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On the constructive investigation of a class of linear boundary value problems for n th order differential equations with deviating arguments
Boundary Value Problems volume 2014, Article number: 53 (2014)
A constructive technique for the study of boundary value problems for n th order differential equations with deviating arguments is described. A computer-assisted proof of the correct solvability of the considered problem is given.
MSC:34B05, 34K06, 65L10, 34K10, 65L70.
Let us cite some facts from the theory of functional differential equations  and the general description of the constructive approach to the investigation of boundary value problems for such equations [2, 3].
Consider the linear boundary value problem
where is a bounded linear operator, is a bounded linear vector functional, and , . denotes the linear space of real columns α, where with the norm ; ; , , denotes the Banach space of measurable functions , , such that
let us fix a collection of points . Put , ; ;
denotes the Banach space of all functions with and the representation:
where ; , with the norm
We assume that the principal boundary value problem
is correctly solvable. Then under these assumptions there exists an fundamental matrix Y for the homogeneous equation:
The following statement holds: the problem (1) is uniquely solvable for any f, α if and only if the matrix Γ,
(each column of the matrix Γ is the result of applying of the functional ℓ to the corresponding column of the matrix Y) is invertible. The problem (2) is correctly solvable for a broad class of equations, including
the ordinary differential equation
and a differential equation with concentrated delays
It should be noted that in the more general case of a differential equation with deviating arguments, the problem (2) does not have this property and some further investigation is required. For example, define the space on the partition and consider the principal boundary value problem
It easy to see that this problem has a unique solution only for .
The key idea of the constructive study of the solvability of the problem (1) is as follows.
Two matrices, aΓ and vΓ, with rational elements are constructed according to a specially developed procedure based on a computer-assisted proof, such that
let denotes the linear space of real -matrices with the norm ; .
The invertibility of the matrix aΓ is verified using exact arithmetic.
If there exists an inverse matrix , it should be checked whether(5)
holds, from which, by the theorem on the inverse operator [, p.207], it follows that the matrix Γ is invertible, i.e., the boundary value problem (1) is correctly solvable.
Further, the suggested general scheme of the constructive investigation will be applied to the boundary value problem for the n th order differential equation with deviating arguments.
A class of functions and operators
The constructive techniques for the study of equations with deviating arguments described below are based on a specific approximation of original problems within the class of computable functions and operators . In what follows, we assume that the spaces and are constructed by means of the partition
where , , are rational numbers. The sets , ; , and the corresponding characteristic functions are defined with respect to the same partition. denotes the Banach space of functions , with , , , and the representation
with the norm
denotes the Banach space of absolutely continuous functions such that , with the norm
Definition 1 A function is said to possess the property (is computable) if its components as well as the components of the functions and take rational values at rational values of their arguments.
Let . The property is satisfied by functions y of the form
where the components , are polynomials with rational coefficients. We denote by the set of all of the form (7).
Definition 2 A function is said to possess the property (is computable) if this function and the functions , and take rational values at rational values of their arguments.
Obviously, the functions with possess the property .
Definition 3 A function is said to be computable over the partition (6) if h possesses the property and for every there exists an integer , , such that as .
An example of a function that is computable over the partition (6) is such that
where , , are rational constants.
Definition 4 A function is said to be computable over the partition (6) in the generalized sense if h possesses the property , and for every , there exists an integer , , such that , as .
An example of a function that is computable over the partition (6) in the generalized sense is such that
where , , are rational constants.
Definition 5 A bounded linear operator is said to possess the property (is computable) if it maps into itself.
An example of an operator that is computable is
if the coefficients are the elements of the set and the functions are computable over the partition (6) in the generalized sense.
Consider the linear boundary value problem
where , , , and , the are continuous and strictly monotonic functions over every , , ,
and , , . denotes the Banach space of functions , defined by the equality
where , , , are continuous functions, and
denotes the adjoint index to p:
Consider the principal boundary value problem corresponding to (8):
under the same assumptions on the problem parameters.
The procedure for the constructive study of the problem (8) consists of the following steps:
approximation of the problem (8) within the class of computable functions and operators,
study of the principal boundary value problem (9),
analysis of its solvability.
Approximation of the problem within the class of computable operators
Fix . Approximate and f on the set by polynomials and with rational coefficients and define the rational error bounds:
for and .
Approximation of the
Find rational approximations of , , , , and define rational vh such that
and define a rational constant . Fix , , and construct the collection of points by the rule:
here the positive integer parameter defines the accuracy of the approximation to , assuming that , . We denote by , , a value of ν such that . For , we denote for the number of values that belong reliably to the inverse image of in the set . We define the elements of the set as follows:
, , if is strictly increasing on ;
, , if is strictly decreasing on .
Next define the pairs of constants , , , such that
We also assume that the following conditions hold:
which can always be satisfied by means of the accuracy of the calculation of the function for the given points. Further, we construct the set , ,
From the order of the points , , it follows that
and denote by the characteristic function of , . We construct the function by the following rule:
Define the set of points by
(excluding duplicate elements). By construction, the , , , are computable over the partition (12) in the generalized sense.
Below suppose that , , , and define the constants and by the equalities
We approximate the functions , by polynomials , with rational coefficients and with rational error bounds , :
Approximation of the functional
The real numbers , , are approximated by rational numbers with rational error bounds . The constants are approximated by rational numbers with rational error bounds , ; ; . On each , , the functions , , are approximated by the polynomials with rational coefficients and with rational error bounds . Define the functions by the equalities
Let us write the boundary value problem approximating the problem (8) as follows:
Note that the operators and are computable by construction.
Study of the principal boundary value problem
The aim of this study is to check whether the problem (9) is correctly solvable, having in mind the computer-assisted proof techniques. This issue is described in detail in . Below suppose that the problem (9) is correctly solvable. Then there exists a fundamental system , of the homogeneous equation
and a fundamental system , , of the homogeneous equation
Every function is defined as a solution of the principal boundary value problem
and every function is defined as a solution of the principal boundary value problem
. Denote by an approximation to the functions and by the approximation error bounds:
A detailed description of the construction of the functions and the estimations of , , is given in .
Analysis of solvability
Denote the elements of the matrices , and as follows:
From definition (18) it follows that
where the matrix Γ is defined by (4). Thus we arrived at the following.
Theorem 1 Let the matrix aΓ defined by (18) be invertible, and let the inequality
hold. Then the boundary value problem (8) is correctly solvable.
Proof Under the condition of the theorem the inequality
holds. By the theorem on the inverse operator the matrix Γ is defined by (4) to be invertible, i.e., the problem (8) is correctly solvable. □
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The author expresses his sincere thanks to the referees for the careful and noteworthy reading of the manuscript and helpful suggestions that improved the manuscript.
The author declares that he has not competing interests.
About this article
- differential equations with deviating arguments
- boundary value problems
- constructive methods
- unique solvability
- approximate solution