Abstract
This paper investigates the wellposedness of a boundary value problem on the semiaxis
for a class of thirdorder operatordifferential equations whose principal part has
multiple real characteristics. We obtain sufficient conditions for the existence and
uniqueness of the solution of a boundary value problem in the Sobolevtype space
MSC: 34G10, 47A50, 47D03, 47N20.
Keywords:
wellposed and unique solvability; operatordifferential equation; multiple characteristic; selfadjoint operator; the Sobolevtype space; intermediate derivatives operators; factorization of pencils1 Introduction
The paper is dedicated to the formulation and study of the wellposedness of a boundary value problem for a class of thirdorder operatordifferential equations with a real and real multiple characteristic. Note that the differential equations whose characteristic equations have real different or real multiple roots find a wide application in modeling problems of mechanics and engineering, such as problems of heat mass transfer and filtration [1], dynamics of arches and rings [2], etc.
Suppose that H is a separable Hilbert space with a scalar product
where
By
Similarly, we define the space
Define the following spaces:
(for more details about these spaces, see [[3], Ch.1]). Here and further, the derivatives are understood in the sense of distributions (see [3]).
The spaces
and
respectively.
Consider the following subspaces of
By the theorem on intermediate derivatives, both of these spaces are complete [3].
Now let us state the boundary value problem under study.
In the Hilbert space H, we consider the following thirdorder operatordifferential equation whose principal part has multiple characteristic:
where
Definition 1.1 If the vector function
Definition 1.2 If for any
then we say that problem (1.1), (1.2) is regularly solvable.
The solvability of boundary value problems for operatordifferential equations has been studied by many authors. Among such works, we should especially mention the papers by Gasymov, Kostyuchenko, Gorbachuk, Dubinskii, Shkalikov, Mirzoev, Jakubov, Aliev and their followers (see, e.g., [411]) that are close to our paper. Allowing to treat both ordinary and partial differential operators from the same point of view, these equations are also interesting from the aspect that the wellposedness of boundary value problems for them is closely related to the spectral theory of polynomial operator pencils [4] (for comprehensive survey, see Shkalikov [12]). And, of course, wellposed solvability of the Cauchy problem and nonlocal boundary value problems for operatordifferential equations as well as related spectral problems (see, e.g., Shkalikov [13], Gorbachuk and Gorbachuk [14], Agarwal et al.[15]) are also of great interest.
In this paper, we obtain conditions for the regular solvability of boundary value problem (1.1) (1.2), which are expressed only in terms of the operator coefficients of equation (1.1). We also show the relationship between these conditions and the exact estimates for the norms of intermediate derivatives operators in the subspaces and with respect to the norm of the operator generated by the principal part of equation (1.1). Mirzoev [16] was the first who paid detailed attention to such relation (for more details about the calculation of the norms of intermediate derivatives operators, see [17]). To estimate these norms, he used the method of factorization of polynomial operator pencils which depend on a real parameter. Further these results have been developed in [18,19].
It should be noted that all the abovementioned works, unlike equation (1.1), consider the operatordifferential equations with a simple characteristic. Although similar matters of solvability and related problems have already been studied for fourthorder operatordifferential equations whose principal parts have multiple characteristic (see, for example, [20,21], also [22] and some references therein), but they have not been studied for odd order operatordifferential equations with multiple characteristic, including those of thirdorder. One of the goals of the present paper is to fill this gap.
2 Equivalent norms and conditional theorem on solvability of boundary value problem (1.1), (1.2)
We show that the norm of the operator generated by the principal part of equation
(1.1) is equivalent to the initial norm
Let
Then the following theorem holds.
Theorem 2.1The operator
Proof First, we note that if
Obviously, the homogeneous equation
where
In fact, such a solution
and the conditions at zero
Let us now show the boundedness of the operator
Since ,
then (2.1) takes the form
Further, by theorems on intermediate derivatives and traces [[3], Ch.1], we obtain
Thus, the operator
Corollary 2.2It follows from Theorem 2.1 that the norm
Before we state the conditional theorem on solvability of boundary value problem (1.1) (1.2), we prove the following lemma.
Lemma 2.3Let the operators
Then
Proof Since for , by virtue of the intermediate derivatives theorem [[3], Ch.1], we obtain
The lemma is proved. □
Theorem 2.4Let the operators
where
Then boundary value problem (1.1), (1.2) is regularly solvable.
Proof We represent boundary value problem (1.1), (1.2) in the form of the operator equation
Under the conditions of Theorem 2.4, we obtain
Therefore, if the inequality
The theorem is proved. □
Naturally, there arises a problem of finding exact values or estimates for the numbers
3 On spectral properties of some polynomial operator pencils and basic equalities
for the functions in the space
W
2
3
(
R
+
;
H
)
Consider the following polynomial operator pencils depending on the real parameter β:
We need to clarify considering naturally arising pencils (3.1). Obviously, for
Since for
then, as a result, we obtain
If we use the Fourier transform in (3.2), then
where
That is why to estimate
The following theorem on factorization of pencils (3.1) holds.
Theorem 3.1Let
where
here
Proof It is clear that
are the characteristic polynomial operator pencils (3.1), where
Since
we obtain
for
where
Now, we state a theorem playing a significant role in the subsequent study. Let us
introduce another notation, which will be used in the proof of that theorem:
Theorem 3.2Let
where
Proof It suffices to prove the theorem for functions
Integrating by parts, we obtain
Making similar calculations for
From (3.8), taking into account (3.9) and applying Theorem 3.1, we get the validity of (3.7). The theorem is proved. □
Corollary 3.3Ifand
Corollary 3.4Ifand
where
4 On the values of the numbers
n
j
,
j
=
1
,
2
It is easy to check that the norms
Lemma 4.1
Proof Passing to the limit as
i.e.,
Note that the procedure of constructing such functions
Remark 4.2 Since , then
Denote by
Theorem 4.3The following relation holds:
Proof If
Now let us consider the Cauchy problem
Since by Theorem 3.1, for
where
This means that
Passing here to the limit as
Continuing the proof of the theorem, we suppose that
Therefore, from (3.11) we find that for
Applying this inequality again to the solution
Taking into account this inequality in (3.11), we obtain
where
Thus,
Remark 4.4 From Theorem 4.3, it becomes clear that to find the numbers
The following theorem holds.
Theorem 4.5
Proof In view of Remark 4.4, in the case
5 Solvability of boundary value problem (1.1), (1.2). Example
The results obtained above allow us to establish exact conditions for regular solvability of boundary value problem (1.1), (1.2). These conditions are expressed in terms of the operator coefficients of equation (1.1).
Theorem 5.1Let the operators
Then boundary value problem (1.1), (1.2) is regularly solvable.
Note that the above conditions for regular solvability of boundary value problem (1.1), (1.2) are easily verified in applications because they are expressed in terms of the operator coefficients of equation (1.1).
Let us illustrate our solvability results with an example of an initialboundary value problem for a partial differential equation.
Example 5.2 On the halfstrip
where
Applying Theorem 5.1, we obtain that under the condition
problem (5.1)(5.3) has a unique solution in the space
Remark 5.3 Using the same procedure, we can obtain similar results for equation (1.1) on the
semiaxis
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
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