On the Existence of Variational Principles for a Class of the Evolutionary Differential-Difference Equations

where D N is a domain of the definition of the operatorN : D N ⊆ U → V , and U,V are normed linear spaces over the field of real numbers R. Later, we will assume that at every point u ∈ D N , there exists the Gâteaux derivative N ′ u ofN defined by the formula d/d N u h | 0 δN u, h N ′ uh. The operator N : D N → V is said to be potential 1 on the set D N relative to a given bilinear form Φ ·, · : V ×U → R, if there exists a functional FN : D FN D N → R such that


Introduction.
We consider the equation: where D N is a domain of the definition of the operator N : D N ⊆ U → V , and U, V are normed linear spaces over the field of real numbers R.
Later, we will assume that at every point u ∈ D N , there exists the Gâteaux derivative N u of N defined by the formula d/d N u h | 0 δN u, h N u h.The operator N : D N → V is said to be potential 1 on the set D N relative to a given bilinear form Φ •, • : V × U → R, if there exists a functional In that case, we also say that the given equation admits the direct variational formulation.
A problem of the construction of the functional F N upon the given operator N is known as the classical inverse problem of the calculus of variations 1 .Note that practically no one has been solving inverse problem of the calculus of variations for partial differential-difference operators until recently 2-4 .Let us also note that for a wide classes of partial differential equations, there has been developped the problem of recongnition of variationality upon the structure of corresponding operators 5, 6 .There is a theoretical and practical interst in the extention of these results on partial differential-difference equations 7-9 .For what follows, we suppose that D N is a convex set, and we also need the following potentiality criterion 1 : Under this condition, the potential F N is given by where u 0 is a fixed element of D(N).
The functional F N is called the potential of the operator N, and in turn the operator N is called the gradient of the functional F N .

Statement of the Problem.
Let us consider the following differential-difference operator equation: Here, P λ : The domain of definition D N is given by the equality: where ϕ i , i 1, 2 are given functions.Under the solution of 2.1 , we mean a function u ∈ D N satisfying the identity: Let us give the following bilinear form: where the bilinear mapping Φ 1 ≡ •, • satisfies the following conditions: 2.5 Our aim is to define the structure of operators P λ λ −1, 0, 1 and Q under which 2.1 allows the solution of the inverse problem of the calculus of variations relative to the bilinear form 2.4 such that D t d/dt is the antisymmetric operator on D N u , that is,

Conditions of Potentiality and the Structure of 2.1 in the Case of Its Variationality.
We denote by K * the operator adjoint to K.

3.1
Proof.Taking into account formula 2.1 , we get The criterion of potentiality takes the following form:

3.4
Bearing into account the condition D * t −D t on the set D N u , from 3.4 , we get or

3.6
Thus, condition 3.4 can be reduced to the following form: This equality is fulfilled identically if and only if for all u ∈ D N .Thus, it is necessary and sufficient that conditions 3.1 hold.
Theorem 3.2.Conditions 3.1 are held if and only if 2.1 has the following form:

3.9
The operators R λ and B depend on P λ t and Q t, u t λτ .Let us consider the following functional:

3.10
It is easy to check that δF N u, h

3.11
Then functional 3.10 is a potential of evolutionary operator 2.1 .

3.12
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.

Examples
Example 4.1.Let us consider the evolutionary differential-difference equation with partial derivatives in the following form: x,t Q .Ω is a bounded domain in R n with piecewise smooth boundary ∂Ω.The domain of definition D N 1 is given by the equality:

4.3
Necessary and sufficient conditions of potentiality take the form:

4.4
From that, we come to the following: That is true if and only if

4.6
Under the fulfilment of that conditions, the corresponding functional is given by

Theorem 3 . 1 .
If D * t −D t on the set D N u , then for the existence of the direct variational formulation for the operator 2.1 on the set D N relative to 2.4 , it is necessary and sufficient that the following conditions hold on the set D N u :

Proof. If D * t −
D t on the set D N 1u and conditions 3.1 are held, then according to Theorem 3.1, operator 2.1 is potential on the set D N relative to a given bilinear form 2.4 .
Let us consider an example when this criterion of potentiality fails.Let us note that this equation is a Korteweg-de Vries' equation τ 0. This equation is equivalent to 4.8 .Let us note that the formula I u ∞ −∞ 1/2 u x t, x − τ u x t, x τ u 2 t, x τ u t, x − τ dx defines the first integral I u const of 4.8 .
a 1 x u t x, t − τ u x, t − a 1 x u t x, t τ u x, t b ij 1 t − τ u x i x, t − τ