05 Nisan 2022,
Bahçeşehir Üniversitesi Analiz ve Uygulamalı Matematik Seminerleri
Some measures of noncompactness and their applications
Department of Mathematics, State University of Novi Pazar, Serbia, Sırbistan
Uygulamalı Matematik İngilizce
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  and by Akhmerov et al.  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.
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