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

Chapter

First Online:

## 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.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.J. Banaś, M. Mursaleen,
*Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations*(Springer, New Delhi, 2014)CrossRefGoogle Scholar - 3.R. Bellman,
*Methods of Nonlinear Analysis II*(Academic, New York, 1973)zbMATHGoogle Scholar - 4.K. Deimling,
*Ordinary Differential Equations in Banach Spaces*. Lecture Notes in Mathematics, vol. 596 (Springer, Berlin, 1977)CrossRefGoogle Scholar - 5.K. Kuratowski, Sur les espaces completes. Fund. Math.
**15**, 301–309 (1930)CrossRefGoogle Scholar - 6.M.N.O. Poreli, On the neural equations of Cowan and Stein. Utilitas Math.
**2**, 305–315 (1972)MathSciNetGoogle Scholar - 7.K. Kuratowski, Sur les espaces complets. Fund. Math.
**15**, 301–309 (1930)CrossRefGoogle Scholar - 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.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.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.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.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.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.G. Darbo, Punti uniti in trasformazioni a codominio non compatto (Italian). Rend. Sem. Mat. Univ. Padova
**24**, 84–92 (1955)MathSciNetzbMATHGoogle Scholar - 15.A. Meir, E. Keeler, A theorem on contraction mappings. J. Math. Anal. Appl.
**28**, 326–329 (1969)MathSciNetCrossRefGoogle Scholar - 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.D.G. Duffy,
*Green’s Function with Applications*(Chapman and Hall/CRC, London, 2001)CrossRefGoogle Scholar - 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.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

## Copyright information

© Springer Nature Singapore Pte Ltd. 2018