On the constructive investigation of a class of linear boundary value problems for n th order differential equations with deviating arguments
© Rumyantsev; licensee Springer. 2014
Received: 12 December 2013
Accepted: 21 February 2014
Published: 12 March 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].
(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 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, a Γ and v Γ, 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 a Γ 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
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.
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 .
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 .
where , , are rational constants.
Definition 5 A bounded linear operator is said to possess the property (is computable) if it maps into itself.
if the coefficients are the elements of the set and the functions are computable over the partition (6) in the generalized sense.
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
for and .
Approximation of the
, , if is strictly increasing on ;
, , if is strictly decreasing on .
(excluding duplicate elements). By construction, the , , , are computable over the partition (12) in the generalized sense.
Approximation of the functional
Note that the operators and are computable by construction.
Study of the principal boundary value problem
A detailed description of the construction of the functions and the estimations of , , is given in .
Analysis of solvability
where the matrix Γ is defined by (4). Thus we arrived at the following.
hold. Then the boundary value problem (8) is correctly solvable.
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. □
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.
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