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Variational principles for the differential difference operator of the second order.

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UDC 517.972.5
Variational Principles for the Differential Difference Operator
of the Second Order
I. A. Kolesnikova
Department Mathematical Analisys And Theory Of Functions
Peoples’ Friendship University of Russia
6, Miklukho-Maklaya str., Moscow, Russia, 117198
The purpose of the present paper is to investigate the potentiality of the operator of
differential difference equations and to construct of the functional, if the given operator is a
potential on a given set relatively to the some bilinear form.
Key words and phrases: differential difference equations, functional differentional
equations, inverse problem of the calculus of variations, variational principles, equations with
deviating arguments.
1.
Introduction
The construction of variational principles in the investigation of the differential
equations  () =  is connected with the inverse problem of the calculus of variations.
The investigation of the solutions of this inverse problem as the construction of the
functional  [] for which the set of critical (extremal or stationary) points coincides
with that of a solution of the given equation  () −  = 0.
Differential difference equations or functional difference equations have already
appeared in mathematical papers of the XVIIIth century, for example, in the Euler
solution of the problem connected with a search of the general form of a line similar to
its evolute.
The search of a functional  that admits some 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 [1], Volterra [2], Santilli [3], Tonti [4], Filippov, Savchin and Shorokhov [5]
and refs. therein).
There is a practical need to develop different approaches to the construction of
integral variational principles for equations with deviating arguments.
It is possible to investigate the problem of the construction of the variational
multiplier if the operator of the given equation is not potential on the given set with
respect to some bilinear form.
The main aim of this paper is to investigate the potentiality of the operator  () of
the differential difference equation and to construct the functional  [], if the operator
 () is potential on the given set relatively to some bilinear form.
2.
Some auxiliary notations and definitions
Let ,  be normed linear spaces over the field of real numbers R, and  ,  be
their zero elements.
Take any operator  : ( ) → ( ), where ( ) ⊆  , ( ) ⊆  . A limit
1
lim [ ( + ℎ) −  ()] =  (, ℎ),
→0 
 ∈ ( ),
( + ℎ) ∈ ( ),
if it exists, is called the G^ateaux differential of  at the point . If it is linear relative
to ℎ, then the operator  (, ·) :  →  is called the G^ateaux derivative of  at 
and will be denoted by ′ . Its domain of definition (′ ) consists of elements ℎ ∈ 
such that ( + ℎ) ∈ ( ) for all  sufficiently small.
Received 2nd July, 2012.
26
Bulletin of PFUR. Series Mathematics. Information Sciences. Physics. No 3, 2012. Pp. 25–34
Let us consider the equation  () =  ,  ∈ ( ) with the G^ateaux differentiable
operator  , and a convex set ( ).
In order to consider the existence of its variational formulation we need a nondegenerate bilinear form
∫︁ ∫︁1
⟨, ⟩ =
(, ), (, )dd.
(1)
Ω 0
Definition. The operator  : ( ) →  is said to be potential on the set ( )
relative to a given bilinear form ⟨·, ·⟩ :  ×  → , if there exists a functional  :
( ) = ( ) →  such that  [, ℎ] = ⟨ (), ℎ⟩ ∀ ∈ ( ), ∀ℎ ∈ (′ ).
The functional  is called the potential of the operator  , and in turn the operator
 is called the gradient of the functional  . As it is known (see Volterra [2]) the
condition for potentiality of the operator  takes the form
⟨′ ℎ, ⟩ = ⟨′ , ℎ⟩
∀ ∈ ( ),
∀ℎ,  ∈ (′ ).
(2)
Under this condition the potential  is given by
∫︁1
 [] =
⟨ (0 + ( − 0 )),  − 0 ⟩ d + const,
0
where 0 is a fixed element of ( ).
Let us consider the differential equation with deviating arguments
 () ≡
1
∑︁
=−1
 (, )
2
2

(,

+

)
−

(,
)
(,  +  )+

2
 
+  (,  + , (,  +  ),  (,  +  ),  (,  +  )) = 0, (3)
where  is an unknown function; (, ) ∈  = Ω × (1 , 2 ); 2 − 1 > 2 ;  (, ) ∈
0,2
2,0
,
(), 
 (, ) ∈ , (), ∀,  = 1, .
The domain of definition ( ) is given by the equality
{︁
2,2
( ) =  ∈  = ,
(Ω × [0 − , 1 +  ]) :
  (, )
= 1 (, ), (, ) ∈ 1 = Ω × [0 − , 0 ],  = 0, 1,

  (, )
= 2 (, ), (, ) ∈ 2 = Ω × [1 , 1 +  ],  = 0, 1,

}︃
⃒
   ⃒⃒
=  ,  = 0, 1 , (4)
 ⃒
Γ
where Ω ⊂ R , Γ = Ω × (0 − , 1 +  ), 10 , 20 ,  — are given sufficiently smooth
  0
, ( = 1, 2;  = 0, 1).
functions,  =

The formulation of the problem:
1) to investigate the potentiality of the operator  of the equation (3) on the set
( ) (4) relatively to the some bilinear form (1);
Kolesnikova I. A. Variational Principles for the Differential Difference Opera . . .
27
2) if the operator  is potantial, then to construct the variational principle for the
operator  of the equation (3).
The investigation of the potentiality of the operator  of the equation (3).
Theorem. For the potentiality of the operator  (3) on the given set ( ) (4)
with respect to the bilinear form (1), it is necessary and sufficient that the following
conditions hold:
 (, ) = − (,  +  ),


 (, ) = − (,  +  )
∀ = −1, 0, 1,
  (, , ,  ,  ) =  (, , ,  ,  ),
 (, )
 (, )
 (,  +  ) − 
 (,  +  ) +  (, , ),


(5)
where  are sufficiently smooth functions,  (, ) = (,  +  ).
  (, , ,  ,  ) =
Proof. We denote
′ ℎ
=
1
∑︁
 (, )
=−1
2ℎ
2ℎ

(,

+

)
−

(,
)
(,  +  )+

2
 
{︂
}︂



ℎ +
+
ℎ+
ℎ (,  +  )

  
∀ ∈ ( ), ∀ℎ,  ∈ (′ ), ∀,  = 1, . (6)
Taking into account formulas (1) and (6), we get
⟨′ ℎ, ⟩
{︃
1 ∫︁ ∫︁1
∑︁
2ℎ
2ℎ
 (, ) 2 (,  +  ) − 
(,  +  )+
=
 (, )

 
=−1 Ω 
0
}︃
{︂
}︂



+
ℎ +
ℎ+
ℎ (,  +  ) (, )dd

  
∀ ∈ ( ), ∀ℎ,  ∈ (′ ),
∀,  = 1, . (7)
We denote in the items of the formula (7) thus
1 ≡
1 ∫︁ ∫︁1
∑︁
 (, )
=−1 Ω 
0
2 ≡
1 ∫︁ ∫︁1
∑︁

 (, )
=−1 Ω 
0
2ℎ
(,  +  )(, )dd,
2
2ℎ
(,  +  )(, )dd,
 
1 ∫︁ ∫︁1
∑︁

3 ≡
ℎ (,  +  )(, )dd,
 
=−1 Ω 
0
1 ∫︁ ∫︁1
∑︁

4 ≡
ℎ (,  +  )(, )dd,

=−1 Ω 
0
28
Bulletin of PFUR. Series Mathematics. Information Sciences. Physics. No 3, 2012. Pp. 25–34
1 ∫︁ ∫︁1
∑︁

5 ≡
ℎ(,  +  )(, )dd.

=−1 Ω 
0
We know that
∫︁0
∫︁
1 +
ℎ(, )d =
0 −
ℎ(, )d = 0,
1
∫︁1
(8)
∫︁
0 +
ℎ(,  −  )d =
1 −
ℎ(,  +  )d = 0 ∀ ∈ Ω,
0
from the formula 1 − 5 we get
1 =
1 ∫︁ ∫︁1
∑︁
ℎ(,  +
 )2 ( (, )(, ))dd
=
×
ℎ(,  +  )×
=−1 Ω 
0
2
=−1 Ω 
0
(︂
1 ∫︁ ∫︁1
∑︁
)︂
 2  (, )
 (, ) (, )  (, )
(, ) + 2
+
 (, ) dd.
2


2
By using the change ′ =  +  , we obtain
1 =
1 ∫︁
∑︁
1∫︁+
′
ℎ(,  )
=−1 Ω  +
0
(︂
 2  (, ′ −  )
(, ′ −  ) +
2
)︂
 (, ′ −  ) (, ′ −  )  2 (, ′ −  )
′
+2
+
 (,  −  ) dd.


2
Denoting ′ by  and taking into account formula (8), we get
1 =
1 ∫︁ ∫︁1
∑︁
(︂
ℎ(, )
=−1 Ω 
0
 2  (,  −  )
(,  −  ) +
2
)︂
 (,  −  ) (,  −  )  2 (,  −  )
+2
+
 (,  −  ) dd


2
or
1 =
1 ∫︁ ∫︁1
∑︁
(︂
ℎ(, )
=−1 Ω 
0
 2 − (,  +  )
(,  +  ) +
2
)︂
− (,  +  ) (,  +  )  2 (,  +  )
+2
+
− (,  +  ) dd. (9)


2
2 =
1 ∫︁ ∫︁1
∑︁
=−1 Ω 
0
ℎ(,  +
 )  (
 (, )(, ))dd
=
1 ∫︁ ∫︁1
∑︁
=−1 Ω 
0
ℎ(,  +  )×
Kolesnikova I. A. Variational Principles for the Differential Difference Opera . . .
(︃
×
 2 
 (, )
 
(, ) + 2

 (, )

29
)︃
(, )  2 (, ) 
+
 (, ) dd.

  
Reasoning by analogy as for 1 , we obtain
2 =
1 ∫︁ ∫︁1
∑︁
(︃
ℎ(, )
=−1 Ω 
0
 2 
− (,  +  )
(,  +  ) +
 
)︃

 2 (,  +  ) 
− (,  +  ) (,  +  )
+2
+
− (,  +  ) dd


 
∀,  = 1, . (10)
∫︁ ∫︁1
1
∑︁
3 = −
ℎ(,  +  )×
=−1 Ω 
0
)︂
 (,  + , (,  +  ),  (,  +  ),  (,  +  ))
(, ) dd =


1
(︂
∫︁ ∫︁
1
∑︁
 (,  + , (,  +  ),  (,  +  ),  (,  +  ))
ℎ(, + )
=−
 (, )+

(︂
×  
=−1 Ω 
0
(︂
+ (, )
 (,  + , (,  +  ),  (,  +  ),  (,  +  ))

)︂)︂
dd.
By using the change ′ =  + 
3 = −
1 ∫︁
∑︁
=−1 Ω
1∫︁+
 (, ′ , (, ′ ),  (, ′ ),  (, ′ ))
 (, ′ −  )+

0 +
(︂
)︂)︂
 (, ′ , (, ′ ),  (, ′ ),  (, ′ ))
′
+ (,  −  )
dd.

′
(︂
ℎ(,  )
Denoting ′ by  and taking into account formula (8), we get
3 = −
1 ∫︁ ∫︁1
∑︁
=−1 Ω 
0
 (, , (, ),  (, ),  (, ))
 (,  −  )+

(︂
)︂)︂
 (, , (, ),  (, ),  (, ))
+ (,  −  )
dd.

(︂
ℎ(, )
or
3 = −
1 ∫︁ ∫︁1
∑︁
=−1 Ω
 (, , (, ),  (, ),  (, ))
 (,  +  )+

0
(︂
)︂)︂
 (, , (, ),  (, ),  (, ))
+ (,  +  )
dd. (11)

(︂
ℎ(, )
30
Bulletin of PFUR. Series Mathematics. Information Sciences. Physics. No 3, 2012. Pp. 25–34
∫︁ ∫︁1
1
∑︁
4 = −
ℎ(,  +  )×
=−1 Ω 
0
)︂
 (,  + , (,  +  ),  (,  +  ),  (,  +  ))
(, ) dd =

1
(︂
∫︁
∫︁
1
∑︁
 (,  + , (,  +  ),  (,  +  ),  (,  +  ))
=−
ℎ(,  +  )
 (, )+

(︂
× 
=−1 Ω 
0
(︂
+ (, )
 (,  + , (,  +  ),  (,  +  ),  (,  +  ))

)︂)︂
dd.
By using the change ′ =  + 
4 = −
1 ∫︁
∑︁
=−1 Ω
1∫︁+
 (, ′ , (, ′ ),  (, ′ ),  (, ′ ))
 (, ′ −  )+

0 +
(︂
)︂)︂
 (, ′ , (, ′ ),  (, ′ ),  (, ′ ))
′
+ (,  −  )
dd.

′
(︂
ℎ(,  )
Denoting ′ by  and taking into account formula (8), we get
4 = −
1 ∫︁ ∫︁1
∑︁
 (, , (, ),  (, ),  (, ))
 (,  −  )+

)︂)︂
(︂
 (, , (, ),  (, ),  (, ))
dd.
+ (,  −  )

(︂
ℎ(, )
=−1 Ω 
0
or
4 = −
1 ∫︁ ∫︁1
∑︁
=−1 Ω
5 =
 (, , (, ),  (, ),  (, ))
 (,  +  )+

0
)︂)︂
(︂
 (, , (, ),  (, ),  (, ))
dd. (12)
+ (,  +  )

(︂
ℎ(, )
1 ∫︁ ∫︁1
∑︁
ℎ(, )
=−1 Ω 
0
 (, , (, ),  (, ),  (, ))
(,  +  )dd.

(13)
Thus from (7) and (9)–(13) we obtain
⟨′ ℎ, ⟩
=
1 ∫︁ ∫︁1
∑︁
=−1 Ω 
0
(︂
−  
{︃(︃
ℎ(, )

 2 − (,  +  )  2 − (,  +  )
−
−
2
 
)︂
(︂
)︂
 (, , (, ),  (, ),  (, ))
 (, , (, ),  (, ),  (, ))
− 
+


Kolesnikova I. A. Variational Principles for the Differential Difference Opera . . .
31
)︃
+
 (, , (, ),  (, ),  (, ))
(,  +  )+

 2 (,  +  )
 2 (,  +  )

−

(,

+

)
−
−
2
 
(︃
)︃

 (, , (, ),  (, ),  (, ))
− (,  +  )
− 2
+
 (,  +  )+


}︃
)︂
(︂
− (,  +  )  (, , (, ),  (, ),  (, ))
−
 (,  +  ) dd
+ 2


+ − (,  +  )
∀ ∈ ( ), ∀ℎ,  ∈ (′ ),
∀,  = 1, . (14)
Using the G^ateaux derivative of the operator (3) , we get
⟨′ , ℎ⟩
{︃
1 ∫︁ ∫︁1
∑︁
 2 (,  +  )
 2 (,  +  )
 (, )
=
− 
+
 (, )
2

 
=−1 Ω 
0
}︃
}︂
{︂



 +
+
 (,  +  ) ℎ(, )dd
+

  
∀ ∈ ( ), ∀ℎ,  ∈ (′ ),
∀,  = 1, . (15)
We mean   (, , ,  ,  ) =  (,  + , (,  +  ),  (,  +  ),  (,  +  )).
The equations (14) and (15) are equal if and only if
  (, , ,  ,  ) =  (, , ,  ,  ),
 (, ) = − (,  +  ),
1 ∫︁ ∫︁1
∑︁
(16)


 (, ) = − (,  +  ) ∀ = −1, 0, 1,
{︃(︃
 2  (, )  2 
 (, )
−
−
2

 
=−1 Ω 
0
(︂
)︂
 (, , (, ),  (, ),  (, ))
−  
−

(︂
)︂)︂
 (, , (, ),  (, ),  (, ))
(,  +  )−
− 

(︃
)︃
}︃
(︂
)︂


 (, )

 (, )
−2
+
 (,  +  ) + 2
−
 (,  +  ) dd = 0




ℎ(, )
∀ ∈ ( ),
∀ℎ,  ∈ (′ ),
∀,  = 1, .
The condition (16) is fulfiled for some periodic functions, for example. For the
potentiality of the operator  of the equation (3) on the set ( ) (4) relatively to
some bilinear form (1), it is necessary and sufficient that the following conditions hold
32
Bulletin of PFUR. Series Mathematics. Information Sciences. Physics. No 3, 2012. Pp. 25–34
for all ℎ and 
⎧

⎪
 (, )
⎪
⎪
⎪

⎪
 + 
= 0,
⎪
⎪

⎪
⎪
⎨
 (, )

−
= 0,
⎪


⎪
⎪
(︃
)︃
⎪
)︂
(︂
⎪

⎪
(,
)



(,
)

⎪


⎪
⎪
−
−  
+
= 0.
⎩ 




(17)
The third condition is a consequence of the first and the second conditions.
  (, , ,  ,  ) =
 (, )
 (,  +  ) +  (, , ,  ),

(18)
where  is sufficiently smooth function
Substituting (18) for  in the second condition of the sistem (17), we get
 (, , ,  ) = −

 (, )
 (,  +  ) +  (, , ),

(19)
where  is sufficiently smooth function.
Thus from equalities (18) and (19), we obtain
  (, , ,  ,  ) =
 (, )
 (, )
 (, + )− 
 (, + )+ (, , ), (20)


The proof of the theorem is completed.
The construction of the functional  [] for the operator  of the equation (3)
We can write (3), making use of (20)
 () ≡
1
∑︁
 (, )
=−1
2
2

(,

+

)
−

(,
)
(,  +  )+

2
 

(, )
 (, )
+
 (,  +  ) − 
 (,  +  ) +  (, , ) = 0. (21)


We can find the functional  []
∫︁1
 [] =
⟨ (0 + ( − 0 )),  − 0 ⟩  =
0
=
1
∑︁
∫︁ ∫︁1 ∫︁1
( (, )[0 + ( − 0 )] − 
 (, )[0  + (  − 0  )]+
=−1 Ω  0
0
+  (, )[0 + ( − 0 )] − 
 (, )[0 + ( − 0 )])( − 0 )ddd+
+
1 ∫︁ ∫︁1 ∫︁1
∑︁
=−1 Ω  0
0
 (, , 
˜)( − 0 )ddd,

˜ = 0 + ( − 0 ).
Kolesnikova I. A. Variational Principles for the Differential Difference Operator . . .
33
We integrate the first integral (21) in the variable  and get
1 ∫︁ ∫︁1 ∫︁1 (︂
∑︁
[︂
]︂
1
 (, ) 0 + ( − 0 ) −
 [] =
2
=−1 Ω  0
0
[︂
]︂
]︂
[︂
1
1
− 
(,
)

(
(
−

)
−
+
−

)
+

(,
)

+
0 
 

0
0 

0

2
2
[︂
]︂)︂
1

−  (, ) 0 + ( − 0 ) ( − 0 )ddd+
2
1 ∫︁ ∫︁1 ∫︁1
∑︁
 (, , 
˜)( − 0 )ddd
+
=−1 Ω  0
0
or
∫︁ ∫︁1
1
1 ∑︁
 [] =
( (, )[0 +  ] − 
 (, )[0  +   ]+
2
=−1 Ω 
0
+  (, )[0 +  ] − 
 (, )[0 +  ])( − 0 )dd+
+
∫︁ ∫︁1 ∫︁1
1
∑︁
=−1 Ω
+
)︂
∫︁ ∫︁1 (︂
1
1 ∑︁
 (, , 
˜)( − 0 )ddd =
  (, )(0 +  )( − 0 ) −
2
=−1 Ω 
0 0
0
(︂
)︂

−   (, )(0 +  )( − 0 ) −  (, )(0 +  )( − 0 )+

 (, )(0
∫︁ ∫︁1 ∫︁1
1
∑︁
+  )( − 0 )dd +
=−1 Ω 
0
 (, , 
˜)( − 0 )ddd.
(22)
0
Thus, the unknown functional for the operator (21) (from the equality (22)) is
obtained in the following form
1 ∫︁ ∫︁1
1 ∑︁
( (, )2 − 
 [] = −
 (, )  )dd+
2
=−1 Ω 
0
+
1 ∫︁ ∫︁1 ∫︁1
∑︁
 (, , 
˜)( − 0 )ddd.
=−1 Ω  0
0
References
1. Helmholtz H. Ueber die physikalische Bedeutung des Prinzips der kleinsten
Wirkung // J. Reine und Angew. Math. — 1887. — Vol. 100. — Pp. 137–166.
2. Volterra V. Leçons sur les Fonctions de Lignes. — Paris: Gautier-Villars, 1913.
3. Santilli R. M. Foundations of Theoretical Mechanics l. The Inverse Problem in
Newtonian Mechanics. — Springer-Verlag, 1978.
4. Tonti E. A General Solution of the Inverse Problem of the Calculus of Variations //
Hadronic J. — 1982. — Vol. 5, No 4. — Pp. 1404–1450.
5. Filippov V. M., Savchin V. M., Shorokhov S. G. Variational Principles for Nonpotential Operators // J. Math. Sci. — 1994. — Vol. 68, No 3. — Pp. 275–398.
34
Bulletin of PFUR. Series Mathematics. Information Sciences. Physics. No 3, 2012. Pp. 25–34
УДК 517.972.5
Структура дифференциально-разностного оператора
второго порядка, допускающего вариационный принцип
И. А. Колесникова
Кафедра математического анализа и теориии функций
Российский университет дружбы народов
ул.Миклухо-Маклая, д.6, Москва, Россия, 117198
В статье исследуется на потенциальность оператор на заданной области определения
и относительно некоторой билинейной формы. В случае потенциальности строится
соответствующий функционал.
Ключевые слова: дифференциально-разностные уравнения, функционально-дифференциальные уравнения, обратная задача вариационного исчисления, вариационный
принцип, уравнения с отклоняющимися аргументами.
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