Abstract

In the paper we formulate a criterion for relative compactness in the space R(J,E) consisting of all regulated functions, defined on the compact interval J and with values in the Banach space E. On the basis of the criterion we construct two arithmetically convenient and regular measures of noncompactness and investigate the connections of these measures with the Hausdorff measure. We show that obtained estimates are optimal. Moreover, we formulate necessary and sufficient conditions for the Nemytskii operator to be acting from the space R(J,E) into itself. Finally, we show the applicability of the mentioned measures in proving the existence of solutions of a nonlinear functional-integral equation.

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