Generalization of some M.A. Krasnosel'skii's results

We prove necessary and sufficient conditions for complete continuity of the Fréchet derivative at a point and the asymptotic derivative (in the case of their existence). For any measure of noncompactness ψ, we introduce two new classes of operators: a locally strongly ψ-condensing operator at a point and a strongly ψ-condensing at infinity on spheres. The new classes contain all completely continuous operators and some noncondensing operators. In addition, we extend some results of M.A. Krasnosel'skii on bifurcation points to a class of vector fields more general than completely continuous.