The problem of existence of variational principles for wide classes of generally nonlinear differential-difference equations with nonpotential operators is investigated.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price excludes VAT (USA)
Tax calculation will be finalised during checkout.
V. M. Filippov, V. M. Savchin, and S. G. Shorokhov, Variational Principles for Nonpotential Operators (VINITI, Moscow, 1992), Itogi Nauki Tekh., Ser.: Sovr. Probl. Mat., Noveishie Dostizh. 40; Engl. transl.: J. Math. Sci. 68 (3), 275–398 (1994).
V. M. Savchin, “An operator approach to Birkhoff’s equations,” Vestn. Ross. Univ. Druzhby Narodov, Mat. 2(2), 111–123 (1995).
V. M. Savchin and S. A. Budochkina, “On the structure of a variational equation of evolution type with the second t-derivative,” Diff. Uravn. 39(1), 118–124 (2003) [Diff. Eqns. 39, 127–134 (2003)].
V. M. Savchin and S. A. Budochkina, “On the existence of a variational principle for an operator equation with second derivative with respect to ‘time’,” Mat. Zametki 80(1), 87–94 (2006) [Math. Notes 80, 83–90 (2006)].
S. A. Budotchkina and V. M. Savchin, “On indirect variational formulations for operator equations,” J. Funct. Spaces Appl. 5(3), 231–242 (2007).
A. D. Myshkis and L. E. El’sgol’ts, “Some results and problems in the theory of differential equations,” Usp. Mat. Nauk 22(2), 21–57 (1967) [Russ. Math. Surv. 22 (2), 19–57 (1967)].
A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications (Birkhäuser, Basel, 1997).
V. M. Filippov and V. M. Savchin, “On the inverse problem of the calculus of variations (IPCV) for functional partial differential equations,” in Function Spaces, Differential Operators, and Problems of Mathematical Education: Abstr. Talks 2nd Int. Conf. Dedicated to the 80th Birthday of L.D. Kudryavtsev (Fizmatlit, Moscow, 2003), pp. 235–236.
A. M. Popov, “Potentiality conditions for differential-difference equations,” Diff. Uravn. 34(3), 422–424 (1998) [Diff. Eqns. 34, 423–426 (1998)].
I. A. Kolesnikova, A. M. Popov, and V. M. Savchin, “On variational formulations for functional differential equations,” J. Funct. Spaces Appl. 5(1), 89–101 (2007).
I. A. Kolesnikova and V. M. Savchin, “On the existence of variational principles for a class of the evolutionary differential-difference equations,” J. Funct. Spaces Appl., 780382 (2012).
Original Russian Text © V.M. Filippov, V.M. Savchin, S.A. Budochkina, 2013, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2013, Vol. 283, pp. 25–39.
About this article
Cite this article
Filippov, V.M., Savchin, V.M. & Budochkina, S.A. On the existence of variational principles for differential-difference evolution equations. Proc. Steklov Inst. Math. 283, 20–34 (2013). https://doi.org/10.1134/S0081543813080038