Fredholm Boundary-value Problems for Linear Delay Systems Defined by Pairwise Permutable Matrices

The paper deals with a Fredholm boundary value problem for a linear delay system with several delays defined by pairwise permutable constant matrices. The initial value condition is given on a finite interval and the boundary condition is given by a linear vector functional. A sufficient condition for the existence of solutions of this type of boundary value problem is proved. Moreover, a family of linearly independent solutions in an explicit general analytic form is constructed under the assumption that the number of boundary conditions (determined by the dimension of linear vector functional) do not coincide with the number of unknowns of the system of the delay differential equations. The proof of this result is based on a representation of solutions by using the so-called multi-delayed matrix exponential and a method of a pseudo-inverse matrix of the Moore–Penrose type.


Introduction
The aim of the paper is to prove an existence result for the following boundary-value problem: ) where τ 1 , . . ., τ n > 0, (n > 0), τ := max{τ 1 , . . ., τ n } and A, B 1 , . . ., B n are N × N constant permutable matrices such that AB i = B i A, B i B j = B j B i for each i, j ∈ {1, . . ., n} and g(t) is an N-dimensional column-vector, with components in the space L p [0, b] (1 < p < ∞) being functions integrable on [0, b]; ψ : R \ [0, b] → R N is a given N-dimensional columnvector function; α is an m-dimensional constant vector-column, l is an m-dimensional linear vector-functional, defined on the space D p [0, b] of n-dimensional vector-functions absolutely continuous on [0, b]: l = col(l 1 , . . ., l m ) : It is not very difficult to prove that in this space such problems for functional-differential equations are of Fredholm's type with nonzero index (see, e.g., [1,4,5]).
First of all we consider initial value problems for a system of linear differential equations with delays defined by pairwise permutable matrices: Using the notations ) it is possible to rewrite initial value problems for (1.3) as an operator equation where (S h i z)(t) is an N-dimensional column-vector and ϕ(t) is an N-dimensional column- vector defined by the formula The operator S h i : D p → L p admits the following representation: where χ h i (t, s) is the characteristic function of the set defined by We will investigate the equation (1.6) assuming that the operator L maps a Banach space Fredholm boundary-value problems for linear delay systems equipped with the standard norms for these spaces.It is well-known [1] that, in the considered spaces, problem (1.6) is equivalent to initial value problem (1.3).The transformations (1.4), (1.5) allow to add the initial function ψ(s), s < 0 to an inhomogeneity and thus to generate an additive and homogeneous operation not depending on ψ, and without a classical assumption regarding the continuous connection of solution z(t) with the initial function ψ(t) at the point t = 0.A solution of differential system (1.6) is defined as a vector-function z(t) if it satisfies the system (1.6) almost everywhere on [0, b].Such a treatment makes it possible to apply to the equation (1.6) with the linear and bounded operator L well developed methods of linear functional analysis.It is well-known (see, e.g., [1,3,4]) that an inhomogeneous operator equation (1.6) with delayed arguments is solvable for an arbitrary right-hand side ϕ(t) ∈ L p [0, b] and has an N-dimensional family of solutions (dim ker L = N) in the form where the kernel K(t, s) of the integral is an (N × N)-dimensional Cauchy matrix K(t, s) being, for every fixed s, a solution of the matrix Cauchy problem: In the following we assume that the matrix A disadvantage of this approach, when investigating the above-formulated problem, is the necessity to find analytically a fundamental X(t) and the Cauchy K(t, s) matrices [5,7].Below we consider the case of a system with delays, when this problem can be directly solved.In this case the problem of how to construct the Cauchy matrix is successfully solved analytically due to a delayed matrix exponential defined in [6] and generalized to the case of several delays in [8].
Using the multi-delayed matrix exponential (2.1) we can represent a solution z(t) of a corresponding linear system (1.6) with multiple delays and pairwise permutable matrices in the form (1.7), where and

Fredholm boundary-value problem
Using the results [3,4], it is easy to derive results for a general boundary-value problem if the number m of boundary conditions does not coincide with the number N of unknowns in a differential system with a delay.We derive such results in an explicit analytical form.We consider the boundary-value problem where α is an m-dimensional constant vector-column, l = col(l 1 , . . ., l m ) : We will derive sufficient and necessary conditions, and a representation of the solutions Substituting the general solution (1.7) of the equation (3.1) into the boundary condition (3.2), in accordance with (2.2), we will have the algebraic system with the constant m × N dimension matrix Preserving the above used notation [4], we have: rank matrix (orthogonal projection) projecting the space R N to the kernel (ker Q) of the matrix Q, P Q * := I m − QQ + in an m × m-dimensional matrix (orthogonal projection) projecting the space R m to the kernel Q * of the transposed matrix The case of rank Q = m is interesting as well.Then the inequality m ≤ N holds, i.e., the boundary-value problem is underdetermined.In this case, Theorem 3.1 has the following corollary.
Finally, combining both particular cases mentioned above, we get the following.
Corollary 3.5.If A = 0 and i = 1 , then from Theorem 3.1 we obtain the result published in [2].

Example
Consider the boundary value problem with two delays [8, p. 3350] where = col(l 1 , l 2 ) is a two-dimensional vector functional: The general solution of the equation (4.1) has the form where Y(t) is the solution of the corresponding homogeneous (4.1) equation on the interval [0, 1] [8, p. 3351] Substituting the general solution (4.3) into the boundary conditions (4.2), we obtain an algebraic equation Then is satisfied, and after the transformation that is of the form where Since P Q = 0, then under the condition (4.5), equation (4.4) has a unique solution which after conversion has the form and the generalized Green matrix, corresponding to the boundary-value problem (4.1), (4.2), has the form  For example, the condition (4.5) will be fulfilled for the inhomogeneities of the following form: On the interval 0 ≤ t < linear vector-functional defined on the space D p [0, b] of N-dimensional vector-functions absolutely continuous on [0, b].As above, we state that, in the spaces considered, this problem is equivalent to problem (1.1), (1.2), where

Corollary 3 . 3 .
If rank Q = m, then the boundary-value problem has an r-dimensional (r = N − m) family of solutions.The inhomogeneous problem (3.1), (3.2) is solvable for arbitrary ϕ(t) ∈ L p [0, b] and α ∈ R m and has an r-parametric family of solutions

Corollary 3 . 4 .
If rank Q = N = m, then the homogeneous problem has only the trivial solution.The inhomogeneous boundary-value problem (3.1), (3.2) is solvable for arbitrary ϕ(t) ∈ L p [0, b] and α ∈ R N , and has a unique solution