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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 780382, 9 pages
Research Article

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

University of Russia, Miklukho-Maklaya Street 6, Moscow 117198, Russia

Received 10 March 2011; Accepted 18 October 2011

Academic Editor: V. Stepanov

Copyright © 2012 I. A. Kolesnikova and V. M. Savchin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.

1. Introduction

We consider the equation:𝑁(𝑢)=0,𝑢𝐷(𝑁),(1.1)

where 𝐷(𝑁) is a domain of the definition of the operator 𝑁𝐷(𝑁)𝑈𝑉, and 𝑈,𝑉 are normed linear spaces over the field of real numbers .

Later, we will assume that at every point 𝑢𝐷(𝑁), there exists the Gâteaux derivative 𝑁𝑢 of 𝑁 defined by the formula (𝑑/𝑑𝜖)𝑁(𝑢+𝜖)|𝜖=0=𝛿𝑁(𝑢,)=𝑁𝑢.

The operator 𝑁𝐷(𝑁)𝑉 is said to be potential [1] on the set 𝐷(𝑁) relative to a given bilinear form Φ(,)𝑉×𝑈𝑅, if there exists a functional 𝐹𝑁𝐷(𝐹𝑁)=𝐷(𝑁)𝑅 such that𝛿𝐹𝑁[]𝑁𝑢,=Φ(𝑁(𝑢),),𝑢𝐷(𝑁),𝐷𝑢.(1.2)

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

A problem of the construction of the functional 𝐹𝑁 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 [24]. 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 [79].

For what follows, we suppose that 𝐷(𝑁) is a convex set, and we also need the following potentiality criterion [1]: Φ𝑁𝑢𝑁,𝑔=Φ𝑢𝑁𝑔,,𝑢𝐷(𝑁),,𝑔𝐷𝑢.(1.3) Under this condition, the potential 𝐹𝑁 is given by𝐹𝑁[𝑢]=10Φ𝑁𝑢0+𝜆𝑢𝑢0,𝑢𝑢0𝑑𝜆+const,(1.4)

where 𝑢0 is a fixed element of D(N).

The functional 𝐹𝑁 is called the potential of the operator 𝑁, and in turn the operator 𝑁 is called the gradient of the functional 𝐹𝑁.

2. Statement of the Problem

Let us consider the following differential-difference operator equation: 𝑁(𝑢)1𝜆=1𝑃𝜆(𝑡)𝑢𝑡𝑡(𝑡+𝜆𝜏)+𝑄(𝑡,𝑢(𝑡+𝜆𝜏))=0,𝑢𝐷(𝑁),𝑡0,𝑡1.(2.1)

Here, 𝑃𝜆𝑈1𝑉1(𝜆=1,0,1) are linear operators depending on 𝑡;𝑄[𝑡0𝜏,𝑡1+𝜏]×𝑈13𝑉1 is in general a nonlinear operator;𝑁𝐷(𝑁)𝑈𝑉;𝑈=𝐶1([𝑡0𝜏,𝑡1+𝜏];𝑈1);𝑉=𝐶([𝑡0𝜏,𝑡1+𝜏];𝑉1), where 𝑈1,𝑉1 are real normed linear spaces, 𝑈1𝑉1.

The domain of definition 𝐷(𝑁) is given by the equality: 𝐷(𝑁)=𝑢𝑈𝑢(𝑡)=𝜑1𝑡(𝑡),𝑡0𝜏,𝑡0,𝑢(𝑡)=𝜑2𝑡(𝑡),𝑡1,𝑡1+𝜏,(2.2) where 𝜑𝑖,(𝑖=1,2) are given functions.

Under the solution of (2.1), we mean a function 𝑢𝐷(𝑁) satisfying the identity: 𝑁(𝑢)1𝜆=1𝑃𝜆(𝑡)𝑢𝑡𝑡(𝑡+𝜆𝜏)+𝑄(𝑡,𝑢(𝑡+𝜆𝜏))=0,𝑡0,𝑡1.(2.3)

Let us give the following bilinear form: Φ(,)𝑡1𝑡0,𝑑𝑡𝑉×𝑈,(2.4) where the bilinear mapping Φ1, satisfies the following conditions: 𝑣1,𝑣2=𝑣2,𝑣1,𝑣1,𝑣2𝑉1,𝐷𝑡𝑣,𝑔=𝐷𝑡𝑣,𝑔+𝑣,𝐷𝑡𝑔,𝑣,𝑔𝑉.(2.5)

Our aim is to define the structure of operators 𝑃𝜆(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 𝐷𝑡=𝑑/𝑑𝑡 is the antisymmetric operator on 𝐷(𝑁𝑢), that is, Φ𝐷𝑡1,2𝐷=Φ𝑡2,1,1,2𝑁𝐷𝑢.(2.6)

3. Conditions of Potentiality and the Structure of (2.1) in the Case of Its Variationality

We denote by 𝐾 the operator adjoint to 𝐾.

Theorem 3.1. If 𝐷𝑡=𝐷𝑡 on the set 𝐷(𝑁𝑢), 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(𝑁𝑢): 𝑃𝜆+𝑃𝜆|𝑡𝑡𝜆𝜏=0,𝜕𝑃𝜆||||𝜕𝑡𝑡𝑡𝜆𝜏+𝑄𝑢(𝑡𝜆𝜏)𝑄𝑢(𝑡+𝜆𝜏)||𝑡𝑡𝜆𝜏𝑡=0,𝜆=1,0,1,𝑢𝐷(𝑁),𝑡0,𝑡1.(3.1)

Proof. Taking into account formula (2.1), we get 𝑁𝑢=1𝜆=1𝑃𝜆𝑡(𝑡+𝜆𝜏)+1𝜆=1𝑄𝑢(𝑡+𝜆𝜏)(𝑡+𝜆𝜏).(3.2) The criterion of potentiality takes the following form: 𝑡1𝑡01𝜆=1𝑃𝜆𝑡(𝑡+𝜆𝜏)+𝑄𝑢(𝑡+𝜆𝜏)=(𝑡+𝜆𝜏),𝑔(𝑡)𝑑𝑡𝑡1𝑡01𝜆=1𝑃𝜆𝑔𝑡(𝑡+𝜆𝜏)+𝑄𝑢(𝑡+𝜆𝜏)𝑔(𝑡+𝜆𝜏),(𝑡)𝑑𝑡,(3.3) or 𝑡1𝑡01𝜆=1𝑃𝜆𝑡(𝑡+𝜆𝜏)+𝑄𝑢(𝑡+𝜆𝜏)(𝑡+𝜆𝜏),𝑔(𝑡)1𝜆=1𝑃𝜆𝑔𝑡(𝑡+𝜆𝜏)+𝑄𝑢(𝑡+𝜆𝜏)𝑁𝑔(𝑡+𝜆𝜏),(𝑡)𝑑𝑡=0,𝑢𝐷(𝑁),𝑔,𝐷𝑢.(3.4) Bearing into account the condition 𝐷𝑡=𝐷𝑡 on the set 𝐷(𝑁𝑢), from (3.4), we get 𝑡1𝑡01𝜆=1𝑃𝜆𝐷𝑡+𝑄𝑢(𝑡+𝜆𝜏)(𝑡+𝜆𝜏),𝑔(𝑡)1𝜆=1𝐷𝑡𝑃𝜆(𝑡)+𝑄𝑢(𝑡+𝜆𝜏)(𝑡),𝑔(𝑡+𝜆𝜏)𝑑𝑡=0,(3.5) or 𝑡1𝑡01𝜆=1𝑃𝜆𝐷𝑡+𝑄𝑢(𝑡𝜆𝜏)(𝑡𝜆𝜏),𝑔(𝑡)1𝜆=1𝜕𝑃𝜆𝜕𝑡𝑃𝜆𝐷𝑡+𝑄𝑢||||𝑡𝑡𝜆𝜏(𝑡𝜆𝜏),𝑔(𝑡)𝑑𝑡=0.(3.6) Thus, condition (3.4) can be reduced to the following form: 𝑡1𝑡01𝜆=1𝑃𝜆𝐷𝑡+𝑄𝑢(𝑡𝜆𝜏)(𝑡𝜆𝜏)𝜕𝑃𝜆𝜕𝑡𝑃𝜆𝐷𝑡+𝑄𝑢||||𝑡𝑡𝜆𝜏𝑁(𝑡𝜆𝜏),𝑔(𝑡)𝑑𝑡=0,𝑢𝐷(𝑁),𝑔,𝐷𝑢.(3.7) This equality is fulfilled identically if and only if 1𝜆=1𝑃𝜆+𝑃𝜆|𝑡𝑡𝜆𝜏𝐷𝑡+𝜕𝑃𝜆||||𝜕𝑡𝑡𝑡𝜆𝜏+𝑄𝑢(𝑡𝜆𝜏)𝑄𝑢(𝑡+𝜆𝜏)|𝑡𝑡𝜆𝜏(𝑡𝜆𝜏)=0,(3.8) for all 𝑢𝐷(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: 𝑁1(𝑢)1𝜆=1𝑅𝜆|𝑡𝑡𝜆𝜏𝑅𝜆𝑢𝑡(𝑡𝜆𝜏)+gradΦ𝐵[𝑢]1𝜆=1𝜕𝑅𝜆𝑁𝜕𝑡𝑢(𝑡+𝜆𝜏)=0,𝑢𝐷1𝑡,𝑡0,𝑡1.(3.9)
The operators 𝑅𝜆 and 𝐵 depend on 𝑃𝜆(𝑡) and 𝑄(𝑡,𝑢(𝑡+𝜆𝜏)).

Proof. If 𝐷𝑡=𝐷𝑡 on the set 𝐷(𝑁1𝑢) 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 the following functional:𝐹𝑁[𝑢]=𝑡1𝑡01𝜆=1𝑅𝜆(𝑡)𝑢(𝑡+𝜆𝜏),𝑢𝑡[𝑢](𝑡)+𝐵𝑑𝑡+𝐹𝑁𝑢0.(3.10) It is easy to check that 𝛿𝐹𝑁[]=𝑢,𝑡1𝑡01𝜆=1𝑅𝜆(𝑡+𝜆𝜏),𝑢𝑡+(𝑡)1𝜆=1𝑅𝜆𝑢(𝑡+𝜆𝜏),𝑡(𝑡)+gradΦ𝐵[𝑢]=,(𝑡)𝑑𝑡𝑡1𝑡01𝜆=1𝑅𝜆|𝑡𝑡𝜆𝜏𝑢𝑡(𝑡𝜆𝜏),(𝑡)1𝜆=1𝐷𝑡𝑅𝜆𝑢(𝑡+𝜆𝜏),(𝑡)+gradΦ𝐵[𝑢]=,(𝑡)𝑑𝑡𝑡1𝑡01𝜆=1𝑅𝜆|𝑡𝑡𝜆𝜏𝑢𝑡(𝑡𝜆𝜏),(𝑡)1𝜆=1𝜕𝑅𝜆𝜕𝑡𝑢(𝑡+𝜆𝜏),(𝑡)1𝜆=1𝑅𝜆𝑢𝑡(𝑡𝜆𝜏),(𝑡)+gradΦ𝐵[𝑢]=,𝑑𝑡𝑡1𝑡01𝜆=1𝑅𝜆|𝑡𝑡𝜆𝜏𝑅𝜆𝑢𝑡(+𝑡𝜆𝜏)gradΦ𝐵[𝑢]1𝜆=1𝜕𝑅𝜆=𝜕𝑡𝑢(𝑡+𝜆𝜏),(𝑡)𝑑𝑡𝑡1𝑡0𝑁𝑁(𝑢),𝑑𝑡,𝑢𝐷(𝑁),𝐷𝑢.(3.11) Then functional (3.10) is a potential of evolutionary operator (2.1).
If 𝐷𝑡=𝐷𝑡 on 𝐷(𝑁𝑢), then𝑃𝜆(𝑡)=𝑅𝜆|𝑡𝑡𝜆𝜏𝑅𝜆,𝑄(𝑡,𝑢(𝑡+𝜆𝜏))=1𝜆=1𝜕𝑅𝜆𝜕𝑡𝑢(𝑡+𝜆𝜏)+gradΦ𝐵[𝑢].(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.

4. Examples

Example 4.1. Let us consider the evolutionary differential-difference equation with partial derivatives in the following form: 𝑁1(𝑢)1𝜆=1𝑎𝜆(𝑥,𝑡)𝜕𝑢𝜕𝑡(𝑥,𝑡+𝜆𝜏)𝑏𝜆𝑖𝑗𝜕(𝑥,𝑡)2𝑢𝜕𝑥𝑖𝜕𝑥𝑗(𝑥,𝑡+𝜆𝜏)=0,(4.1) where (𝑥,𝑡)𝑄=Ω×(𝑡0,𝑡1),𝑡1𝑡0>2𝜏,𝑖,𝑗=1,𝑛,𝑎𝜆(𝑥,𝑡)𝐶0,1𝑥,𝑡(𝑄),𝑏𝜆𝑖𝑗(𝑥,𝑡)𝐶2,0𝑥,𝑡(𝑄).
Ω is a bounded domain in 𝑛 with piecewise smooth boundary 𝜕Ω.
The domain of definition 𝐷(𝑁1) is given by the equality: 𝐷𝑁1=𝑢𝑈=𝐶2,1𝑥,𝑡𝑡Ω×0𝜏,𝑡1+𝜏𝑢(𝑥,𝑡)=𝜑1(𝑥,𝑡),(𝑥,𝑡)𝐸1=𝑡Ω×0𝜏,𝑡0,𝑢(𝑥,𝑡)=𝜑2(𝑥,𝑡),(𝑥,𝑡)𝐸2=𝑡Ω×1,𝑡1,𝜕+𝜏𝜇𝑢𝜕𝑛𝜇𝑥||||𝜕Γ𝜏=𝜓𝜇,(𝑥,𝑡),𝜇=0,1(4.2) where Γ𝜏=𝜕Ω×[𝑡0𝜏,𝑡1+𝜏], 𝜑𝑖𝐶(𝐸𝑖),(𝑖=0,1),𝜓𝜇𝐶(Γ𝜏),(𝜇=0,1) are given functions.
We investigate the existence of variational principle for (4.1) relative to a given bilinear form (2.4).
For (4.1), we get 𝑁1𝑢=1𝜆=1𝑎𝜆(𝑥,𝑡)𝑡(𝑥,𝑡+𝜆𝜏)𝑏𝜆𝑖𝑗(𝑥,𝑡)𝑥𝑖𝑥𝑗(𝑥,𝑡+𝜆𝜏).(4.3)
Necessary and sufficient conditions of potentiality take the form: Φ𝑁1𝑢𝑁,𝑔Φ1𝑢𝑔,𝑡1𝑡0Ω1𝜆=1𝐷𝑡𝑎𝜆|||||(𝑥,𝑡)𝑔(𝑥,𝑡)𝑡𝑡𝜆𝜏+1𝜆=1𝐷𝑥𝑖𝑥𝑗𝑏𝜆𝑖𝑗(|||||𝑥,𝑡)𝑔(𝑥,𝑡)𝑡𝑡𝜆𝜏(𝑥,𝑡)𝑑𝑥𝑑𝑡𝑡1𝑡0Ω1𝜆=1𝑎𝜆(𝑥,𝑡)𝑔𝑡(𝑥,𝑡+𝜆𝜏)𝑏𝜆𝑖𝑗(𝑥,𝑡)𝑔𝑥𝑖𝑥𝑗𝑁(𝑥,𝑡+𝜆𝜏)(𝑥,𝑡)𝑑𝑥𝑑𝑡=0,𝑢𝐷1𝑁,𝑔,𝐷1𝑢.(4.4)
From that, we come to the following: 1𝜆=1𝐷𝑡𝑎𝜆𝑔(𝑥,𝑡)𝐷𝑥𝑖𝑥𝑗𝑏𝜆𝑖𝑗|||𝑔(𝑥,𝑡)𝑡𝑡𝜆𝜏=1𝜆=1𝑎𝜆𝑔𝑡(𝑥,𝑡+𝜆𝜏)𝑏𝜆𝑖𝑗𝑔𝑥𝑖𝑥𝑗,𝑁(𝑥,𝑡+𝜆𝜏)𝑔𝐷1𝑢.(4.5)
That is true if and only if 𝑎𝜆(𝑥)=𝑎𝜆(𝑥),𝜆=1,1,𝑎0(𝑥)=0,𝑏𝜆𝑖𝑗||(𝑡)𝑡𝑡𝜆𝜏=𝑏𝑖𝑗𝜆(𝑡),𝜆=1,0,1,𝑖,𝑗=1,𝑛,(𝑥,𝑡)𝑄.(4.6)
Under the fulfilment of that conditions, the corresponding functional is given by 𝐹𝑁1[𝑢]=12𝑡1𝑡0Ω𝑎1(𝑥)𝑢𝑡(𝑥,𝑡𝜏)𝑢(𝑥,𝑡)𝑎1(𝑥)𝑢𝑡(𝑥,𝑡+𝜏)𝑢(𝑥,𝑡)+𝑏1𝑖𝑗(𝑡𝜏)𝑢𝑥𝑖(𝑥,𝑡𝜏)𝑢𝑥𝑗(𝑥,𝑡)+𝑏0𝑖𝑗(𝑡)𝑢𝑥𝑖(𝑥,𝑡)𝑢𝑥𝑗(𝑥,𝑡)+𝑏𝑖𝑗1(𝑡+𝜏)𝑢𝑥𝑖(𝑥,𝑡+𝜏)𝑢𝑥𝑗(𝑥,𝑡)𝑑𝑥𝑑𝑡+const.(4.7)
Let us consider an example when this criterion of potentiality fails.

Example 4.2. Consider the equation: 𝑁2(𝑢)=𝑢𝑡(𝑡,𝑥)2𝑢(𝑡,𝑥+2𝜏)𝑢𝑥(𝑡,𝑥+2𝜏)2𝑢𝑥(𝑡,𝑥)𝑢(𝑡,𝑥2𝜏)2𝑢(𝑡,𝑥)𝑢𝑥+1(𝑡,𝑥2𝜏)2𝑢𝑥𝑥𝑥1(𝑡,𝑥+2𝜏)+2𝑢𝑥𝑥𝑥(𝑡,𝑥2𝜏)=0,(𝑡,𝑥)𝑄𝑇=(0,𝑇)×(,+).(4.8) Let us note that this equation is a Korteweg-de Vries’ equation 𝜏=0.
We denote 𝐷𝑁2=𝑢𝑈𝐶1,3𝑡,𝑥((0,𝑇)×(,+))𝑢|𝑡=0=𝑢0(𝑥),𝑢|𝑡=𝑇=𝑢1(𝑥),lim|𝑥|+𝐷𝑛𝑥𝑢=0𝑛=.0,3(4.9) It is easy to be convinced that operator (4.8) is not potential on set (4.9) relative to the bilinear form: Φ(𝑣,𝑔)=𝑇0+𝑣(𝑡,𝑥)𝑔(𝑡,𝑥)𝑑𝑥𝑑𝑡,𝑣𝑉,𝑔𝑈.(4.10) Here, 𝑉={𝑣𝐶0,3𝑡,𝑥((0,𝑇)×(,+))lim|𝑥|+𝐷𝑛𝑥𝑣=0,(𝑛=0,3)}.
We define the integrating operator 𝑀 as 𝑀𝑣(𝑡,𝑥)=𝑥𝑣(𝑡,𝑦)𝑑𝑦.
Then, the operator, 𝑁(𝑢)𝑀𝑁2(𝑢)=𝑥𝑢𝑡(𝑡,𝑦)𝑑𝑦𝑢2+1(𝑡,𝑥+2𝜏)2𝑢(𝑡,𝑥)𝑢(𝑡,𝑥2𝜏)2𝑢𝑥𝑥1(𝑡,𝑥+2𝜏)+2𝑢𝑥𝑥(𝑡,𝑥2𝜏)𝑑𝑦(4.11)
is potential on set (4.9) relative to bilinear form (4.10). The corresponding functional 𝐹𝑁2[𝑢] has the form: 𝐹𝑁2[𝑢]=12𝑇0+𝑢(𝑡,𝑥)𝑥𝑢𝑡(𝑡,𝑦)𝑑𝑦𝑢𝑥(𝑡,𝑥𝜏)𝑢𝑥(𝑡,𝑥+𝜏)2𝑢2(𝑡,𝑥+𝜏)𝑢(𝑡,𝑥𝜏)𝑑𝑦𝑑𝑥𝑑𝑡.(4.12)
Indeed, using (4.12), we find that 𝛿𝐹𝑁2[]=𝑢,𝑇0+𝑥𝑢𝑡1(𝑡,𝑦)𝑑𝑦+2𝑢𝑥𝑥1(𝑡,𝑥+2𝜏)+2𝑢𝑥𝑥(𝑡,𝑥2𝜏)2𝑢(𝑡,𝑥)𝑢(𝑡,𝑥2𝜏)𝑢2𝑁(𝑡,𝑥+2𝜏)𝑑𝑦(𝑡,𝑥)𝑑𝑥𝑑𝑡,𝑢𝐷(𝑁),𝐷𝑢.(4.13)
From the condition 𝛿𝐹𝑁[𝑢,]=0,𝑢𝐷(𝑁), for all 𝐷(𝑁𝑢), we obtain 𝑁(𝑢)𝑀𝑁2(𝑢)𝑥𝑢𝑡(𝑡,𝑦)𝑑𝑦𝑢21(𝑡,𝑥+2𝜏)2𝑢(𝑡,𝑥)𝑢(𝑡,𝑥2𝜏)+2𝑢𝑥𝑥+1(𝑡,𝑥+2𝜏)2𝑢𝑥𝑥(𝑡,𝑥2𝜏)𝑑𝑦=0,𝑢𝐷(𝑁),(4.14) this equation is equivalent to (4.8).
Let us note that the formula 𝐼[𝑢]=+(1/2)𝑢𝑥(𝑡,𝑥𝜏)𝑢𝑥(𝑡,𝑥+𝜏)+𝑢2(𝑡,𝑥+𝜏)𝑢(𝑡,𝑥𝜏)𝑑𝑥 defines the first integral 𝐼[𝑢]=const of (4.8).


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


  1. V. M. Filippov, V. M. Savchin, and S. G. Shorokhov, “Variational principles for nonpotential operators,” Journal of Mathematical Sciences, vol. 68, no. 3, pp. 275–398, 1994. View at Publisher · View at Google Scholar
  2. I. A. Kolesnikova, A. M. Popov, and V. M. Savchin, “On variational formulations for functional differential equations,” Journal of Function Spaces and Applications, vol. 5, no. 1, pp. 89–101, 2007. View at Google Scholar · View at Zentralblatt MATH
  3. V. M. Savchin, “Helmholtz’s conditions of potentiality for PDE with deviating arguments,” in Proceedings of the 35th Scientific Conference of the Departement of Physico-Mathematical and Natural Science, Part 2. Mathematical Sections, p. 25, M.: RUDN, May 1994.
  4. A. M. Popov, “Potentiality conditions for differential-difference equations,” Differentsial'nye Uravneniya, vol. 34, no. 3, pp. 422–424, 1998. View at Google Scholar
  5. V. M. Savchin and S. A. Budochkina, “On the structure of a variational equation of evolution type with a second derivative with respect to t-derivative,” Differentsial'nye Uravneniya, vol. 39, no. 1, pp. 118–124, 2003. View at Publisher · View at Google Scholar
  6. V. M. Savchin, “An operator approach to Birkhoff’s equation,” Vestnik RUDN, vol. 2, no. 2, pp. 111–123, 1995. View at Google Scholar
  7. L. E. El’sgol’tz, Qualitative Methods in Mathematical Analysis, vol. 1 of Translations of Mathematical Monographs, American Mathematical Society, 1964.
  8. A. L. Skubachevskii, Elliptic Functional-Differential Equations and Applications, Birkhäuser, Basel, Switzerland, 1997.
  9. G. A. Kamenskiĭ, “Variational and boundary value problems with deviating argument,” Differentsial'nye Uravneniya, vol. 69, no. 8, pp. 1349–1358, 1970. View at Google Scholar