Applications of Measure of Noncompactness and Operator Type Contraction for Existence of Solution of Functional Integral Equations

  • Bipan HazarikaEmail author
  • Reza Arab
  • M. Mursaleen


The objective of this paper is to study the existence of solution of functional integral equations using a measure of noncompactness argument combined with generalized coupled fixed point theorem with the help of operator type contraction mapping in Banach spaces. We present an example to illustrate our results.


Operator type contraction Coupled fixed point theorem Functional integral equations Measure of noncompactness 

Mathematics Subject Classification

34A34 46B45 47H08 47H10 



The authors thank the editor and the referees for their valuable comments and helpful suggestions for better readability of the article. Research of the third author (M. Mursaleen) was supported by SERB Core Research Grant, DST, New Delhi, under grant No. EMR/2017/000340.


  1. 1.
    Agarwal, R.P., O’Regan, D.: Fixed Point Theory and Applications. Cambridge University Press, Cambridge (2004)Google Scholar
  2. 2.
    Aghajani, A., Allahyari, R., Mursaleen, M.: A generalization of Darbo’s theorem with application to the solvability of systems of integral equations. J. Comput. Appl. Math. 260, 68–77 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aghajani, A., Banaś, J., Sabzali, N.: Some generalization of Darbo fixed point theorem and applications. Bull. Belg. Math. Soc. Simon Stevin 20(2), 345–358 (2013)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Aghajani, A., Sabzali, N.: Existence of coupled fixed points via measure of noncompactness and applications. J. Nonlinear Convex Anal. 14(5), 941–952 (2014)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Aghajani, A., Mursaleen, M., Shole Haghighi, A.: Fixed point theorems for Meir–Keeler condensing operators via measure of noncompactness. Acta. Math. Sci. 35(3), 552–566 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Aghajani, A., Pourhadi, E.: Application of measure of noncompactness to \(\ell _{1}\)-solvability of infinite systems of second order differential equations. Bull. Belg. Math. Soc. Simon Stevin 22, 105–118 (2015)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Allahyari, R., Arab, R., Shole Haghighi, A.: Existence of solutions for some classes of integro-differential equations via measure of non-compactness. Electron. J. Qual. Theory Differ. Equ. 41, 1–18 (2015)CrossRefzbMATHGoogle Scholar
  8. 8.
    Alotaibi, A., Mursaleen, M., Mohiuddine, S.A.: Application of measure of noncompactness to infinite system of linear equations in sequence spaces. Bull. Iran. Math. Soc. 41, 519–527 (2015)MathSciNetzbMATHGoogle Scholar
  9. 9.
    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
  10. 10.
    Arab, R., Allahyari, R., Haghighi, A.S.: Existence of solutions of infinite systems of integral equations in two variables via measure of noncompactness. Appl. Math. Comput. 246, 283–291 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Arab, R., Rabbani, M., Mollapourasl, R.: On solution of a nonlinear integral equation with deviating argument based the on fixed point technique. Appl. Comput. Math. 14(1), 38–49 (2015)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Arab, R., Allahyari, R., Shole Haghighi, A.: Construction of a Measure of non-compactness on \(BC(\Omega )\) and its application to Volterra integral equations. Mediter. J. Math. 13(3), 1197–1210 (2016)CrossRefzbMATHGoogle Scholar
  13. 13.
    Arab, R.: The existence of fixed points via the measure of noncompactness and its application to functional-integral equations. Mediter. J. Math. 13(2), 759–773 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Banaś, J., Goebel, K.: Measure of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, vol. 60. Marcel Dekker, New York (1980)zbMATHGoogle Scholar
  15. 15.
    Banaś, J., Lecko, M.: Solvability of infinite systems of differential equations in Banach sequence spaces. J. Comput. Appl. Math. 137, 363–375 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chang, S.S., Cho, Y.J., Huang, N.J.: Coupled fixed point theorems with applications. J. Korean Math. Soc. 33(3), 575–585 (1996)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Darbo, G.: Punti uniti in trasformazioni a codominio non compatto. Rend. Sem. Mat. Univ. Padova 24, 84–92 (1955) (in Italian)Google Scholar
  18. 18.
    Das, A., Hazarika, B., Arab, R., Mursaleen, M.: Solvability of the infinite system of integral equations in two variables in the sequence spaces \(c_0\) and \(\ell _1,\). J. Comput. Appl. Math. 326, 183–192 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hazarika, B., Karapınar, E., Arab, R., Rabbani, M.: Metric-like spaces to prove existence of solution for nonlinear quadratic integral equation and numerical method to solve it. J. Comput. Appl. Math. 328(15), 302–313 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kuratowski, K.: Sur les espaces complets. Fund. Math. 15, 301–309 (1930)CrossRefzbMATHGoogle Scholar
  21. 21.
    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
  22. 22.
    Mursaleen, M., Mohiuddine, S.A.: Applications of measures of noncompactness to the infinite system of differential equations in \(\ell _p\) spaces. Nonlinear Anal. 75, 2111–2115 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mursaleen, M., Alotaibi, A.: Infinite system of differential equations in some BK-spaces. Abst. Appl. Anal. 2012, 20 (2012)MathSciNetzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsRajiv Gandhi UniversityRono Hills, DoimukhIndia
  2. 2.Department of MathematicsGauhati UniversityGuwahatiIndia
  3. 3.Department of Mathematics, Sari BranchIslamic Azad UniversitySariIran
  4. 4.Department of MathematicsAligarh Muslim UniversityAligarhIndia

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