Tripartite Coincidence-Best Proximity Points and Convexity in Generalized Metric Spaces

  • M. Norouzian
  • A. AbkarEmail author


We consider a triple (KST) consisting of three nonlinear mappings defined on the union \(A\cup B\cup C\) of closed subsets of a generalized metric space. After introducing a notion of convex structure in the generalized metric space, we introduce the notion of tripartite contractions, tripartite semi-contractions, tripartite coincidence points, as well as tripartite best proximity points for the triple (KST). We establish theorems on the existence and convergence of tripartite coincidence-best proximity points. Examples are given to support the new findings.


Coincidence point Best proximity point Cyclic contraction Noncyclic contraction G-metric space Uniformly convex G-metric space 

Mathematics Subject Classification

47H10 47H09 54H25 



  1. Abkar, A., Gabeleh, M.: Best proximity points for cyclic mappings in ordered metric spaces. J. Optim. Theory Appl. 150, 188–193 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Al-Thagafi, M.A., Shahzad, N.: Convergence and existence results for best proximity points. Nonlinear Anal. 70, 3665–3671 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Borcut, M., Berinde, V.: Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces. Nonlinear Anal. 74, 4889–4897 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Cho, Y.J., Gupta, A., Karapinar, E., Kumam, P., Sintunawarat, W.: Tripled best proximity point theorem in metric spaces. Math. Inequal. Appl. 16, 1197–1216 (2013)MathSciNetzbMATHGoogle Scholar
  5. De la Sen, M.: Some results on fixed and best proximity points of multivalued cyclic self mappings with a partial order. Abstr. Appl. Anal. 2013, 11 (2013) (Article ID 968492) Google Scholar
  6. De la Sen, M., Agarwal, R.P.: Some fixed point-type results for a class of extended cyclic self mappings with a more general contractive condition. Fixed Point Theory Appl. 59, 14 (2011). MathSciNetzbMATHGoogle Scholar
  7. Di Bari, C., Suzuki, T., Verto, C.: Best proximity points for cyclic Meir–Keeler contractions. Nonlinear Anal. 69, 3790–3794 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Eldred, A.A., Veeramani, P.: Existence and convergence of best proximity points. J. Math. Anal. Appl. 323, 1001–1006 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Eldred, A.A., Kirk, W.A., Veeramani, P.: Proximal normal structure and relatively nonexpansive mappings. Studia Math. 171, 283–293 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Espinola, R., Gabeleh, M., Veeramani, P.: On the structure of minimal sets of relatively nonexpansive mappings. Numer. Funct. Anal. Optim. 34, 845–860 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Fukhar-ud-din, H., Khan, A.R., Akhtar, Z.: Fixed point results for a generalized nonexpansive map in uniformly convex metric spaces. Nonlinear Anal. 75, 4747–4760 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Gabeleh, M., Lakzian, H., Shahzad, N.: Best proximity points for asymptotic pointwise contractions. J. Nonlinear Convex Anal. 16, 83–93 (2015)MathSciNetzbMATHGoogle Scholar
  13. Gabeleh, M., Shahzad, N., Olela Otafudu, O.: Coincidence best proximity points in convex metric spaces. Filomat 32(7), 2451–2463 (2018)MathSciNetGoogle Scholar
  14. Garcia Falset, J., Mlesinte, O.: Coincidence problems for generalized contractions. Appl. Anal. Discrete Math. 8, 1–15 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Hussain, N., Latif, A., Salimi, P.: Best proximity point results in \(G\)-metric spaces. Abstr. Appl. Anal., Article ID 837943 (2014).
  16. Karapinar, E.: Best proximity points of Kannan type cyclic weak \(\phi \)-contractions in ordered metric spaces. An. St. Univ. Ovidius Constanta 20, 51–64 (2012)MathSciNetzbMATHGoogle Scholar
  17. Kirk, W.A., Srinivasan, P.S., Veeramani, P.: Fixed points for mappings satisfying cyclic contractive conditions. Fixed Point Theory 4(1), 79–86 (2003)MathSciNetzbMATHGoogle Scholar
  18. Lashkaripour, R., Hamzehnejadi, J.: Generalization of the best proximity point. J. Inequal. Spec. Funct. 4, 136–147 (2017)MathSciNetGoogle Scholar
  19. Leon, A.F., Gabeleh, M.: Best proximity pair theorems for noncyclic mappings in Banach and metric spaces. Fixed Point Theory 17, 63–84 (2016)MathSciNetzbMATHGoogle Scholar
  20. Mustafa, Z.: A new structure for generalized metric spaces with applications to fixed point theory, Ph.D. thesis. The University of Newcastle, New South Wales (2005)Google Scholar
  21. Mustafa, Z., Sims, B.: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 7, 289–297 (2006)MathSciNetzbMATHGoogle Scholar
  22. Mustafa, Z., Obiedat, H., Awawdeh, F.: Some fixed point theorem for mapping on complete \(G\)-metric spaces. Fixed Point Theory Appl., Article ID 189870 (2008).
  23. Norouzian, M., Abkar, A.: Coincidence quasi-best proximity points for quasi-cyclic-noncyclic mappings in convex metric spaces. Iran. J. Math. Sci. Inform. (2019a) (to appear) Google Scholar
  24. Norouzian, M., Abkar, A.: Coincidence-best proximity points of contraction pairs in uniformly convex metric spaces. J. Math. Ext. (2019b) (to appear) Google Scholar
  25. Pragadeeswarar, V., Marudai, M.: Best proximity points: approximation and optimization in partially ordered metric spaces. Optim. Lett. 7, 1883–1892 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. Shimizu, T., Takahashi, W.: Fixed points of multivalued mappings in certian convex metric spaces. Topol. Methods Nonlinear Anal. 8, 197–203 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Suzuki, T., Kikkawa, M., Vetro, C.: The existence of best proximity points in metric spaces with to property UC. Nonlinear Anal. 71, 2918–2926 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Takahashi, W.: A convexity in metric space and nonexpansive mappings. Kodai Math. Sem. Rep. 22, 142–149 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  29. Van An, T., Van An, N., Le Hang, V.O.T.: A new approach to fixed point theorems on \(G\)-metric spaces. Topol. Appl. 160, 1486–1493 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Sociedade Brasileira de Matemática 2019

Authors and Affiliations

  1. 1.Department of Pure Mathematics, Faculty of ScienceImam Khomeini International UniversityQazvinIran

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