Journal of Fixed Point Theory and Applications

, Volume 19, Issue 3, pp 1953–1976 | Cite as

Recent contributions to fixed point theory of monotone mappings

Article

Abstract

In this manuscript, we discuss the latest fixed point results of monotone mappings. The fixed point theory of such mappings has seen a tremendous interest in the last decade since the publication of Ran and Reurings paper in 2004. Fixed point theory for monotone mappings is useful and has many applications. For example when one is looking for a positive or negative solution, the use of the classical fixed point results is not adapted in this situation.

Keywords

Fixed point Integral delay equation Krasnoselskii iteration Lebesgue measure Monotone mapping Nonexpansive mapping Partially ordered 

Mathematics Subject Classification

Primary 46B20 45D05 Secondary 47E10 34A12 

References

  1. 1.
    Abdou, A.N., Khamsi, M.A.: On monotone pointwise contractions in Banach and metric spaces. Fixed Point Theory Appl. 2015, 135 (2015). doi:10.1186/s13663-015-0381-7 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alfuraidan, M.R.: Fixed points of monotone nonexpansive mappings with a graph. Fixed Point Theory Appl. 2015, 49 (2015). doi:10.1186/s13663-015-0299-0 MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Alfuraidan, M.R., Bachar, M., Khamsi, M.A.: On monotone contraction mappings in modular function spaces. Fixed Point Theory Appl. 2015, 28 (2015). doi:10.1186/s13663-015-0274-9 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Alfuraidan, M.R., Khamsi, M.A.: Fixed points of monotone nonexpansive mappings on a hyperbolic metric space with a graph. Fixed Point Theory Appl. 2015, 44 (2015). doi:10.1186/s13663-015-0294-5 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Aksoy, A.G., Khamsi, M.A.: Nonstandard Methods in Fixed Point Theory. Springer-Verlag, New York (1990)CrossRefMATHGoogle Scholar
  6. 6.
    Argoubi, H., Jleli, M., Samet, B.: The study of fixed points for multivalued mappings in a Menger probabilistic metric space endowed with a graph. Fixed Point Theory Appl. 2015, 113 (2015). doi:10.1186/s13663-015-0361-y MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bachar, M., Magal, P.: Existence of periodic solution for a class of delay differential equations with impulses. Topics in functional differential and difference equations (Lisbon, 1999), 37–49. Fields Inst. Commun. 29 Amer. Math. Soc., Providence, RI (2001)Google Scholar
  8. 8.
    Bachar, M., Khamsi, M.A.: Delay differential equation in metric spaces: a partial ordered sets approach. Fixed Point Theory Appl. 2014, 193 (2014). doi:10.1186/1687-1812-2014-193 CrossRefGoogle Scholar
  9. 9.
    Bachar, M., Khamsi, M.A.: Fixed points of monotone mappings and application to integral equations. Fixed Point Theory Appl. 2015, 110 (2015). doi:10.1186/s13663-015-0362-x MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bachar, M., Khamsi, M.A.: On monotone Ćirić quasi-contraction mappings. J. Math. Inequal. 10(2), 511–519 (2016). doi:10.7153/jmi-10-40 MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bachar, M., Khamsi, M.A.: Properties of fixed point sets of monotone nonexpansive mappings in Banach spaces. Numer Func Anal Opt. 37(3), 277–283 (2016). doi:10.1080/01630563.2015.1136889 MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Banach, S.: Sur les opérations dans les ensembles abstraits et leurs applications. Fund. Math. 3, 133–181 (1922)MATHGoogle Scholar
  13. 13.
    Bhaskar, T.G., Lakshmikantham, V.: Fixed point theory in partially ordered metric spaces and applications. Nonlinear Anal. 65, 1379–1393 (2006)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Beauzamy, B.: Introduction to Banach Spaces and Their Geometry. North-Holland, Amsterdam (1985)MATHGoogle Scholar
  15. 15.
    Ben-El-Mechaiekh, H.: The Ran-Reurings fixed point theorem without partial order: a simple proof. J. Fixed Point Theory Appl. (2015). doi:10.1007/s11784-015-0218-3 MathSciNetMATHGoogle Scholar
  16. 16.
    Bin Dehaish, B.A., Khamsi, M.A.: Mann iteration process for monotone nonexpansive mappings. Fixed Point Theory Appl. 2015, 177 (2015). doi:10.1186/s13663-015-0416-0
  17. 17.
    Bin Dehaish, B.A., Khamsi, M.A.: Browder and Göhde fixed point theorem for monotone nonexpansive mappings. Fixed Point Theory Appl. 2016, 20 (2016). doi:10.1186/s13663-016-0505-8
  18. 18.
    Brezis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88–3, 486–490 (1983)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Browder, F.E.: Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA 54, 1041–1044 (1965)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Bruck, R.E.: Properties of fixed-point sets of nonexpansive mappings in Banach spaces. Trans. Am. Math. Soc. 179, 251–262 (1973)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Ćirić, L.B.: A generalization of Banach’s contraction principle. Proc. Am. Math. Soc. 45, 267–273 (1974)MathSciNetMATHGoogle Scholar
  22. 22.
    Diestel, J.: Geometry of Banach Spaces—Selected Topics. Springer Lecture Notes in Math. No. 485. Springer, Berlin (1957)Google Scholar
  23. 23.
    Dugundji, J., Granas, A.: Fixed Point Theory. PWN-Polish Scientific Publ, Warszawa, Polska Akademia Nauk, Instytut Matematyczny (1982)MATHGoogle Scholar
  24. 24.
    El-Sayed, S.M., Ran, A.C.M.: On an iteration method for solving a class of nonlinear matrix equations. SIAM J. Matrix Anal. Appl. 23(3), 632–645 (2002)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Goebel, K., Kirk, W.A.: Iteration processes for nonexpansive mappings. Contemp. Math. 21, 115–123 (1983)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)CrossRefMATHGoogle Scholar
  27. 27.
    Göhde, D.: Zum Prinzip der kontraktiven Abbildung. Math. Nachr. 30, 251–258 (1965)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Ishikawa, S.: Fixed points and iteration of a nonexpansive mapping in a Banach space. Proc. Am. Math. Soc. 59, 65–71 (1976)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Jachymski, J.: The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 136, 1359–1373 (2008)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Jleli, M., Samet, B.: Remarks on Nieto and Rodriguez-López fixed point theorem (personal communication) Google Scholar
  31. 31.
    Kelisky, R.P., Rivlin, T.J.: Iterates of Bernstein polynomials. Pacific J. Math. 21, 511–520 (1967)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Khamsi, M.A., Khan, A.R.: On monotone nonexpansive mappings in \(L_1([0,1])\). Fixed Point Theory Appl. 2015, 94 (2015). doi:10.1186/s13663-015-0346-x MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Khamsi, M.A., Kirk, W.A.: An Introduction to Metric Spaces and Fixed Point Theory. John Wiley, New York (2001)CrossRefMATHGoogle Scholar
  34. 34.
    Kirk, W.A.: A fixed point theorem for mappings which do not increase distances. Am. Math. Monthly 72, 1004–1006 (1965)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Kirk, W.A.: Fixed points of asymptotic contractions. J. Math. Anal. Appl. 277, 645–650 (2003)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Kirk, W.A.: Asymptotic pointwise contractions. In: Plenary Lecture, the 8th International Conference on Fixed Point Theory and Its Applications, Chiang Mai University, Thailand, July 16–22, 2007Google Scholar
  37. 37.
    Krasnoselskii, M.A.: Two observations about the method of successive approximations. Uspehi Mat. Nauk 10, 123–127 (1955)MathSciNetGoogle Scholar
  38. 38.
    Nadler Jr., S.B.: Multi-valued contraction mappings. Maruzen, Tokyo. 30, 475–488 (1950)MathSciNetMATHGoogle Scholar
  39. 39.
    Nieto, J.J., Rodríguez-López, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22(3), 223–239 (2005)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Pǎcurar, M: Iterative methods for fixed point approximation. Ph.D. Thesis, Babeş-Bolyai University, Cluj-Napoca (2009)Google Scholar
  42. 42.
    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(5), 1435–1443 (2004)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Rus, I.A.: Iterates of Bernstein operators, via contraction principle. J. Math. Anal. Appl. 292, 259–261 (2004)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Samet, B.: Ran-Reurings fixed point theorem is an immediate consequence of the Banach contraction principle. J. Nonlinear Sci. Appl. 9, 873–875 (2016)MathSciNetMATHGoogle Scholar
  45. 45.
    Smith, J.L.: Monotone dynamical systems. An introduction to the theory of competitive and cooperative systems. Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995. A.M.S. vol. 41 (1995)Google Scholar
  46. 46.
    Sultana, A., Vetrivel, V.: Fixed points of Mizoguchi Takahashi contraction on a metric space with a graph and applications. J. Math. Anal. Appl. 417, 336–344 (2014)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Turinici, M.: Fixed points for monotone iteratively local contractions. Dem. Math. 19, 171–180 (1986)MathSciNetMATHGoogle Scholar
  48. 48.
    Turinici, M.: Ran and Reurings theorems in ordered metric spaces. J. Indian Math Soc. 78, 207–2014 (2011)MathSciNetMATHGoogle Scholar
  49. 49.
    Zeidler, E.: Nonlinear Functional Analysis and its Applications I: Fixed-Point Theorems. Springer-Verlag, New York (1986)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of Mathematics, College of SciencesKing Saud UniversityRiyadhSaudi Arabia
  2. 2.Department of Mathematical SciencesThe University of Texas at El PasoEl PasoUSA
  3. 3.Department of Mathematics and StatisticsKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia

Personalised recommendations