market equilibrium aspects to economics.


fuzzy metric spaces, fixed point, controlled fuzzy metric spaces, fuzzy $\Theta _f$-contractive mapping, dynamic market equilibrium


54H25, 47H10, A11


  1. H. Aydi, M. Bota, E. Karapinar and S.Moradi: A common fixed point for weak $\phi$-contractions on $b$-metric spaces. Fixed Point Theory 13 (2012), 337-346.   CrossRef
  2. H. Afshari, M. Atapour and H. Aydi: Generalized $\alpha-\psi-$Geraghty multivalued mappings on $b$-metric spaces endowed with a graph. J. Appl. Eng. Math. 7 (2017), 248-260.   CrossRef
  3. H. Aydi, R.Banković, I.Mitrović and M. Nazam: Nemytzki-Edelstein-Meir-Keeler type results in b-metric spaces. Discret. Dyn. Nat. Soc. (2018), 4745764.   DOI:10.1155/2018/4745764
  4. N. Alharbi, H. Aydi, A. Felhi, C. Ozel and S. Sahmim: $\alpha$-Contractive mappings on rectangular b-metric spaces and an application to integral equations. J. Math. Anal. 9 (2018), 47-60.   CrossRef
  5. S. Banach: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrals. Fund. Math. 3 (1922), 133-181.   DOI:10.4064/fm-3-1-133-181
  6. I. A. Bakhtin: The contraction mapping principle in almost metric spaces. Funct. Anal. 30 (1989), 26-37.   DOI:10.1039/ap9892600037
  7. M. Boriceanu, A. Petrusel. and I. A. Rus: Fixed point theorems for some multivalued generalized contraction in b-metric spaces. Int. J. Math. Statist. 6 (2010), 65-76.   CrossRef
  8. S. Czerwik: Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostrava 1 (1993), 5-11.
  9. T. E. Dowling: Introduction to Mathematical Economics. Schaum's Outline Series, 2001.   CrossRef
  10. A. George and P. Veeramani: On some results in fuzzy metric spaces. Fuzzy Sets Systems 64 (1994), 395-399.   DOI:10.1016/0165-0114(94)90162-7
  11. D. Gopal: Contributions to fixed point theory of fuzzy contractive mappings. Adv. Metric Fixed Point Theory Appl. (2021), 241-282.   DOI:10.1007/978-981-33-6647-3\_11
  12. D. Gopal and T.Došenović: Fixed point theory for fuzzy contractive mappings. Metric Struct. Fixed Point Theory (2021), 199-244.   DOI:10.4324/9781003139607-6
  13. D. Gopal and C. Vetro: Some new fixed point theorems in fuzzy metric spaces. Iranian J. Fuzzy Systems 11(2014), 3, 95-107.   DOI:10.4324/9781003139607-6
  14. M. Grabiec: Fixed points in fuzzy metric spaces. Fuzzy Sets Systems 27 (1988), 385-389.   DOI:10.1016/0165-0114(88)90064-4
  15. Y. Hao and H. Guan: On some common fixed point results for weakly contraction mappings with application. J. Funct. Spaces 2021 (2021), 5573983.   DOI:10.1155/2021/5573983
  16. J. K. Kim: Common fixed point theorems for non-compatible self-mappings in b-fuzzy metric spaces. J. Comput. Anal. Appl. 22 (2017), 336-345.   CrossRef
  17. I. Kramosil and J. Michálek: Fuzzy metric and statistical metric spaces. Kybernetika 11 (1975), 326-334.
  18. F. Mehmood, R. Ali, C. Ionescu and T. Kamran: Extended fuzzy b-metric spaces. J. Math. Anal. 8 (2017), 124-131.
  19. S. Melliani and A. Moussaoui: Fixed point theorem using a new class of fuzzy contractive mappings. J. Univer. Math. 1 (2018), 2, 148-154.   CrossRef
  20. D. Mihet: Fuzzy $\psi$-contractive mappings in non-archimedean fuzzy metric spaces. Fuzzy Sets Systems 159 (2008), 6, 739-744.   DOI:10.1016/j.fss.2007.07.006
  21. N. Mlaiki, H. Aydi, N. Souayah and T. Abdeljawad: Controlled metric type spaces and the related contraction principle. Math. Molecul. Divers. Preservat. Int. 6 (2018), 1-7.   DOI:10.3390/math6100194
  22. S. N\u{a}d\u{a}ban: Fuzzy b-metric spaces. Int. J. Comput. Commun. Control 11 (2016), 273-281.   DOI:10.15837/ijccc.2016.2.2443
  23. H. S. Nasr, M. Imdad, I. Khan and M. Hasanuzzaman: Fuzzy $\Theta_f$-contractive mappings and their fixed points with applications. J. Intell. Fuzzy Systems (2020), 1-10.   DOI:10.3233/jifs-200319
  24. M. S. Sezen: Controlled fuzzy metric spaces and some related fixed point results. Numer. Part. Different. Equations (2020), 1-11.   DOI:10.1002/num.22541
  25. S. Shukla, D. Gopal and W. Sintunavarat: A new class of fuzzy contractive mappings and fixed point theorems. Fuzzy Sets Systems 350 (2018), 85-94.   DOI:10.1016/j.fss.2018.02.010
  26. B. Schweizer and A. Sklar: Statistical metric spaces. Paciffc J. Math. 10 (1960), 313-334.   DOI:10.2140/pjm.1960.10.313
  27. D. Wardowski: Fuzzy contractive mappings and fixed points in fuzzy metric spaces. Fuzzy Sets Systems 222 (2013), 108-114.   DOI:10.1016/j.fss.2013.01.012
  28. A. L. Zadeh: Fuzzy sets. Inform. Control 8 (1965), 338-353.   DOI:10.1016/\\S0019-9958(65)90241-X