On variational formulations for functional differential equations

Necessary and sufficient conditions for the existence of integral variational principles for boundary value problems for given ordinary and partial functional differential equations are obtained. Examples are given illustrating the results.


Introduction
By the problem of construction of integral variational principles for a system of equations of some given model we mean the construction of functionals for which the set of critical (extremal or stationary) points coincides with the set of the solutions of the given system.
The search of a functional F , that admits the given equations as its Euler-Lagrange equations is known as the classical inverse problem of the calculus of variations.Since the end of the XIXth century there has been a great deal of activity in this field (see Helmholtz [4], Volterra [13], Santilli [9], Tonti [12], Filippov, Savchin and Shorokhov [2] and refs.therein).
In the course of a long time advantages of the variational principles have been mainly used for ordinary and partial differential equations with the so-called potential operators [2].
There is a practical need to develop different approaches to the construction of integral variational principles for equations with deviating arguments.
The main aim of this paper is to find out necessary and sufficient conditions under which the given ordinary and partial functional differential equations with appropriate boundary conditions admit variational formulation.
The reader should keep in mind that most of the results can be formulated under weaker smoothness conditions.

Certain auxiliary notations and definitions
Let U, V be normed linear spaces over the field of real numbers R, and O U , O V be their zero elements.
Take any operator N : if it exists, is called the Gâteaux differential of N at the point u .If it is linear relative to h, then the operator δN (u, •) : U → V is called the Gâteaux derivative of N at u and will be denoted by N u .Its domain of definition D(N u ) consists of the elements h ∈ U such that (u + h) ∈ D(N ) for all sufficiently small.Let us consider the equation with the Gâteaux differentiable operator N , and a convex set D(N ).
In order to consider the existence of its variational formulation we need a non-degenerate bilinear form Definition.The operator N : D(N ) → V is said to be potential on the set D(N ) relative to a given bilinear form 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 .As it is known Volterra [13] the condition for potentiality of the operator N takes the form Under this condition the potential F N is given by where u 0 is a fixed element of D(N).

The potentiality conditions for ordinary functional differential equations
Let us consider the equation where ω 0 (t) ≡ t, γ 0 (t) ≡ t; ω i (t) and γ i (t) are mutually inverse functions; We assume that β 1 < α 2 .The functions ϕ i (i = 1, 2), belonging to the class of functions C 2 , are given on the sets Let us denote by U the set of all absolutely continuous functions on [α 1 , β 2 ] for which the first and the second derivatives exist and are almost everywhere bounded on [α 1 , β 2 ].
The domain of definition of the given operator N is defined by the equality where ϕ j (t) are certain given functions, ϕ j (t) ∈ C 2 .
We define the bilinear form by the relation Theorem 1.In order that the operator N (4) be potential on D(N ) (5) relative to the bilinear form (6), it is necessary and sufficient that ∀t ∈ (t 1 , t 2 ) where Proof.We shall verify that under the conditions ( 7)-( 9) the Volterra's criterion of potentiality (2), where (10) will be fulfilled.
Using the Gâteaux derivative of the operator (4), we get Let us perform the following transformations in the right-hand side of the equality ( 11): (a) in the first group of terms we use the change of variable of the form s = ω i (γ k (t)); (b) in the second group of terms we use the change of the variable of integration, then the formula of integration by parts; (c) in the third group of terms we use the change of the variable of integration, then twice the formula of integration by parts.
Bearing in mind the equality (10) for h and g , and going back to the initial notation of the variable of integration, we get

Now we get
Comparing the expressions ( 11) and ( 13), and taking into account the above transformations, we conclude that for the fulfillment of the potentiality criterion it is necessary and sufficient that the relations ( 7)-( 9) hold.
The proof of the theorem is completed.
In accordance with the accepted notations (4) here we have ω Let us note that if a 6 ≡ 0 then the equation ( 14) is of neutral type, and if a 6 = 0 , then it is of advanced type (see Kamenskii [6]).It is easy to check that the given operator (14) satisfies the conditions of potentiality ( 7)- (9).Applying the formula (3) for the potential F N of the operator N we get Integrating by parts, we diminish the second order of derivatives in the integrand, and finally we come to the following potential

On variational formulation for partial differential difference equations
Let us consider the following equation where u is an unknown function, f is a given smooth function, The domain of definition D(N ) is given by the equality Here ϕ 10 , ϕ 20 , ψ ν are given sufficiently smooth functions, ϕ jk = The numbers l 0 and s 0 depend on l and s, respectively.If l, s are even, then l 0 = l 2 − 1, s 0 = s 2 − 1.For odd l, s we set l 0 = l+1 2 − 1, s 0 = s+1 2 − 1.We shall use the notations and Assume that there is given the bilinear form Theorem 2. For the potentiality of the operator (15) on the given set (16) with respect to the bilinear form (17), it is necessary and sufficient that the following conditions hold where the notation (. ..)| t→t+λτ means that in the expression within the parentheses one has to change t by t + λτ.
Proof.Taking into account formulas (15),( 17), we get By integrating by parts and taking into account that the functions g, h ∈ D(N u ) are subject to the conditions we obtain from ( 19) where D k t is the total derivative of order k with respect to the variable t; D α is the total derivative corresponding to the multiindex α.
From here by using the Leibnitz formula there follows that By using the change t − λτ = t , from (20) we get Denoting t by t, we obtain from ( 21) Taking into account that the equality (22) can be written as Bearing in mind the equality (19) we obtain β (x, t − λτ ) h(x, t)dxdt.
By using ( 23), (24) we get Since g and h are arbitrary functions from D(N u ) then N u h, g − N u g, h = 0 ∀u ∈ D(N ), ∀h, g ∈ D(N u ) if and only if conditions (18) hold.
Example 2. Let us consider the equation Here u = u(x, t) is an unknown function, x,t (Q), Ω is a bounded domain in R m with a piecewise smooth boundary ∂Ω; Ω is the closure of Ω in R m , and repeated indices of factors situated at different levels denote summation, i, j = 1, m.
Let us define the domain of definition of N 1 by setting ( 26) First, let us study the existence of the classical variational formulation of problem (25), (26).For that aim, we introduce the notation V = C(Q τ ) and use the bilinear form (17).
The conditions of potentiality (18) for the given case take the form If this relations are hold the given problem (25), (26) allows the classical variational formulation and the corresponding functional has the form where a, b are constants,u is an unknown function, Let us definite the domain of definition of N 2 by setting (28) where It is easy to check that operator (27) is not potential on the domain (28) with respect to the classical bilinear form (17).In that connection we search for the function M = M (x, t, u, u t ) = 0 and the functional Using the theorem 2 it is easy to find that M = e u(x,t) .The operator N (u) = e u(x,t) N 2 (u) is potential.
Remark 1. Differential equations with several variable deviating arguments occur for example in the technical cybernetics (see [11]).
Remark 3. Different approaches to the definition of solutions of different types of functional differential equations are discussed by Hale [3].
Remark 4. The study of the inverse problems of the calculus of variations for equations with deviating arguments has been initiated by Savchin [10].