Advertisement

A Fixed Point Result on the Interesting Abstract Space: Partial Metric Spaces

  • Erdal Karapınar
Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 179)

Abstract

In this chapter, we shall investigate the existence of fixed point of certain mappings via simulation functions in the framework of an interesting abstract space, namely, partial metric spaces. The main results of this manuscript not only extend, but also generalize, improve and unify several existing results on the literature of metric fixed point theory.

References

  1. Abedeljawad, T., Karapınar, E., Taş, K.: Existence and uniqueness of common fixed point on partial metric spaces. Appl. Math. Lett. 24, 1894–1899 (2011)MathSciNetCrossRefGoogle Scholar
  2. Abdeljawad, T., Karapınar, E., Tas, K.: A generalized contraction principle with control functions on partial metric spaces. Comput. Math. Appl. 63(3), 716–719 (2012)MathSciNetCrossRefGoogle Scholar
  3. Agarwal, R.P., Alghamdi, M.A., Shahzad, N.: Fixed point theory for cyclic generalized contractions in partial metric spaces. Fixed Point Theory Appl. 2012, 40 (2012)MathSciNetCrossRefGoogle Scholar
  4. Ali, M.U., Kamran, T., Karapınar, E.: On (\(\alpha,\phi,\eta \))-contractive multivalued mappings. Fixed Point Theory Appl. 2014, 7 (2014)MathSciNetCrossRefGoogle Scholar
  5. Alsulami, H.H., Karapınar, E., Khojasteh, F., Roldán-López-de-Hierro, A.F.: A proposal to the study of contractions in quasi-metric spaces. Discret. Dyn. Nat. Soc. (2014), Article ID 269286Google Scholar
  6. Altun, I., Acar, O.: Fixed point theorems for weak contractions in the sense of Berinde on partial metric spaces. Topol. Appl. 159, 2642–2648 (2012)MathSciNetCrossRefGoogle Scholar
  7. Altun, I., Erduran, A.: Fixed point theorems for monotone mappings on partial metric spaces. Fixed Point Theory Appl. 2011 (2011) Article ID 508730MathSciNetCrossRefGoogle Scholar
  8. Altun, I., Simsek, H.: Some fixed point theorems on dualistic partial metric spaces. J. Adv. Math. Stud. 1, 1–8 (2008)MathSciNetzbMATHGoogle Scholar
  9. Altun, I., Sola, F., Simsek, H.: Generalized contractions on partial metric spaces. Topol. Appl. 157, 2778–2785 (2010)MathSciNetCrossRefGoogle Scholar
  10. Aydi, H.: Some coupled fixed point results on partial metric spaces. Int. J. Math. Math. Sci. (2011a), Article ID 647091Google Scholar
  11. Aydi, H.: Some fixed point results in ordered partial metric spaces. J. Nonlinear Sci. Appl. 4(2), 210–217 (2011b)MathSciNetCrossRefGoogle Scholar
  12. Aydi, H.: Fixed point results for weakly contractive mappings in ordered partial metric spaces. J. Adv. Math. Stud. 4(2), 1–12 (2011c)MathSciNetzbMATHGoogle Scholar
  13. Aydi, H.: Fixed point theorems for generalized weakly contractive condition in ordered partial metric spaces. J. Nonlinear Anal. Optim.: Theory Appl. 2(2), 33–48 (2011d)Google Scholar
  14. Aydi, H.: Common fixed point results for mappings satisfying \((\phi,\phi )\)-weak contractions in ordered partial metric space. Int. J. Math. Stat. 12(2), 53–64 (2012)MathSciNetzbMATHGoogle Scholar
  15. Aydi, H., Karapınar, E.: A Meir-Keeler common type fixed point theorem on partial metric spaces. Fixed Point Theory Appl. 2012, 26 (2012)MathSciNetCrossRefGoogle Scholar
  16. Aydi, H., Karapınar, E., Shatanawi, W.: Coupled fixed point results for \((\phi,\varphi )\)- weakly contractive condition in ordered partial metric spaces. Comput. Math. Appl. 62(12), 4449–4460 (2011)MathSciNetCrossRefGoogle Scholar
  17. Aydi, H., Abbas, M., Vetro, C.: Partial Hausdorff metric and Nadler’s fixed point theorem on partial metric spaces. Topol. Appl. 159, 3234–3242 (2012a)MathSciNetCrossRefGoogle Scholar
  18. Aydi, H., Vetro, C., Sintunavarat, W., Kumam, P.: Coincidence and fixed points for contractions and cyclical contractions in partial metric spaces. Fixed Point Theory Appl. 2012, 124 (2012b)MathSciNetCrossRefGoogle Scholar
  19. Aydi, H., Vetro, C., Karapınar, E.: On Ekeland’s variational principle in partial metric spaces. Appl. Math. Inf. Sci. 9(1), 257–262 (2015a)MathSciNetCrossRefGoogle Scholar
  20. Aydi, H., Bilgili, N., Karapınar, E.: Common fixed point results from quasi-metric spaces to \(G\)-metric spaces. J. Egypt. Math. Soc. 23(2), 356–361 (2015b)MathSciNetCrossRefGoogle Scholar
  21. Aydi, H., Jellali, M., Karapınar, E.: Common fixed points for generalized \(\alpha \)-implicit contractions in partial metric spaces: consequences and application. RACSAM 109(2), 367–384 (2015c)MathSciNetCrossRefGoogle Scholar
  22. Banach, S.: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3, 133–181 (1922)CrossRefGoogle Scholar
  23. Bukatin, M., Kopperman, R., Matthews, S., Pajoohesh, H.: Partial metric spaces. Am. Math. Mon. 116(8), 708–718 (2009)MathSciNetCrossRefGoogle Scholar
  24. Chen, C.-M., Karapınar, E.: Fixed point results for the \(\alpha \)-Meir-Keeler contraction on partial Hausdorff metric spaces. J. Inequal. Appl. 2013, 410 (2013)MathSciNetCrossRefGoogle Scholar
  25. Chi, K.P., Karapınar, E., Thanh, T.D.: A generalized contraction principle in partial metric spaces. Math. Comput. Model. 55, 1673–1681 (2012).  https://doi.org/10.1016/j.mcm.2011.11.005MathSciNetCrossRefzbMATHGoogle Scholar
  26. Ćirić, L.j., 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)MathSciNetCrossRefGoogle Scholar
  27. Ćirić, L.B.: On some maps with a nonunique fixed point. Publications de L’Institut Mathématique 17, 52–58 (1974a)MathSciNetzbMATHGoogle Scholar
  28. Ćirić, L.B.: A generalization of Banach’s contraction principle. Proc. Am. Math. Soc. 45(2), 267–273 (1974b)MathSciNetCrossRefGoogle Scholar
  29. Escardo, M.H.: PCF extended with real numbers. Theor. Comput. Sci. 162, 79–115 (1996)MathSciNetCrossRefGoogle Scholar
  30. Frechét, M.R.: Sur quelques points du calcul fonctionnel. Rend. Circ. Mat. Palermo 22, 174 (1906)CrossRefGoogle Scholar
  31. Gulyaz, S., Karapınar, E.: Coupled fixed point result in partially ordered partial metric spaces through implicit function. Hacet. J. Math. Stat. 42(4), 347–357 (2013)MathSciNetzbMATHGoogle Scholar
  32. Haghi, R.H., Rezapour, Sh, Shahzad, N.: Be careful on partial metric fixed point results. Topol. Appl. 160(3), 450–454 (2013)MathSciNetCrossRefGoogle Scholar
  33. Heckmann, R.: Approximation of metric spaces by partial metric spaces. Appl. Categ. Struct. 7, 71–83 (1999)MathSciNetCrossRefGoogle Scholar
  34. Hitzler, P., Seda, A.: Mathematical Aspects of Logic Programming Semantics. CRC Press, Taylor and Francis Group, Boca Raton, Studies in Informatics Series. Chapman and Hall (2011)zbMATHGoogle Scholar
  35. Hos̆ková-Mayerová, S̆., Maturo, F., Kacprzyk, J.: Recent Trends in Social Systems: Quantitative Theories and Quantitative Models Edition: Studies in System, Decision and Control 66. Springer International Publishing AG, Switzerland (2016), 426 p. ISSN 2198-4182. ISBN 978-3-319-40583-4Google Scholar
  36. Hos̆ková-Mayerová, S̆., Maturo, F., Kacprzyk, J.: Mathematical-Statistical Models and Qualitative Theories for Economic and Social Sciences. Springer International Publishing, New York (2017), 437 p. ISBN 978-3-319-54819-7Google Scholar
  37. Ilić, D., Pavlović, V.: Rakoc̆ević, V.: Some new extensions of Banachs contraction principle to partial metric space. Appl. Math. Lett. 24(8), 1326–1330 (2011)MathSciNetCrossRefGoogle Scholar
  38. Ilić, D., Pavlović, V.: Rakoc̆ević, V.: Extensions of the Zamfirescu theorem to partial metric spaces. Original Research Article. Math. Comput. Model. 55(34), 801–809 (2012)CrossRefGoogle Scholar
  39. Jleli, M., Karapınar, E., Samet, B.: Best proximity points for generalized \(\alpha -\phi \)-proximal contractive type mappings, J. Appl. Math. (2013a) Article ID 534127Google Scholar
  40. Jleli, M., Karapınar, E., Samet, B.: Fixed point results for \(\alpha -\phi _\lambda \) contractions on gauge spaces and applications. Abstract Appl. Anal. (2013b) Article ID 730825Google Scholar
  41. Jleli, M., Karapınar, E., Samet, B.: Further remarks on fixed point theorems in the context of partial metric spaces. Abstract Appl. Anal. (2013c) Article Id: 715456Google Scholar
  42. Kannan, R.: Some results on fixed points. Bull. Calcutta Math. Soc. 60, 71–76 (1968)MathSciNetzbMATHGoogle Scholar
  43. Karapınar, E.: Generalizations of Caristi Kirk’s theorem on partial metric spaces. Fixed Point Theory Appl. 2011, 4 (2011a)Google Scholar
  44. Karapınar, E.: A note on common fixed point theorems in partial metric spaces. Miskolc Math. Notes 12(2), 185–191 (2011b)MathSciNetzbMATHGoogle Scholar
  45. Karapınar, E.: Some fixed point theorems on the class of comparable partial metric spaces on comparable partial metric spaces. Appl. General Topol. 12(2), 187–192 (2011c)MathSciNetzbMATHGoogle Scholar
  46. Karapınar, E.: Weak \(\phi \)-contraction on partial metric spaces. J. Comput. Anal. Appl. 14(2), 206–210 (2012a)MathSciNetzbMATHGoogle Scholar
  47. Karapınar, E.: Ćirić types nonunique fixed point theorems on partial metric spaces. J. Nonlinear Sci. Appl. 5, 74–83 (2012b)MathSciNetCrossRefGoogle Scholar
  48. Karapınar, E., Erhan, I.M.: Fixed point theorems for operators on partial metric spaces. Appl. Math. Lett. 24, 1900–1904 (2011)MathSciNetCrossRefGoogle Scholar
  49. Karapınar, E., Erhan, I.M.: Fixed point theorem for cyclic maps on partial metric spaces. Appl. Math. Inf. Sci. 6, 239–244 (2012)MathSciNetGoogle Scholar
  50. Karapınar, E., Romaguera, S.: Nonunique fixed point theorems in partial metric spaces. Filomat 27(7), 1305–1314 (2013)MathSciNetCrossRefGoogle Scholar
  51. Karapınar, E., Samet, B.: Generalized \((\alpha -\phi )\)-contractive type mappings and related fixed point theorems with applications. Abstract Appl. Anal. (2012), Article ID 793486Google Scholar
  52. Karapınar, E., Yuksel, U.: Some common fixed point theorems in partial metric spaces. J. Appl. Math. (2011) Article ID 263621Google Scholar
  53. Karapınar, E., Shobkolaei, N., Sedghi, S., Vaezpour, S.M.: A common fixed point theorem for cyclic operators on partial metric spaces. Filomat 26(2), 407–414 (2012)MathSciNetCrossRefGoogle Scholar
  54. Karapınar, E., Erhan, I., Ozturk, A.: Fixed point theorems on quasi-partial metric spaces. Math. Comput. Model. 57(9–10), 2442–2448 (2013a)MathSciNetCrossRefGoogle Scholar
  55. Karapınar, E., Kuman, P., Salimi, P.: On \(\alpha -\phi \)-Meri-Keeler contractive mappings. Fixed Point Theory Appl. 2013, 94 (2013b)MathSciNetCrossRefGoogle Scholar
  56. Karapınar, E., Alsulami, H.H., Noorwali, M.: Some extensions for Geragthy type contractive mappings. J. Inequal. Appl. 2015, 303 (2015)MathSciNetCrossRefGoogle Scholar
  57. Karapınar, E., Taş, K., Rakočević, V.: Advances on fixed point results on partial metric spaces. In: Tas, K., Tenreiro, J.A., Baleanu, D. (eds.) Mathematical Methods in Engineering: Theory, pp. 1–59. Springer, Berlin (2018)Google Scholar
  58. Khojasteh, F., Shukla, S., Radenović, S.: A new approach to the study of fixed point theorems via simulation functions. Filomat 29(6), 1189–1194 (2015)MathSciNetCrossRefGoogle Scholar
  59. Kirk, W.A., Srinivasan, P.S., Veeramani, P.: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory 4(1), 79–89 (2003)MathSciNetzbMATHGoogle Scholar
  60. Kopperman, R.D., Matthews, S.G., Pajoohesh, H.: What do partial metrics represent? Notes distributed at the 19th Summer Conference on Topology and Its Applications, University of CapeTown (2004)Google Scholar
  61. Kramosil, O., Michalek, J.: Fuzzy metric and statistical metric spaces. Kybernetika 11, 326–334 (1975)MathSciNetzbMATHGoogle Scholar
  62. Künzi, H.P.A., Pajoohesh, H., Schellekens, M.P.: Partial quasi-metrics. Theor. Comput. Sci. 365(3), 237–246 (2006)MathSciNetCrossRefGoogle Scholar
  63. Matthews, S.G.: Partial metric topology. Research report 212. Department of Computer Science. University of Warwick (1992)Google Scholar
  64. Matthews, S.G.: Partial metric topology. In: Proceedings of the 8th Summer of Conference on General Topology and Applications (Ann. N.Y. Acad. Sci. 728, 183–197) (1994)MathSciNetCrossRefGoogle Scholar
  65. Mohammadi, B., Rezapour, Sh, Shahzad, N.: Some results on fixed points of \(\alpha \)-\(\phi \)-Ciric generalized multifunctions. Fixed Point Theory Appl. 2013, 24 (2013)MathSciNetCrossRefGoogle Scholar
  66. Oltra, S., Valero, O.: Banach’s fixed point theorem for partial metric spaces. Rend. Istid. Math. Univ. Trieste 36, 17–26 (2004)MathSciNetzbMATHGoogle Scholar
  67. Paesano, D., Vetro, P.: Suzuki’s type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces. Topol. Appl. 159(3), 911–920 (2012)MathSciNetCrossRefGoogle Scholar
  68. Popa, V.: Fixed point theorems for implicit contractive mappings. Stud. Cerc. St. Ser. Mat. Univ. Bacau 7, 129–133 (1997)Google Scholar
  69. Popescu, O.: Some new fixed point theorems for \(\alpha \)-Geraghty contractive type maps in metric spaces. Fixed Point Theory Appl. 2014, 190 (2014)CrossRefGoogle Scholar
  70. Ran, A.C.M., Reurings, M.C.B.: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 132, 1435–1443 (2003)MathSciNetCrossRefGoogle Scholar
  71. Reich, S.: Kannans fixed point theorem. Boll. Un. Mat. Ital. 4(4), 111 (1971)MathSciNetGoogle Scholar
  72. Roldan, A., Martinez-Moreno, J., Roldan, C., Karapınar, E.: Multidimensional fixed point theorems in partially ordered complete partial metric spaces under (\(\psi ,\varphi \))-contractivity conditions. Abstract Appl. Anal. (2013) Article Id: 634371Google Scholar
  73. Roldán-López-de-Hierro, A.F., Karapınar, E., Roldán-López-de-Hierro, C., Martínez-Moreno, J.: Coincidence point theorems on metric spaces via simulation functions. J. Comput. Appl. Math. 275, 345–355 (2015)MathSciNetCrossRefGoogle Scholar
  74. Romaguera, S.: A Kirk type characterization of completeness for partial metric spaces. Fixed Point Theory Appl. (2010), Article ID 493298Google Scholar
  75. Romaguera, S.: Matkowskis type theorems for generalized contractions on (ordered) partial metric spaces. Appl. General Topol. 12(2), 213–220 (2011)MathSciNetzbMATHGoogle Scholar
  76. Romaguera, S.: Fixed point theorems for generalized contractions on partial metric spaces. Topol. Appl. 159, 194–199 (2012)MathSciNetCrossRefGoogle Scholar
  77. Romaguera, S., Schellekens, M.: Duality and quasi-normability for complexity spaces. Appl. General Topol. 3, 91–112 (2002)MathSciNetCrossRefGoogle Scholar
  78. Romaguera, S., Schellekens, M.: Partial metric monoids and semivaluation spaces. Topol. Appl. 153(5–6), 948–962 (2005)MathSciNetCrossRefGoogle Scholar
  79. Romaguera, S., Valero, O.: A quantitative computational model for complete partial metric spaces via formal balls. Math. Struct. Comput. Sci. 19(3), 541–563 (2009)MathSciNetCrossRefGoogle Scholar
  80. Rus, I.A.: Generalized Contractions and Applications. Cluj University Press, Cluj-Napoca (2001)zbMATHGoogle Scholar
  81. Samet, B., Rajović, M., Lazović, R., Stoiljković, R.: Common fixed point results for nonlinear contractions in ordered partial metric spaces. Fixed Point Theory Appl. 2011, 71 (2011)MathSciNetCrossRefGoogle Scholar
  82. Samet, B., Vetro, C., Vetro, P.: Fixed point theorem for \(\alpha -\phi \) contractive type mappings. Nonlinear Anal. 75, 2154–2165 (2012)MathSciNetCrossRefGoogle Scholar
  83. Samet, B., Vetro, C., Vetro, F.: From metric spaces to partial metric spaces. Fixed Point Theory Appl. 2013, 5 (2013)MathSciNetCrossRefGoogle Scholar
  84. Schellekens, M.P.: A characterization of partial metrizability: domains are quantifiable. Theor. Comput. Sci. 305(13), 409–432 (2003)MathSciNetCrossRefGoogle Scholar
  85. Schellekens, M.P.: The correspondence between partial metrics and semivaluations. Theor. Comput. Sci. 315(1), 135–149 (2004)MathSciNetCrossRefGoogle Scholar
  86. Sehgal, V.M.: Some fixed and common fixed point theorems in metric spaces. Can. Math. Bull. 17(2), 257–259 (1974)MathSciNetCrossRefGoogle Scholar
  87. Shatanawi, W., Samet, B., Abbas, M.: Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces. Math. Comput. Model. 55(3–4), 680–687 (2012)MathSciNetCrossRefGoogle Scholar
  88. Shobkolaei, N., Vaezpour, S.M., Sedghi, S.: A common fixed point theorem on ordered partial metric spaces. J. Basic Appl. Sci. Res. 1(12), 3433–3439 (2011)Google Scholar
  89. Stoy, J.E.: Denotational Semantics: The Scott-Strachey Approach to Programming Language Theory. MIT Press, Cambridge (1981)zbMATHGoogle Scholar
  90. Valero, O.: On Banach fixed point theorems for partial metric spaces. Appl. General Topol. 6(2), 229–240 (2005)MathSciNetCrossRefGoogle Scholar
  91. Vetro, C., Vetro, F.: Common fixed points of mappings satisfying implicit relations in partial metric spaces. J. Nonlinear Sci. Appl. 6(3), 152–161 (2013)MathSciNetCrossRefGoogle Scholar
  92. Vetro, C., Vetro, F.: Metric or partial metric spaces endowed with a finite number of graphs: a tool to obtain fixed point results. Topol. Appl. 164, 125–137 (2014)MathSciNetCrossRefGoogle Scholar
  93. Vetro, F., Radenović, S.: Nonlinear \(\phi \)-quasi-contractions of Ćirić-type in partial metric spaces. Appl. Math. Comput. 219(4), 1594–1600 (2012)MathSciNetzbMATHGoogle Scholar
  94. Waszkiewicz, P.: Quantitative continuous domains. Appl. Categ. Struct. 11, 4167 (2003)MathSciNetCrossRefGoogle Scholar
  95. Waszkiewicz, P.: Partial metrisability of continuous posets. Math. Struct. Comput. Sci. 16(2), 359–372 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Atilim UniversityIncek, AnkaraTurkey

Personalised recommendations