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Measures of noncompactness are very useful tools in functional analysis, for in-stance in metric fixed point theory and the theory of operator equations in Banach spaces. They are also used in the studies of functional equations, ordinary and partial differential equations, fractional partial differential equations, integral and integro-differential equa-tions, optimal control theory, and in the characterizations of compact operators between Ba-nach spaces. We present an axiomatic introduction to measures of noncompactness on bounded subsets of complete metric spaces [6, 4, 5, 3], and also the alternative axiomatic approaches by Banaś and Goebel [2] and by Akhmerov et al. [1] for measures of noncompact-ness in Banach spaces. As examples, we consider the Kuratowski, Hausdorff and separation measures of noncompactness and their most important properties. The Kuratowski measure of noncompactness is used in Darbo’s fixed point theorem. Furthermore we study the notion of measures of noncompactness of operators between Banach spaces and some of their prop-erties. Finally we give a few applications to the characterization of compact linear operators between certain BK spaces and to some results concerning the solvability of integral equa-tions.
References:
[1] R.R. Akhmerov, M.I. Kamenskii, A.S. Potapov, A.E. Rodkina, and B.N. Sadovskii. Measures of Noncompactness and Condensing Operators. Birkhäuser Verlag, Basel, 1992.
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[2] J. Banaś and K. Goebel. Measures of Noncompactness in Banach Spaces, volume 60 of Lecture Notes in Pure and Applied Mathematics. Marcel Dekker Inc., New York and Basel, 1980.
[3] B. de Malafosse, E. Malkowsky, and V. Rakočević. Operators Between Sequence Spaces and Applications. Springer, 2021.
[4] E. Malkowsky and V. Rakočević. An introduction into the theory of sequence spaces and measures of noncompactness, volume 9(17) of Zbornik radova, Matematčki in-stitut SANU, pages 143–234. Mathematical Institute of SANU, Belgrade, 2000.
[5] E. Malkowsky and V. Rakočević. Advanced Functional Analysis. Taylor and Francis, 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487, USA, 2019.
[6] J.M. Ayerbe Toledano, T. Dominguez Benavides, and G. Lopez Acedo. Measures of Noncompactness in Metric Fixed Point Theory, volume 99 of Operator Theory Ad-vances and Applications. Birkhäuser Verlag, Basel, Boston, Berlin, 1997.