Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 593834, 20 pages
doi:10.1155/2010/593834
Research Article
Boundary Value Problems for
Delay Differential Systems
A. Boichuk,
1, 2
J. Dibl
´
ık,
3, 4
D. Khusainov,
5
and M. R
˚
u
ˇ
zi
ˇ
ckov
´
a
1
1
Department of Mathematics, Faculty of Science, University of
ˇ
Zilina,
Univerzitn
´
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 conditions defined by a linear vector functional not 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
˙z
t
− A
t
z
h
t
g
t
is a given vector function with components in L
p
a, b.Usingthe
denotations
S
h
z
t
:
⎧
⎨
⎩
z
h
t
, if h
t
∈
a, b
,
h
t
, if h
t
/
∈
a, b
,
1.4
where θ is an n-dimensional zero column vector, and assuming t ∈ a, b, it is possible to
rewrite 1.1, 1.2 as
Lz
t
: ˙z
t
− A
t
h
t
∈ L
p
a, b
.
1.6
We will investigate 1.5 assuming that the operator L maps a Banach space D
p
a, b of
absolutely continuous functions z : a, b → R
n
into a Banach space L
p
a, b1 ≤ p<∞
of function ϕ : a, b → R
n
integrable on a, b with the degree p ; the operator S
h
maps
the space D
p
a, b into the space L
p
a, b. Transformations of 1.3, 1.4 make it possible to
add the initial vector function ψs, s<ato nonhomogeneity, thus generating an additive and
t
c
b
a
K
t, s
ϕ
s
ds, ∀c ∈ R
n
, 1.7
where the kernel Kt, s is an n × n Cauchy matrix defined in the square a, b × a, b which
is, for every s ≤ t, a solution of the matrix Cauchy problem:
LK
·,s
t
:
∂K
t, s
Consider a Cauchy problem for a linear nonhomogeneous differential system with constant
coefficients and with a single delay τ
˙z
t
Az
t − τ
g
t
, 2.1
z
s
ψ
s
, if s ∈
−τ,0
2.2
with n × n constant matrix A, g : 0, ∞ → R
n
Denote by e
At
τ
a matrix function called a delayed matrix exponential see 7 and
defined as
e
At
τ
:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
1!
A
2
t − τ
2
2!
, if τ ≤ t<2τ,
···
I A
t
1!
··· A
k
t −
k − 1
τ
k
k!
, if
k − 1
τ ≤ t<kτ,
··· .
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 exponential Xt : e
At−τ
τ
satisfies the relations
˙
X
t
AX
t − τ
, for t ≥ 0,X
s
0, for s ∈
τ,0
,X
0
I. 2.7
By integrating the delayed matrix exponential, we get
t
t
0
e
As
τ
ds A
−1
·
e
At−τ
τ
− e
Aτ
τ
.
2.9
Advances in Difference Equations 5
Delayed matrix exponential e
At
τ
, t>0 is an infinitely many times continuously differentiable
function except for the nodes kτ, k 0, 1, where there is a discontinuity of the derivative
of order k 1:
lim
t → kτ−0
e
ψ
−τ
0
−τ
e
At−τ−s
τ
ψ
s
ds.
2.11
B A particular solution of a nonhomogeneous system 2.1 with a single delay satisfying the
zero initial condition zs0 if s ∈ −τ, 0 can be represented in the form
z
t
t
0
e
At−τ−s
At−τ
τ
c
t
0
e
At−τ−s
τ
g
s
ds.
2.14
3. Main Results
Without loss of generality, let a 0. The problem 2.1, 2.2 can be transformed ht : t − τ
to an equation of type 1.1see 1.5:
˙z
t
− A
S
h
z
t
,
θ, if t − τ
/
∈
0,b
,
ϕ
t
g
t
Aψ
h
t
∈ L
p
0,b
,
ψ
h
s
c ∈ R
n
, if s ∈
−τ,0
3.3
has the form 1.7:
z
t
X
t
c
b
0
K
t, s
ϕ
s
At−τ−s
τ
, if 0 ≤ s<t≤ b,
K
t, s
≡ Θ, if 0 ≤ t<s≤ b.
3.6
Obviously,
K
t, 0
e
At−τ
τ
X
t
,K
0, 0
e
A−τ
τ
X
Advances in Difference Equations 7
3.1. Fredholm Boundary Value Problem
Using the results in 8, 9, it is easy to derive statements 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 single delay.
We consider a boundary value problem
˙z
t
− Az
t − τ
g
t
, if t ∈
0,b
,
z
s
: ψ
s
,t∈
0,b
, 3.11
z α ∈ R
m
, 3.12
where α is an m-dimensional constant vector column, and : D
p
0,b → R
m
is a linear vector
functional. It is well known that, for functional differential equations, such problems are of
Fredholm’s type see, e.g., 1, 9. We will derive the necessary and sufficient conditions and
a representation in an explicit analytical form of the solutions z ∈ D
p
0,b, ˙z ∈ L
p
0,b of
the boundary value problem 3.11, 3.12.
We recall that, because of properties 3.6–3.7, a general solution of system 3.11 has
the form
z
t
e
At−τ
τ
3.14
derived by substituting 3.13 into boundary condition 3.12; the constant matrix
Q : X
·
e
A·−τ
τ
3.15
has a size of m × n. Denote
rank Q n
1
, 3.16
8 Advances in Difference Equations
where, obviously, n
1
≤ minm, n. Adopting the well-known notation e.g., 9, we define an
n × n-dimensional matrix
P
Q
: I − Q
Q 3.17
which is an orthogonal projection projecting space R
n
to ker Q of the matrix Q where I is an
n × n identity matrix and an m × m-dimensional matrix
P
Q
∗
rank Q n
1
, we will denote by P
Q
∗
d
a d × m-dimensional matrix constructed
from d linearly independent rows of the matrix P
Q
∗
. Moreover, taking into account the
property
rank P
Q
n − rank Q r n − n
1
, 3.20
we will denote by P
Q
r
an n × r-dimensional matrix constructed from r linearly independent
columns of the matrix P
Q
.
Then see 9, page 79, formulas 3.43, 3.44 the condition
P
Q
∗
d
α −
b
0
K
·,s
ϕ
s
ds
, ∀c
r
∈ R
r
. 3.22
Substituting the constant c ∈ R
n
defined by 3.22 into 3.13, we get a formula for a
general solution of problem 3.11, 3.12:
z
t
z
Advances in Difference Equations 9
where Gϕt is a generalized Green operator. If the vector functional satisfies the relation
9, page 176
b
0
K
·,s
ϕ
s
ds
b
0
K
·,s
ϕ
s
ds,
3.24
which is assumed throughout the rest of the paper, then the generalized Green operator takes
− e
At−τ
τ
Q
K
·,s
3.26
is a generalized Green matrix, corresponding to the boundary value problem 3.11, 3.12,
and the Cauchy matrix Kt, s has the form of 3.6. Therefore, the following theorem holds
see 10.
Theorem 3.1. Let Q be defined by 3.15 and rank Q n
1
. Then the homogeneous problem
˙z
t
− A
S
h
z
t
θ, t ∈
r
c
r
, ∀c
r
∈ R
r
.
3.28
Nonhomogeneous problem 3.11, 3.12 is solvable if and only if ϕ ∈ L
p
0,b and α ∈ R
m
satisfy
d linearly independent conditions 3.21. In that case, this problem has an r-dimensional family of
linearly independent solutions represented in an explicit analytical form 3.23.
The case of rank Q n implies the inequality m ≥ n.Ifm>n, the boundary value
problem is overdetermined, the number of boundary conditions is more than the number of
unknowns, and Theorem 3.1 has the following corollary.
Corollary 3.2. If rankQ n, then the homogeneous problem 3.27 has only the trivial solution.
Nonhomogeneous problem 3.11, 3.12 is solvable if and only if ϕ ∈ L
p
0,b and α ∈ R
m
satisfy d
linearly independent conditions 3.21 where d m − n. Then the unique solution can be represented
as
z
t
P
Q
r
c
r
e
At−τ
τ
P
Q
r
c
r
, ∀c
r
∈ R
r
.
3.30
Nonhomogeneous problem 3.11, 3.12 is solvable for arbitrary ϕ ∈ L
p
0,b and α ∈ R
m
and has an
r-parametric family of solutions
z
t, c
r
Q
−1
), then the homogeneous problem 3.27 has only
the trivial solution. The nonhomogeneous problem 3.11, 3.12 is solvable for arbitrary ϕ ∈ L
p
0,b
and α ∈ R
n
and has a unique solution
z
t
Gϕ
t
X
t
Q
−1
α,
3.32
where
At−τ
τ
Q
−1
K
·,s
3.34
is a related Green matrix, corresponding to the problem 3.11, 3.12.
4. Perturbed Boundary Value Problems
As an example of application of Theorem 3.1, we consider the problem of bifurcation from
point ε 0 of solutions z : 0,b → R
n
, b>0 satisfying, for a.e. t ∈ 0,b, systems of ordinary
differential equations
˙z
t
Az
h
0
t
ε
k
0,b,
Advances in Difference Equations 11
ε is a small parameter, delays h
i
: 0,b → R are measurable on 0,b, h
i
t ≤ t, t ∈ 0,b,
i 0, 1, ,k, g : 0,b → R, g ∈ L
p
0,b, and satisfying the initial and boundary conditions
z
s
ψ
s
, if s<0,z α, 4.2
where α ∈ R
m
, ψ : R \ 0,b → R
n
is a given vector function with components in L
p
a, b,
and : D
p
0,b → R
m
,z α. 4.3
In 4.3 we specify h
0
: 0,b → R as a single delay defined by formula h
0
t : t − τ τ>0;
S
h
z
t
col
S
h
1
z
t
, ,
S
h
k
z
t
t
.
4.5
It is easy to see that ϕ ∈ L
p
0,b. The operator S
h
maps the space D
p
into the space
L
N
p
L
p
×···×L
p
k times
,
4.6
that is, S
h
: D
p
→ L
N
p
. Using denotation 1.3 for the operator S
ds χ
h
i
t, 0
z
0
,
4.7
where
χ
h
i
t, s
⎧
⎨
⎩
1, if
t, s
∈ Ω
i
,
}
,i 1, 2, ,k. 4.9
12 Advances in Difference Equations
Assume that nonhomogeneities ϕt, 0 ∈ L
p
0,b and α ∈ R
m
are such that the shortened
boundary value problem
˙z
t
A
S
h
0
z
t
ϕ
t, 0
,lz α, 4.10
being a particular case of 4.3 for ε 0, does not have a solution. In such a case, according to
Theorem 3.1, the solvability criterion 3.21 does not hold for problem 4.10. Thus, we arrive
at the following question.
s
ds
b
0
H
s
k
i1
B
i
s
S
h
i
XP
Q
r
s
ds,
τ
,
4.12
constructed by using the coefficients of the problem 4.3.
Using the Vishik and Lyusternik method 11 and the theory of generalized inverse
operators 9, we can find bifurcation conditions. Below we formulate a statement proved
using 8 and 9, page 177 which partially answers the above problem. Unlike an earlier
result 9, this one is derived in an explicit analytical form. We remind that the notion of a
solution of a boundary value problem was specified in part 1.
Theorem 4.1. Consider system
˙z
t
Az
t − τ
ε
k
i1
B
i
t
z
h
z
s
ψ
s
, if s<0,z α, 4.14
where α ∈ R
m
, ψ : R \ 0,b → R
n
is a given vector function with components in L
p
a, b, and
: D
p
0,b → R
m
is a linear vector functional, and assume that
ϕ
t, 0
g
t
Aψ
t, 0
,z α 4.16
does not have a solution. If
rank B
0
d or P
B
∗
0
: I
d
− B
0
B
0
0,
4.17
then the boundary value problem 4.13, 4.14 has a set of ρ : n − m linearly independent solutions
in the form of the series
z
t, ε
∞
i−1
∈ C
0,ε
∗
,
4.18
converging for fixed ε ∈ 0,ε
∗
,whereε
∗
is an appropriate constant characterizing the domain of the
convergence of the series 4.18, and z
i
t, c
ρ
are suitable coefficients.
Remark 4.2. Coefficients z
i
t, c
ρ
, i −1, ,∞,in4.18 can be determined. The procedure
describing the method of their deriving is a crucial part of the proof of Theorem 4.1 where we
give their form as well.
Proof. Substitute 4.18 into 4.3 and equate the terms t hat are multiplied by the same powers
of ε. For ε
−1
, we obtain the homogeneous boundary value problem
˙z
Q
r
tc
−1
where the r-dimensional
column vector c
−1
∈ R
r
can be determined from the solvability condition of the problem for
z
0
t.
For ε
0
, we get the boundary value problem
˙z
0
t
A
S
h
0
z
0
t
∗
d
α −
b
0
H
s
ϕ
s, 0
B
s
S
h
XP
Q
r
s
c
ds.
4.22
The right-hand side of 4.22 is nonzero only in the case that the shortened problem does not
have a solution. The system 4.22 is solvable for arbitrary ϕt, 0 ∈ L
p
0,b and α ∈ R
m
if the
condition 4.17 is satisfied 9, page 79. In this case, system 4.22 becomes resolvable with
respect to c
−1
∈ R
r
up to an arbitrary constant vector P
B
0
c ∈ R
r
from the null-space of matrix
B
0
and
c
−1
−B
0
P
Q
0
. 4.23
This solution can be rewritten in the form
c
−1
c
−1
P
B
ρ
c
ρ
, ∀c
ρ
∈ R
ρ
,
4.24
where
c
−1
−B
0
P
Q
∗
d
z
−1
t, c
ρ
z
−1
t,
c
−1
X
t
P
Q
r
P
B
ρ
c
ρ
, ∀c
ρ
∈ R
ρ
P
Q
r
c
0
X
t
Q
α
b
0
G
t, s
ϕ
s, 0
B
s
0
is an r-dimensional constant vector, which is determined at the next step from
the solvability condition of the boundary value problem for z
1
t.
Advances in Difference Equations 15
For ε
1
, we get the boundary value problem
˙z
1
t
A
S
h
0
z
1
t
B
t
S
h
. The solvability criterion for the problem 4.29 has the
form in computations below we need a composition of operators and the order of operations
is following the inner operator S
h
which acts to matrices and vector function having an
argument denoted by ” · ” and the outer operator S
h
which acts to matrices having an
argument denoted by ” ”
b
0
H
s
k
i1
B
i
s
ψ
h
i
s
Q
α
b
0
G
, s
1
ϕ
s
1
, 0
B
s
S
h
z
−1
·,
B
0
c
0
−
b
0
H
s
k
i1
B
i
s
ψ
h
i
s
ds
−
b
ϕ
s
1
, 0
B
s
1
S
h
z
−1
·,
c
−1
X
·
P
Q
r
P
B
b
0
H
s
B
s
S
h
b
0
G
, s
1
B
s
1
S
h
X
−B
0
b
0
H
s
k
i1
B
i
s
ψ
h
i
s
ds
− B
0
b
s
1
, 0
B
s
1
S
h
z
−1
·,
c
−1
s
1
ds
1
s
ds.
∈ R
ρ
, 4.34
where
z
0
t,
c
0
X
t
P
Q
r
c
0
X
t
Q
α
b
0
X
t
P
Q
r
I
r
− B
0
b
0
H
s
B
s
S
h
b
s
ds
b
0
G
t, s
B
s
S
h
X
·
P
Q
r
s
S
h
z
0
·,
c
0
X
0
·
P
B
ρ
c
ρ
s
ds.
4.36
Here, c
1
,z
2
0. 4.37
The solvability criterion for the problem 4.37 has the form
b
0
H
s
B
s
S
h
X
P
Q
r
c
1
b
c
ρ
s
1
ds
1
s
ds 0
4.38
Advances in Difference Equations 17
or, equivalently, the form
B
0
c
1
−
b
0
H
s
B
X
0
·
P
B
ρ
c
ρ
s
1
ds
1
s
ds.
4.39
Under condition 4.17, the last equation has the ρ-parametric family of solutions
c
1
c
1
s
1
S
h
X
0
·
s
1
ds
1
s
ds
P
B
ρ
c
ρ
B
s
1
S
h
z
0
·,
c
0
s
1
ds
1
s
ds. 4.41
So, for the coefficient z
1
t, c
1
z
∈ R
ρ
,
4.42
where
z
1
t,
c
1
X
t
P
Q
r
c
1
b
0
G
t, s
B
− B
0
b
0
H
s
B
s
S
h
b
0
G
, s
1
B
s
1
s
S
h
X
0
·
s
ds.
4.43
Continuing this process, by assuming that 4.17 holds, it follows by induction that the
coefficients z
i
t, c
i
z
i
t, c
ρ
of the series 4.18 can be determined, from the relevant
boundary value problems as follows:
z
i
where
z
i
t,
c
i
X
t
P
Q
r
c
1
b
0
G
t, s
B
s
S
b
0
G
, s
1
B
s
1
S
h
z
i−1
·,
c
i−1
s
1
ds
1
s
S
h
b
0
G
, s
1
B
s
1
S
h
X
i−1
·
s
1
s
ds, i 0, 1, 2, ,
4.45
and
X
−1
tXtP
Q
r
.
The convergence of the series 4.18 can be proved by traditional methods of
majorization 9, 11.
In the case m n, the condition 4.17 is equivalent with det B
0
/
0, and problem 4.13,
4.14 has a unique solution.
Example 4.3. Consider the linear boundary value problem for the delay differential equation
˙z
t
z
t − τ
ε
k
s
ψ
s
, if s<0, and z
0
z
T
,
4.46
where, as in the above, B
i
,g,ψ ∈ L
p
0,T and h
i
t are measurable functions. Using the
symbols S
h
i
and ψ
h
i
see 1.3, 1.4, 1.6,and4.7, we arrive at the following operator
T
0,
4.47
where BtB
1
t, ,B
k
t is an n × N matrix N nk,and
ϕ
t, ε
g
t
ψ
h
0
t
ε
k
i1
B
i
− e
IT−τ
τ
0,
P
Q
P
Q
∗
I
r n, d m n
,
K
t, s
⎧
⎨
⎩
e
It−τ−s
τ
I, if 0 ≤ s ≤ t ≤ T,
Θ, if s>t,
K
·,s
h
i
t, 0
I I ·
⎧
⎨
⎩
1, if 0 ≤ h
i
t
≤ T,
0, if h
i
t
< 0.
4.49
Then the n × n matrix B
0
has the form
B
0
T
I
s
ds
−
k
i1
T
0
B
i
s
χ
h
i
s, 0
ds.
4.50
If det B
0
/
0, problem 4.46 has a unique solution zt, ε with the properties
z
i
where 0 < Δ
i
const <T, i 1, ,k, then
χ
h
i
t, 0
⎧
⎨
⎩
1, if 0 ≤ h
i
t
t − Δ
i
≤ T,
0, if h
i
t
t − Δ
i
< 0,
T
0
B
i
s
χ
h
i
s, 0
ds −
k
i1
T
Δ
i
B
i
s
ds,
4.54
20 Advances in Difference Equations
and the boundary value problem 4.46 is uniquely solvable if
References
1 N. V. Azbelev and V. P. Maksimov, “Equations with retarded argument,” Journal of Difference Equations
and Applications, vol. 18, no. 12, pp. 1419–1441, 1983, translation from Differentsial’nye Uravneniya,vol.
18, no. 12, pp. 2027–2050, 1982.
2
ˇ
S. Schwabik, M. Tvrd
´
y, and O. Vejvoda, Differential and Integral Equations, Boundary Value Problems
and Adjoint, Reidel, Dordrecht, The Netherlands, 1979.
3 V. P. Maksimov and L. F. Rahmatullina, “A linear functional-differential equation that is solved with
respect to the derivative,” Differentsial’nye Uravneniya, vol. 9, pp. 2231–2240, 1973 Russian.
4 N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatullina, Introduction to the Theory of Functional
Differential Equations: Methods and Applications, vol. 3 of Contemporary Mathematics and Its Applications,
Hindawi Publishing Corporation, New York, NY, USA, 2007.
5 J. Hale, Theory of Functional Differential Equations, vol. 3 of Applied Mathematical Sciences, Springer, New
York, NY, USA, 2nd edition, 1977.
6 J. Mallet-Paret, “The Fredholm alternative for functional-differential equations of mixed type,” Journal
of Dynamics and Differential Equations, vol. 11, no. 1, pp. 1–47, 1999.
7 D. Ya. Khusainov and G. V. Shuklin, “On relative controllability in systems with pure delay,”
International Applied Mechanics, vol. 41, no. 2, pp. 210–221, 2005, translation from Prikladnaya
Mekhanika, vol. 41, no. 2, pp.118–130, 2005.
8 A. A. Boichuk and M. K. Grammatikopoulos, “Perturbed Fredholm boundary value problems for
delay differential systems,” Abstract and Applied Analysis, no. 15, pp. 843–864, 2003.
9 A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value
Problems, VSP, Utrecht, The Netherlands, 2004.
10 A. A. Boichuk, J. Dibl
´
ık, D. Ya. Khusainov, and M. R
Copyright: Tài liệu đại học ©