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We consider a generalization of an inequality from papers by Yu. A. Dubinskii, J.L. Lions and E. Magenes. This inequality is of great importance for the proof of solvability of nonlinear elliptic and parabolic equations. In contrast to their works, we do not require the compactness of the embedding. We suggest a new approach to the problem of narrow operators. In particular, we find a new application of measures of non-compactness. Mathematics subject classification (2010): Primary 47H08, Secondary 46B50, 46E30, 47J20, 47A50.

Applications of one inequality to measures of non-compactness and narrow operators