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Journal of Fixed Point Theory and Applications

, Volume 18, Issue 3, pp 645–671 | Cite as

Fixed point theorems in new generalized metric spaces

  • Erdal Karapınar
  • Donal O’Regan
  • Antonio Francisco Roldán López de Hierro
  • Naseer Shahzad
Article

Abstract

The aim of our paper is to present new fixed point theorems under very general contractive conditions in generalized metric spaces which were recently introduced by Jleli and Samet in [Fixed Point Theory Appl. 2015 (2015), doi: 10.1186/s13663-015-0312-7]. Although these spaces are not endowed with a triangle inequality, these spaces extend some well known abstract metric spaces (for example, b-metric spaces, Hitzler–Seda metric spaces, modular spaces with the Fatou property, etc.). We handle several types of contractive conditions. The main theorems we present involve a reflexive and transitive binary relation that is not necessarily a partial order. We give a counterexample to a recent fixed point result of Jleli and Samet. Our results extend and unify recent results in the context of partially ordered abstract metric spaces.

Keywords

Generalized metric space b-metric space fixed point contractive mapping 

Mathematics Subject Classification

Primary 47H10 46T99 Secondary 47H09 54H25 

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References

  1. 1.
    Aage C. T., Salunke J. N.: The results on fixed points in dislocated and dislocated quasi-metric space. Appl. Math. Sci. 2, 2941–2948 (2008)MathSciNetzbMATHGoogle Scholar
  2. 2.
    R. P. Agarwal, E. Karapınar, D. O’Regan and A. F. Roldán-López-de-Hierro, Fixed Point Theory in Metric Type Spaces. Springer, Switzerland, 2015.Google Scholar
  3. 3.
    Ahmad M. A., Zeyada F. M., Hassan G. F.: Fixed point theorems in generalized types of dislocated metric spaces and its applications. Thai J. Math. 11, 67–73 (2013)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Akkouchi M.: Common fixed point theorems for two selfmappings of a b-metric space under an implicit relation. Hacet. J. Math. Stat. 40, 805–810 (2011)MathSciNetzbMATHGoogle Scholar
  5. 5.
    S. Alshehri, I. Aranđelović and N. Shahzad, Symmetric spaces and fixed points of generalized contractions. Abstr. Appl. Anal. 2014 (2014), Art. ID 763547, 8 pages.Google Scholar
  6. 6.
    Banach S.: Sur les opérations dans les ensembles abstraits et leur application aux équations intégerales. Fund. Math. 3, 133–181 (1922)zbMATHGoogle Scholar
  7. 7.
    Berinde V.: Sequences of operators and fixed points in quasimetric spaces. Stud. Univ. Babeş-Bolyai Math. 41, 23–27 (1996)MathSciNetzbMATHGoogle Scholar
  8. 8.
    V. Berinde, Contracţii Generalizate şi Aplicaţii. Editura Cub Press 22, Baia Mare, Romania, 1997.Google Scholar
  9. 9.
    M. Boriceanu, M. Bota and A. Petruşel, Multivalued fractals in b-metric spaces. Cent. Eur. J. Math. 8 (2010), 367–377 Google Scholar
  10. 10.
    Ćirić L. B.: A generalization of Banach's contraction principle. Proc. Amer. Math. Soc. 45, 267–273 (1974)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Czerwik S.: Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostraviensis 1, 5–11 (1993)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Czerwik S., Dlutek K., Singh S. L.: Round-off stability of iteration procedures for set-valued operators in b-metric spaces. J. Natur. Phys. Sci. 15, 1–8 (2001)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Domínguez Benavides T., Khamsi M. A., Samadi S.: Uniformly Lipschitzian mappings in modular function spaces. Nonlinear Anal. 46, 267–278 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Haghi R. H., Rezapour Sh., Shahzad N.: Some fixed point generalizations are not real generalizations. Nonlinear Anal. 74, 1799–1803 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    A. Hajji and E. Hanebaly, Fixed point theorem and its application to perturbed integral equations in modular function spaces. Electron. J. Differential Equations, No. 105 (2005), 11 pages.Google Scholar
  16. 16.
    P. Hitzler, Generalized metrics and topology in logic programming semantics. Dissertation, Faculty of Science, National University of Ireland, University College, Cork, 2001.Google Scholar
  17. 17.
    Hitzler P., Seda A. K.: Dislocated topologies. J. Electr. Eng. 51, 3–7 (2000)zbMATHGoogle Scholar
  18. 18.
    M. Jleli and B. Samet, A generalized metric space and related fixed point theorems. Fixed Point Theory Appl. 2015 (2015), doi: 10.1186/s13663-015-0312-7, 14 pages.
  19. 19.
    E. Karapınar, A. Roldán, N. Shahzad and W. Sintunavarat, Discussion of coupled and tripled coincidence point theorems for \({\varphi}\)-contractive mappings without the mixed g-monotone property. Fixed Point Theory Appl. 2014 (2014), doi: 10.1186/1687-1812-2014-92, 16 pages.
  20. 20.
    E. Karapınar and P. Salimi, Dislocated metric space to metric spaces with some fixed point theorems. Fixed Point Theory Appl. 2013 (2013), doi: 10.1186/1687-1812-2013-222, 19 pages.
  21. 21.
    Khamsi M. A.: Nonlinear semigroups in modular function spaces. Math. Jpn. 37, 291–299 (1992)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Khamsi M. A., Kozlowski W. M., Reich S.: Fixed point theory in modular function spaces. Nonlinear Anal. 14, 935–953 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    W. Kirk and N. Shahzad, b-Metric spaces. In: Fixed Point Theory in Distance Spaces, Springer, Cham, 2014, 113–131.Google Scholar
  24. 24.
    W. M. Kozlowski, Modular Function Spaces. Monographs and Textbooks in Pure and Applied Mathematics 122, Marcel Dekker, Inc., New York, 1988.Google Scholar
  25. 25.
    M. A. Kutbi, A. Roldán, W. Sintunavarat, J. Martínez-Moreno and C. Roldán, F-closed sets and coupled fixed point theorems without the mixed monotone property. Fixed Point Theory Appl. 2013 (2013), doi: 10.1186/1687-1812-2013-330, 11 pages.
  26. 26.
    Musielak J., Orlicz W.: On modular spaces. Studia Math. 18, 49–65 (1959)MathSciNetzbMATHGoogle Scholar
  27. 27.
    H. Nakano, Modular Semi-Ordered Spaces. Tokyo, Japan, 1959.Google Scholar
  28. 28.
    Nieto J. J., Rodríguez-López R.: Contractive mapping theorem in partially ordered sets and applications to ordinary differential equations. Order 22, 223–239 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Popović B., Radenović S., Shukla S.: Fixed point results to tvs-cone b-metric spaces. Gulf J. Math. 1, 51–64 (2013)Google Scholar
  30. 30.
    Ran A. C. M., Reurings M. C. B.: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Amer. Math. Soc. 132, 1435–1443 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Razani A., Pour S. H., Nabizadeh E., Mohamadi M. B.: A new version of the Ćirić quasi-contraction principle in the modular space. Novi Sad J. Math. 43, 1–9 (2003)zbMATHGoogle Scholar
  32. 32.
    A. F. Roldán López de Hierro and N. Shahzad, Some fixed/coincidence point theorems under (\({\psi, \varphi}\))-contractivity conditions without an underlying metric structure. Fixed Point Theory Appl. 2014 (2014), doi: 10.1186/1687-1812-2014-218, 24 pages.
  33. 33.
    Roldán A., Martínez-Moreno J., Roldán C., Cho Y. J.: Multidimensional coincidence point results for compatible mappings in partially ordered fuzzy metric spaces. Fuzzy Sets and Systems 251, 71–82 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    A. F. Roldán López de Hierro and N. Shahzad, New fixed point theorem under R-contractions. Fixed Point Theory Appl. 2015 (2015), doi: 10.1186/s13663-015-0345-y, 18 pages.
  35. 35.
    B. Samet, The class of (\({\alpha, \psi}\))-type contractions in b-metric spaces and fixed point theorems. Fixed Point Theory Appl.2015 (2015), doi: 10.1186/s13663-015-0344-z, 17 pages.
  36. 36.
    B. Samet, Fixed points for \({\alpha-\psi}\) contractive mappings with an application to quadratic integral equations. Electron. J. Differential Equations, No. 152 (2014), 1–18.Google Scholar
  37. 37.
    N. Shahzad, E. Karapınar and A. F. Roldán-López-de-Hierro, On some fixed point theorems under (\({\alpha, \psi, \phi}\))-contractivity conditions in metric spaces endowed with transitive binary relations. Fixed Point Theory Appl. 2015 (2015), doi: 10.1186/s13663-015-0359-5, 24 pages.

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Erdal Karapınar
    • 1
    • 2
  • Donal O’Regan
    • 3
  • Antonio Francisco Roldán López de Hierro
    • 4
    • 5
  • Naseer Shahzad
    • 6
  1. 1.Department of MathematicsAtilim Universityİncek, AnkaraTurkey
  2. 2.Nonlinear Analysis and Applied Mathematics Research Group (NAAM)King Abdulaziz UniversityJeddahSaudi Arabia
  3. 3.School of Mathematics, Statistics and Applied MathematicsNational University of IrelandGalwayIreland
  4. 4.Department of Quantitative Methods for Economics and BusinessUniversity of GranadaGranadaSpain
  5. 5.PAIDI Research Group FQM-268University of JaénJaénSpain
  6. 6.Operator Theory and Applications Research Group, Department of MathematicsKing Abdulaziz UniversityJeddahSaudi Arabia

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