Skip to main content

Fixed Points of Monotone Mappings via Generalized-Measure of Noncompactness


In this article, the notions of partially continuous mappings, partially bounded, partially compact and partially closed sets of an ordered E-metric space are defined. These notions are used to introduce partial E-measure of noncompactness and prove the fixed point results of monotone mappings and sum of two monotone mappings. In this way we generalize many results including the well known results of Schauder, Darbo and Krasnoselskii, in the settings of ordered E-metric and ordered Banach spaces. We also provide nontrivial examples and existence results for a class of integral equations to validate the significance of our theory and results.

This is a preview of subscription content, access via your institution.


  1. Akhmerov, R.R., Kamenskii, M.I., Potapov, A.S., Rodkina, A.E., Sadovskii, B.N.: Measures of Noncompactness and Condensing Operators. Operator Theory: Advances and Applications, vol. 55. Birkhäuser, Basel (1992)

    Book  Google Scholar 

  2. Al-Rawashdeh, A., Shatanawi, W., Khandaqji, M.: Normed ordered and E-metric spaces. Int. J. Math. Math. Sci. 2012, 272137 (2012)

    MathSciNet  Article  Google Scholar 

  3. Banaś, J., Mursaleen, M.: Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations. Springer, New Delhi (2014)

    Book  Google Scholar 

  4. Banaś, J., Jleli, M., Mursaleen, M., Samet, B., Vetro, C. (eds.): Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness. Springer, Singapore (2017)

    MATH  Google Scholar 

  5. Banaś, J.: On measures of noncompactness in Banach spaces. Comment. Math. Univ. Carol. 21, 131–143 (1980)

    MathSciNet  MATH  Google Scholar 

  6. Chen, C. -M., Karapınar, E., Chen, G.-T.: On the Meir-Keeler-Khan set contractions. J. Nonlinear Sci. Appl. 9, 5271–5280 (2016)

    MathSciNet  Article  Google Scholar 

  7. Chen, C.-M., Karapınar, E.: Fixed point results for the α-Meir-Keeler contraction on partial Hausdorff metric spaces. J. Inequal. Appl. 2013, 410 (2013)

    MathSciNet  Article  Google Scholar 

  8. Darbo, G.: Punti uniti in trasformazioni a codominio non compatto. Rend. Semin. Mat. Univ. Padova 24, 84–92 (1955)

    MathSciNet  MATH  Google Scholar 

  9. Dhage, B.C.: Partially condensing mappings in partially ordered normed linar spaces and applications to functional integral equations. Tamkang J. Math. 45, 397–426 (2014)

    MathSciNet  Article  Google Scholar 

  10. Deng, G., Huang, H., Cvetković, M., Radenović, S.: Cone valued measure of noncompactness and related fixed point theorems. Bull. Int. Math. Virtual Inst. 8, 233–243 (2018)

    MathSciNet  MATH  Google Scholar 

  11. Goebel, K.: Thickness of Sets in Metric Spacea and Its Applicationa to the Fixed Point Theory. Habil. Thesia, Lublin (1970)

    Google Scholar 

  12. Gokhberg, I.T., Goldstein, I.S., Markus, A.S.: Investigation of some properties of bounded linear operators in connection with their q-norm. Uch zap Kishinevsk In a 29, 29–36 (1957)

    Google Scholar 

  13. Huang, H.: Topological properties of E-metric spaces with applications to fixed point theory. Mathematics 7, 1222 (2019)

    Article  Google Scholar 

  14. Jleli, M., Mursaleen, M., Sadarangani, K., Samet, B.: A cone measure of noncompactness and some generalizations of Darbo’s theorem with applications to functional integral equations. J. Funct. Spaces 2016, 9896502 (2016)

    MathSciNet  MATH  Google Scholar 

  15. Kapeluszny, J., Kuczumow, T., Reich, S.: The Denjoy–Wolff theorem for condensing holomorphic mappings. J. Funct. Anal. 167, 79–93 (1999)

    MathSciNet  Article  Google Scholar 

  16. Krasnosel’skii, M.A., Zabreĭko, P.P.: Geometrical Methods of Nonlinear Analysis. Springer, Berlin (1984)

    Book  Google Scholar 

  17. Kuratowski, K.: Sur les espaces complets. Fundam. Math. 15, 301–309 (1930)

    Article  Google Scholar 

  18. Mehmood, N., Al Rawashdeh, A., Radenović, S.: New fixed point results for E-metric spaces. Positivity 23, 1101–1111 (2019)

    MathSciNet  Article  Google Scholar 

  19. Reich, S.: Fixed points in locally covex spaces. Math. Z. 125, 17–31 (1972)

    MathSciNet  Article  Google Scholar 

  20. Reich, S.: Fixed points of condensing functions. J. Math. Anal. Appl. 41, 460–467 (1973)

    MathSciNet  Article  Google Scholar 

  21. Sadovskii, B.N.: A fixed-point principle. Funct. Anal. Appl. 1, 151–153 (1967)

    MathSciNet  Article  Google Scholar 

  22. Todorčević, V.: Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics. Springer, Cham (2019)

    Book  Google Scholar 

Download references


The first author gratefully acknowledges with thanks the Department of Research Affairs at UAEU. This article is supported by the grant: UPAR-2019, Fund No. 31S397.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Nayyar Mehmood.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ahmad, N., Al-Rawashdeh, A., Mehmood, N. et al. Fixed Points of Monotone Mappings via Generalized-Measure of Noncompactness. Vietnam J. Math. 50, 275–285 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • E-metric space
  • Partially continuous
  • Partially compact
  • Partially bounded
  • Partially closed
  • Fixed points
  • Sum of monotone mappings

Mathematics Subject Classification (2010)

  • Primary 46S40
  • 47H10
  • 54H25