Advertisement

Meir–Keeler Sequential Contractions and Applications

  • Mihai TuriniciEmail author
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 151)

Abstract

Some fixed point results are given for a class of Meir–Keeler sequential contractions acting on relational metric spaces. The connections with a related statement in Turinici [MDMFPT, Paper-3-3, Pim, Iaşi, 2016] are also being discussed. Finally, an application of the obtained facts to integral equations theory is given.

AMS Subject Classification

47H17 (Primary); 54H25 (Secondary) 

References

  1. 1.
    M. Abtahi, Fixed point theorems for Meir-Keeler type contractions in metric spaces. Fixed Point Theory 17, 225–236 (2016)MathSciNetzbMATHGoogle Scholar
  2. 2.
    M. Abtahi, Z. Kadelburg, S. Radenović, Fixed points of Cirić-Matkowski-type contractions in ν-generalized metric spaces. RACSAM. https://doi.org/10.1007/s13398-016-0275-5 zbMATHCrossRefGoogle Scholar
  3. 3.
    R.P. Agarwal, M.A. El-Gebeily, D. O’Regan, Generalized contractions in partially ordered metric spaces. Appl. Anal. 87, 109–116 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    H. Argoubi, B. Samet, C. Vetro, Nonlinear contractions involving simulation functions in a metric space with a partial order. J. Nonlinear Sci. Appl. 8, 1082–1094 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund. Math. 3, 133–181 (1922)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    P. Bernays, A system of axiomatic set theory: part III. Infinity and enumerability analysis. J. Symb. Log. 7, 65–89 (1942)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    M. Berzig, Coincidence and common fixed point results on metric spaces endowed with an arbitrary binary relation and applications. J. Fixed Point Theory Appl. 12, 221–238 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    A. Bielecki, Une remarque sur la méthode de Banach-Cacciopoli-Tikhonov dans la théorie des équations différentielles ordinaires. Bull. Acad. Pol. Sci. 4, 261–264 (1956)MathSciNetzbMATHGoogle Scholar
  9. 9.
    D.W. Boyd, J.S.W. Wong, On nonlinear contractions. Proc. Am. Math. Soc. 20, 458–464 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    I. Cabrera, J. Harjani, K. Sadarangani, A fixed point theorem for contractions of rational type in partially ordered metric spaces. Ann. Univ. Ferrara 59, 251–258 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    S. Chandok, B.S. Choudhury, N. Metiya, Fixed point results in ordered metric spaces for rational type expressions with auxiliary functions. J. Egyptian Math. Soc. 23, 95–101 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    B.S. Choudhury, A. Kundu, A Kannan-like contraction in partially ordered spaces. Demonstr. Math. 46, 327–334 (2013)MathSciNetzbMATHGoogle Scholar
  13. 13.
    L.B. Cirić, A new fixed-point theorem for contractive mappings. Publ. Inst. Math. 30(44), 25–27 (1981)MathSciNetzbMATHGoogle Scholar
  14. 14.
    P.J. Cohen, Set Theory and the Continuum Hypothesis (Benjamin, New York, 1966)zbMATHGoogle Scholar
  15. 15.
    C. Corduneanu, Sur certain équations fonctionnelles de Volterra. Funkc. Ekv. 9, 119–127 (1966)zbMATHGoogle Scholar
  16. 16.
    C. Di Bari, C. Vetro, Common fixed point theorems for weakly compatible maps satisfying a general contractive condition. Int. J. Math. Math. Sci. 2008, 891375 (2008)MathSciNetzbMATHGoogle Scholar
  17. 17.
    W.-S. Du, F. Khojasteh, New results and generalizations for approximate fixed point property and their applications. Abstr. Appl. Anal. 2014, 581267 (2014)MathSciNetzbMATHGoogle Scholar
  18. 18.
    A. Dubickas, Partitions of positive integers into sets without infinite progressions. Math. Commun. 13, 115–122 (2008)MathSciNetzbMATHGoogle Scholar
  19. 19.
    P.N. Dutta, B.S. Choudhury, A generalization of contraction principle in metric spaces. Fixed Point Theory Appl. 2008, 406368 (2008)zbMATHCrossRefGoogle Scholar
  20. 20.
    M.A. Geraghty, On contractive mappings. Proc. Am. Math. Soc. 40, 604–608 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    M.A. Geraghty, An improved criterion for fixed points of contraction mappings. J. Math. Anal. Appl. 48, 811–817 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    P.R. Halmos, Naive Set Theory (Van Nostrand Reinhold, New York, 1960)zbMATHGoogle Scholar
  23. 23.
    J. Harjani, B. Lopez, K. Sadarangani, A fixed point theorem for mappings satisfying a contractive condition of rational type on a partially ordered metric space. Abstr. Appl. Anal. 2010, 190701 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    P. Hitzler, Generalized metrics and topology in logic programming semantics. Ph.D. Thesis, National University of Ireland, University College Cork (2001)Google Scholar
  25. 25.
    J. Jachymski, Common fixed point theorems for some families of mappings. Indian J. Pure Appl. Math. 25, 925–937 (1994)MathSciNetzbMATHGoogle Scholar
  26. 26.
    J. Jachymski, Equivalent conditions for generalized contractions on (ordered) metric spaces. Nonlinear Anal. 74, 768–774 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    K. Jha, K.P.R. Rao, D. Panthi, On development of Meir-Keeler type fixed point theorems. Nepal J. Sci. Tech. 10, 141–147 (2009)CrossRefGoogle Scholar
  28. 28.
    M. Jleli, B. Samet, A new generalization of the Banach contraction principle. J. Inequal. Appl. 2014, 38 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    C.F.K. Jung, On generalized complete metric spaces. Bull. Am. Math. Soc. 75, 113–116 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    E. Karapinar, A. Roldan, N. Shahzad, W. Sintunaravat, Discussion on coupled and tripled coincidence point theorems for φ-contractive mappings without the mixed g-monotone property. Fixed Point Theory Appl. 2014, 92 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    S. Kasahara, On some generalizations of the Banach contraction theorem. Publ. Res. Inst. Math. Sci. Kyoto Univ. 12, 427–437 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    M.S. Khan, M. Swaleh, S. Sessa, Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 30, 1–9 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    F. Khojasteh, V. Rakočević, Some new common fixed point results for generalized contractive multi-valued non-self-mappings. Appl. Math. Lett. 25, 287–293 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    F. Khojasteh, S. Shukla, S. Radenović, Formulization of many contractions via the simulation functions. Arxiv, 1109-3021-v2 (2013)Google Scholar
  35. 35.
    W.A. Kirk, Fixed points of asymptotic contractions. J. Math. Anal. Appl. 277, 645–650 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    S. Leader, Fixed points for general contractions in metric spaces. Math. Japon. 24, 17–24 (1979)MathSciNetzbMATHGoogle Scholar
  37. 37.
    S. Leader, Equivalent Cauchy sequences and contractive fixed points in metric spaces. Stud. Math. 66, 63–67 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    S. Leader, S.L. Hoyle, Contractive fixed points. Fund. Math. 87, 93–108 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    T.-C. Lim, On characterizations of Meir-Keeler contractive maps. Nonlinear Anal. 46, 113–120 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    W.A.J. Luxemburg, On the convergence of successive approximations in the theory of ordinary differential equations. Indag. Math. 20, 540–546 (1958)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    J. Matkowski, Integrable Solutions of Functional Equations. Dissertationes Mathematicae, vol. 127 (Polish Scientific Publishers, Warsaw, 1975)Google Scholar
  42. 42.
    J. Matkowski, Fixed point theorems for contractive mappings in metric spaces. Časopis Pest. Mat. 105, 341–344 (1980)MathSciNetzbMATHGoogle Scholar
  43. 43.
    A. Meir, E. Keeler, A theorem on contraction mappings. J. Math. Anal. Appl. 28, 326–329 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    G.H. Moore, Zermelo’s Axiom of Choice: Its Origin, Development and Influence (Springer, New York, 1982)zbMATHCrossRefGoogle Scholar
  45. 45.
    Y. Moskhovakis, Notes on Set Theory (Springer, New York, 2006)Google Scholar
  46. 46.
    A. Nastasi, P. Vetro, Fixed point results on metric and partial metric spaces via simulation functions. J. Nonlinear Sci. Appl. 8, 1059–1069 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    J.J. Nieto, R. Rodriguez-Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, 223–239 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    P.D. Proinov, Fixed point theorems in metric spaces. Nonlinear Anal. 64, 546–557 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    A.C.M. Ran, M.C. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 132, 1435–1443 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    S. Reich, Fixed points of contractive functions. Boll. Un. Mat. Ital. 5, 26–42 (1972)MathSciNetzbMATHGoogle Scholar
  51. 51.
    B.E. Rhoades, A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc. 226, 257–290 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    B.E. Rhoades, Some theorems on weakly contractive maps. Nonlinear Anal. 47, 2683–2693 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    A.-F. Roldán, N. Shahzad, New fixed point theorems under R-contractions. Fixed Point Theory Appl. 2015, 98 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    A.-F. Roldán, E. Karapinar, C. Roldán, J. Martinez-Moreno, Coincidence point theorems on metric spaces via simulation method. Fixed Point Theory Appl. 2015, 98 (2015)zbMATHCrossRefGoogle Scholar
  55. 55.
    I.A. Rus, Generalized Contractions and Applications (Cluj University Press, Cluj-Napoca, 2001)zbMATHGoogle Scholar
  56. 56.
    A.S. Saluja, R.A. Rashwan, D. Magarde, P.K. Jhade, Some result in ordered metric spaces for rational type expressions. Facta Univ. Niš 31, 125–138 (2016)MathSciNetzbMATHGoogle Scholar
  57. 57.
    B. Samet, M. Turinici, Fixed point theorems on a metric space endowed with an arbitrary binary relation and applications. Commun. Math. Anal. 13, 82–97 (2012)MathSciNetzbMATHGoogle Scholar
  58. 58.
    E. Schechter, Handbook of Analysis and its Foundation (Academic Press, New York, 1997)zbMATHGoogle Scholar
  59. 59.
    T. Suzuki, Fixed point theorem for asymptotic contractions of Meir-Keeler type in complete metric spaces. Nonlinear Anal. 64, 971–978 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    T. Suzuki, A definitive result on asymptotic contractions. J. Math. Anal. Appl. 335, 707–715 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    T. Suzuki, C. Vetro, Three existence theorems for weak contractions of Matkowski type. Int. J. Math. Stat. 6, 110–120 (2010)MathSciNetCrossRefGoogle Scholar
  62. 62.
    A. Tarski, Axiomatic and algebraic aspects of two theorems on sums of cardinals. Fund. Math. 35, 79–104 (1948)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    M. Turinici, Fixed points of implicit contraction mappings. An. Şt. Univ. Al. I. Cuza Iaşi 22, 177–180 (1976)MathSciNetzbMATHGoogle Scholar
  64. 64.
    M. Turinici, Abstract comparison principles and multivariable Gronwall-Bellman inequalities. J. Math. Anal. Appl. 117, 100–127 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    M. Turinici, Function pseudometric VP and applications. Bul. Inst. Polit. Iaşi 53(57), 393–411 (2007)Google Scholar
  66. 66.
    M. Turinici, Ran-Reurings theorems in ordered metric spaces. J. Indian Math. Soc. 78, 207–214 (2011)MathSciNetzbMATHGoogle Scholar
  67. 67.
    M. Turinici, Kannan spectral contractive maps in relational metric spaces. Modern Directions Metrical FPT, Paper-3-3, Pim Edit. House, Iaşi (2016)Google Scholar
  68. 68.
    S.S. Wagstaff Jr., Sequences not containing an infinite arithmetic progression. Proc. Am. Math. Soc. 36, 395–397 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012, 94 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    E.S. Wolk, On the principle of dependent choices and some forms of Zorn’s lemma. Can. Math. Bull. 26, 365–367 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    R.N. Yadava, R. Shrivastava, S.S. Yadav, Rational type contraction mapping in T-orbitally complete metric space. Math. Theory Model. 4, 115–133 (2014)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.A. Myller Mathematical SeminarA. I. Cuza UniversityIaşiRomania

Personalised recommendations