#### Abstract

By using the methods of operator theory, all solvable extensions of minimal operator generated by first order pantograph-type delay differential-operator expression in the Hilbert space of vector-functions on finite interval have been considered. As a result, the exact formula for the spectrums of these extensions is presented. Applications of obtained results to the concrete models are illustrated.

#### 1. Introduction

The quantitative and qualitative theory of linear pantograph differential equations, sometimes known as pantograph-type delay differential equations, was first studied in detail by T. Kato and J. B. McLeod [1], L. Fox et al. [2], and A. Iserles [3] in the nineteen seventies.

These equations arose as a mathematical model of an industrial problem involving wave motion in the overhead supply line to an electrified railway system, so they are often called pantograph equations.

In industrial applications in works [2, 4] and in studies on biology and economics, control and electrodynamics in works [5–7] have been researched (for more comprehensive list of features see [3]).

Since an analytical computation of solutions, eigenvalues, and eigenfunctions of corresponding problems is very difficult theoretically and technically, then in this theory methods of numerical analysis play a significant role (for more information see [8–13]).

Let us remember that an operator in Hilbert space is called solvable, if is one-to-one, , and .

In this work, by using methods of operator theory all solvable extensions of minimal operator generated by pantograph-type delay differential-operator expression for first order in the Hilbert space of vector-functions at a finite interval have been described in terms of boundary values in Section 2. Consequently, the resolvent operators of these extensions can be written clearly.

The exact formula for the spectrums of these extensions has been given in Section 3. Applications of obtained results to the concrete models have been illustrated in Section 4.

#### 2. Description of Solvable Extensions

In the Hilbert space of vector-functions consider a linear pantograph differential-operator expression for first order in the form where (1) is a separable Hilbert space with inner product and norm ,(2)operator-function , , is continuous on the uniform operator topology,(3), , .

On the other hand, here the following differential expression will be considered: in the Hilbert space corresponding to (1).

It is clear that the formally adjoint expression of (2) is of the form Now define operator on the dense in set of vector-functions , as .

The closure of in is called the minimal operator generated by differential-operator expression (2) and is denoted by .

In a similar way, the minimal operator in corresponding to differential expression (3) can be defined.

The adjoint operator of in is called the maximal operator generated by (2)((3)) and is denoted by .

Now define an operator in in the form Then for and for it is obtained that Therefore we have and , .

In this situation the following defined operator is a linear bounded operator in .

Throughout this work the following defined operators will be called the minimal and maximal operators corresponding to differential expression (1) in , respectively.

Now let , , be the family of evolution operators corresponding to the homogeneous differential equation The operator , , is linear continuous, boundedly invertible in and (for more detailed analysis of this concept see [14]).

Let us introduce the operator In this case it is easy to see that, for the differentiable vector-function , satisfies the following relation: From this . Hence it is clear that if is some extension of the minimal operator , that is, , then For example, prove the validity of the last relation. It is known that If , then ; that is, . From the last relation . Contrarily, if a vector-function , then that is, . From last relation ; that is, . Hence, .

Theorem 1. *Each solvable extension of the minimal operator in is generated by the pantograph differential-operator expression (1) and boundary condition
**
where and is an identity operator in . The operator is determined uniquely by the extension ; that is, .**On the contrary, the restriction of the maximal operator to the manifold of vector-functions satisfies condition (16) for some bounded operator is a solvable extension of the minimal operator in the .*

*Proof. *Firstly, all solvable extensions of the minimal operator in in terms of boundary values are described.

Consider the following so-called Cauchy extension
of the minimal operator . It is clear that is a solvable extension of and
Now assume that is a solvable extension of the minimal operator in . In this case it is known that the domain of can be written in direct sum in the form
where [15, 16]. Therefore for each it is true that
That is, , , . Hence , . Hence , and from these relations it is obtained that
On the other hand, uniqueness of operator follows from [15]. Therefore, . This completes the necessary part of this assertion.

On the contrary, if is an operator generated by differential expression (2) and boundary condition (21), then is bounded, boundedly invertible, and
Consequently, all solvable extensions of the minimal operator in are generated by differential expression (2) and boundary condition (21) with any linear bounded operator .

Now consider the general case. For this in the introduce an operator in the form
From the properties of the family of evolution operators , , it is implied that an operator is linear bounded and has a bounded inverse and
On the other hand, from the relations
it is implied that an operator is one-to-one between sets of solvable extensions of minimal operators and in .

The extension of the minimal operator is solvable in if and only if the operator is an extension of the minimal in . Then, if and only if
that is, . This proves the validity of the claims in the theorem.

Corollary 2. *In particular the resolvent operator , , of any solvable extension of the minimal operator , generated by pantograph-type delay differential expression
**
with boundary condition in ,
**
is of the form
**.*

*Remark 3. *Note that in the general case , for any .

Indeed, if
then for and

Corollary 4. *Assume that for any and any **
In this case, all solvable extensions of minimal operator are generated by the following differential expression
**
and boundary condition
**
in the Hilbert and vice versa.**Note that the series in the right side of the last equality is convergent, because for any *

Corollary 5. *All solvable extensions of the minimal operator generated by pantograph differential expression , , are described with boundary conditions
**
in the Hilbert space .*

Corollary 6. *It can be proved that all the solvable extensions of the minimal operator are generated by pantograph-type differential-operator expressions for first order
**
in generated by and boundary condition
**
in .*

*Remark 7. *Theorem 1 can be generalized in the differential expression
where , , , , , , .

Theorem 8. *All solvable extensions of minimal operator corresponding to pantograph-type delay differential-operator expression in Hilbert space are described by and boundary condition
**
where and , , is a family of evolution operators corresponding to the homogeneous differential equation
**
with boundary condition and vice versa.*

#### 3. Spectrum of Solvable Extensions

In this section, the spectrum structure of solvable extensions of minimal operator in will be investigated.

Firstly, prove the following fact.

Theorem 9. *If is a solvable extension of a minimal operator and corresponds to the solvable extension of a minimal operator , then the spectrum of these extensions is true .*

*Proof. *Consider a problem to the spectrum for a solvable extension of a minimal operator generated by pantograph differential-operator expression (1); that is,
From this it is obtained that
or . Hence .

Therefore, the validity of the theorem is clear.

Now prove the following result for the spectrum of solvable extension.

Theorem 10. *If is a solvable extension of the minimal operator in the space , then spectrum of has the form
*

*Proof. *Firstly, the spectrum of the solvable extension of the minimal operator in will be investigated.

Consequently, consider the following problem for the spectrum; that is, , , . Then
It is clear that a general solution of the above differential equation in has the form
Therefore, from the boundary condition it is obtained that
For , , from the last relation it is established that
Consequently, in this case the resolvent operator of is in the form
On the other hand, it is clear that , .

Now assume that , , . Then using (47) we have
Therefore, if and only if
In this case since ,
Then
Later on, using the last relation and Theorem 9 the validity of the claim in the theorem is proved.

Now one result on the asymptotic behavior of eigenvalues of solvable extensions in special cases will be proved.

Theorem 11. *Let , , and . In addition, assume that there exists such that for any are true and . Then as . (i.e., there exist and ).*

*Proof. *In this case for
Since for any
then
Therefore, for any is true
On the other hand, for any
Consequently, for any
This means that as .

#### 4. Applications

*Example 12. *Let
By Theorem 1, all solvable extensions of minimal operator generated by , , in are described with and boundary condition
In addition, the resolvent operator of these extensions is in the form
, and for spectrum of this extension is in the form

*Example 13. *Let
Consider the following pantograph functional-differential expression in the form
in . Then by Theorem 1, all solvable extensions of minimal operator are generated by and boundary condition
and vice versa.

Moreover, the resolvent operator of these extensions is
On the other hand, by Theorem 9 for the spectrum of solvable extension is in the form
Now consider the following differential equation
with initial-boundary value problem
in the Hilbert space , where , , and .

In order to solve this problem change the function by

Then the considered problem transforms the following problem:
where .

The last problem can be written in the form
where .

Then solution of the above Cauchy problem by Corollary 2 can be analytically expressed in the form
Another approach to this problem has been investigated in [17].

*Example 14. *Consider the following integrodifferential equation for first order in the form
in Hilbert space . Changing the unknown function by
the following initial-value problem for integrodifferential equation is obtained:
where , in .

The last equation can be rewritten in the form
It is easy to see that the analytical solution of this problem is in the form
Consequently, for

*Example 15. *All solvable extensions of minimal operator generated by differential expression
in the Hilbert space are described by this and boundary condition
where and , , is a solution of operator equation
where , .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The authors would like to thank Professor A. Ashyralyev (Fatih University, Istanbul, Turkey) for his various comments and suggestions.