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Necessary and sufficient conditions for the existence of variational principles for a given wide class of evolutionary differential-difference operator equation are obtained. The theoretic results are illustrated by two examples.

We consider the equation:

where

Later, we will assume that at every point

The operator

In that case, we also say that the given equation admits the direct variational formulation.

A problem of the construction of the functional

For what follows, we suppose that

where

The functional

Let us consider the following differential-difference operator equation:

Here,

The domain of definition

Under the solution of (

Let us give the following bilinear form:

Our aim is to define the structure of operators

We denote by

If

Taking into account formula (

Conditions (

The operators

If

Let us consider the following functional:

If

This theorem shows the structure of the given kind of differential-difference operator, which admits the solution of the inverse problem of the calculus of variations.

Let us consider the evolutionary differential-difference equation with partial derivatives in the following form:

The domain of definition

We investigate the existence of variational principle for (

We investigate the existence of variational principle for (

For (

Necessary and sufficient conditions of potentiality take the form:

From that, we come to the following:

That is true if and only if

Under the fulfilment of that conditions, the corresponding functional is given by

Let us consider an example when this criterion of potantiality fails.

Consider the equation:

We denote

We define the integrating operator

Then, the operator,

is potential on set (

Indeed, using (

From the condition

This equation is equivalent to (

Let us note that the formula

The work of V. M. Sovchin was partially supported by the Russian Foundation for Basic Research (Projects 09-01-00093 and 09-01-00586).