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Well-posedness of a boundary value problem for a class of third-order operator-differential equations

Araz R Aliev12 and Ahmed L Elbably13*

Author Affiliations

1 Baku State University, 23 Z. Khalilov St., Baku, 1148, Azerbaijan

2 Institute of Mathematics and Mechanics of NAS of Azerbaijan, 9 B. Vahabzadeh St., Baku, 1141, Azerbaijan

3 Helwan University, Ain Helwan, Cairo, 11795, Egypt

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Boundary Value Problems 2013, 2013:140  doi:10.1186/1687-2770-2013-140

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/140


Received:26 July 2012
Accepted:9 May 2013
Published:30 May 2013

© 2013 Aliev and Elbably; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates the well-posedness of a boundary value problem on the semiaxis for a class of third-order operator-differential 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 Sobolev-type space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M1">View MathML</a>. These conditions are expressed in terms of the operator coefficients of the investigated equation. We find relations between the estimates of the norms of intermediate derivatives operators in the subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M1">View MathML</a> and the solvability conditions. Furthermore, we calculate the exact values of these norms. The results are illustrated with an example of the initial-boundary value problems for partial differential equations.

MSC: 34G10, 47A50, 47D03, 47N20.

Keywords:
well-posed and unique solvability; operator-differential equation; multiple characteristic; self-adjoint operator; the Sobolev-type space; inter-mediate derivatives operators; factorization of pencils

1 Introduction

The paper is dedicated to the formulation and study of the well-posedness of a boundary value problem for a class of third-order operator-differential 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M4">View MathML</a>, A is a self-adjoint positive definite operator on H (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M6">View MathML</a>, E is the identity operator), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M7">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M8">View MathML</a>) is the scale of Hilbert spaces generated by the operator A, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M11">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M12">View MathML</a> we consider that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M15">View MathML</a>. Here the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M16">View MathML</a> is determined from the spectral decomposition of the operator A, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M17">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M18">View MathML</a> is the resolution of the identity for A.

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M19">View MathML</a> we denote the Hilbert space of all vector-functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M20">View MathML</a> defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M21">View MathML</a> with values in H and the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M22">View MathML</a>

Similarly, we define the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M23">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M24">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M25">View MathML</a>

Define the following spaces:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M26">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M27">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M1">View MathML</a> become Hilbert spaces with respect to the norms

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M29">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M30">View MathML</a>

respectively.

Consider the following subspaces of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M1">View MathML</a>:

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 third-order operator-differential equation whose principal part has multiple characteristic:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M33">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M34">View MathML</a>, A is the self-adjoint positive definite operator defined above, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M36">View MathML</a> are linear, in general, unbounded operators on H. Assuming <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M37">View MathML</a>, we attach to equation (1.1) boundary conditions at zero of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M38">View MathML</a>

(1.2)

Definition 1.1 If the vector function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M37">View MathML</a> satisfies equation (1.1) almost everywhere in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M40">View MathML</a>, then it is called a regular solution of equation (1.1).

Definition 1.2 If for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M34">View MathML</a> there exists a regular solution of equation (1.1) satisfying the boundary conditions (1.2) in the sense of relation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M43">View MathML</a> and the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M44">View MathML</a>

then we say that problem (1.1), (1.2) is regularly solvable.

The solvability of boundary value problems for operator-differential 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., [4-11]) 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 well-posedness 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, well-posed solvability of the Cauchy problem and non-local boundary value problems for operator-differential 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 above-mentioned works, unlike equation (1.1), consider the operator-differential equations with a simple characteristic. Although similar matters of solvability and related problems have already been studied for fourth-order operator-differential 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 operator-differential equations with multiple characteristic, including those of third-order. 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M47">View MathML</a> on the space .

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M49">View MathML</a> denote the operator acting from the space to the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M51">View MathML</a> as follows:

Then the following theorem holds.

Theorem 2.1The operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M49">View MathML</a>is an isomorphism between the spacesand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M23">View MathML</a>.

Proof First, we note that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M56">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M57">View MathML</a> and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M58">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M59">View MathML</a> (see, e.g., [23]), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M60">View MathML</a> is the strongly continuous semi-group of bounded operators generated by the operator −A.

Obviously, the homogeneous equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M61">View MathML</a> has only the trivial solution in . But the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M63">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M34">View MathML</a> has a solution of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M66">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M67">View MathML</a>

In fact, such a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M20">View MathML</a> satisfies the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M69">View MathML</a>

and the conditions at zero <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M70">View MathML</a>, therefore, it belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M1">View MathML</a> (see, e.g., [4,5]).

Let us now show the boundedness of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M49">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M73">View MathML</a>

(2.1)

Since ,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M75">View MathML</a>

then (2.1) takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M76">View MathML</a>

Further, by theorems on intermediate derivatives and traces [[3], Ch.1], we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M77">View MathML</a>

Thus, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M49">View MathML</a> is bounded and bijective from the space to the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M23">View MathML</a>. Therefore, due to the Banach inverse operator theorem, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M49">View MathML</a> is an isomorphism between these spaces. The theorem is proved. □

Corollary 2.2It follows from Theorem 2.1 that the norm<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M82">View MathML</a>on the spaceis equivalent to the initial norm<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M84">View MathML</a>.

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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86">View MathML</a>be bounded onHand let the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M87">View MathML</a>act from the spaceto the space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M23">View MathML</a>as follows:

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M87">View MathML</a>is also bounded.

Proof Since for , by virtue of the intermediate derivatives theorem [[3], Ch.1], we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M93">View MathML</a>

The lemma is proved. □

Theorem 2.4Let the operators<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M94">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86">View MathML</a>be bounded onHand the following inequality hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M96">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M98">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M99">View MathML</a>, . Since by Theorem 2.1 the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M49">View MathML</a> has the bounded inverse <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M102">View MathML</a> which acts from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M23">View MathML</a> to , then, after the replacement <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M105">View MathML</a>, we obtain the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M106">View MathML</a> in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M23">View MathML</a>.

Under the conditions of Theorem 2.4, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M108">View MathML</a>

Therefore, if the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M109">View MathML</a> holds, then the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M110">View MathML</a> is invertible and we can define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M20">View MathML</a> by the formula <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M112">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M113">View MathML</a>

The theorem is proved. □

Naturally, there arises a problem of finding exact values or estimates for the numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86">View MathML</a>, and it is very important for extending the class of operator-differential equations of the form (1.1) for which our boundary value problem is solvable. We will make the calculations for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86">View MathML</a> in Section 4.

3 On spectral properties of some polynomial operator pencils and basic equalities for the functions in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M118">View MathML</a>

Consider the following polynomial operator pencils depending on the real parameter β:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M119">View MathML</a>

(3.1)

We need to clarify considering naturally arising pencils (3.1). Obviously, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M120">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M121">View MathML</a>

Since for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M122">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M123">View MathML</a>

then, as a result, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M124">View MathML</a>

(3.2)

If we use the Fourier transform in (3.2), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M125">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M126">View MathML</a> is the Fourier transform of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M20">View MathML</a>. Therefore, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M128">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M129">View MathML</a>

That is why to estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86">View MathML</a>, it is necessary to study some properties of pencils (3.1).

The following theorem on factorization of pencils (3.1) holds.

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M132">View MathML</a>. Then polynomial operator pencils (3.1) are invertible on the imaginary axis and the following representations are true:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M133">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M134">View MathML</a>

here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M136">View MathML</a>, the numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M138">View MathML</a>are positive and satisfy the following systems of equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M139">View MathML</a>

(3.3)

Proof It is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M140">View MathML</a>

(3.4)

are the characteristic polynomial operator pencils (3.1), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M141">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M142">View MathML</a> denotes the spectrum of the operator A). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M144">View MathML</a>. Then characteristic polynomials (3.4) satisfy the following relations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M145">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M146">View MathML</a>

we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M147">View MathML</a>

(3.5)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M132">View MathML</a>. Inequalities (3.5) imply that polynomials (3.4) have no roots on the imaginary axis for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M149">View MathML</a>. Besides, it can be seen from (3.4) that each of the characteristic polynomial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M150">View MathML</a> has exactly three roots in the left half-plane for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M151">View MathML</a>. Since polynomials (3.4) are homogeneous with respect to the arguments λ and σ, then the following factorization is true for them:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M152">View MathML</a>

(3.6)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M153">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M136">View MathML</a>, and the numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M138">View MathML</a> are positive according to Vieta’s formulas and satisfy systems of equations (3.3) derived from (3.6) during the comparison of same degree coefficients. Further, using the spectral decomposition of the operator A, from equalities (3.6) we obtain the assertions of the theorem. The theorem is proved. □

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: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M158">View MathML</a> will denote the linear set of infinitely differentiable functions with values in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M159">View MathML</a> and compact support in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M40">View MathML</a>. As is well known, the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M158">View MathML</a> is everywhere dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M1">View MathML</a> (see [[3], Ch.1]).

Theorem 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M132">View MathML</a>. Then, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M164">View MathML</a>, the following relation holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M165">View MathML</a>

(3.7)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M166">View MathML</a>

Proof It suffices to prove the theorem for functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M167">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M168">View MathML</a>

Integrating by parts, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M169">View MathML</a>

(3.8)

Making similar calculations for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M170">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M171">View MathML</a>

(3.9)

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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M132">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M174">View MathML</a>

(3.10)

Corollary 3.4Ifand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M132">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M177">View MathML</a>

(3.11)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M178">View MathML</a>is obtained from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M179">View MathML</a>by discarding the first two rows and columns, here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M180">View MathML</a>.

4 On the values of the numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86">View MathML</a>

It is easy to check that the norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M183">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M184">View MathML</a> are equivalent on . Then it follows from the theorem on intermediate derivatives [[3], Ch.1] that the following numbers are finite:

Lemma 4.1<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M187">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86">View MathML</a>.

Proof Passing to the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M189">View MathML</a> in (3.10), we see that, for any function , the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M191">View MathML</a>

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M192">View MathML</a>. We need to show that here we have the equality. To do this, it suffices to show that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M193">View MathML</a>, there exists a function such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M195">View MathML</a>

Note that the procedure of constructing such functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M196">View MathML</a> is thoroughly described in [17] (in addition, the one for fourth-order equations with multiple characteristic is available in [20]). This method is applicable to our case, too. Therefore, we omit the respective part of the proof. So lemma is proved. □

Remark 4.2 Since , then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M198">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86">View MathML</a>. Therefore, there arises the question: When do we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M200">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86">View MathML</a>?

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M202">View MathML</a> the root of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M203">View MathML</a> in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M204">View MathML</a> if such exists.

Theorem 4.3The following relation holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M205">View MathML</a>

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M200">View MathML</a>, then it follows from equation (3.11) that for any function and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M132">View MathML</a>, the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M209">View MathML</a>

(4.1)

Now let us consider the Cauchy problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M210">View MathML</a>

(4.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M211">View MathML</a>

(4.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M212">View MathML</a>

(4.4)

Since by Theorem 3.1, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M132">View MathML</a>, the polynomial operator pencil <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M214">View MathML</a> is of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M215">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M216">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M136">View MathML</a>, then Cauchy problem (4.2)-(4.4) has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M218">View MathML</a>, which can be expressed as:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M219">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M220">View MathML</a> are uniquely determined by the conditions at zero (4.3), (4.4). Therefore, if we rewrite the inequality (4.1) for function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M221">View MathML</a>, then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M132">View MathML</a> we will have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M223">View MathML</a>

This means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M224">View MathML</a>, and therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M225">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M226">View MathML</a>. Now let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M225">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M226">View MathML</a>. Then it follows from (3.11) that for any function and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M149">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M231">View MathML</a>

Passing here to the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M189">View MathML</a>, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M233">View MathML</a>, and hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M200">View MathML</a>.

Continuing the proof of the theorem, we suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M235">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M236">View MathML</a>. Note that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M237">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M238">View MathML</a>

Therefore, from (3.11) we find that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M237">View MathML</a>, the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M240">View MathML</a>

Applying this inequality again to the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M20">View MathML</a> of Cauchy problem (4.2)-(4.4), we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M225">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M243">View MathML</a>. On the other hand, it follows from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M114">View MathML</a> that, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M245">View MathML</a>, there exists a function such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M247">View MathML</a>

Taking into account this inequality in (3.11), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M248">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M249">View MathML</a>. Therefore, there exists a vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M250">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M251">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M252">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M245">View MathML</a>. And since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M254">View MathML</a> is a continuous function of the argument β in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M255">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M256">View MathML</a>. It follows from these arguments that the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M203">View MathML</a> has a root in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M258">View MathML</a>. Now let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M203">View MathML</a> have a root in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M204">View MathML</a>. This means that the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M225">View MathML</a> cannot be satisfied for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M132">View MathML</a>. Therefore, according to our earlier reasonings in the proof of this theorem, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M263">View MathML</a>. Obviously, for the root <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M264">View MathML</a> of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M203">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M266">View MathML</a>, because the proof of the theorem for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M243">View MathML</a> implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M225">View MathML</a>. And since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M256">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M270">View MathML</a>. The theorem is proved. □

Remark 4.4 From Theorem 4.3, it becomes clear that to find the numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86">View MathML</a>, we must solve the equations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M203">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86">View MathML</a>, together with systems (3.3) respectively. In this case, it is necessary to take into account the properties of the numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M275">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M276">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86">View MathML</a>.

The following theorem holds.

Theorem 4.5<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M278">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M279">View MathML</a>.

Proof In view of Remark 4.4, in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M280">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M278">View MathML</a> due to the negativity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M282">View MathML</a>, despite <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M283">View MathML</a>. In the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M284">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M285">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M286">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M287">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M288">View MathML</a>, respectively. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M289">View MathML</a>. The theorem is proved. □

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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M290">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M86">View MathML</a>be bounded onHand the following inequality hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M292">View MathML</a>

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 initial-boundary value problem for a partial differential equation.

Example 5.2 On the half-strip <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M293">View MathML</a>, consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M294">View MathML</a>

(5.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M295">View MathML</a>

(5.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M296">View MathML</a>

(5.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M297">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M298">View MathML</a> are bounded functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M299">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M300">View MathML</a>. Note that problem (5.1)-(5.3) is a special case of boundary value problem (1.1), (1.2). In fact, here we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M301">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M302">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M303">View MathML</a>. The operator A is defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M304">View MathML</a> by the relation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M305">View MathML</a> and the conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M306">View MathML</a>.

Applying Theorem 5.1, we obtain that under the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M307">View MathML</a>

problem (5.1)-(5.3) has a unique solution in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M308">View MathML</a>.

Remark 5.3 Using the same procedure, we can obtain similar results for equation (1.1) on the semiaxis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M40">View MathML</a> with boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M310">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/140/mathml/M311">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

References

  1. Barenblatt, GI, Zheltov, YP, Kochina, IN: On the fundamental representations of the theory of filtration of homogeneous fluids in fissured rocks. Prikl. Mat. Meh.. 24(5), 852–864 (J. Appl. Math. Mech.) (1960)

  2. Pilipchuk, VN: On essentially nonlinear dynamics of arches and rings. Prikl. Mat. Meh.. 46(3), 461–466 (J. Appl. Math. Mech.) (1982)

  3. Lions, JL, Magenes, E: Non-Homogeneous Boundary Value Problems and Applications, Dunod, Paris (1968) Mir, Moscow (1971); Springer, Berlin (1972)

  4. Gasymov, MG: On the theory of polynomial operator pencils. Dokl. Akad. Nauk SSSR. 199(4), 747–750 (Sov. Math. Dokl.) (1971)

  5. Gasymov, MG: The solubility of boundary-value problems for a class of operator-differential equations. Dokl. Akad. Nauk SSSR. 235(3), 505–508 (Sov. Math. Dokl.) (1977)

  6. Gorbachuk, ML: Completeness of the system of eigenfunctions and associated functions of a nonself-adjoint boundary value problem for a differential-operator equation of second order. Funct. Anal. Appl.. 7(1), 68–69 (translated from Funkc. Anal. Prilozh. 7(1), 58-59 (1973)) (1973)

  7. Dubinskii, YA: On some differential-operator equations of arbitrary order. Mat. Sb.. 90(132)(1), 3–22 (Math. USSR Sb.) (1973)

  8. Jakubov, SJ: Correctness of a boundary value problem for second order linear evolution equations. Izv. Akad. Nauk Azerb. SSR, Ser. Fiz.-Teh. Mat. Nauk. 2, 37–42 (1973)

  9. Kostyuchenko, AG, Shkalikov, AA: Self-adjoint quadratic operator pencils and elliptic problems. Funct. Anal. Appl.. 17(2), 38–61 (translated from Funkc. Anal. Prilozh. 17(2), 109-128 (1983)) (1983)

  10. Mirzoev, SS: Multiple completeness of root vectors of polynomial operator pencils corresponding to boundary-value problems on the semiaxis. Funct. Anal. Appl.. 17(2), 84–85 (translated from Funkc. Anal. Prilozh. 17(2), 151-153 (1983)) (1983)

    (translated from Funkc. Anal. Prilozh. 17(2), 151-153 (1983))

    Publisher Full Text OpenURL

  11. Aliev, AR, Babayeva, SF: On the boundary value problem with the operator in boundary conditions for the operator-differential equation of the third order. J. Math. Phys. Anal. Geom.. 6(4), 347–361 (2010)

  12. Shkalikov, AA: Elliptic equations in Hilbert space and associated spectral problems. J. Sov. Math.. 51(4), 2399–2467 (translated from Tr. Semin. Im. I.G. Petrovskogo 14, 140-224 (1989)) (1990)

    (translated from Tr. Semin. Im. I.G. Petrovskogo 14, 140-224 (1989))

    Publisher Full Text OpenURL

  13. Shkalikov, AA: Strongly damped pencils of operators and solvability of the corresponding operator-differential equations. Math. USSR Sb.. 63(1), 97–119 (translated from Mat. Sb. 135(177)(1), 96-118 (1988)) (1989)

    (translated from Mat. Sb. 135(177)(1), 96-118 (1988))

    Publisher Full Text OpenURL

  14. Gorbachuk, ML, Gorbachuk, VI: On well-posed solvability in some classes of entire functions of the Cauchy problem for differential equations in a Banach space. Methods Funct. Anal. Topol.. 11(2), 113–125 (2005)

  15. Agarwal, RP, Bohner, M, Shakhmurov, VB: Linear and nonlinear nonlocal boundary value problems for differential-operator equations. Appl. Anal.. 85(6-7), 701–716 (2006). Publisher Full Text OpenURL

  16. Mirzoev, SS: Conditions for the well-defined solvability of boundary-value problems for operator differential equations. Dokl. Akad. Nauk SSSR. 273(2), 292–295 (Sov. Math. Dokl.) (1983)

  17. Mirzoyev, SS: On the norms of operators of intermediate derivatives. Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci.. 23(1), 157–164 (2003)

  18. Aliev, AR: Boundary-value problems for a class of operator differential equations of high order with variable coefficients. Math. Notes - Ross. Akad.. 74(6), 761–771 (translated from Mat. Zametki 74(6), 803-814 (2003)) (2003)

  19. Aliev, AR, Mirzoev, SS: On boundary value problem solvability theory for a class of high-order operator-differential equations. Funct. Anal. Appl.. 44(3), 209–211 (translated from Funkc. Anal. Prilozh. 44(3), 63-65 (2010)) (2010)

    (translated from Funkc. Anal. Prilozh. 44(3), 63-65 (2010))

    Publisher Full Text OpenURL

  20. Aliev, AR, Gasymov, AA: On the correct solvability of the boundary-value problem for one class operator-differential equations of the fourth order with complex characteristics. Bound. Value Probl.. 2009, Article ID 710386 (2009)

  21. Aliev, AR: On a boundary-value problem for one class of differential equations of the fourth order with operator coefficients. Azerb. J. Math.. 1(1), 145–156 (2011)

  22. Favini, A, Yakubov, Y: Regular boundary value problems for elliptic differential-operator equations of the fourth order in UMD Banach spaces. Sci. Math. Jpn.. 70(2), 183–204 (2009)

  23. Gorbachuk, VI, Gorbachuk, ML: Boundary Value Problems for Operator-Differential Equations, Naukova Dumka, Kiev (1984)