Open Access

On the Correct Solvability of the Boundary-Value Problem for One Class Operator-Differential Equations of the Fourth Order with Complex Characteristics

Boundary Value Problems20092009:710386

DOI: 10.1155/2009/710386

Received: 19 February 2009

Accepted: 26 August 2009

Published: 27 September 2009

Abstract

Sufficient coefficient conditions for the correct and unique solvability of the boundary-value problem for one class of operator-differential equations of the fourth order with complex characteristics, which cover the equations arising in solving the problems of stability of plastic plates, are obtained in this paper. Exact values of the norms of operators of intermediate derivatives, which are involved in the perturbed part of the operator-differential equation under investigation, are found along with these in subspaces https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq1_HTML.gif in relation to the norms of the operator generated by the main part of this equation. It is noted that this problem has its own mathematical interest.

1. Introduction

It is well known that a number of problems in mechanics lead to studying the completeness of all or part of the eigenvectors and joint vectors of certain polynomial operator groups and the completeness of elementary solutions of the operator-differential equations corresponding to these groups (see, e.g., [1, 2], and their references). In this case, it is first necessary to investigate the correct solvability of Cauchy or boundary-value problems for these equations, and only after this it will be possible to proceed to the abovementioned problems. The present paper is dedicated to the problem of correct solvability of the boundary-value problem for one class of operator-differential equations of the fourth order, considered on a semiaxis.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq2_HTML.gif be a separable Hilbert space and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq3_HTML.gif be a self-adjoint positively defined operator in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq4_HTML.gif .

Let us consider the following operator-differential equation of the fourth order:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ1_HTML.gif
(11)
with the boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ2_HTML.gif
(12)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq5_HTML.gif , are linear and generally unbounded operators in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq6_HTML.gif . Under https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq7_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq8_HTML.gif , the following Hilbert spaces can be described:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ3_HTML.gif
(13)

(see [35]).

Definition 1.1.

If the vector function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq9_HTML.gif satisfies ( 1.1 ) almost everywhere in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq10_HTML.gif , then it is called a regular solution of ( 1.1 ).

Definition 1.2.

If for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq11_HTML.gif , there exists a regular solution of ( 1.1 ) which satisfies boundary condition ( 1.2 ) in the sense that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ4_HTML.gif
(14)
and the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ5_HTML.gif
(15)

holds, then it can be said that problem ( 1.1 ), ( 1.2 ) is regularly solvable.

Let us define the following subspaces of the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq12_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ6_HTML.gif
(16)

It should be noted that the solvability theory for the Cauchy problem and the boundary-value problems for first- and second-order operator-differential equations have been studied in more detail elsewhere. In addition to books [6, 7], these problems have been considered also by Agmon and Nirenberg [8], Gasymov and Mirzoev [9], Kostyuchenko and Shkalikov [10], and in works in their bibliographies. Other papers in which issues of the solvability of various problems for operator-differential equations of higher order have been studied have appeared alongside these works, and sufficiently interesting results have been obtained. Among these papers are those by Gasymov [11, 12], Dubinskii [13], Mirzoev [14], Shakhmurov [15], Shkalikov [16], Aliev [17, 18], Agarwal et al. [19], Favini and Yakubov [20], the book by Yakubov [7], and other works listed in their bibliographies.

Sufficient coefficient conditions for regular solvability of the boundary-value problem stated in (1.1) and (1.2) are presented in this paper. To obtain these conditions, the main challenge is to find the exact values of the norms of operators of intermediate derivatives in subspaces https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq13_HTML.gif , the norms of which are expressed by the main part of (1.1). This problem has its own mathematical interest (see, e.g., [21, 22], and works given in their bibliographies). Estimation of the norms of operators of intermediate derivatives, which are involved in the perturbed part of (1.1), is performed with the help of a factorization method for one class of polynomial operator groups of eighth order, depending on a real parameter. A similar approach has been presented in [9, 14], which makes it possible to formulate solvability theorems for the boundary-value problems, with conditions which can be easily checked.

It should be noted that if the main part of the equation has the operator in the form https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq14_HTML.gif , then a biharmonic equation results, which is of mathematical interest not only theoretically, and also from a practical point of view. Many problems of elasticity theory (e.g., the theory of bending of thin elastic slabs [23]) can be reduced to studying the boundary-value problems for such equations. Much research has been performed to investigate the solvability of such problems, for example, that reported in [24]. Operator-differential equations, which are studied in the present paper, include the fourth-order equations which arise when solving the stability problems of plates made of plastic material (see [25, pages 185–196]). It is very difficult to solve such problems because the differential equation must be solved in a more complete form, that is, when the main part of the equation has terms containing https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq15_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq16_HTML.gif . As a result, the equation has more complex characteristics, and (1.1) is of this type.

Furthermore, let us denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq17_HTML.gif the spectrum of the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq18_HTML.gif .

2. Auxiliary Results

First, let us study the main part of (1.1):

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ7_HTML.gif
(21)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq19_HTML.gif .

The following theorem is true.

Theorem 2.1.

Operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq20_HTML.gif , acting from the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq21_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq22_HTML.gif in the following way:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ8_HTML.gif
(22)

is an isomorphism between the spaces https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq23_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq24_HTML.gif .

Proof.

It holds that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq25_HTML.gif has a solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq26_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq27_HTML.gif . In fact, the vector function
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ9_HTML.gif
(23)
satisfies the equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ10_HTML.gif
(24)
in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq28_HTML.gif almost everywhere. Let us prove that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq29_HTML.gif . As is made clear here, this means that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ11_HTML.gif
(25)
From the Plancherel theorem, it follows that it is sufficient to show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq30_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq31_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq32_HTML.gif is the Fourier transform of the vector function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq33_HTML.gif . From the spectral theory of self-adjoint operators,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ12_HTML.gif
(26)
Here https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq34_HTML.gif is the Fourier transform of the vector function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq35_HTML.gif . Analogously, it is possible to prove that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq36_HTML.gif . Consequently, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq37_HTML.gif . Furthermore, let us denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq38_HTML.gif the narrowing of the vector function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq39_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq40_HTML.gif . It is clear that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq41_HTML.gif . Now,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ13_HTML.gif
(27)
where the vectors https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq42_HTML.gif , and are defined by the condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq43_HTML.gif . This is why the following system of equations can be obtained relatively to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq44_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ14_HTML.gif
(28)
From this, it is possible to obtain the operator equation,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ15_HTML.gif
(29)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ16_HTML.gif
(210)

Because https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq45_HTML.gif , then from the theorem on trace [35], it follows that all elements of the vector https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq46_HTML.gif belong to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq47_HTML.gif . Continuing this process, it is apparent that the operator matrix https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq48_HTML.gif is boundedly invertible in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq49_HTML.gif . Therefore, all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq50_HTML.gif . Consequently, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq51_HTML.gif . In the same way, it can be established that the equation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq52_HTML.gif has only a trivial solution.

Operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq53_HTML.gif is bounded, because
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ17_HTML.gif
(211)
because for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq54_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ18_HTML.gif
(212)
The theorem on intermediate derivatives [35] can be used to obtain the last inequality, with the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ19_HTML.gif
(213)
assumed. Moreover, the Bunyakovsky-Schwartz and Young inequalities,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ20_HTML.gif
(214)

are used in the expression https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq55_HTML.gif .

As a result, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq56_HTML.gif is bounded and acts mutually and uniquely from the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq57_HTML.gif to the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq58_HTML.gif . Then, taking into account the Banach theorem on the inverse operator, it can be established that the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq59_HTML.gif carries out the isomorphism from the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq60_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq61_HTML.gif . Thus, the theorem is proved.

Denoting by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq62_HTML.gif the operator which acts from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq63_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq64_HTML.gif in the following way:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ21_HTML.gif
(215)

the following statement results.

Lemma 2.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq65_HTML.gif be bounded operators in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq66_HTML.gif . Then the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq67_HTML.gif is a bounded operator from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq68_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq69_HTML.gif .

Proof.

Because for any vector function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq70_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ22_HTML.gif
(216)
then, from the theorem on intermediate derivatives [35], and from (2.16), we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ23_HTML.gif
(217)

Thus, the lemma is proved.

Now certain properties of polynomial operator groups will be investigated, which will have in the future a special role.

Let the following hold:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ24_HTML.gif
(218)
Consider the following polynomial operator groups which depend on the parameter https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq71_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ25_HTML.gif
(219)

The following can then be established.

Lemma 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq72_HTML.gif . Then the polynomial operator groups https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq73_HTML.gif , are invertible on the imaginary axis and can be represented as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ26_HTML.gif
(220)
moreover,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ27_HTML.gif
(221)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq74_HTML.gif and the numbers https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq75_HTML.gif satisfy the following systems of equations:

(1)for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq76_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ28_HTML.gif
(222)
  1. (2)
    for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq77_HTML.gif
    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ29_HTML.gif
    (223)
     

(3)for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq78_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ30_HTML.gif
(224)

Proof.

Characteristic polynomials of the operator groups https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq79_HTML.gif , are
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ31_HTML.gif
(225)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq80_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq81_HTML.gif . Then it is clear that for these characteristic polynomials, the following correlations are true:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ32_HTML.gif
(226)
Because
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ33_HTML.gif
(227)
then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ34_HTML.gif
(228)
for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq82_HTML.gif . From (2.28), it becomes clear that the polynomials https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq83_HTML.gif do not have roots on the imaginary axis for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq84_HTML.gif . Each of the characteristic polynomials https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq85_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq86_HTML.gif has exactly four roots from the left semiplane. Because these polynomials are homogeneous with respect to the arguments https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq87_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq88_HTML.gif , they can be stated in the following form:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ35_HTML.gif
(229)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ36_HTML.gif
(230)

and moreover https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq89_HTML.gif , and the numbers https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq90_HTML.gif satisfy the systems of equations shown in Lemma 2.3, which are obtained from (2.29) in the process of comparing the coefficients for the same degrees. Then, from the spectral decomposition of operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq91_HTML.gif , the proof of the lemma can be obtained from (2.29). Thus, the lemma is proved.

The next step is to prove the theorem, which will play an important role in future investigations and will show the special importance of the spectral properties of the polynomial operator groups https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq92_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq93_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq94_HTML.gif .

Theorem 2.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq95_HTML.gif . Then for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq96_HTML.gif , the following equality is true:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ37_HTML.gif
(231)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ38_HTML.gif
(232)

Proof.

First define the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq97_HTML.gif as the set of infinitely differentiable functions with values in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq98_HTML.gif , having compact support in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq99_HTML.gif . Because the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq100_HTML.gif is dense in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq101_HTML.gif (see [35]), it is sufficient to prove the theorem for the vector functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq102_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ39_HTML.gif
(233)
After integration by parts,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ40_HTML.gif
(234)
Calculating https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq103_HTML.gif analogously to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq104_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ41_HTML.gif
(235)

Substituting (2.35) into (2.34), from Lemma 2.3, (2.31) can be obtained. Thus, the theorem is proved.

From Theorem 2.4, it follows that:

Corollary 2.5.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq105_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq106_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ42_HTML.gif
(236)
Note that from Theorem 2.1, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq107_HTML.gif is the norm in the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq108_HTML.gif , which is equivalent to the initial norm https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq109_HTML.gif . Because the operators of the intermediate derivatives
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ43_HTML.gif
(237)

are continuous [35], then the norms of these operators can be estimated using https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq110_HTML.gif . It is also easy to demonstrate that the norms https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq111_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq112_HTML.gif are equivalent in the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq113_HTML.gif .

3. Norms of the Operators of Intermediate Derivatives

The rest of this paper will be related to the calculation of the following numbers:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ44_HTML.gif
(31)

First, let us calculate https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq114_HTML.gif .

Lemma 3.1.

It holds that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq115_HTML.gif

Proof.

As (2.36) goes to the limit as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq116_HTML.gif , it is apparent that for any vector function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq117_HTML.gif , the following inequality:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ45_HTML.gif
(32)
is true. Thus, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq118_HTML.gif . Furthermore, it is necessary to show that here the equalities https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq119_HTML.gif also hold. This can be done by taking an arbitrary number https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq120_HTML.gif and showing that there exists a vector function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq121_HTML.gif such that the following holds functional:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ46_HTML.gif
(33)
Let the vector https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq122_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq123_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq124_HTML.gif be the numeral function; moreover, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq125_HTML.gif . Then using the Parseval equality, it is possible to obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ47_HTML.gif
(34)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq126_HTML.gif .

It will next be shown that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq127_HTML.gif for a given vector https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq128_HTML.gif has negative values in some interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq129_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq130_HTML.gif is an eigenvalue of the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq131_HTML.gif , and if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq132_HTML.gif is its eigenvector, then it is obvious that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ48_HTML.gif
(35)
and, as can be seen from the properties of the polynomial https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq133_HTML.gif , is negative for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq134_HTML.gif for sufficiently small https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq135_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq136_HTML.gif is not the eigenvalue, then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq137_HTML.gif is close to an eigenvalue, that is, there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq138_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq139_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ49_HTML.gif
(36)

because in this case, for sufficiently small https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq140_HTML.gif , the smallest value is negative for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq141_HTML.gif . Then there exists an interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq142_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq143_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq144_HTML.gif .

Now consider the four times differentiable function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq145_HTML.gif , support of which comes from the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq146_HTML.gif . Then from (3.4) and from the negativity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq147_HTML.gif in the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq148_HTML.gif , it can be determined that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ50_HTML.gif
(37)

Consequently, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq149_HTML.gif , and the lemma is proved.

Because https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq150_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq151_HTML.gif . It is necessary to note that, for any vector function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq152_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq153_HTML.gif , the equality
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ51_HTML.gif
(38)

is true, where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq154_HTML.gif is obtained from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq155_HTML.gif by removing the first three rows and columns, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq156_HTML.gif . The correctness of (3.8) follows directly from Theorem 2.4.

The following statement indicates when the numbers https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq157_HTML.gif , can be equal to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq158_HTML.gif .

Lemma 3.2.

To establish the condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq159_HTML.gif , it is necessary and sufficient that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq160_HTML.gif be positive for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq161_HTML.gif .

Proof.

Necessity will be shown first. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq162_HTML.gif . Then, from (3.8), for any vector function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq163_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq164_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ52_HTML.gif
(39)
Because the polynomial operator group https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq165_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq166_HTML.gif has the form
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ53_HTML.gif
(310)
(see Lemma 2.3), where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq167_HTML.gif , then the Cauchy problem,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ54_HTML.gif
(311)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ55_HTML.gif
(312)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ56_HTML.gif
(313)
has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq168_HTML.gif , which can be presented in the form
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ57_HTML.gif
(314)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq169_HTML.gif are uniquely determined from the conditions at zero in (3.12) and (3.13). As a result, writing inequality (3.9) for the vector function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq170_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq171_HTML.gif . Necessity is thereby proved.

Now sufficiency must be proved. If for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq172_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq173_HTML.gif is positive, then from (3.8), it follows that for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq174_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq175_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ58_HTML.gif
(315)

As this expression goes to the limit as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq176_HTML.gif , it can be observed that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq177_HTML.gif , and from this, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq178_HTML.gif . Sufficiency is thereby proved, and thus the lemma is completely proved.

It is interesting that for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq179_HTML.gif , it may occur that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq180_HTML.gif .

Lemma 3.3.

It holds that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq181_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq182_HTML.gif has a solution in the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq183_HTML.gif ; moreover, this root is equal to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq184_HTML.gif .

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq185_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq186_HTML.gif . From (3.8), for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq187_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ59_HTML.gif
(316)
Substituting the solution of (3.11)–(3.13) into the last inequality, the result is that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq188_HTML.gif is positive for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq189_HTML.gif . From the definition of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq190_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq191_HTML.gif , there exists a vector function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq192_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ60_HTML.gif
(317)
From the last inequality in (3.8), it is possible to obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ61_HTML.gif
(318)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ62_HTML.gif
(319)

Thus, there exists a vector https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq193_HTML.gif such that for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq194_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq195_HTML.gif . Because https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq196_HTML.gif is a continuous function of the argument https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq197_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq198_HTML.gif , and this means that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq199_HTML.gif has a root in the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq200_HTML.gif .

Inversely, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq201_HTML.gif has a root in the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq202_HTML.gif , then this means that for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq203_HTML.gif the number https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq204_HTML.gif cannot be positive. This is why, from Lemma 3.2, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq205_HTML.gif . Denoting the root of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq206_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq207_HTML.gif , it can be seen that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq208_HTML.gif , because from the proof of the lemma, it was obtained that for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq209_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq210_HTML.gif is positive. Moreover, because https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq211_HTML.gif , it can be determined that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq212_HTML.gif . The lemma is thereby proved.

By generalizing the last two lemmas, the following theorem can be derived.

Theorem 3.4.

The following equality is true:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ63_HTML.gif
(320)

Remark 3.5.

In the same way, it is possible to determine the results for boundary-value problems of the form ( 1.1 ), ( 1.2 ) for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq213_HTML.gif having any three values from the collection https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq214_HTML.gif .

By considering concretely the cases https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq215_HTML.gif , the following statement results.

Theorem 3.6.

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq216_HTML.gif

Proof.

Taking into account the abovementioned procedure for finding the numbers https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq217_HTML.gif , it is necessary to solve the systems from the proof of Lemma 2.3 together with the equation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq218_HTML.gif .

In the case https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq219_HTML.gif , it can be determined that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq220_HTML.gif . This is why https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq221_HTML.gif . To find the number https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq222_HTML.gif , it is necessary to solve the system from Lemma 2.3 for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq223_HTML.gif together with the equation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq224_HTML.gif . In this case, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq225_HTML.gif , and consequently https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq226_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq227_HTML.gif . As a result, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq228_HTML.gif . In the case https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq229_HTML.gif , it is found that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq230_HTML.gif . Then, from the corresponding system, it can be obtained that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq231_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq232_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq233_HTML.gif . It is clear that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq234_HTML.gif . From the other side, if in the equation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq235_HTML.gif , it is assumed that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq236_HTML.gif , then the result is that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq237_HTML.gif , which has only one real root, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq238_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq239_HTML.gif , and the theorem is proved.

4. Solvability Conditions for the Boundary-Value Problem (1.1), (1.2)

The results obtained make it possible to determine sufficient coefficient conditions of regular solvability for the boundary-value problem (1.1), (1.2). In particular, the following main theorem is true.

Theorem 4.1.

Let the operators https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq240_HTML.gif , be bounded in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq241_HTML.gif so that the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ64_HTML.gif
(41)

is satisfied, where the numbers https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq242_HTML.gif , are as defined in Theorem 3.6. Then the boundary-value problem (1.1), (1.2) is regularly solvable.

Proof.

The boundary-value problem (1.1), (1.2) can be presented in the form of the operator equation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq243_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq244_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq245_HTML.gif . From Theorem 2.1, it follows that the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq246_HTML.gif has a bounded inverse operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq247_HTML.gif which acts from the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq248_HTML.gif into the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq249_HTML.gif . Then, after substitution of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq250_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq251_HTML.gif , the equation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq252_HTML.gif results. Now it must be shown that whenever the conditions of the theorem are met, the norm of the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq253_HTML.gif is less than one. Assuming Theorem 3.6, the following can be obtained:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ65_HTML.gif
(42)
As a result:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ66_HTML.gif
(43)
Then, in this case, the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq254_HTML.gif has an inverse in the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq255_HTML.gif , and it is possible to determine https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq256_HTML.gif from the following formula:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ67_HTML.gif
(44)
Moreover,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ68_HTML.gif
(45)

Thus, the theorem is proved.

Remark 4.2.

The conditions of regular solvability obtained here for the boundary-value problem (1.1), (1.2) are not improvable in terms of the operator coefficients of (1.1).

Following is an example in which the conditions of Theorem 4.1 are verified. Consider the following problem on the semi-axis https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq257_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ69_HTML.gif
(46)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq258_HTML.gif are bounded on segment https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq259_HTML.gif functions, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq260_HTML.gif , which is a partial case of problem (1.1), (1.2). On the condition that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_Equ70_HTML.gif
(47)

the given problem has a unique solution in the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F710386/MediaObjects/13661_2009_Article_876_IEq261_HTML.gif .

Declarations

Acknowledgment

The authors have dedicated this paper, in gratitude for useful consultation in their work, as a sign of deep respect to the memory of the Academician of the Azerbaijan National Academy of Sciences, Professor M. G. Gasymov.

Authors’ Affiliations

(1)
Institute of Mathematics and Mechanics of NAS of Azerbaijan
(2)
Baku State University

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Copyright

© A.R. Aliev and A.A. Gasymov. 2009

This article is published under license to BioMed Central Ltd. 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.