• Tas, Nihal (Department of Mathematics Balikesir University) ;
  • Ozgur, Nihal Yilmaz (Department of Mathematics Balikesir University)
  • Received : 2017.07.14
  • Accepted : 2017.12.21
  • Published : 2018.07.31


Our aim is to introduce the notion of a parametric $N_b-metric$ and study some basic properties of parametric $N_b-metric$ spaces. We give some fixed-point results on a complete parametric $N_b-metric$ space. Some illustrative examples are given to show that our results are valid as the generalizations of some known fixed-point results. As an application of this new theory, we prove a fixed-circle theorem on a parametric $N_b-metric$ space.


parametric $N_b-metric$;${\acute{C}}iri{\acute{c}}^{\prime}s$ fixed-point result;Kannan's fixed-point result;Chatterjea's fixed-point result;expansive mapping;fixed circle


  1. I. A. Bakhtin, The contraction mapping principle in almost metric space, in Functional analysis, No. 30 (Russian), 26-37, Ulyanovsk. Gos. Ped. Inst., Ulyanovsk, 1989.
  2. S. K. Chatterjea, Fixed-point theorems, C. R. Acad. Bulgare Sci. 25 (1972), 727-730.
  3. Lj. B. Ciric, A generalization of Banach's contraction principle, Proc. Amer. Math. Soc. 45 (1974), 267-273.
  4. N. Hussain, S. Khaleghizadeh, P. Salimi, and A. A. N. Abdou, A new approach to fixed point results in triangular intuitionistic fuzzy metric spaces, Abstr. Appl. Anal. 2014 (2014), Art. ID 690139, 16 pp.
  5. N. Hussain, P. Salimi, and V. Parvaneh, Fixed point results for various contractions in parametric and fuzzy b-metric spaces, J. Nonlinear Sci. Appl. 8 (2015), no. 5, 719-739.
  6. R. Kannan, Some results on fixed points. II, Amer. Math. Monthly 76 (1969), 405-408.
  7. Y. Ozgur and N. Tas, Some fixed-circle theorems on metric spaces, Bull. Malays. Math. Sci. Soc. (2017);
  8. N. Y. Ozgur, N. Tas, and U. Celik, New fixed-circle results on S-metric spaces, Bull. Math. Anal. Appl. 9 (2017), no. 2, 10-23.
  9. B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226 (1977), 257-290.
  10. S. Sedghi and N. V. Dung, Fixed point theorems on S-metric spaces, Mat. Vesnik 66 (2014), no. 1, 113-124.
  11. S. Sedghi, A. Gholidahneh, T. Dosenovic, J. Esfahani, and S. Radenovic, Common fixed point of four maps in Sb-metric spaces, J. Linear Topol. Algebra 5 (2016), no. 2, 93-104.
  12. S. Sedghi, N. Shobe, and A. Aliouche, A generalization of fixed point theorems in S-metric spaces, Mat. Vesnik 64 (2012), no. 3, 258-266.
  13. N. Tasand N. Y. Ozgur, On parametric S-metric spaces and fixed-point type theorems for expansive mappings, J. Math. 2016 (2016), Art. ID 4746732, 6 pp.
  14. M. Ughade, D. Turkoglu, S. K. Singh, and R. D. Daheriya, Some fixed point theorems in Ab-metric space, British J. Math. & Computer Science 19 (2016), no. 6, 1-24.
  15. S. Z.Wang, B. Y. Li, Z. M. Gao, and K. Iseki, Some fixed point theorems on expansion mappings, Math. Japon. 29 (1984), no. 4, 631-636.
  16. Wolfram Research, Inc., Mathematica, Trial Version, Champaign, IL, 2017.