ON OPERATORS WITH THE SPHERICAL PROPERTY
Abstract
Properties of continuous positively homogeneous operators of degree via various functions (e.g. measures of noncompactness) on all bounded subsets of a Banach space are studied. Necessary and sufficient conditions for these functions to vanish on the image of the unit ball under positively homogeneous operators are given. In particular, we give criteria for the complete continuity of the Fréchet derivative in an arbitrary Banach space and criteria for operators, acting in regular spaces, to be improving.
References
1. Akhmerov R.R., Kamenskii M.I., Potapov A.S., Rodkina A.E., Sadovskii B.N. Measures of Noncompactness and Condensing Operators. Basel, Boston, Berlin: Birkhäser Verlag, 1992. – 255 p.
2. Krasnoselskii M.A. Topological methods in the theory of nonlinear integral equations. Oxford: Pergamon Press, 1964. 392 p.
3. Melamed V.B., Perov A.I. A generalization of a theorem of M.A. Krasnosel’skii on the complete continuity of the Frechet derivative of a completely continuous operator. Sibirsk. Mat. Zh. Vol. 4. No. 3. 1963. Pp. 702–704.
4. Erzakova N.A. On locally condensing operators. Nonlinear Analysis: Theory Methods& Applications, Vol. 75. Issue 8. 2012. Pp. 3552–3557.
5. Erzakova N.A. A near-ring of locally condensing operators. Scientific Bulletin of MSTUCA. Vol. 184 (10). 2012. Pp. 78–86.
6. Erzakova N.A. On strongly condensing operators at infinity. Scientific Bulletin of MSTUCA. 2014. Vol. 207 (09). Pp. 110–118.
7. Yerzakova N.A. On a Criterion for the Complete Continuity of the Fréchet Derivative. Funct. Anal. Appl. Vol. 49. No. 4. 2015. Pp. 304–306.
8. Erzakova N.A. Generalization of some M.A. Krasnosel'skii's results. J. Math. Anal. Appl. Vol. 428. 2015. Pp. 1368–1376.
9. Erzakova N.A. On strongly condensing operators. Contemporary methods theory of functions and associated problems: Proceedings of International Conference, Voronezh Winter Mathematical School (January 27 – February 2, 2015). Voronezh: VSU, 2015. Pp. 42–43.
10. Erzakova N.A. On strongly condensing operators on spheres and balls. Modern methods, problems and applications of operator theory and harmonic analysis – V: Proceedings of International Scientific Conference. Section I. Rostov-on-Don: DSTU, 2015. P. 29.
11. Erzakova N.A. On bifurcation points of strongly condensing operators. Scientific Bulletin of MSTUCA. Vol. 220 (10). 2015. Pp. 105–113.
12. Erzakova N.A. On semi-homogeneous maps of degree k . URL: http://de.arxiv.org/ pdf/1508.04215
13. Krasnoselskii M.A., Zabreiko P.P., Pustylnik E.I., Sobolevskii P.E. Integral operators in spaces of summable functions. Noordhoff, Leyden. 1976. 500 p.
14. Erzakova N.A. Measures of Noncompactness in Regular Spaces. Canad. Math. Bull. Vol. 57. 2014. Pp. 780–793.
15. Erzakova N.A. Measure of noncompactness in Lorentz spaces. Scientific Bulletin of MSTUCA. Vol. 207 (09). 2014. Pp. 110–118.
16. Erzakova N.A. Measure of noncompactness in spaces Lp . Scientific Bulletin of MSTUCA. Vol. 195. 2013. Pp. 65–73.
Review
For citations:
Erzakova N.A. ON OPERATORS WITH THE SPHERICAL PROPERTY. Civil Aviation High Technologies. 2016;(224):88-96. (In Russ.)
Views:
414
Email this article 
