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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 Ï, we introduce two new classes of operators: a locally strongly Ï-condensing operator at a point and a strongly Ï-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.
Journal: Journal of Mathematical Analysis and Applications - Volume 428, Issue 2, 15 August 2015, Pages 1368-1376