Geodesic
On a Criterion for the Complete Continuity of the Fréchet Derivative
Funkcionalʹnyj analiz i ego priloženiâ, Tome 49 (2015) no. 4, pp. 79-82
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Keywords: Hausdorff measure of noncompactness, condensing maps, asymptotically linear operator, rotation of vector fields, Hammerstein operator
Mots-clés : Fréchet derivative, bifurcation point, Lebesgue spaces. 
@article{FAA_2015_49_4_a6,
     author = {N. A. Yerzakova},
     title = {On a {Criterion} for the {Complete} {Continuity} of the {Fr\'echet} {Derivative}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {79--82},
     year = {2015},
     volume = {49},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2015_49_4_a6/}
}
TY  - JOUR
AU  - N. A. Yerzakova
TI  - On a Criterion for the Complete Continuity of the Fréchet Derivative
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2015
SP  - 79
EP  - 82
VL  - 49
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/FAA_2015_49_4_a6/
LA  - ru
ID  - FAA_2015_49_4_a6
ER  - 
%0 Journal Article
%A N. A. Yerzakova
%T On a Criterion for the Complete Continuity of the Fréchet Derivative
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2015
%P 79-82
%V 49
%N 4
%U http://geodesic.mathdoc.fr/item/FAA_2015_49_4_a6/
%G ru
%F FAA_2015_49_4_a6
N. A. Yerzakova. On a Criterion for the Complete Continuity of the Fréchet Derivative. Funkcionalʹnyj analiz i ego priloženiâ, Tome 49 (2015) no. 4, pp. 79-82. http://geodesic.mathdoc.fr/item/FAA_2015_49_4_a6/

[1] R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina, B. N. Sadovskii, Mery nekompaktnosti i uplotnyayuschie operatory, Nauka, Novosibirsk, 1986 | MR

[2] M. A. Krasnoselskii, Topologicheskie metody v teorii nelineinykh integralnykh uravnenii, Gostekhizdat, M., 1956 | MR

[3] N. A. Erzakova, Nonlinear Anal., 75:8 (2012), 3552–3557 | DOI | MR | Zbl

[4] N. A. Erzakova, Nauchnyi vestnik MGTU GA, 207 (2014), 110–117

[5] V. B. Melamed, A. I. Perov, Sib. matem. zh., 4:3 (1963), 702–704 | MR | Zbl

[6] M. A. Krasnoselskii, P. P. Zabreiko, E. I. Pustylnik, P. E. Sobolevskii, Integralnye operatory v prostranstvakh summiruemykh funktsii, Nauka, M., 1966 | MR

[7] N. A. Erzakova, Izv. vuzov. Matem., 9 (2011), 1–8 | MR