Fixed Points of (ψ, φ)Ω-Contractive Mappings in Ordered P-Metric Spaces

Document Type : Research Paper


1 Department of Mathematics, Gilan-E-Gharb Branch, Islamic Azad University, Gilan-E-Gharb, Iran

2 Department of Mathematics, Takestan Branch, Islamic Azad University, Takestan, Iran


In this paper, we introduce the notion of an extended metric space (p-metric space) as a new generalization of the concept of b-metric space. Also, we present the concept of (ψ, φ)Ω-contractive mappings and we establish some fixed point results for this class of mappings in ordered complete p-metric spaces. Our results generalize several well-known comparable results in the literature. Finally, examples support our results. 


  1. A. Aghajani, M. Abbas and J.R. Roshan, Common fixed point of generalized weak contractive mappings in partially ordered b-metric spaces, Math. Slovaca, 64(4), 941–960, (2014).
  2. A. Aghajani, S. Radenovi´c and J.R. Roshan, Common fixed point results for four mappings satisfying almost generalized (S, T)-contractive condition in partially ordered metric spaces, Appl. Math. Comput., 218, 5665–5670, (2012).
  3. H. Aydi, M.-F. Bota, E. Karapınar and S. Moradi, A common fixed point for weak ϕ-contractions on b−metric spaces, Fixed Point Theory Appl., 13(2), 337–346, (2012).
  4. H. Aydi, M.-F. Bota, E. Karapınar and S. Mitrovi´c, A fixed point theorem for set-valued quasi-contractions in b-metric spaces, Fixed Point Theory Appl., doi:10.1186/1687-1812-2012-88, (2012).
  5. S. Banach, Sur les operations dans les ensembles et leur application aux equation sitegrales, Fund. Math., 3, 133–181, (1922).
  6. M. Boriceanu, Strict fixed point theorems for multivalued operators in b−metric spaces, Int. J. Modern Math., 4(3), 285–301, (2009).
  7. Lj. Ciri´c, A generalization of Banach’s contraction principle, Proc. Amer. Math. Soc., 45, 265–273, (1974).
  8. Lj. Ciri´c, On contractive type mappings, Math. Balkanica, 1, 52–57, (1971).
  9. S. Czerwik, Nonlinear set-valued contraction mappings in b−metric spaces, Atti Sem. Mat. Fis. Univ. Modena, 46(2), 263–276, (1998).
  10. C. Di Bari and P. vetro, ϕ-paris and common fixed points in cone metric spaces, Rendiconti del Circolo Matematico di Palermo, 57, 279–285, (2008).
  11. P.N. Dutta and B.S. Choudhury, A generalization of contraction principle in metric spaces, Fixed point Theory Appl., Article ID 406368, (2008).
  12. D. Dori´c, Common fixed point for generalized (ψ, φ)-weak contraction, Appl. Math. Lett., 22, 1896–1900, (2009).
  13. N. Hussain and M.H. Shah, KKM mappings in cone b−metric spaces, Comput. Math. Appl., 62, 1677–1684, (2011).
  14. Z. Kadelburg, Z. Pavlovi´c and S. Radenovi´c, Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces, Comput. Math. Appl., 59, 3148–3159, (2010).
  15. E. Karapinar and K. Sadarangani, Fixed point theory for cyclic (ψ, φ)-contractions, Fixed Point Theory Appl., 2011:69, (2011).
  16. M.A. Khamsi, Remarks on cone metric spaces and fixed point theorems of contractive mappings, Fixed Point Theory Appl., Article ID 315398, doi:10.1155/2010/315398, (2010).
  17. M.S. Khan, M. Swaleh and S. Sessa, Fixed point theorems by altering distances between the points, Bull. Aust. Math. Soc., 30, 1–9, (1984).
  18. N.H. Nashine, Z. Kadelburg and S. Radenovi´c, Common fixed point theorems for weakly isotone increasing mappings in ordered partial metric spaces, Math. Comput. Modelling, doi:10.1016/j.mcm.2011.12.019, (2011).
  19. N.H. Nashine and B. Samet, Fixed point results for mappings satisfying (ψ, φ)-weakly contractive condition in partially ordered metric spaces, Nonlinear Anal., 74, 2201–2209, (2011).
  20. M. Pacurar, Sequences of almost contractions and fixed points in b−metric spaces, Analele Universitatii de Vest, Timisoara Seria Matematica Informatica XLVIII, 3, 125–137 (2010).
  21. O. Popescu, Fixed points for (ψ, φ)-weak contractions, Appl. Math. lett., 24, 1–4, (2011).
  22. J.R. Roshan, N. Shobkolaei, S. Sedghi and M. Abbas, Common fixed point of four maps in b-metric spaces, Hacettepe J. Math. Stat., 43(4), 613–624, (2014).
  23. W. Shatanawi and A. Al-Rawashdeh, Common fixed points of almost generalized (ψ, φ)−contractive mappings in ordered metric spaces, Fixed Point Theory Appl., 2012:80, (2012).
  24. W. Shatanawi and B. Samet, On (ψ, φ)-weakly contractive condition in partially ordered metric spaces, Comput. Math. Appl., 62, 3204–3214, (2011).
  25. J.J. Nieto and R.R. Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22, 223–239, (2005).
  26. S.L. Singh and B. Prasad, Some coincidence theorems and stability of iterative procedures, Comput. Math. Appl., 55, 2512–2520, (2008).