Skip to main content
SpringerLink
Log in

Various generalizations of metric spaces and fixed point theorems

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas Aims and scope Submit manuscript

Abstract

There have been many attempts to generalize the definition of a metric space in order to obtain possibilities for more general fixed point results. In this paper, we give a survey of recent results on reducing fixed point theorems on generalized metric spaces to fixed point theorems on metric spaces and then investigate this fact in other generalized metric spaces. We show that many generalized metric spaces are topologically equivalent to certain metric spaces or to previously generalized metric spaces. Also, the fixed point theory in these generalized metric spaces may be a consequence of the fixed point theory in certain metric spaces or in previously generalized metric spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abbas, M., Jungck, G.: Common fixed point results for noncommuting mappings without continuity in cone metric spaces. J. Math. Anal. Appl. 341, 416–420 (2008)

    MATH  MathSciNet  Google Scholar 

  2. Abbas, M., Nazir, T., Radenović, S.: Some periodic point results in generalized metric spaces. Appl. Math. Comput. 217, 4094–4099 (2010)

    MATH  MathSciNet  Google Scholar 

  3. Abbas, M., Rajić, V. C., Nazir, T., Radenović, S.: Common fixed point of mappings satisfying rational inequalities in ordered complex valued generalized metric spaces. Afr. Mat. 1–14 (2014). doi:10.1007/s13370-013-0185-z

  4. Abbas, M., Rhoades, B.E.: Common fixed point results for noncommuting mappings without continuity in generalized metric spaces. Appl. Math. Comput. 215, 262–269 (2009)

    MATH  MathSciNet  Google Scholar 

  5. Abdeljawad, T.: Fixed points for generalized weakly contractive mappings in partial metric spaces. Math. Comput. Model. 54, 2923–2927 (2011)

    MATH  MathSciNet  Google Scholar 

  6. Abdeljawad, T., Karapinar, E., Tas, K.: Existence and uniqueness of a common fixed point on partial metric spaces. Comput. Math. Appl. 24, 1900–1904 (2011)

    MATH  MathSciNet  Google Scholar 

  7. Abdeljawad, T., Karapinar, E., Tas, K.: A generalized contraction principle with control functions on partial metric spaces. Comput. Math. Appl. 63, 716–719 (2012)

    MATH  MathSciNet  Google Scholar 

  8. Agarwal, R.P., Kadelburg, Z., Radenović, S.: On coupled fixed point results in asymmetric \(G\)-metric spaces. J. Inequal. Appl. 2013(528), 1–12 (2013)

    Google Scholar 

  9. Agarwal, R.P., Karapinar, E.: Remarks on some coupled fixed point theorems in \(G\)-metric spaces. Fixed Point Theory Appl. 2013(2), 1–33 (2013)

    Google Scholar 

  10. Agarwal, R.P., Meehan, M., O’Regan, D.: Fixed Point Theory and Applications. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  11. Aghajani, A., Abbas, M., Roshan, J.R.: Common fixed point of generalized weak contractive mappings in partially ordered \(G_b\)-metric spaces. Filomat (2014) (accepted paper)

  12. Ahmed, A.E.S., Omran, S., Asad, A.J.: Fixed point theorems in quaternion-valued metric spaces. Abstr. Appl. Anal. 2014, 1–9 (2014)

    Google Scholar 

  13. Akkouchi, M.: Common fixed points of four maps using generalized weak contractivity and well-posedness. Int. J. Nonlinear Anal. Appl. 2, 73–81 (2011)

    MATH  Google Scholar 

  14. Alghamdi, M.A., Hussain, N., Salimi, P.: Fixed point and coupled fixed point theorems on \(b\)-metric-like spaces. J. Inequal. Appl. 2013(402), 1–25 (2013)

    MathSciNet  Google Scholar 

  15. Aliouche, A., Simpson, C.: Fixed points and lines in 2-metric spaces. Adv. Math. 229, 668–690 (2012)

    MATH  MathSciNet  Google Scholar 

  16. Altun, I., Durmaz, G.: Weak partial metric spaces and some fixed point results. Appl. Gen. Topol. 13, 179–191 (2012)

    MATH  MathSciNet  Google Scholar 

  17. Altun, I., Sola, F., Simsek, H.: Generalized contractions on partial metric spaces. Topol. Appl. 157, 2778–2785 (2010)

    MATH  MathSciNet  Google Scholar 

  18. Amini-Harandi, A.: Metric-like spaces, partial metric spaces and fixed points. Fixed Point Theory Appl. 2012(204), 1–11 (2012)

    Google Scholar 

  19. Amini-Harandi, A., Fakhar, M.: Fixed point theory in cone metric spaces obtained via the scalarization method. Comput. Math. Appl. 59, 3529–3534 (2010)

    MATH  MathSciNet  Google Scholar 

  20. An, T.V., Dung, N.V., Hang, V.T.L.: A new approach to fixed point theorems on \(G\)-metric spaces. Topol. Appl. 160, 1486–1493 (2013)

    MATH  MathSciNet  Google Scholar 

  21. Asadi, M., Karapinar, E., Salimi, P.: A new approach to \(G\)-metric and related fixed point theorems. J. Inequal. Appl. 2013(454), 1–14 (2013)

    Google Scholar 

  22. Asadi, M., Karapinar, E., Salimi, P.: New extension of \(p\)-metric spaces with some fixed-point results on \(M\)-metric spaces. J. Inequal. Appl. 2014(18), 1–9 (2014)

    Google Scholar 

  23. Asadi, M., Vaezpour, S.M., Rhoades, B.E., Soleimani, H.: Metrizability of cone metric spaces via renorming the Banach spaces. Nonlinear Anal. Appl. 2012, 1–5 (2012)

    Google Scholar 

  24. Aydi, H., Postolache, M., Shatanawi, W.: Coupled fixed point results for \((\psi,\phi )\)-weakly contractive mappings in ordered \(G\)-metric spaces. Comput. Math. Appl. 63, 298–309 (2012)

    MATH  MathSciNet  Google Scholar 

  25. Aydi, H., Shatanawi, W., Vetro, C.: On generalized weakly \(G\)-contraction mapping in \(G\)-metric spaces. Comput. Math. Appl. 62, 4222–4229 (2011)

    MATH  MathSciNet  Google Scholar 

  26. Azam, A., Arshad, M., Beg, I.: Banach contraction principle on cone rectangular metric spaces. Appl. Anal. Discrete Math. 3, 236–241 (2009)

    MATH  MathSciNet  Google Scholar 

  27. Azam, A., Fisher, B., Khan, M.: Common fixed point theorems in complex valued metric spaces. Numer. Funct. Anal. Optim. 32, 243–253 (2011)

    MATH  MathSciNet  Google Scholar 

  28. Bakhtin, I.A.: The contraction principle in quasimetric spaces. Func. An. Ulianowsk Gos. Ped. Ins. 30, 26–37 (1989). (in Russian)

  29. Banach, S.: Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales. Fundam. Math. 3, 133–181 (1922)

    MATH  Google Scholar 

  30. Beg, I., Abbas, M., Nazir, T.: Generalized cone metric spaces. J. Nonlinear Sci. Appl. 3, 21–31 (2010)

    MATH  MathSciNet  Google Scholar 

  31. Beg, I., Azam, A., Arshad, M.: Common fixed points for maps on topological vector space valued cone metric spaces. Int. J. Math. Math. Sci. 2009, 1–8 (2009)

    MathSciNet  Google Scholar 

  32. Berinde, V.: A common fixed point theorem for compatible quasi contractive self mappings in metric spaces. Appl. Math. Comput. 213, 348–354 (2009)

    MATH  MathSciNet  Google Scholar 

  33. Bianchini, R.M.T.: Su un problema di S. Reich aguardante la teoria dei punti fissi. Boll. Un. Mat. Ital. 5, 103–108 (1972)

    MATH  MathSciNet  Google Scholar 

  34. Branciari, A.: A fixed point theorem of Banach–Caccioppoli type on a class of generalized metric spaces. Publ. Math. Debrecen 57, 31–37 (2000)

    MATH  MathSciNet  Google Scholar 

  35. Bukatin, M., Kopperman, R., Matthews, S., Pajoohesh, H.: Partial metric spaces. Am. Math. Mon. 116, 708–718 (2009)

    MATH  MathSciNet  Google Scholar 

  36. Bumbariu, O.: An acceleration technique for slowly convergent fixed point iterative methods. Miskolc Math. Notes 13, 271–281 (2012)

    MATH  MathSciNet  Google Scholar 

  37. Caristi, J.: Fixed point theorems for mappings satisfying inwardness condition. Trans. Am. Math. Soc. 215, 241–251 (1976)

    MATH  MathSciNet  Google Scholar 

  38. Chaipunya, P., Kumam, P.: On the distance between three arbitrary points. J. Funct. Spaces Appl. 2013, 1–7 (2013)

    MathSciNet  Google Scholar 

  39. Chandok, S., Kumar, D.: Some common fixed point results for rational type contraction mappings in complex valued metric spaces. J. Oper. 2013, 1–6 (2013)

    Google Scholar 

  40. Cho, Y.J., Khan, M.S., Singh, S.L.: Common fixed points of weakly commuting mappings. Univ. u Novom Sadu, Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 18, 129–142 (1988)

    MATH  MathSciNet  Google Scholar 

  41. Choudhurya, B.S., Konar, P., Rhoades, B.E., Metiya, N.: Fixed point theorems for generalized weakly contractive mappings. Nonlinear Anal. 74, 2116–2126 (2011)

    MathSciNet  Google Scholar 

  42. Ćirić, L.B.: A generalization of Banach’s contraction principle. Proc. Am. Math. Soc. 45, 267–273 (1974)

    MATH  Google Scholar 

  43. Ćirić, L.B., Samet, B., Aydi, H., Vetro, C.: Common fixed points of generalized contractions on partial metric spaces and an application. Appl. Math. Comput. 218, 2398–2406 (2011)

    MATH  MathSciNet  Google Scholar 

  44. Collaço, P., Silva, J.C.E.: A complete comparison of 25 contraction conditions. Nonlinear Anal. 30, 471–476 (1997)

    MATH  MathSciNet  Google Scholar 

  45. Czerwik, S.: Contraction mappings in \(b\)-metric spaces. Acta Math. Univ. Ostrav. 1, 5–11 (1993)

    MATH  MathSciNet  Google Scholar 

  46. Das, K.M., Naik, K.V.: Common fixed point theorems for commuting maps on a metric space. Proc. Am. Math. Soc. 77, 369–373 (1979)

    MATH  MathSciNet  Google Scholar 

  47. Dhage, B.C.: A study of some fixed point theorems. Ph.D. thesis, Marathwada, Aurangabad (1984)

  48. Dhage, B.C.: On generalized metric spaces and topological structure II. Pure Appl. Math. Sci. XXXX(1–2), 37–41 (1994)

  49. Dhage, B.C.: Generalized metric spaces and topological structure I. An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 46, 3–24 (2000)

  50. Ding, H.S., Jovanović, M., Kadelburg, Z., Radenović, S.: Common fixed point results for generalized quasicontractions in \(tvs\)-cone metric spaces. J. Comput. Appl. Math. 15, 463–470 (2013)

    MATH  Google Scholar 

  51. Du, W.S.: A note on cone metric fixed point theory and its equivalence. Nonlinear Anal. 72, 2259–2261 (2010)

    MATH  MathSciNet  Google Scholar 

  52. Du, W.S., Karapinar, E.: A note on \(b\)-cone metric and its related results: generalizations or equivalence? Fixed Point Theory Appl. 2013(210), 1–6 (2013)

    Google Scholar 

  53. Dung, N.V.: Remarks on quasi-metric spaces. Miskolc Math. Notes (2014) (accepted paper)

  54. Dung, N.V., Hang, V.T.L.: Remarks on partial \(b\)-metric spaces and fixed point theorems. Mat. Vesnik (2014) (submitted)

  55. Dung, N.V., Hang, V.T.L.: A note on partial rectangular metric spaces. Math. Morav. 18, 1–8 (2014)

    Google Scholar 

  56. Dung, N.V., Hang, V.T.L., Sedghi, S.: Remarks on metric-type spaces and applications. Asian Bull. Math. (submitted)

  57. Dung, N.V., Hieu, N.T., Hang, V.T.L.: A simple approach to fixed point theorems on metric-like spaces. Le Matematiche (submitted)

  58. Dung, N.V., Hieu, N.T., Ly, N.T.T., Thinh, V.D.: Remarks on the fixed point problem of 2-metric spaces. Fixed Point Theory Appl. 2013(167), 1–7 (2013)

    MathSciNet  Google Scholar 

  59. Durmaz, G., Acar, Ö., Altun, I.: Some fixed point results on weak partial metric spaces. Filomat 27, 317–326 (2013)

    MathSciNet  Google Scholar 

  60. Dutta, P.N., Choudhury, B.S.: A generalisation of contraction principle in metric spaces. Fixed Point Theory Appl. 2008, 1–8 (2008)

    MathSciNet  Google Scholar 

  61. Edelstein, M.: An extension of Banach’s contraction principle. Proc. Am. Math. Soc. 12, 7–10 (1961)

    MATH  MathSciNet  Google Scholar 

  62. Edelstein, M.: On fixed and periodic points under contractive mappings. J. Lond. Math. Soc. 37, 74–79 (1962)

    MATH  MathSciNet  Google Scholar 

  63. Feng, Y., Mao, W.: The equivalence of cone metric spaces and metric spaces. Fixed Point Theory 11, 259–264 (2010)

    MATH  MathSciNet  Google Scholar 

  64. Fréchet, M.: La notion d’écart et le calcul fonctionel. C. R. Acad. Sci. Paris 140, 772–774 (1905)

    MATH  Google Scholar 

  65. Gähler, V.S.: 2-Metrische Räume und ihre topologische struktur. Math. Nachr. 26, 115–118 (1963/1964)

    Google Scholar 

  66. Geraghty, M.A.: On contractive mappings. Proc. Am. Math. Soc. 40(2), 604–608 (1973)

    MATH  MathSciNet  Google Scholar 

  67. Haghi, R.H., Rezapour, S., Shahzad, N.: Be careful on partial metric fixed point results. Topol. Appl. 160, 450–454 (2013)

    MATH  MathSciNet  Google Scholar 

  68. Heckmann, R.: Approximation of metric spaces by partial metric spaces. Appl. Categ. Struct. 7, 71–83 (1999)

    MATH  MathSciNet  Google Scholar 

  69. Hitzler, P., Seda, A.: Mathematical Aspects of Logic Programming Semantics. Chapman & Hall/CRC Studies in Informatic Series, CRC Press, Boca Raton (2011)

    MATH  Google Scholar 

  70. Hsiao, C.R.: A property of contractive type mappings in 2-metric spaces. Jnanabha 16, 223–239 (1986)

    MATH  MathSciNet  Google Scholar 

  71. Huang, L.G., Zhang, X.: Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 332, 1468–1476 (2007)

    MATH  MathSciNet  Google Scholar 

  72. Hussain, N., Roshan, J.R., Parvaneh, V., Latif, A.: A unification of G-metric, partial metric and b-metric spaces. Abstr. Appl. Anal. 2014, 1–9 (2014) (article ID 180698)

  73. Hussain, N., Shah, M.H.: KKM mappings in cone b-metric spaces. Comput. Math. Appl. 62, 1677–1684 (2011)

    MATH  MathSciNet  Google Scholar 

  74. Huy, N.B., Thanh, T.D.: Fixed point theorems and the Ulam–Hyers stability in non-Archimedean cone metric spaces. J. Math. Anal. Appl. 414, 10–20 (2014)

    MathSciNet  Google Scholar 

  75. Ilić, D., Rakočević, V.: Common fixed points for maps on a cone metric space. J. Math. Anal. Appl. 341, 876–882 (2008)

    MATH  MathSciNet  Google Scholar 

  76. Iseki, K.: Fixed point theorems in 2-metric spaces, Math. Seminar Notes, Kobe Univ. 3, 133–136 (1975)

  77. Janković, S., Kadelburg, Z., Radenović, S.: On cone metric spaces: a survey. Nonlinear Anal. 74, 2591–2601 (2011)

  78. Jleli, M., Karapinar, E., Samet, B.: Further remarks on fixed-point theorems in the context of partial metric spaces. Abstr. Appl. Anal. 2013, 1–6 (2013)

    Google Scholar 

  79. Jleli, M., Samet, B.: Remarks on \(G\)-metric spaces and fixed point theorems. Fixed Point Theory Appl. 2012(201), 1–10 (2012)

    Google Scholar 

  80. Jungck, G., Radenović, S., Radojević, S., Rakočević, V.: Common fixed point theorems for weakly compatible pairs on cone metric spaces. Fixed Point Theory Appl. 2009, 1–13 (2009)

    Google Scholar 

  81. Kadelburg, Z., Nashine, H.K., Radenović, S.: Common coupled fixed point results in partially ordered \(G\)-metric spaces. Bull. Math. Anal. Appl. 2, 51–63 (2012)

    Google Scholar 

  82. Kadelburg, Z., Radenović, S.: On generalized metric spaces: a survey. TWMS J. Pure Appl. Math. 5(1), 3–13 (2014)

    MATH  MathSciNet  Google Scholar 

  83. Kadelburg, Z., Radenović, S., Rakočević, V.: Topological vector space-valued cone metric spaces and fixed point theorems. Fixed Point Theory Appl. 2010, 1–17 (2010)

    Google Scholar 

  84. Kadelburg, Z., Radenović, S., Rakočević, V.: A note on the equivalence of some metric and cone metric fixed point results. Appl. Math. Lett. 24, 370–374 (2011)

    MATH  MathSciNet  Google Scholar 

  85. Kannan, R.: Some results on fixed points II. Am. Math. Mon. 76, 405–408 (1969)

    MATH  Google Scholar 

  86. Karapinar, E., Agarwal, R.P.: Further remarks on \(G\)-metric spaces. Fixed Point Theory Appl. 2013(154), 1–14 (2013)

    Google Scholar 

  87. Karapinar, E., Erhan, I.M., Öztürk, A.: Fixed point theorems on quasi-partial metric spaces. Math. Comput. Model. 57, 2442–2448 (2013)

    MATH  Google Scholar 

  88. Karapinar, E., Yüksel, U.: Some common fixed point theorems in partial metric spaces. J. Appl. Math. 2011, 1–16 (2011)

    Google Scholar 

  89. Khamsi, M.A.: Remarks on cone metric spaces and fixed point theorems of contractive mappings. Fixed Point Theory Appl. 2010, 1–7 (2010)

    Google Scholar 

  90. Khamsi, M.A., Hussain, N.: KKM mappings in metric type spaces. Nonlinear Anal. 7(9), 3123–3129 (2010)

    MathSciNet  Google Scholar 

  91. Khamsi, M.A., Wojciechowski, J.: On the additivity of the Minkowski functionals. Numer. Funct. Anal. Optim. 34, 635–647 (2013)

    MATH  MathSciNet  Google Scholar 

  92. Khani, M., Pourmahdian, M.: On the metrizability of cone metric spaces. Topol. Appl. 158, 190–193 (2011)

    MATH  MathSciNet  Google Scholar 

  93. Kopperman, R., Matthews, S., Pajoohesh, H.: Partial metrizability in value quantales. Appl. Gen. Topol. 5(1), 115–127 (2004)

    MATH  MathSciNet  Google Scholar 

  94. Kumam, P., Dung, N.V.: Remarks on generalized metric spaces in the Branciari’s sense. Sarajevo J. Math. (2014) (accepted paper)

  95. Kumam, P., Dung, N.V., Hang, V.T.L.: Some equivalences between cone \(b\)-metric spaces and \(b\)-metric spaces. Abstr. Appl. Anal. 2013, 1–8 (2013)

    Google Scholar 

  96. Künzi, H.-P.A., Pajoohesh, H., Schellekens, M.P.: Partial quasi-metrics. Theoret. Comput. Sci. 365, 237–246 (2006)

    MATH  MathSciNet  Google Scholar 

  97. Kurepa, Đ.R.: Tableaux ramifi’es d’ensembles. C. R. Acad. Sci. Paris 198, 1563–1565 (1934)

  98. Lahiri, B.K., Das, P., Dey, L.K.: Cantor’s theorem in 2-metric spaces and its applications to fixed point problems. Taiwan. J. Math. 15, 337–352 (2011)

    MATH  MathSciNet  Google Scholar 

  99. Malhotra, S.K., Shukla, S., Sen, R., Verma, N.: Generalizaton of fixed point theorems in partial cone metric spaces. Int. J. Math. Arch. 2, 610–616 (2011)

    Google Scholar 

  100. Matthews, S.G.: Partial metric topology, Papers on general topology and applications. In: Proceedings of 8th Summer Conference. Queen’s College (1992)

  101. Matthews, S.G.: Partial metric topology. Ann. N. Y. Acad. Sci. 728, 183–197 (1994)

    MathSciNet  Google Scholar 

  102. Mukheimer, A.A.: Some common fixed point theorems in complex valued \(b\)-metric spaces. Sci. World J. 2014, 1–6 (2014) (article ID 587825)

  103. Mustafa, Z., Aydi, H., Karapinar, E.: On common fixed points in \(G\)-metric spaces using (E.A) property. Comput. Math. Appl. 64, 1944–1956 (2012)

    MATH  MathSciNet  Google Scholar 

  104. Mustafa, Z., Obiedat, H., Awawdeh, F.: Some fixed point theorem for mapping on complete \(G\)-metric spaces. Fixed Point Theory Appl. 2008, 1–12 (2008)

    MathSciNet  Google Scholar 

  105. Mustafa, Z., Parvaneh, V., Roshan, J.R., Kadelburg, Z.: \(b_2\)-metric spaces and some fixed point theorems (submitted)

  106. Mustafa, Z., Roshan, J.R., Parvaneh, V.: Coupled coincidence point results for \((\psi,\varphi )\)-weakly contractive mappings in partially ordered \(G_{b}\)-metric spaces. Fixed Point Theory Appl. 2013(206), 1–19 (2013)

    Google Scholar 

  107. Mustafa, Z., Roshan, J.R., Parvaneh, V., Kadelburg, Z.: Some common fixed point results in ordered partial b-metric spaces. J. Inequal. Appl. 2013(562), 1–26 (2013)

    MathSciNet  Google Scholar 

  108. Mustafa, Z., Sims, B.: Some remarks concerning \(D\)-metric spaces. In: Proceedings of the International Conferences on Fixed Point Theory and Applications (Valencia, Spain), pp. 189–198 (2003)

  109. Mustafa, Z., Sims, B.: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 7(2), 289–297 (2006)

    MATH  MathSciNet  Google Scholar 

  110. Mustafa, Z., Sims, B.: Fixed point theorems for contractive mappings in complete \(G\)-metric spaces. Fixed Point Theory Appl. 2009, 1–10 (2009)

    MathSciNet  Google Scholar 

  111. Naidu, S.V.R., Prasad, J.R.: Fixed point theorems in 2-metric spaces. Indian J. Pure App. Math. 17, 974–993 (1986)

    MATH  Google Scholar 

  112. Naidu, S.V.R., Rao, K.P.R., Rao, N.S.: On the concepts of balls in a \(D\)-metric space. Int. J. Math. Math. Sci. 2005, 133–141 (2005)

    MATH  MathSciNet  Google Scholar 

  113. Nashine, H.K., Kadelburg, Z., Radenović, S.: Fixed point theorems via various cyclic contractive conditions in partial metric spaces. Publ. Inst. Math. (Beograd) (N.S.) 93, 69–93 (2013)

    MathSciNet  Google Scholar 

  114. Popović, B., Radenović, S., Shukla, S.: Fixed point results to \(tvs\)-cone \(b\)-metric spaces. Gulf J. Math 1, 51–64 (2013)

    Google Scholar 

  115. Proinov, P.D.: A unified theory of cone metric spaces and its applications to the fixed point theory. Fixed Point Theory Appl. 2013(103), 1–41 (2013)

    MathSciNet  Google Scholar 

  116. Radenović, S., Kadelburg, Z.: Quasi-contractions on symmetric and cone symmetric spaces. Banach J. Math. Anal. 5, 38–50 (2011)

    MATH  MathSciNet  Google Scholar 

  117. Rakotch, E.: A note on contractive mappings. Proc. Am. Math. Soc 13, 459–465 (1962)

    MATH  MathSciNet  Google Scholar 

  118. Rezapour, S., Hamlbarani, R.: Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”. J. Math. Anal. Appl. 345, 719–724 (2008)

    MATH  MathSciNet  Google Scholar 

  119. Rhoades, B.E.: A comparison of various definition of contractive mappings. Trans. Am. Math. Soc. 226, 257–290 (1977)

    MATH  MathSciNet  Google Scholar 

  120. Roldán, A., Karapinar, E.: Some multidimensional fixed point theorems on partially preordered \(G^{*}\)-metric spaces under \((\psi,\varphi )\)-contractivity conditions. Fixed Point Theory Appl. 2013(158), 1–21 (2013)

    Google Scholar 

  121. Romaguera, S.: A Kirk type characterization of completeness for partial metric spaces. Fixed Point Theory Appl. 2010, 1–6 (2010)

    MathSciNet  Google Scholar 

  122. Romaguera, S., Schellekens, M.: Quasi-metric properties of complexity spaces. Topol. Appl. 98, 311–322 (1999)

    MATH  MathSciNet  Google Scholar 

  123. Rouzkard, F., Imdad, M.: Some common fixed point theorems on complex valued metric spaces. Comput. Math. Appl. 64, 1866–1874 (2012)

    MATH  MathSciNet  Google Scholar 

  124. Rus, I.A.: Fixed point theory in partial metric spaces. An. Univ. Vest Timiş Ser. Mat. Inform. XLVI, 149–160 (2008)

    MathSciNet  Google Scholar 

  125. Salimi, P., Vetro, P.: A result of Suzuki type in partial \(G\)-metric spaces. Acta Math. Sci. Ser. B Engl. Ed. 34B, 274–284 (2014)

    MathSciNet  Google Scholar 

  126. Samet, B., Vetro, C., Vetro, F.: Remarks on \(G\)-metric spaces. Int. J. Anal. 2013, 1–6 (2013)

    MathSciNet  Google Scholar 

  127. Samet, B., Vetro, C., Vetro, F.: From metric spaces to partial metric spaces. Fixed Point Theory Appl. 2013(5), 1–11 (2013)

    MathSciNet  Google Scholar 

  128. Sastry, K.P.R., Naidu, G.A., Bekeshie, T.: Metrizability of complex valued metric spaces and some remarks on fixed point theorems in complex valued metric spaces. Int. J. Pure Appl. Math. 3, 2686–2690 (2012)

    Google Scholar 

  129. Schellekens, M.: On upper weightable spaces. Ann. N. Y. Acad. Sci. 806, 348–363 (2002)

    MathSciNet  Google Scholar 

  130. Sedghi, S., Shobe, N., Aliouche, A.: A generalization of fixed point theorem in \(S\)-metric spaces. Mat. Vesnik 64, 258–266 (2012)

    MATH  MathSciNet  Google Scholar 

  131. Sedghi, S., Shobe, N., Zhou, H.: A common fixed point theorem in \(D^*\)-metric spaces. Fixed Point Theory Appl. 2007, 1–13 (2007)

    MathSciNet  Google Scholar 

  132. Shah, M.H., Hussain, N.: Nonlinear contractions in partially ordered quasi \(b\)-metric spaces. Commun. Korean Math. Soc. 27, 117–128 (2012)

    MATH  MathSciNet  Google Scholar 

  133. Shatanawi, W.: Fixed point theory for contractive mapping satisfying \(\Phi \)-maps in \(G\)-metric spaces. Fixed Point Theory Appl. 2010, 1–9 (2010)

    MathSciNet  Google Scholar 

  134. Shukla, S.: Partial rectangular metric spaces and fixed point theorems. Sci. World J. 2014, 1–7 (2014)

    Google Scholar 

  135. Shukla, S.: Partial \(b\)-metric spaces and fixed point theorems. Mediterr. J. Math. 11, 703–711 (2014)

    MATH  MathSciNet  Google Scholar 

  136. Sintunavarat, W., Kumam, P.: Generalized common fixed point theorems in complex valued metric spaces and applications. J. Inequal. Appl. 2012(84), 1–12 (2012)

    Google Scholar 

  137. Sitthikul, K., Saejung, S.: Some fixed point theorems in complex valued metric space. Fixed Point Theory Appl. 2012(189), 1–15 (2012)

    MathSciNet  Google Scholar 

  138. Smyth, M.B.: Completeness of quasi-uniform and syntopological spaces. J. Lond. Math. Soc. 49, 385–400 (1994)

    MATH  MathSciNet  Google Scholar 

  139. Turinici, M.: Contractive maps in Mustafa–Sims metric spaces. 1–12 (2013). arXiv:1309.3702v1

  140. Vandergraft, J.S.: Newton’s method for convex operators in partially ordered spaces. SIAM J. Numer. Anal. 4, 406–432 (1967)

    MATH  MathSciNet  Google Scholar 

  141. Zabrejko, P.P.: \(K\)-metric and \(K\)-normed linear spaces: survey. Collect. Math. 48, 825–859 (1997)

    MathSciNet  Google Scholar 

  142. Zand, M.R.A., Nezhad, A.D.: A generalization of partial metric spaces. J. Contemp. Appl. Math. 24, 86–93 (2011)

    Google Scholar 

  143. Zhu, L., Zhu, C.X., Chen, C.F., Stojanović, Ž.: Multidimensional fixed points for generalized \(\psi \)-quasi-contractions in quasi-metric-like spaces. J. Inequal. Appl. 2014(27), 1–19 (2014)

    Google Scholar 

Download references

Acknowledgments

The authors sincerely thank referees for their valuable comments on revising the paper. Also, the authors sincerely thank The Dong Thap Seminar on Mathematical Analysis and Applications for discussing partly this article and thank Prof. S. Sedghi, Islamic Azad University, Iran; Prof. D. Wardowski, University of Łodź, Poland; Prof. P. Kumam, King Mongkut’s University of Technology Thonburi, Thailand; Prof. Z. Mustafa, The Hashemite University, Jordan; Prof. E. Karapinar, Atilim University, Turkey for their helps. The third and fourth authors are grateful to Ministry of Education, Science and Technological Development of Serbia.

Author information

Authors and Affiliations

  1. Department of Mathematics, Vinh University, Vinh, Nghe An, Viet Nam

    Tran Van An

  2. Faculty of Mathematics and Information Technology Teacher Education, Dong Thap University, Cao Lanh, Dong Thap , 871200, Viet Nam

    Nguyen Van Dung

  3. Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 , Beograd, Serbia

    Zoran Kadelburg

  4. Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 , Beograd, Serbia

    Stojan Radenović

Authors
  1. Tran Van An
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nguyen Van Dung
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zoran Kadelburg
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Stojan Radenović
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zoran Kadelburg.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Van An, T., Van Dung, N., Kadelburg, Z. et al. Various generalizations of metric spaces and fixed point theorems. RACSAM 109, 175–198 (2015). https://doi.org/10.1007/s13398-014-0173-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13398-014-0173-7

Keywords

Mathematics Subject Classification (2010)

Search

Navigation