On Some Results Using Measures of Noncompactness

Abstract

In this chapter, we review some results on measures of noncompactness, including an axiomatic introduction, a study of the Kuratowski, Hausdorff, inner Hausdorff and separation measures of noncompactness, a survey of their most important properties and their roles in fixed-point theory and operator theory. We also determine the Hausdorff measure of noncompactness of matrix operators and, more generally, of bounded linear operators between some BK spaces. Among other things we present many of our results (with co-authors) involving the classical sequence spaces, matrix domains of triangles in certain sequence spaces and spaces of strongly summable and bounded sequences.