By R. R. Akhmerov, A. S. Potapov, A. E. Rodkina, B. N. Sadovskii

A condensing (or densifying) operator is a mapping below which a dead ringer for any set is in a definite feel extra compact than the set itself. The measure of noncompactness of a collection is measured by way of 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 really compact one; mostly textual content the time period "compact" might 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'll 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 by means of the truth that just about all in their spectrum is integrated in a disc of radius smaller than one. The examples given above express that condensing operators are a sufficiently regular phenomenon in numerous functions of practical research, for instance, within the conception of differential and imperative equations. As is seems, the condensing operators have homes just like the compact ones.

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