doi:10.1155/2010/593834
Research Article
Boundary Value Problems for
Delay Differential Systems
A. Boichuk,
1, 2J. Dibl´ık,
3, 4D. Khusainov,
5and M. R ˚u ˇziˇckov ´a
11Department of Mathematics, Faculty of Science, University of ˇZilina,
Univerzitn´a 8215/1, 01026 ˇZilina, Slovakia
2Institute of Mathematics, National Academy of Sciences of Ukraine,
Tereshchenkovskaya Str. 3, 01601 Kyiv, Ukraine
3Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering,
Brno University of Technology, Veveˇr´ı331/95, 60200 Brno, Czech Republic
4Department of Mathematics, Faculty of Electrical Engineering and Communication,
Brno University of Technology, Technick´a 8, 61600 Brno, Czech Republic
5Department of Complex System Modeling, Faculty of Cybernetics, Taras,
Shevchenko National University of Kyiv, Vladimirskaya Str. 64, 01033 Kyiv, Ukraine
Correspondence should be addressed to A. Boichuk,[email protected]
Received 16 January 2010; Revised 27 April 2010; Accepted 12 May 2010
Academic Editor: A ˘gacik Zafer
Copyrightq2010 A. Boichuk et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Conditions are derived of the existence of solutions of linear Fredholm’s boundary-value problems for systems of ordinary differential equations with constant coefficients and a single delay, assuming that these solutions satisfy the initial and boundary conditions. Utilizing a delayed matrix exponential and a method of pseudoinverse by Moore-Penrose matrices led to an explicit and analytical form of a criterion for the existence of solutions in a relevant space and, moreover, to the construction of a family of linearly independent solutions of such problems in a general case with the number of boundary conditionsdefined by a linear vector functionalnot coinciding with the number of unknowns of a differential system with a single delay. As an example of application of the results derived, the problem of bifurcation of solutions of boundary-value problems for systems of ordinary differential equations with a small parameter and with a finite number of measurable delays of argument is considered.
1. Introduction
First we mention auxiliary results regarding the theory of differential equations with delay. Consider a system of linear differential equations with concentrated delay
˙
assuming that
zs:ψs, if s /∈a, b, 1.2
where A is an n × n real matrix, and g is an n-dimensional real column vector, with components in the spaceLpa, b where p ∈ 1,∞of functions integrable ona, bwith the degree p; the delay ht ≤ t is a function h : a, b → R measurable on a, b; ψ : R\ a, b → Rn is a given vector function with components in L
pa, b. Using the denotations
Shzt:
⎧ ⎨ ⎩
zht, ifht∈a, b,
θ, ifht/∈a, b, 1.3
ψht: ⎧ ⎨ ⎩
θ, ifht∈a, b,
ψht, ifht/∈a, b, 1.4
whereθ is ann-dimensional zero column vector, and assuming t ∈ a, b, it is possible to rewrite1.1,1.2as
Lzt:z˙t−AtShzt ϕt, t∈a, b, 1.5
whereϕis ann-dimensional column vector defined by the formula
ϕt:gt Atψht∈Lpa, b. 1.6
We will investigate 1.5 assuming that the operator L maps a Banach space Dpa, b of absolutely continuous functionsz : a, b → Rn into a Banach spaceL
pa, b 1 ≤ p < ∞ of functionϕ : a, b → Rn integrable ona, b with the degreep; the operatorS
h maps the spaceDpa, binto the spaceLpa, b. Transformations of1.3,1.4make it possible to add the initial vector functionψs,s < ato nonhomogeneity, thus generating an additive and homogeneous operation not depending onψ, and without the classical assumption regarding the continuous connection of solutionztwith the initial functionψtatta.
A solution of differential system1.5is defined as ann-dimensional column vector functionz∈ Dpa, b, absolutely continuous ona, bwith a derivative ˙z in a Banach space Lpa, b 1≤p <∞of functions integrable ona, bwith the degreep,satisfying1.5almost everywhere ona, b. Throughout this paper we understand the notion of a solution of a differential system and the corresponding boundary value problem in the sense of the above definition.
Such treatment makes it possible to apply the well-developed methods of linear functional analysis to1.5with a linear and bounded operatorL. It is well knownsee, e.g.,
the spaceDpa, b for an arbitrary right-hand side ϕ ∈ Lpa, b and has ann-dimensional family of solutionsdim kerLnin the form
zt Xtc b
a
Kt, sϕsds, ∀c∈Rn, 1.7
where the kernelKt, sis ann×nCauchy matrix defined in the squarea, b×a, bwhich is, for everys≤t,a solution of the matrix Cauchy problem:
LK·, st: ∂Kt, s
∂t −AtShK·, st Θ, Ks, s I, 1.8
whereKt, s≡Θifa≤t < s≤b, andΘis then×nnull matrix. A fundamentaln×nmatrix Xtfor the homogeneousϕ≡θ 1.5has the formXt Kt, a,Xa I.
A serious disadvantage of this approach, when investigating the above-formulated problem, is the necessity to find the Cauchy matrixKt, s 5,6. It exists but, as a rule, can only be found numerically. Therefore, it is important to find systems of differential equations with delay such that this problem can be solved directly. Below, we consider the case of a system with what is called a single delay7. In this case, the problem of how to construct the Cauchy matrix is solvedanalyticallythanks to a delayed matrix exponential, as defined below.
2. A Delayed Matrix Exponential
Consider a Cauchy problem for a linear nonhomogeneous differential system with constant coefficients and with a single delayτ
˙
zt Azt−τ gt, 2.1
zs ψs, ifs∈−τ,0 2.2
withn×nconstant matrixA,g :0,∞ → Rn,ψ : −τ,0 → Rn,τ >0 and an unknown vector solutionz:−τ,∞ → Rn. Together with a nonhomogeneous problem2.1,2.2, we consider a related homogeneous problem
˙
zt Azt−τ, 2.3
Denote by eAt
τ a matrix function called a delayed matrix exponential see7and defined as
This definition can be reduced to the following expression:
eAt
wheret/τis the greatest integer function. The delayed matrix exponential equals a unit matrix I on−τ,0 and represents a fundamental matrix of a homogeneous system with a single delay.
We mention some of the properties ofeAt
τ given in7. Regarding the system without delayτ 0, the delayed matrix exponential does not have the form of a matrix series, but it is a matrix polynomial, depending on the time interval in which it is considered. It is easy to prove directly that the delayed matrix exponentialXt:eAτt−τsatisfies the relations
˙
Xt AXt−τ, fort≥0, Xs 0, fors∈τ,0, X0 I. 2.7
By integrating the delayed matrix exponential, we get
Delayed matrix exponentialeAt
τ ,t >0 is an infinitely many times continuously differentiable function except for the nodeskτ,k 0,1, . . .where there is a discontinuity of the derivative of orderk 1:
lim t→kτ−0
eAtτ k 10, lim t→kτ 0
eAtτ k 1Ak 1. 2.10
The following resultsproved in7and being a consequence of1.7withKt, s eτAt−τ−s as wellhold.
Theorem 2.1. AThe solution of a homogeneous system2.3with a single delay satisfying the initial condition2.4whereψsis an arbitrary continuously differentiable vector function can be represented in the form
zt eAtτ ψ−τ 0
−τ
eAτt−τ−sψsds. 2.11
BA particular solution of a nonhomogeneous system2.1with a single delay satisfying the zero initial conditionzs 0ifs∈−τ,0can be represented in the form
zt t
0
eAτt−τ−sgsds. 2.12
CA solution of a Cauchy problem of a nonhomogeneous system with a single delay2.1 satisfying a constant initial condition
zs ψs c∈Rn, ifs∈−τ,0 2.13
has the form
zt eAτt−τc
t
0
eAτt−τ−sgsds. 2.14
3. Main Results
Without loss of generality, leta0. The problem2.1,2.2can be transformedht:t−τ to an equation of type1.1 see1.5:
˙
where, in accordance with1.3,1.4,
Shzt
⎧ ⎨ ⎩
zt−τ, ift−τ ∈0, b,
θ, ift−τ /∈0, b,
ϕt gt A ψht∈Lp0, b,
ψht ⎧ ⎨ ⎩
θ, if t−τ ∈0, b,
ψt−τ, if t−τ /∈0, b.
3.2
A general solution of a Cauchy problem for a nonhomogeneous system3.1with a single delay satisfying a constant initial condition
zs ψs c∈Rn, ifs∈−τ,0 3.3
has the form1.7:
zt Xtc b
0
Kt, sϕsds, ∀c∈Rn, 3.4
where, as can easily be verifiedin view of the above-defined delayed matrix exponentialby substituting into3.1,
Xt eAτt−τ, X0 eτ−Aτ I 3.5
is a normal fundamental matrix of the homogeneous system related to3.1 or2.1with the initial dataX0 I, and the Cauchy matrixKt, shas the form
Kt, s eτAt−τ−s, if 0≤s < t≤b,
Kt, s≡Θ, if 0≤t < s≤b.
3.6
Obviously,
Kt,0 eτAt−τXt, K0,0 eτA−τX0 I, 3.7
and, therefore, the initial problem3.1for systems of ordinary differential equations with constant coefficients and a single delay, satisfying a constant initial condition, has an n-parametric family of linearly independent solutions
zt eAτt−τc
t
0
eAτt−τ−sϕsds, ∀c∈Rn. 3.8
3.1. Fredholm Boundary Value Problem
Using the results in8,9, it is easy to derive statements for a general boundary value problem if the numbermof boundary conditions does not coincide with the numbernof unknowns in a differential system with a single delay.
We consider a boundary value problem
˙
zt−Azt−τ gt, ift∈0, b,
zs:ψs, ifs /∈0, b, 3.9
assuming that
z α∈Rm 3.10
or, using3.2, in an equivalent form
˙
zt−AShzt ϕt, t∈0, b, 3.11
z α∈Rm, 3.12
whereαis anm-dimensional constant vector column, and:Dp0, b → Rmis a linear vector functional. It is well known that, for functional differential equations, such problems are of Fredholm’s typesee, e.g.,1,9. We will derive the necessary and sufficient conditions and a representationin anexplicit analyticalformof the solutionsz ∈ Dp0, b, z˙ ∈Lp0, bof the boundary value problem3.11,3.12.
We recall that, because of properties3.6–3.7, a general solution of system3.11has the form
zt eAτt−τc
b
0
Kt, sϕsds, ∀c∈Rn. 3.13
In the algebraic system
Qcα− b
0
K·, sϕsds, 3.14
derived by substituting3.13into boundary condition3.12; the constant matrix
Q:X· eAτ·−τ 3.15
has a size ofm×n. Denote
where, obviously,n1≤minm, n. Adopting the well-known notatione.g.,9, we define an
n×n-dimensional matrix
PQ:I−Q Q 3.17
which is an orthogonal projection projecting spaceRnto kerQof the matrixQwhereIis an n×nidentity matrix and anm×m-dimensional matrix
PQ∗:Im−QQ 3.18
which is an orthogonal projection projecting spaceRmto kerQ∗of the transposed matrixQ∗
QTwhereI
mis anm×midentity matrix andQ is ann×m-dimensional matrix pseudoinverse to them×n-dimensional matrixQ. Using the property
rankPQ∗m−rankQ∗d:m−n1, 3.19
where rankQ∗rankQn1, we will denote byPQ∗dad×m-dimensional matrix constructed from d linearly independent rows of the matrix PQ∗. Moreover, taking into account the property
rankPQ n−rankQrn−n1, 3.20
we will denote byPQr ann×r-dimensional matrix constructed fromr linearly independent
columns of the matrixPQ.
Thensee9, page 79, formulas3.43,3.44the condition
PQ∗d
α−
b
0
K·, sϕsds
θd 3.21
is necessary and sufficient for algebraic system3.14to be solvable whereθdisthroughout the paperad-dimensional column zero vector. If such condition is true, system3.14has a solution
cPQrcr Q
α− b
0
K·, sϕsds
, ∀cr ∈Rr. 3.22
Substituting the constantc∈ Rn defined by3.22into3.13, we get a formula for a general solution of problem3.11,3.12:
zt zt, cr:XtPQrcr
whereGϕtis a generalized Green operator. If the vector functionalsatisfies the relation
9, page 176
b
0
K·, sϕsds b
0
K·, sϕsds, 3.24
which is assumed throughout the rest of the paper, then the generalized Green operator takes the form
Gϕt: b
0
Gt, sϕsds, 3.25
where
Gt, s:Kt, s−eτAt−τQ K·, s 3.26
is a generalized Green matrix, corresponding to the boundary value problem3.11,3.12, and the Cauchy matrixKt, shas the form of3.6. Therefore, the following theorem holds
see10.
Theorem 3.1. LetQbe defined by3.15andrankQn1. Then the homogeneous problem
˙
zt−AShzt θ, t∈0, b,
z θm∈Rm
3.27
corresponding to the problem3.11,3.12has exactlyrn−n1linearly independent solutions
zt, cr XtPQrcr e
At−τ
τ PQrcr, ∀cr ∈R
r. 3.28
Nonhomogeneous problem3.11,3.12is solvable if and only ifϕ ∈ Lp0, bandα ∈Rm satisfy dlinearly independent conditions3.21. In that case, this problem has anr-dimensional family of linearly independent solutions represented in an explicit analytical form3.23.
The case of rankQ nimplies the inequalitym ≥ n. Ifm > n, the boundary value problem is overdetermined, the number of boundary conditions is more than the number of unknowns, andTheorem 3.1has the following corollary.
Corollary 3.2. IfrankQ n, then the homogeneous problem 3.27has only the trivial solution. Nonhomogeneous problem3.11,3.12is solvable if and only ifϕ∈Lp0, bandα∈Rmsatisfyd
linearly independent conditions3.21wheredm−n.Then the unique solution can be represented as
The case of rankQ mis interesting as well. Then the inequalitym ≤ n, holds. If m < nthe boundary value problem is not fully defined. In this case, Theorem 3.1has the following corollary.
Corollary 3.3. If rankQ m, then the homogeneous problem3.27 has anr-dimensionalr n−mfamily of linearly independent solutions
zt, cr XtPQrcr e
At−τ
τ PQrcr, ∀cr ∈R
r. 3.30
Nonhomogeneous problem3.11,3.12is solvable for arbitraryϕ∈Lp0, bandα∈Rmand has an r-parametric family of solutions
zt, cr XtPQrcr
Gϕt XtQ α, ∀cr ∈Rr. 3.31
Finally, combining both particular cases mentioned in Corollaries3.2and3.3, we get a noncritical case.
Corollary 3.4. IfrankQ mn(i.e.,Q Q−1), then the homogeneous problem3.27has only
the trivial solution. The nonhomogeneous problem3.11,3.12is solvable for arbitraryϕ∈Lp0, b
andα∈Rnand has a unique solution
zt Gϕt XtQ−1α, 3.32
where
Gϕt: b
0
Gt, sϕsds 3.33
is a Green operator, and
Gt, s:Kt, s−eτAt−τQ−1K·, s 3.34
is a related Green matrix, corresponding to the problem3.11,3.12.
4. Perturbed Boundary Value Problems
As an example of application ofTheorem 3.1, we consider the problem of bifurcation from pointε0 of solutionsz:0, b → Rn,b >0 satisfying, for a.e.t∈0, b, systems of ordinary differential equations
˙
zt Azh0t ε k
i1
Bitzhit gt, 4.1
εis a small parameter, delayshi : 0, b → Rare measurable on0, b,hit ≤ t,t ∈ 0, b, i0,1, . . . , k,g :0, b → R,g∈Lp0, b, and satisfying the initial and boundary conditions
zs ψs, ifs <0, zα, 4.2
whereα∈ Rm,ψ :R\0, b → Rnis a given vector function with components inL pa, b, and :Dp0, b → Rmis a linear vector functional. Using denotations1.3,1.4, and1.6, it is easy to show that the perturbed nonhomogeneous linear boundary value problem4.1,
4.2can be rewritten as
˙
zt ASh0zt εBtShzt ϕt, ε, zα. 4.3
In4.3we specifyh0:0, b → Ras a single delay defined by formulah0t:t−τ τ >0;
Shzt colSh1zt, . . . ,Shkzt 4.4
is anN-dimensional column vector, andϕt, εis ann-dimensional column vector given by
ϕt, ε gt A ψh0t ε
k
i1
Bitψhit. 4.5
It is easy to see thatϕ∈Lp0, b. The operatorShmaps the spaceDpinto the space
LN
p Lp× · · · ×Lp
ktimes
,
4.6
that is,Sh :Dp → LNp. Using denotation1.3for the operatorShi :Dp → Lp, we have the
following representation:
Shizt b
0
χhit, sz˙sds χhit,0z0, 4.7
where
χhit, s ⎧ ⎨ ⎩
1, ift, s∈Ωi,
0, ift, s/∈Ωi
4.8
is the characteristic function of the set
Assume that nonhomogeneities ϕt,0∈Lp0, band α ∈ Rm are such that the shortened boundary value problem
˙
zt ASh0zt ϕt,0, lzα, 4.10
being a particular case of4.3forε0, does not have a solution. In such a case, according to
Theorem 3.1, the solvability criterion3.21does not hold for problem4.10. Thus, we arrive
at the following question.
Is it possible to make the problem4.10solvable by means of linear perturbations and, if this is possible, then of what kind should the perturbationsBiand the delayshi,i1,2, . . . , kbe for the
boundary value problem4.3to be solvable?
We can answer this question with the help of thed×r-matrix
B0:
b
0
HsBsShXPQr
sds
b
0
Hs k
i1
Bis
ShiXPQr
sds, 4.11
where
Hs:PQ∗dK·, s PQd∗eAτ·−τ−s, Xt: eτAt−τ, Q: XeτA·−τ, 4.12
constructed by using the coefficients of the problem4.3.
Using the Vishik and Lyusternik method11and the theory of generalized inverse operators9, we can find bifurcation conditions. Below we formulate a statementproved using8and 9, page 177which partially answers the above problem. Unlike an earlier result 9, this one is derived in an explicit analyticalform. We remind that the notion of a solution of a boundary value problem was specified in part 1.
Theorem 4.1. Consider system
˙
zt Azt−τ ε k
i1
Bitzhit gt, 4.13
whereAisn×nconstant matrix,Bt B1t, . . . , Bktis ann×Nmatrix,Nnk, consisting
ofn×nmatricesBi:0, b → Rn×n,i1,2, . . . , k, having entries inLp0, b,εis a small parameter,
delayshi :0, b → Rare measurable on0, b,hit≤t,t∈0, b,g :0, b → R,g ∈Lp0, b,
with the initial and boundary conditions
zs ψs, ifs <0, zα, 4.14
whereα ∈ Rm,ψ : R\0, b → Rn is a given vector function with components inL
pa, b, and :Dp0, b → Rmis a linear vector functional, and assume that
(satisfyingϕ∈Lp0, b) andαare such that the shortened problem
˙
zt ASh0zt ϕt,0, zα 4.16
does not have a solution. If
rankB0d or PB∗0:Id−B0B0 0, 4.17
then the boundary value problem4.13,4.14has a set ofρ:n−mlinearly independent solutions in the form of the series
zt, ε
∞
i−1
εizi
t, cρ
,
z·, ε∈Dp0, b, z˙·, ε∈Lp0, b, zt,·∈C0, ε∗,
4.18
converging for fixedε∈0, ε∗, whereε∗is an appropriate constant characterizing the domain of the
convergence of the series4.18, andzit, cρare suitable coefficients.
Remark 4.2. Coefficientszit, cρ,i −1, . . . ,∞, in4.18can be determined. The procedure describing the method of their deriving is a crucial part of the proof ofTheorem 4.1where we give their form as well.
Proof. Substitute4.18into4.3and equate the terms that are multiplied by the same powers ofε. Forε−1, we obtain the homogeneous boundary value problem
˙
z−1t ASh0z−1t, z−10, 4.19
which determinesz−1t.
ByTheorem 3.1, the homogeneous boundary value problem4.19has anr-parametric
r n−n1family of solutionsz−1t :z−1t, c−1 XtPQrtc−1where ther-dimensional
column vectorc−1 ∈Rr can be determined from the solvability condition of the problem for
z0t.
Forε0, we get the boundary value problem
˙
z0t ASh0z0t BtShz−1t ϕt,0, z0α, 4.20
which determinesz0t:z0t, c0.
It follows fromTheorem 3.1that the solvability criterion3.21for problem4.20has the form
PQ∗dα−
b
0
Hsϕs,0 BsShXPQr
sc−1
from which we receive, with respect toc−1∈Rr, an algebraic system
B0c−1PQ∗dα−
b
0
Hsϕs,0ds. 4.22
The right-hand side of4.22is nonzero only in the case that the shortened problem does not have a solution. The system4.22is solvable for arbitraryϕt,0∈Lp0, bandα∈Rmif the condition4.17is satisfied9, page 79. In this case, system4.22becomes resolvable with respect toc−1∈Rr up to an arbitrary constant vectorPB0c∈Rr from the null-space of matrix B0and
c−1−B0
PQ∗dα−
b
0
Hsϕs,0ds
PB0c
PB0Ir −B0B0
. 4.23
This solution can be rewritten in the form
c−1 c−1 PBρcρ, ∀cρ∈R
ρ, 4.24
where
c−1−B0
PQ∗dα−
b
0
Hsϕs,0ds
, 4.25
and PBρ is an r × ρ-dimensional matrix whose columns are a complete set of ρ linearly
independent columns of ther×r-dimensional matrixPB0with
ρ:rankPB0r−rankB0r−dn−m. 4.26
So, for the solutions of the problem3.14, we have the following formulas:
z−1
t, cρ
z−1t, c−1 XtPQrPBρcρ, ∀cρ∈R
ρ,
z−1t, c−1 XtPQrc−1.
4.27
Assuming that3.24and4.17hold, the boundary value problem4.20has ther-parametric family of solutions
z0t, c0 XtPQrc0 XtQ α b
0
Gt, sϕs,0 BsSh
z−1·, c−1 X·PQrPBρcρ s
ds.
4.28
Here,c0is anr-dimensional constant vector, which is determined at the next step from
Forε1, we get the boundary value problem formin computations below we need a composition of operators and the order of operations is following the inner operator Sh which acts to matrices and vector function having an argument denoted by ” ·” and the outer operator Sh which acts to matrices having an argument denoted by ””
Assuming that4.17holds, the algebraic system4.31has the following family of solutions:
where
So, for the ρ-parametric family of solutions of the problem 4.20, we have the following formula:
Again, assuming that4.17holds, the boundary value problem4.29has ther-parametric family of solutions
Here,c1is anr-dimensional constant vector, which is determined at the next step from the
solvability condition of the boundary value problem forz2t:
˙
z2t ASh0z2t BtShz1t, z20. 4.37
The solvability criterion for the problem4.37has the form
or, equivalently, the form
Under condition4.17, the last equation has theρ-parametric family of solutions
c1c1
Continuing this process, by assuming that 4.17 holds, it follows by induction that the coefficients zit, ci zit, cρ of the series 4.18 can be determined, from the relevant boundary value problems as follows:
where
The convergence of the series 4.18 can be proved by traditional methods of majorization9,11.
In the casemn, the condition4.17is equivalent with detB0/0, and problem4.13, 4.14has a unique solution.
Example 4.3. Consider the linear boundary value problem for the delay differential equation
˙
see that, in this case,Xt eIτt−τ Iis a normal fundamental matrix for the homogeneous
Now the matrixB0turns into
and the boundary value problem4.46is uniquely solvable if
det
−k
i1
T
Δi
Bisds
/
0. 4.55
Acknowledgments
The authors highly appreciate the work of the anonymous referee whose comments and suggestions helped them greatly to improve the quality of the paper in many aspects. The first author was supported by Grant 1/0771/08 of the Grant Agency of Slovak RepublicVEGA and Project APVV-0700-07 of Slovak Research and Development Agency. The second author was supported by Grant 201/08/0469 of Czech Grant Agency and by the Council of Czech Government MSM 0021630503, MSM 0021630519, and MSM 0021630529. The third author was supported by Project M/34-2008 of Ukrainian Ministry of Education. The fourth author was supported by Grant 1/0090/09 of the Grant Agency of Slovak RepublicVEGAand project APVV-0700-07 of Slovak Research and Development Agency.
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