On a Criterion for the Complete Continuity of the Fréchet Derivative

In this paper we introduce, by means of the Hausdorff measure of noncompactness $\chi$, two new classes of operators (not necessarily linear): operators locally strongly $\chi$-condensing at a point and operators strongly $\chi$-condensing at infinity (on spherical interlayers). These classes include all completely continuous operators and some noncondensing operators. Necessary and sufficient conditions for the complete continuity of a Fréchet derivative at a point and of an asymptotic derivative (if they exist) are proved. M. A. Krasnoselżskii's theorem on asymptotic bifurcation points for completely continuous vector fields is generalized to the class of vector fields strongly $\chi$-condensing at infinity.