In this paper we determine theHausdorff measure of noncompactness on the sequence space n(ϕ) ofW. L. C. Sargent. Further we apply the technique of measures of noncompactness to the theory of infinite systems of differential equations in the Banach sequence spaces n(ϕ) and m(ϕ). Our aim is to present some existence results for infinite systems of differential equations formulated with the help of measures of noncompactness.
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M. Mursaleen (2011) studied this question.