On the Construction of a Variational Principle for a Certain Class of Differential-Difference Operator Equations

Cover Page

Cite item

Full Text

Abstract

In this paper, we obtain necessary and sufficient conditions for the existence of variational principles for a given first-order differential-difference operator equation with a special form of the linear operator Pλ(t) depending on t and the nonlinear operator Q. Under the corresponding conditions the functional is constructed. These conditions are obtained thanks to the well-known criterion of potentiality. Examples show how the inverse problem of the calculus of variations is constructed for given differentialdifference operators.

About the authors

I. A. Kolesnikova

Peoples’ Friendship University of Russia (RUDN University)

Author for correspondence.
Email: kolesnikova-ia@rudn.ru
Moscow, Russia

References

  1. Беллман Р., Кук К. Дифференциально-разностные уравнения. - М.: Мир, 1967.
  2. Каменский Г. А. Вариационные и краевые задачи с отклоняющимся аргументом// Дифф. уравн. - 1970. - 69, № 8. - С. 1349-1358.
  3. Попов А. М. Условия потенциальности дифференциально-разностных уравнений// Дифф. уравн. - 1998. - 34, № 3. - С. 422-424.
  4. Савчин В. М. Условия потенциальности Гельмгольца для ДУЧП с отклоняющимися аргументами// Тез. докл. XXXII научной конф. ф-та физ.-мат. и естеств. наук. - М.: РУДН, 1994. - С. 25.
  5. E` l’sgol’c L. E`. Qualitative methods in mathematical analysis. - Am. Math. Soc., 1964.
  6. Filippov V. M., Savchin V. M., Shorokhov S. G. Variational principles for nonpotential operators// J. Math. Sci. - 1994. - 68, № 3. - С. 275-398.
  7. Kolesnikova I. A., Popov A. M., Savchin V. M. On variational formulations for functional differential equations// J. Funct. Spaces Appl. - 2007. - 5, № 1. - С. 89-101.
  8. Savchin V. M. An operator approach to Birkhoff’s equation// Вестн. РУДН. Сер. Мат. - 1995. - 2, № 2. - С. 111-123.
  9. Skubachevskii A. L. Elliptic functional differential equations and applications. - Ba¨sel-Boston-Berlin: Birkha¨user, 1997.

Copyright (c) 2021 Contemporary Mathematics. Fundamental Directions

License URL: https://creativecommons.org/licenses/by-nc-nd/4.0/deed.en

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies