Functional Analysis and Its Applications

, Volume 49, Issue 4, pp 304–306 | Cite as

On a criterion for the complete continuity of the Fréchet derivative



In this paper we introduce, by means of the Hausdorff measure of noncompactness χ, two new classes of operators (not necessarily linear): operators locally strongly χ-condensing at a point and operators strongly χ-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. Krasnosel′skii’s theorem on asymptotic bifurcation points for completely continuous vector fields is generalized to the class of vector fields strongly χ-condensing at infinity.


Hausdorff measure of noncompactness condensing maps Fréchet derivative asymptotically linear operator bifurcation point rotation of vector fields Hammerstein operator Lebesgue spaces 


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  1. [1]
    R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskii, Measures of Noncompactness and Condensing Operators, Birkhauser, Basel, 1992.CrossRefMATHGoogle Scholar
  2. [2]
    M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, New York, 1964.Google Scholar
  3. [3]
    N. A. Erzakova, Nonlinear Anal., 75:8 (2012), 3552–3557.CrossRefMathSciNetMATHGoogle Scholar
  4. [4]
    N. A. Erzakova, Nauchn. Vestn. Mosk. Gos. Tekh. Univ. Grazhdanskoi Aviatsii [in Russian], 207 (2014), 110–117.Google Scholar
  5. [5]
    V. B. Melamed and A. I. Perov, Sibirsk. Mat. Zh., 4:3 (1963), 702–704.MathSciNetMATHGoogle Scholar
  6. [6]
    M. A. Krasnoselskii, P. P. Zabreiko, E. I. Pustylnik, and P. E. Sobolevskii, Integral Operators in Spaces of Summable Functions, Noordhoff International Publishing, Leyden, 1976.CrossRefGoogle Scholar
  7. [7]
    N. A. Erzakova, Izv. Vuzov Matem., 9 (2011), 1–8; English transl.: Russian Mathematics (Iz. VUZ), 55:9 (2011), 37–42.MathSciNetMATHGoogle Scholar

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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Moscow State Technical University of Civil AviationMoscowRussia

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