By R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina, B. N. Sadovskii
A condensing (or densifying) operator is a mapping lower than which similar to any set is in a undeniable feel extra compact than the set itself. The measure of noncompactness of a collection is measured via capabilities known as measures of noncompactness. The contractive maps and the compact maps [i.e., during this creation, the maps that ship any bounded set right into a rather compact one; in general textual content the time period "compact" may be reserved for the operators that, as well as having this estate, are non-stop, i.e., within the authors' terminology, for the thoroughly non-stop operators] are condensing. For contractive maps you will take as degree of noncompactness the diameter of a suite, whereas for compact maps can take the indicator functionality of a kin of non-relatively com pact units. The operators of the shape F( x) = G( x, x), the place G is contractive within the first argument and compact within the moment, also are condensing with appreciate to a couple usual measures of noncompactness. The linear condensing operators are characterised through the truth that just about all in their spectrum is integrated in a disc of radius smaller than one. The examples given above exhibit that condensing operators are a sufficiently general phenomenon in quite a few functions of sensible research, for instance, within the concept of differential and crucial equations. As is seems, the condensing operators have houses just like the compact ones.