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Application of Measure of Noncompactness to the Infinite Systems of Second-Order Differential Equations in Banach Sequence Spaces \(c,\ell _p\), and \(c_{0}^{\beta }\)

  • Anupam Das
  • Bipan HazarikaEmail author
Chapter

Abstract

In this paper, we establish the existence of solutions of infinite systems of second-order differential equations in Banach sequence spaces by using techniques associated with measures of noncompactness in a combination of Meir–Keeler condensing operators. We illustrate our results with the help of some examples.

References

  1. 1.
    J. Banaś, K. Goebel, Measure of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, vol. 60 (Marcel Dekker, New York, 1980)Google Scholar
  2. 2.
    J. Banaś, M. Mursaleen, Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations (Springer, New Delhi, 2014)CrossRefGoogle Scholar
  3. 3.
    R. Bellman, Methods of Nonlinear Analysis II (Academic, New York, 1973)zbMATHGoogle Scholar
  4. 4.
    K. Deimling, Ordinary Differential Equations in Banach Spaces. Lecture Notes in Mathematics, vol. 596 (Springer, Berlin, 1977)CrossRefGoogle Scholar
  5. 5.
    K. Kuratowski, Sur les espaces completes. Fund. Math. 15, 301–309 (1930)CrossRefGoogle Scholar
  6. 6.
    M.N.O. Poreli, On the neural equations of Cowan and Stein. Utilitas Math. 2, 305–315 (1972)MathSciNetGoogle Scholar
  7. 7.
    K. Kuratowski, Sur les espaces complets. Fund. Math. 15, 301–309 (1930)CrossRefGoogle Scholar
  8. 8.
    M. Mursaleen, S.A. Mohiuddine, Applications of measures of noncompactness to the infinite system of differential equations in \(\ell _p\) spaces. Nonlinear Anal. 75, 2111–2115 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    S.A. Mohiuddine, H.M. Srivastava, A. Alotaibi, Application of measures of noncompactness to the infinite system of second-order differential equations in \(\ell _p\) spaces. Adv. Difference Equ. 2016, Article 317 (2016)Google Scholar
  10. 10.
    A. Alotaibi, M. Mursaleen, S.A. Mohiuddine, Application of measure of noncompactness to infinite system of linear equations in sequence spaces. Bull. Iranian Math. Soc. 41, 519–527 (2015)MathSciNetzbMATHGoogle Scholar
  11. 11.
    M. Mursaleen, A. Alotaibi, Infinite system of differential equations in some BK-spaces. Abst. Appl. Anal. 2012, Article ID 863483, 20 (2012)Google Scholar
  12. 12.
    Józef Banaś, Millenia Lecko, Solvability of infinite systems of differential equations in Banach sequence spaces. J. Comput. Appl. Math. 137, 363–375 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    R.R. Akhmerov , M.I. Kamenskii, A.S. Potapov, A.E. Rodkina, B.N. Sadovskii, Measure of noncompactness and condensing operators, in Operator Theory: Advances and Applications, (Translated from the 1986 Russian original by A. Iacob), vol. 55 (Birkhäuser Verlag; Basel, 1992), pp. 1–52Google Scholar
  14. 14.
    G. Darbo, Punti uniti in trasformazioni a codominio non compatto (Italian). Rend. Sem. Mat. Univ. Padova 24, 84–92 (1955)MathSciNetzbMATHGoogle Scholar
  15. 15.
    A. Meir, E. Keeler, A theorem on contraction mappings. J. Math. Anal. Appl. 28, 326–329 (1969)MathSciNetCrossRefGoogle Scholar
  16. 16.
    A. Aghajani, M. Mursaleen, A.S. Haghighi, Fixed point theorems for Meir-Keeler condensing operators via measure of noncompactness. Acta. Math. Sci. 35(3), 552–566 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    D.G. Duffy, Green’s Function with Applications (Chapman and Hall/CRC, London, 2001)CrossRefGoogle Scholar
  18. 18.
    M. Mursaleen, S.M.H. Rizvi, 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)MathSciNetCrossRefGoogle Scholar
  19. 19.
    A. Aghajani, E. Pourhadi, 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)Google Scholar

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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of MathematicsRajiv Gandhi UniversityDoimukhIndia
  2. 2.Department of MathematicsGauhati UniversityGuwahatiIndia

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