On fixed point theorems of monotone functions in Ordered metric spaces


The purpose of this paper is to establish some coincidence point results for f-nondecreasing self-mapping satisfying certain rational type contractions in the frame work of a metric space endowed with partial order. Some consequences of the main result are given by involving integral type contractions in the space. Some examples are illustrated to support our results. Further, as an application we prove the existence of a unique solution of integral equation by the method of upper and lower solutions.

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  1. 1.

    Banach, S. 1922. Sur les operations dans les ensembles abstraits et leur application aux equations untegrales. Foundations of Mathematics 3: 133–181.

    Article  Google Scholar 

  2. 2.

    Dass, B.K., and S. Gupta. 1975. An extension of Banach contraction principle through rational expression. Indian Journal of Pure and Applied Mathematics 6: 1455–1458.

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Chetterjee, S.K. 1972. Fixed point theorems. C.R. Academie Bulgare des Sciences 25: 727–730.

    MathSciNet  Google Scholar 

  4. 4.

    Edelstein, M. 1962. On fixed points and periodic points under contraction mappings. Journal of the London Mathematical Society 37: 74–79.

    MathSciNet  Article  Google Scholar 

  5. 5.

    Hardy, G.C., and T. Rogers. 1973. A generalization of fixed point theorem of S. Reich. Canadian Mathematical Bulletin 16: 201–206.

    MathSciNet  Article  Google Scholar 

  6. 6.

    Jaggi, D.S. 1977. Some unique fixed point theorems. Indian Journal of Pure and Applied Mathematics 8: 223–230.

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Kannan, R. 1969. Some results on fixed points - II. American Mathematical Monthly 76: 71–76.

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Reich, S. 1971. Some remarks concerning contraction mappings. Canadian Mathematical Bulletin 14: 121–124.

    MathSciNet  Article  Google Scholar 

  9. 9.

    Sharma, P.L., and A.K. Yuel. 1984. A unique fixed point theorem in metric space. Bulletin of Calcutta Mathematical Society 76: 153–156.

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Smart, D.R. 1974. Fixed Point Theorems. Cambridge: Cambridge University Press.

    Google Scholar 

  11. 11.

    Wong, C.S. 1973. Common fixed points of two mappings. Pacific Journal of Mathematics 48: 299–312.

    MathSciNet  Article  Google Scholar 

  12. 12.

    Agarwal, R.P., M.A. El-Gebeily, and D. O’Regan. 2008. Generalized contractions in partially ordered metric spaces. Applied Analysis 87: 1–8.

    MathSciNet  Article  Google Scholar 

  13. 13.

    Amini-Harandi, A., and H. Emami. 2010. A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Anal. Theory, Methods and Applications 72: 2238–2242.

    MathSciNet  Article  Google Scholar 

  14. 14.

    Ankush, C., B. Damjanovic, and D. Lakshmi Kanta. 2017. Fixed point results on metric spaces via simulation functions. FILOMAT 31 (11): 3365–3375. https://doi.org/10.2298/FIL1711365C.

    MathSciNet  Article  Google Scholar 

  15. 15.

    Bhaskar, T.G., and V. Lakshmikantham. 2006. Fixed point theory in partially ordered metric spaces and applications. Nonlinear Analysis, Theory, Methods and Applications 65: 1379–1393.

    Article  Google Scholar 

  16. 16.

    Chandok, S. 2013. Some common fixed point results for generalized weak contractive mappings in partially ordered metrix spaces. Journal of Nonlinear Analysis and Optics 4: 45–52.

    MATH  Google Scholar 

  17. 17.

    Chandok, S. 2013. Some common fixed point results for rational type contraction mappings in partially ordered metric spaces. Mathematica Bohemica 138 (4): 407–413.

    MathSciNet  Article  Google Scholar 

  18. 18.

    Harjani, J., B. Lopez, and K. Sadarangani. 2010. A fixed point theorem for mappings satisfying a contractive condition of rational type on a partially ordered metric space. Abstract and Applied Analysis. Article ID 190701, 8 pages.

  19. 19.

    Hong, S. 2010. Fixed points of multivalued operators in ordered metric spaces with applications. Nonlinear Analysis, Theory, Methods and Applications 72: 3929–3942.

    MathSciNet  Article  Google Scholar 

  20. 20.

    Liu Xiao, I., Mi Zhou, and B. Damjanovic. 2018. Nonlinear Operators in Fixed point theory with Applications to Fractional Differential and Integral Equations. Journal of Function spaces 2018, Article ID 9863267, 11 pages. https://doi.org/10.1155/2018/9063267.

  21. 21.

    Liz, E., and J.J. Nieto. 1994. Periodic boundary value problem for integro-differentail equations with general kernel. Dynamical Systems and Applications 3: 297–304.

    MATH  Google Scholar 

  22. 22.

    Zhou, M., X. Liu, B. Damjanovic, and A.H. Ansari. 2018. Fixed point theorems for several types of Meir Keeler contraction mappings in MS metric spaces. Journal Computational Analysis and Applications 25 (7): 1337–1353.

    MathSciNet  Google Scholar 

  23. 23.

    Nieto, J.J. 1997. An abstract monotone iterative technique. Nonlinear Analysis, Theory Methods and Applications. 28 (12): 1923–1933.

    MathSciNet  Article  Google Scholar 

  24. 24.

    Nieto, J.J., and R.R. Lopez. 2005. Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22: 223–239.

    MathSciNet  Article  Google Scholar 

  25. 25.

    Nieto, J.J., and R.R. Lopez. 2007. Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equation. Acta Mathematica Sinica, English Series 23 (12): 2205–2212.

    MathSciNet  Article  Google Scholar 

  26. 26.

    Ran, A.C.M., and M.C.B. Reurings. 2004. A fixed point theorem in partially ordered sets and some application to matrix equations. Proceedings of the American Mathematical Society 132: 1435–1443.

    MathSciNet  Article  Google Scholar 

  27. 27.

    Seshagiri Rao, N., and K. Kalyani. 2020. Generalized Contractions to Coupled Fixed Point Theorems in Partially Ordered Metric Spaces. Journal of Siberian Federal University Mathematics & Physics 13 (4): 492–502. https://doi.org/10.17516/1997-1397-2020-13-4-492-502.

    MathSciNet  Article  Google Scholar 

  28. 28.

    Seshagiri Rao, N., and K. Kalyani. 2020. Coupled fixed point theorems with rational expressions in partially ordered metric spaces. The Journal of Analysis. 11 pages. https://doi.org/10.1007/s41478-020-00236-y

  29. 29.

    Seshagiri Rao, N., K. Kalyani, and Kejal Khatri. 2020. Contractive mapping theorems in Partially ordered metric spaces. CUBO, A Mathematical Journal 22 (2): 203–214.

    MathSciNet  Article  Google Scholar 

  30. 30.

    Seshagiri Rao, N., and K. Kalyani. 2020. Unique fixed point theorems in partially ordered metric spaces. Heliyon 6: e05563. https://doi.org/10.1016/j.heliyon.2020.e05563.

    Article  Google Scholar 

  31. 31.

    Sperb, R. 1981. Maximum Principles and Their Application. New York: Academic Press.

    Google Scholar 

  32. 32.

    Wolk, E.S. 1975. Continuous convergence in partially ordered sets. General Topology and its Applications 5: 221–234.

    MathSciNet  Article  Google Scholar 

  33. 33.

    Zhang, X. 2010. Fixed point theorems of multivalued monotone mappings in ordered metric spaces. Applied Mathematics Letters 23: 235–240.

    MathSciNet  Article  Google Scholar 

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The authors thank the editor and anonymous referees for their valuable suggestion which improved the content of the paper.


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All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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Correspondence to K. Kalyani.

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Kalyani, K., Seshagiri Rao, N. & Mitiku, B. On fixed point theorems of monotone functions in Ordered metric spaces. J Anal (2021). https://doi.org/10.1007/s41478-021-00308-7

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  • Partially ordered metric space
  • rational contractions
  • compatible and weakly compatible mappings
  • monotone f-nondecreasing

Mathematics Subject Classification

  • 41A50
  • 47H10
  • 26A42
  • 46T99