Equations in a Hilbert space whose solution sets are invariant with respect to a group isomorphic to a one-parameter group of unitary operators

  • V. Yu. Slyusarchuk Nat. Univ. farms and nature management, Rivne
Keywords: Hilbert space, group of unitary operators

Abstract

We construct classes of equations in a Hilbert space whose sets of solution are invariant with respect to a group isomorphic to a one-parameter group of unitary operators. It is shown that the unbounded solutions of these equations are unstable.
The applications of the obtained results to nonlinear mechanics are presented.

 

Author Biography

V. Yu. Slyusarchuk, Nat. Univ. farms and nature management, Rivne

Василь Юхимович

References

Moren, Kristof. Методы Гильбертова пространства. (Russian) [[Methods of Hilbert space]] Translated from the Polish by V. É. Ljance. Edited by Ju. M. Berezanskiĭ and E. A. Gorin Izdat. ``Mir'', Moscow 1965 570 pp. MR0200693

Pontryagin, L. S. Непрерывные группы. (Russian) [[Continuous groups]] Fourth edition. ``Nauka'', Moscow, 1984. 520 pp. MR0767087

A. N. Godunov, О теореме Пеано в банаховых пространствах (Russian) [[The Peano theorem in Banach spaces]], Funkczion. analiz i ego pril., 9, vy`p. 1, 59 – 60 (1975). MR364797

Daletskiĭ, Yu. L.; Kreĭn, M. G. Устой чивость решений дифференциальных уравнений в банаховом пространстве. (Russian) [[Stability of solutions of differential equations in Banach space]] Nonlinear Analysis and its Applications Series. Izdat. ``Nauka'', Moscow, 1970. 534 pp. MR0352638

Myshkis, A. D. Лекции по высшей математике. (Russian) [[Lectures on higher mathematics]] Fourth, stereotyped edition. Izdat. ``Nauka'', Moscow, 1973. 640 pp. MR0349292

V. Yu. Slyusarchuk, Математична модель Сонячної системи з урахуванням швидкостi гравiтацiї (Russian) [[ Matematichna model` Sonyachnoyi sistemi z urakhuvannyam shvidkosti gravitacziyi]], Nelinijni kolivannya, 21, № 2, 238 – 261 (2018).

V. Yu. Slyusarchuk, Некеплеровiсть та нестiйкiсть руху двох тiл, спричиненi скiнченнiстю швидкостi гравiтацiї (Russian) [[ Nekeplerovist` ta nestijkist` rukhu dvokh til, sprichineni skinchennistyu shvidkosti gravitacziyi]], Nelinijni kolivannya, 21, no 3, 397 – 419 (2018).

V. I. Arnol`d, V. V. Kozlov, A. N. Nejshtadt, Математические аспекты классической и небесной механики (Russian) [[Mathematical aspects of classical and celestial mechanics]], URSS, Moskva (2002). MR833508

S. M. Kopejkin, E`. Fomalont, Фундаментальный предел скорости гравитации и его измерение (Russian) [[Fundamental`ny`j predel skorosti gravitaczii i ego izmerenie]], Zemlya i Vselennaya, 3 (2004).

Published
15.01.2020
How to Cite
SlyusarchukV. Y. “Equations in a Hilbert Space Whose Solution Sets Are Invariant With Respect to a Group Isomorphic to a One-Parameter Group of Unitary Operators”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 1, Jan. 2020, pp. 86-99, http://umj.imath.kiev.ua/index.php/umj/article/view/687.
Section
Research articles