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References [1] Akhmerov , R. R. , Kamenskii , M. I. , Potapov , A. S. , Rodkina , A. E. , Sadovskii , B. N. 1992 Measures of Noncompactness and Condensing Operators

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of the measure of noncompactness to some nonlinear functional integral equations in C[0, a] , Adv. Math. Sci. Appl. , 23 ( 2 ) ( 2013 ), 575 – 584 . [7] B anás , J. , C

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measures of noncompactness with respect to differentiation and integration of vector-valued functions , Nonlinear Anal. , 7 ( 12 ) ( 1983 ), 1351 – 1371 . 6 O lszowy , L

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Abstract  

In this paper, we characterize classes of matrix transformations from BK spaces into spaces of bounded sequences and their subclasses of infinite matrices that define compact operators. Furthermore, using these results and the solvability of certain infinite linear systems we give necessary and sufficient conditions for A to be a compact operator on spaces that are strongly α-bounded or summable.

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1535 1544 Banás, J. and Goebel, K. , Measure of Noncompactness in Banach Spaces , Lecture Notes in Pure and Applied Mathematics, 60 , Marcel Dekker

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In 1930 Kuratowski introduced the measure of non-compactness for complete metric spaces in order to measure the discrepancy a set may have from being compact.Since then several variants and generalizations concerning quanti .cation of topological and uniform properties have been studied.The introduction of approach uniform spaces,establishes a unifying setting which allows for a canonical quanti .cation of uniform concepts,such as completeness,which is the subject of this article.

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Abstract  

We define two properties of sequences in Banach spaces that may be related to measures of noncompactness of subsets of these spaces. The first one concerns properties of sequences related to the strong topology, and the second one is related to the weak topology. Given a Banach space X, we introduce a new Banach space such that we can find a subset E in it that may be identified with the balls in the first one. We use compactness in this new space to characterize our sequential properties. In particular, we prove a general form of the Eberlein-Smulian theorem.

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223 Wang, J. and Wei, W. , An application of measure of noncompactness in the study of integrodifferential evolution equations with nonlocal conditions, Proc. A. Razmadze Math

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