ON MEASURES OF NONCOMPACTNESS IN INEQUALITIES
Measures of noncompactness are numerical characteristics of bounded subsets of metric space, equal to zero on relatively compact subsets. The quantitative characteristic of measure of noncompactness of metric space subset was intro- duced by K. Kuratovskiy in 1930 in connection with problems of general typology. Different measures of noncompactness exist. Measures of noncompactness are a simple and useful instrument for any problem solving. So the theory of measures of noncompactness is still developing and it finds more and more new applications in different branches of mathematics. In this article measures of noncompactness are used to study inequalities, more exactly the extension of an equality, studied in many works and having wide application. For example in the works by Yu.A. Dubinskiy, J.-L. Lions and E. Magenes this inequality is proved for embedding operators in Banach spaces (a particular case of metric spaces). Then it is used to prove the solvability of nonlinear elliptic and parabolic equations. In contrast to these authors in this work the compactness of the embedding operator is not assumed in the study of the inequality. Furthermore, in metric space for the analogue of the ine- quality, written via any numerical characteristics of bounded subsets (not necessarily measures of noncompactness), the needed and sufficient conditions of the correctness of this analogue are received. In case if numerical characteristic of a set is a measure of noncompactness, the conclusion of this result is a new criterion of compactness of the operator (not neces- sarily linear) under the condition of compactness of another one.The results of this work generalize some results achieved by the author previously.