Abstract
This article is committed to deal with measure of non-compactness of operators in Banach spaces. Firstly, the collection \(\mathcal{C}(X)\) (consisting of all nonempty closed bounded convex sets of a Banach space X endowed with the uaual set addition and scaler multiplication) is a normed semigroup, and the mapping J from \(\mathcal{C}(X)\) onto \(\mathcal{F}(\Omega)\) is a fully order-preserving positively linear surjective isometry, where Ω is the closed unit ball of X* and \(\mathcal{F}(\Omega)\) the collection of all continuous and w*-lower semicontinuous sublinear functions on X* but restricted to Ω. Furthermore, both \(E_\mathcal{C}=\overline{J\mathcal{C}-J\mathcal{C}}\) and \(E_\mathcal{K}=\overline{J\mathcal{K}-J\mathcal{K}}\) are Banach lattices and \(E_\mathcal{K}\) is a lattice ideal of \(E_\mathcal{C}\). The quotient space \(E_\mathcal{C}/E_\mathcal{K}\) is an abstract M space, hence, order isometric to a sublattice of C(K) for some compact Haudorspace K, and \((FQJ)\mathcal{C}\) which is a closed cone is contained in the positive cone of C(K), where \(Q:E_\mathcal{C}\rightarrow{E_\mathcal{C}/E_\mathcal{K}}\) is the quotient mapping and \(F:E_\mathcal{C}/E_\mathcal{K}\rightarrow{C(K)}\) is a corresponding order isometry. Finally, the representation of the measure of non-compactness of operators is given: Let BX be the closed unit ball of a Banach space X, then
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References
Kuratowski K. Sur les espaces complets. Fund Math, 1930, 1(15): 301–309
Gokhberg I T, Goldstein I S, Markus A S. Investigation of some properties of bounded linear operators in connection with their q-norm. Uch zap Kishinevsk In a, 1957, 29: 29–36
Golden<stein L S, Markus A S. On a measure of noncompactness of bounded sets and linear operators. Studies in Algebra and Mathematical Analysis, 1965: 45–54
Sadovskii B N. A fixed-point principle. Functional Analysis and Its Applications, 1967, 1(2): 151–153
Goebel K. Thickness of sets in metric spacea and its applicationa to the fixed point theory, Habilit. Lublin: Thesia, 1970
Aghajani A, Banaś J, Jalilian Y. Existence of solutions for a class of nonlinear volterra singular integral equations. Computers & Mathematics with Applications, 2011, 62(3): 1215–1227
Banaś J. Measures of noncompactness in the study of solutions of nonlinear differential and integral equations. Open Mathematics, 2012, 10(6): 2003–2011
Agarwal R P, Benchohra M, Seba D. On the application of measure of noncompactness to the existence of solutions for fractional differential equations. Results in Mathematics, 2009, 55(3/4): 221
Malafosse B D, Malkowsky E. On the measure of noncompactness of linear operators in spaces of strongly α-summable and bounded sequences. Periodica Mathematica Hungarica, 2007, 55(2): 129–148
Malafosse B D, Malkowsky E, Rakocevic V. Measure of noncompactness of operators and matrices on the spaces c and c0. International Journal of Mathematics and Mathematical Sciences, 2006
Banaś J. On measures of noncompactness in banach spaces. Commentationes Mathematicae Universitatis Carolinae, 1980, 21(1): 131–143
Mallet-Paret J, Nussbaum R D. Inequivalent measures of noncompactness. Annali di Matematica Pura ed Applicata, 2011, 190(3): 453–488
Cheng L X, Cheng Q J, Shen Q R, Tu K, Zhang W. A new approach to measures of noncompactness of banach spaces. Studia Mathematica, 2018, 240: 21–45
Mallet-Paret J, Nussbaum R. Inequivalent measures of noncompactness and the radius of the essential spectrum. Proceedings of the American Mathematical Society, 2011, 139(3): 917–930
Kirk WA. A fixed point theorem for mappings which do not increase distances. The AmericanMathematical Monthly, 1965, 72(9): 1004–1006
Rådström H. An embedding theorem for spaces of convex sets. Proceedings of the American Mathematical Society, 1952, 3(1): 165–169
Cheng L X, Cheng Q J, Wang B, Zhang W. On super-weakly compact sets and uniformly convexifiable sets. Studia Mathematica, 2010, 2(199): 145–169
Cheng L X, Luo Z H, Zhou Y. On super weakly compact convex sets and representation of the dual of the normed semigroup they generate. Canadian Mathematical Bulletin, 2013, 56(2): 272–282
Cheng L X, Cheng Q J, Zhang J C. On super fixed point property and super weak compactness of convex subsets in banach spaces. Journal of Mathematical Analysis and Applications, 2015, 428(2): 1209–1224
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The project supported in part by the National Natural Science Foundation of China (11801255)
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Shen, Q. A Viewpoint to Measure of Non-Compactness of Operators in Banach Spaces. Acta Math Sci 40, 603–613 (2020). https://doi.org/10.1007/s10473-020-0301-8
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DOI: https://doi.org/10.1007/s10473-020-0301-8