SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

This article is part of the series A Tribute to Professor Ivan Kiguradze.

Open Access Research

Linear overdetermined boundary value problems in Hilbert space

Vladimir P Maksimov

Author Affiliations

Perm State University, Bukirev Street 15, Perm 614990, Russia

Boundary Value Problems 2014, 2014:140  doi:10.1186/s13661-014-0140-4


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/140


Received:13 December 2013
Accepted:26 May 2014
Published:11 July 2014

© 2014 Maksimov; licensee Springer

Open Access 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 credited.

Abstract

The general linear boundary value problem for an abstract functional differential equation is considered in the case that the number of boundary conditions is greater than the dimension of the null-space to the corresponding homogeneous equation. Sufficient conditions of the solvability of the problem are obtained. A case of a functional differential system with aftereffect is considered separately.

Keywords:
functional differential equations; boundary value problems; ill-posed problems

Introduction

Linear boundary value problems (BVPs) for differential equations with ordinary derivatives that lack the everywhere and unique solvability are met with in various applications. Among these applications are some problems in oscillation theory (see, for examples, [[1]]) and economic dynamics [[2]]. Results on the solvability and solutions representation for these BVPs are widely used as an instrument of investigating weakly nonlinear BVPs [[3]]. General results concerning linear BVPs for an abstract functional differential equation (AFDE) are given in [[4]]. In this paper, we consider a case that the number of linearly independent boundary conditions is greater than the dimension of the null-space of the corresponding homogeneous equation and obtain sufficient conditions of the solvability without recourse to the adjoint BVP and an extension of the original BVP. Our approach is based in essence on the assumption that the derivative of the solution does belong to a Hilbert space. Then we consider a system of functional differential equations that, formally speaking, is a concrete realization of the AFDE and, on the other hand, covers many kinds of dynamic models with aftereffect (integro-differential, delayed differential, differential difference) [[5]–[7]]. For this case sufficient conditions are derived in an explicit form.

Preliminaries

In this section, we give some necessary facts from the theory of AFDE [[4], [8], [9]]. The linear abstract functional differential equation is the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M1">View MathML</a>

(1)
where is a linear bounded operator, D and B are Banach spaces such that D is isomorphic to the direct product . Let us denote by an isomorphism and let .

A linear operator acting from the direct product of the Banach spaces B and into a Banach space D is defined by a pair of linear operators and in such a way that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M2">View MathML</a>

(2)

A linear operator acting from a space D into a direct product is defined by a pair of linear operators and so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M3">View MathML</a>

(3)

Under the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M4">View MathML</a>

(4)
the space is Banach (here and in what follows, denotes a norm in ). If the bounded operator is the inverse to the bounded operator , then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M5">View MathML</a>

(5)
, , .

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M6">View MathML</a>

(6)
where I is the identity operator. We will identify the finite-dimensional operator with a vector , , such that , .

Denote the components of the vector functional r by . If is a linear vector functional, and is a vector with components , then lX denotes the -matrix, whose columns are the values of the vector functional l on the components of , ; .

Applying ℒ to both parts of (2.4), we obtain the decomposition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M7">View MathML</a>

(7)
where is the principal part, and is the finite-dimensional part of ℒ. Similarly, by application of to the two parts of (2.4), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M8">View MathML</a>

(8)
where is a linear bounded vector functional.

Let be a linear bounded vector functional with linearly independent components, . The system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M9">View MathML</a>

(9)
is called a linear boundary value problem.

Taking into account (2.5) and (2.6), we can rewrite BVP (2.7) in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M10">View MathML</a>

(10)
The operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M11">View MathML</a>

(11)
is the adjoint one to the operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M12">View MathML</a>

(12)
Taking into account the isomorphism between the spaces and , we therefore call the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M13">View MathML</a>

(13)
the adjoint equation to the problem (2.8).

In the sequel it is assumed that the so-called principal BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M14">View MathML</a>

(14)
is uniquely solvable for any , . Recall that in such a case we have the representation [[8]] (Theorem 1.16, p.11)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M15">View MathML</a>

(15)
to the solution of (2.10) with X called the fundamental vector and G called the Green operator.

Problem (2.7) covers a wide class of BVPs for ordinary differential systems, differential delay systems, some singular and impulsive systems [[7]]. This problem is well-posed if . In such a situation, BVP (2.7) is uniquely solvable for any and if and only if the matrix

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M16">View MathML</a>

(16)
where is the jth element of X, is nonsingular, i.e..

In the case that BVP (2.7) lacks the everywhere and unique solvability, namely, it is solvable if and only if the right-hand side is orthogonal to all the solutions of the homogeneous adjoint equation (2.9), i.e. [[8]] (Corollary 1.15, p.11).

In what follows we derive conditions of solvability for (2.7) in a more explicit form without recourse to the adjoint BVP. Our approach is based in essence on the assumption that the space B is a Hilbert space H with an inner product .

A case of AFDE

Consider BVP (2.7) under the assumption that and the system , can be split into two subsystems and such that the BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M17">View MathML</a>

(17)
is uniquely solvable. Without loss of generality we will consider that is formed by the first n components of and the elements of in (3.1) are the corresponding components of γ. Thus will stand for the final components of , and elements of are defined as the final components of γ. For , , we put . Thus in the cases that a vector V is expressed by a complicated formula we will use instead of to indicate the jth component of V.

Define the vector functional , by the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M18">View MathML</a>

(18)
and preserve the symbol for an element of H that generates the functional : for any

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M19">View MathML</a>

(19)

Let us define the -matrix by the equalities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M20">View MathML</a>

(20)

Theorem 1

LetWbe nonsingular. Then BVP (2.7) is solvable for anyof the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M21">View MathML</a>

(21)
where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M22">View MathML</a>

(22)
andis arbitrary element that is orthogonal to each, .

Proof

The general solution of the equation has the representation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M23">View MathML</a>

(23)
with an arbitrary . Apply to both parts of (3.2):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M24">View MathML</a>

(24)
By the unique solvability of BVP (3.1) the condition holds, therefore the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M25">View MathML</a>

(25)
is uniquely solvable with respect to α:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M26">View MathML</a>

(26)
Hence, for any ,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M27">View MathML</a>

(27)
is a solution to BVP (3.1). Now we shall search for such that the corresponding x of the form (3.4) satisfies the equality . For this purpose, apply to both parts of (3.4):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M28">View MathML</a>

(28)
Rewrite this as the equation with respect to :

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M29">View MathML</a>

(29)
The left-hand side of (3.5) defines a linear bounded vector functional λ over the space H:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M30">View MathML</a>

(30)
, with components that are linear bounded functionals. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M31">View MathML</a>

(31)
Thus, for any , the representation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M32">View MathML</a>

(32)
holds, where , are constants and φ is orthogonal to : for any . Let us use the substitution (3.7) as applied to (3.5):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M33">View MathML</a>

(33)
Put . Then (3.8) takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M34">View MathML</a>

(34)
and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M35">View MathML</a>

(35)
To complete the proof, it remains now to substitute c into (3.7). □

A case of systems with aftereffect

In this section, we consider a system of functional differential equations with aftereffect that, formally speaking, is a concrete realization of the AFDE, and, on the other hand, it covers many kinds of dynamic models with aftereffect (integro-differential, delayed differential, differential difference) [[2], [6], [10]].

Despite the case considered in Sections 2, 3 is more general, we derive here conditions of the solvability in detail since the corresponding transformations are based on the properties of operators and spaces as applied to the case under consideration.

Let us introduce the functional spaces where operators and equations are considered. Fix a segment . By we denote the Hilbert space of square summable functions endowed with the inner product (⋅ is the symbol of transposition). The space is the space of absolutely continuous functions such that with the norm , where stands for the norm of . Thus we have here , , , and , , , , (see (2.2)-(2.4)).

Consider the functional differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M36">View MathML</a>

(36)
where the linear bounded operator is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M37">View MathML</a>

(37)
the elements of the kernel are measurable on the set and such that , , , -matrix A has elements that are square summable on . Therefore, we have here , (see (2.5)).

Recall that, under some natural assumptions, the following equations can be rewritten in the form (4.1):

the differential equation with concentrated delay

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M38">View MathML</a>

(38)
(here, for any measurable function such that , , stands for a given function if );

the differential equation with distributed delay

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M39">View MathML</a>

(39)
(with the Stieltjes integral);

the integro-differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M40">View MathML</a>

(40)

In what follows we will use some results from [[5], [8], [11], [12]] concerning (4.1). The homogeneous equation (4.1) (, ) has the fundamental -matrix :

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M41">View MathML</a>

(41)
where is the identity -matrix, each column of the -matrix is a unique solution to the Cauchy problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M42">View MathML</a>

(42)
where is the ith column of A.

The solution of (4.1) with the initial condition has the representation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M43">View MathML</a>

(43)
where is the Cauchy matrix of the operator ℒ. This matrix can be defined (and constructed) as the solution to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M44">View MathML</a>

(44)
under the condition .

The matrix is expressed in terms of the resolvent kernel of the kernel . Namely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M45">View MathML</a>

(45)
Thus , and the above equation for is the well-known relationship between the kernel and its resolvent kernel .

The general solution of (4.1) has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M46">View MathML</a>

(46)
with an arbitrary .

The general linear BVP is the system (4.1) supplemented by linear boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M47">View MathML</a>

(47)
where is a linear bounded vector functional. Let us recall the representation of :

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M48">View MathML</a>

(48)
Here Ψ is a constant -matrix, Φ is -matrix with elements that are square summable on . We assume that the components , of are linearly independent.

BVP (4.1), (4.4) is well-posed if . In such a situation, the BVP is uniquely solvable for any and if and only if the matrix

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M49">View MathML</a>

(49)
where is the jth column of X, is nonsingular, i.e.. It should be noted that this condition cannot be verified immediately because X cannot be (as a rule) evaluated explicitly. In addition, even if X were known, then the elements of ℓX, generally speaking, could not be evaluated explicitly. By the theorem about inverse operators, the matrix ℓX is invertible if one can find an invertible matrix Γ such that . As has been shown in [[13]], such a matrix Γ for the invertible matrix ℓX always can be found among the matrices , where is a vector functional near , and is an approximation of X. That is why the basis of the so-called constructive study of linear BVPs includes a special technique of approximate constructing the solutions to FDE with guaranteed explicit error bounds as well as the reliable computing experiment (RCE) [[2], [10], [13]] which opens a way to the computer-assisted study of BVPs.

We assume in the sequel that and the system , can be split into two subsystems and such that the BVP

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M50">View MathML</a>

(50)
is uniquely solvable. Without loss of generality we will consider that is formed by the first n components of and the elements of in (4.6) are the corresponding components of γ. Thus will stand for the final components of , and elements of are defined as the final components of γ. Let us write in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M51">View MathML</a>

(51)
where and are the corresponding rows of and Ψ, respectively. Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M52">View MathML</a>

(52)

Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M53">View MathML</a>

(53)
and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M54">View MathML</a>

(54)

Theorem 2

Let the matrix, whereFis defined by (4.10), be nonsingular. Then BVP (4.1), (4.4) is solvable for allof the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M55">View MathML</a>

(55)
where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M56">View MathML</a>

(56)
andis an arbitrary function that is orthogonal to each column of:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M57">View MathML</a>

(57)

Proof

Let us apply to both parts of (4.3):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M58">View MathML</a>

(58)
In virtue of the unique solvability of BVP (4.6), the condition holds, therefore the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M59">View MathML</a>

(59)
is uniquely solvable with respect to α:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M60">View MathML</a>

(60)
Hence, for any ,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M61">View MathML</a>

(61)
is a solution to BVP (4.6). Now we shall search for such that the corresponding x of the form (4.11) satisfies the equality . For this purpose, apply to both parts of (4.11):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M62">View MathML</a>

(62)
or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M63">View MathML</a>

(63)
Now we show that the left-hand side of the latter equality can be written, for all , in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M64">View MathML</a>

(64)
with a -matrix F whose columns belong to .

An explicit form of F is simple to derive by elementary transformations taking into account (4.5) and the properties of the Cauchy matrix. To do this, first note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M65">View MathML</a>

(65)
This follows from (4.2) and the equality . Next, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M66">View MathML</a>

(66)
Notice that the interchangeability of the order of integration in the iterated integrals above is proved in [[11]]. In a similar way,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M67">View MathML</a>

(67)
Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M68">View MathML</a>

(68)
Now it remains to find such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M69">View MathML</a>

(69)
As is well known, any can be represented in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/140/mathml/M70">View MathML</a>

(70)
with and such that . By virtue of the condition , we obtain after substitution of (4.14) into (4.13) that the vector gives the corresponding f (see (4.14)) that solves (4.13). This completes the proof. □

In view of Theorem 2, the solvability of BVP (4.1), (4.4) can be investigated on the base of the reliable computing experiment [[2], [10], [13]]. A somewhat different approach to the study of BVP (4.1), (4.4) with is proposed in [[14]].

Competing interests

The author declares that he has no competing interests.

Acknowledgements

The author thanks the referees for their careful reading of the manuscript and useful comments. The author acknowledges the support by the company Prognoz, Perm.

References

  1. Nayfeh, AN, Mook, DT: Nonlinear Oscillations, Wiley, New York (1970)

  2. Maksimov, VP, Rumyantsev, AN: Boundary value problems and problems of pulse control in economic dynamics: constructive study. Russ. Math.. 37, 48–62 (1993)

  3. Boichuk, AA: Constructive Methods of Analysis of Boundary Value Problems, Naukova Dumka, Kyiv (1990)

  4. Azbelev, NV, Rakhmatullina, LF: Theory of linear abstract functional differential equations and applications. Mem. Differ. Equ. Math. Phys.. 8, 1–102 (1996)

  5. Azbelev, NV, Maksimov, VP, Rakhmatullina, LF: Introduction to the Theory of Functional Differential Equations, Nauka, Moscow (1991)

  6. Maksimov, VP: Theory of functional differential equations and some problems in economic dynamics. In: Agarwal R, Perera K (eds.) Proceedings of the Conference on Differential and Difference Equations and Applications, pp. 757–765. Hindawi Publishing Corporation, New York (2006)

  7. Azbelev, NV, Maksimov, VP, Simonov, PM: Theory of functional differential equations and applications. Int. J. Pure Appl. Math.. 69, 203–235 (2011)

  8. Azbelev, NV, Maksimov, VP, Rakhmatullina, LF: Introduction to the Theory of Functional Differential Equations: Methods and Applications, Hindawi Publishing Corporation, New York (2007)

  9. Azbelev, NV, Maksimov, VP, Rakhmatullina, LF: Elements of the Contemporary Theory of Functional Differential Equations. Methods and Applications, Institute of Computer-Assisted Studies, Moscow (2002)

  10. Maksimov, VP, Rumyantsev, AN: Reliable computing experiment in the study of generalized controllability of linear functional differential systems. In: Uvarova L, Latyshev A (eds.) Mathematical Modelling. Problems, Methods, Applications, pp. 91–98. Kluwer Academic, New York (2002)

  11. Maksimov, VP: Cauchy’s formula for a functional-differential equation. Differ. Equ.. 13, 405–409 (1977)

  12. Maksimov, VP: Questions of the General Theory of Functional Differential Equations, Perm State University, Perm (2003)

  13. Rumyantsev, AN: Reliable Computing Experiment in the Study of Boundary Value Problems, Perm State University, Perm (1999)

  14. Maksimov, VP, Chadov, AL: The constructive investigation of boundary-value problems with approximate satisfaction of boundary conditions. Russ. Math.. 54, 71–74 (2010). Publisher Full Text OpenURL