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Application of measure of noncompactness for solvability of the infinite system of integral equations in two variables in \(\ell _{p}\left( 1<p< \infty \right) \)

  • Anupam Das
  • Bipan HazarikaEmail author
  • M. Mursaleen
Original Paper
  • 103 Downloads

Abstract

In this article, we establish the existence of solution of infinite systems of integral equations in two variables in the sequence space \(\ell _{p}(1<p<\infty )\) by using Meir–Keeler condensing operators. We explain the results with the help of simple examples.

Keywords

Systems of integral equations Measure of noncompactness Hausdorff measure of noncompactness Condensing operators Fixed point 

Mathematics Subject Classification

34A34 46B45 47H10 

Notes

Acknowledgements

The authors would like to thank Prof. Manuel López-Pellicer, Editor in Chief and the referees for his/her much encouragement, constructive criticism, careful reading and making a useful comment which improved the presentation and the readability of the paper.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no competing interests.

References

  1. 1.
    Aghajani, A., Mursaleen, M., Shole, A.: Haghighi, fixed point theorems for Meir–Keeler condensing operators via measure of noncompactness. Acta. Math. Sci. 35(3), 552–566 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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
  3. 3.
    Aghajani, A., Haghighi, A.S.: Existence of solutions for a system of integral equations via measure of noncompactness. Novi Sad J. Math. 44(1), 59–73 (2014)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Aghajani, A., Haghighi, A.S.: Existence of solutions for a class of functional integral equations of Volterra type in two variables via measure of noncompactness. Iran. J. Sci. Technol. 38(1), 1–8 (2014)MathSciNetGoogle Scholar
  5. 5.
    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
  6. 6.
    Akhmerov, R.R., Kamenskii, M.I., Potapov, A.S., Rodkina, A.E., Sadovskii, B.N.: Measure of Noncompactness and Condensing Operators, Operator Theory: Advances and Applications, vol. 55. Birkhäuser Verlag, Basel (1992). Translated from the 1986 Russian original by A. IacobCrossRefzbMATHGoogle Scholar
  7. 7.
    Allahyari, R., Arab, R., Haghighi, A.S.: Existence of solutions of infinite systems of integral equations in the Fréchet spaces. Int. J. Nonlinear Anal. Appl. 7(2), 205–216 (2016)zbMATHGoogle 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.
    Arab, R.: The existence of fixed points via the measure of noncompactness and its application to functional-integral equations. Mediterr. J. Math. 13(2), 759–773 (2016)MathSciNetCrossRefzbMATHGoogle 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.
    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
  12. 12.
    Banaś, J., Mursaleen, M.: Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations. Springer, New Delhi (2014)CrossRefzbMATHGoogle Scholar
  13. 13.
    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
  14. 14.
    Banaś, J., Lecko, M.: An existence theorem for a class of infinite systems of integral equations. Math. Comput. Model. 34, 533–539 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bellman, R.: Methods of Nonlinear Analysis II. Academic Press, New York (1973)zbMATHGoogle Scholar
  16. 16.
    Darbo, Gabriele: Punti uniti in trasformazioni a codominio non compatto (Italian). Rend. Sem. Mat. Univ. Padova 24, 84–92 (1955)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Deimling, K.: Ordinary Differential Equations in Banach Spaces. Lecture Notes in Mathematics, vol. 596. Springer, Berlin (1977)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kuratowski, K.: Sur les espaces complets. Fund. Math. 15, 301–309 (1930)CrossRefzbMATHGoogle Scholar
  19. 19.
    Meir, A., Keeler, Emmett: A theorem on contraction mappings. J. Math. Anal. Appl. 28, 326–329 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    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
  21. 21.
    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
  22. 22.
    Mursaleen, M., Abdullah, A.: Infinite system of differential equations in some BK-spaces. Abst. Appl. Anal. 863483, 20 (2012)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Oguzt Poreli, M.N.: Util. Math. On the neural equations of Cowan and Stein 2, 305–315 (1972)Google Scholar
  24. 24.
    Rzepka, R., Sadarangani, K.: On solutions of an infinite system of singular integral equations. Math. Comput. Model. 45, 1265–1271 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l. 2017

Authors and Affiliations

  1. 1.Department of MathematicsRajiv Gandhi UniversityDoimukhIndia
  2. 2.Department of MathematicsGauhati UniversityGuwahatiIndia
  3. 3.Department of MathematicsAligarh Muslim UniversityAligarhIndia

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