Existence of solutions to nonlinear functional-integral equations via the measure of noncompactness

  • Hemant Kumar Nashine
  • Reza Arab


The purpose of this work is to present two new notions of \(\mu \)-set contraction of a bounded subset of a Banach space and establish some fixed point and coupled fixed point results in the direction of Darbo (Rend Sem Math Univ Padova 4:84–92, 1995). We apply our work to get existence of solutions to nonlinear functional-integral equations followed by an illustration. Our work generalizes many existing results in the literature.


Fixed point coupled fixed point measure of noncompactness functional-integral equation 

Mathematics Subject Classification

34A08 54H25 47H10 



The authors express their gratitude to the referees for careful reading of the manuscript. The first author is thankful to the United States-India Education Foundation, New Delhi, India, and IIE/CIES, Washington, DC, USA, for Fulbright-Nehru PDF Award (no. 2052/FNPDR/2015).


  1. 1.
    Aghajani, A., Banaś, J., Sabzali, N.: Existence of solution for a class of nonlinear Volterra singular integral equations. Comput. Math. Appl. 62, 1215–1227 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aghajani, A., Banaś, J., Sabzali, N.: Some generalizations of Darbo fixed point theorem and applications. Bull. Belg. Math. Soc. Simon Stevin. 20(2), 345–358 (2013)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Aghajani, A., Sabzali, N.: Existence of coupled fixed points via measure of noncompactness and applications. J. Nonlinear Convex Anal. 15(5), 941–952 (2014)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Aghajani, A., Allahyari, R., Mursaleen, M.: A generalization of Darbos theorem with application to the solvability of systems of integral equations. J. Comput. Appl. Math. 260, 68–77 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Altun, I., Turkoglu, D.: A fixed point theorem for mappings satisfying a general contractive condition of operator type. J. Comput. Anal. Appl. 9(1), 9–14 (2007)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Arab, R.: Some fixed point theorems in generalized darbo fixed point theorem and the existence of solutions for system of integral equations. J. Korean Math. Soc. 52(1), 125–139 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Arab, R.: The existence of fixed points via the measure of noncompactness and its application to functional-integral equations. Mediterr. J. Math. 13, 759–773 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Banaś, J.: Measures of noncompactness in the space of continuous tempered functions. Demonstr. Math. 14, 127–133 (1981)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Banaś, J., Goebel, K.: Measures of noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, vol. 60. Dekker, New York (1980)zbMATHGoogle Scholar
  10. 10.
    Burton, T.A.: Krasnoselskii’s inversion principle and fixed points. Nonlinear Anal. 30, 3975–3986 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Burton, T.A., Kirk, C.: A fixed point theorem of Krasnoselskii–Schaefer type. Math. Nachrichten 189, 23–31 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Darbo, G.: Punti uniti in transformazioni a condominio non compatto. Rend. Sem. Math. Univ. Padova 4, 84–92 (1995)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Geraghty, M.: On contractive mappings. Proc. Am. Math. Soc. 40, 604–608 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Guo, D., Lakshmikantham, V., Liu, X.: Nonlinear Integral Equations in Abstract Spaces, Mathematics and Its Applications, vol. 373. Kluwer Academic Publishers, Dordrecht (1996)CrossRefGoogle Scholar
  15. 15.
    Kuratowski, K.: Sur les espaces completes. Fund. Math. 15, 301–309 (1930)CrossRefzbMATHGoogle Scholar
  16. 16.
    Mizoguchi, N., Takahashi, W.: Fixed point theorems for multivalued mappings on complete metric spaces. J. Math. Anal. Appl. 141(1), 177–188 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mursaleen, M., Mohiuddine, S.A.: Applications of measures of noncompactness to the infinite system of differential equations in \(l_p\) spaces. Nonlinear Anal. TMA 75, 2111–2115 (2012)CrossRefzbMATHGoogle Scholar
  18. 18.
    Mursaleen, M., Rizvi, S.M.H.: Solvability of infinite systems of second order differential equations in \(c_0\) and \(\ell _1\) by Meir-Keeler condensing operators. Proc. Am. Math. Soc. 144(10), 4279–4289 (2016)CrossRefzbMATHGoogle Scholar
  19. 19.
    Reich, S.: Fixed points in locally convex spaces. Math. Z. 125(1), 17–31 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Reich, S.: Fixed points of condensing functions. J. Math. Anal. Appl. 41, 460–467 (1973)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsTexas A & M UniversityKingsvilleUSA
  2. 2.Department of MathematicsSari Branch Islamic Azad UniversitySariIran

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