Potentiality conditions for differential-difference equations

From the text (translated from the Russian): "In solving problems by variational methods, it is necessary to construct functionals whose critical points coincide with the solutions of the original equations. The investigation of the problem of constructing the desired functionals begins with verifying the potentiality conditions for the corresponding operators. There are efficient methods [V. M. Filippov, V. M. Savchin and S. G. Shorokhov, in {it Current problems in mathematics. Newest results. Vol. 40} (Russian), Ross. Akad. Nauk, Vseross. Inst. Nauchn. i Tekhn. Inform., Moscow, 1992; [msn] MR1314224 (95m:58048) [/msn]] that allow us to verify the potentiality of the corresponding operators for ordinary differential equations. In this paper we obtain potentiality conditions for second-order differential-difference equations of neutral type. We consider a linear equation. Under the assumption that the potentiality conditions are satisfied for it, we construct the potential of the corresponding operator. par "Since differential equations with deviating argument can be integrated in closed form only in exceptional cases, the application of variational methods for finding an approximate solution of the equations considered becomes important. In addition to obtaining approximate analytic solutions of the equations, variational principles have other important consequences, which have been considered in detail by Filippov, Savchin and Shorokhov [op. cit.]."

Popov A.M.
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Sabinin L.V., Goldberg Vladislav
Algebras, Groups and Geometries. Vol. 15. 1998. P. 127-153