On the Cauchy problem for two-dimensional systems of linear functional differential equations with monotone operator
We establish new efficient conditions sufficient for the unique solvability of the Cauchy problem for twodimensional systems of linear functional differential equations with monotone operators.
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2007
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irk-123456789-1772152021-02-12T01:25:56Z On the Cauchy problem for two-dimensional systems of linear functional differential equations with monotone operator Šremr, J. Hakl, R. We establish new efficient conditions sufficient for the unique solvability of the Cauchy problem for twodimensional systems of linear functional differential equations with monotone operators. Знайдено новi ефективнi умови, що є достатнiми для iснування єдиного розв’язку задачi Кошi для двовимiрних систем лiнiйних функцiонально-диференцiальних рiвнянь з монотонними операторами. 2007 Article On the Cauchy problem for two-dimensional systems of linear functional differential equations with monotone operator / J. Šremr, R. Hakl // Нелінійні коливання. — 2007. — Т. 10, № 4. — С. 560-573. — Бібліогр.: 19 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177215 517.9 en Нелінійні коливання Інститут математики НАН України |
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We establish new efficient conditions sufficient for the unique solvability of the Cauchy problem for twodimensional systems of linear functional differential equations with monotone operators. |
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Article |
| author |
Šremr, J. Hakl, R. |
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Šremr, J. Hakl, R. On the Cauchy problem for two-dimensional systems of linear functional differential equations with monotone operator Нелінійні коливання |
| author_facet |
Šremr, J. Hakl, R. |
| author_sort |
Šremr, J. |
| title |
On the Cauchy problem for two-dimensional systems of linear functional differential equations with monotone operator |
| title_short |
On the Cauchy problem for two-dimensional systems of linear functional differential equations with monotone operator |
| title_full |
On the Cauchy problem for two-dimensional systems of linear functional differential equations with monotone operator |
| title_fullStr |
On the Cauchy problem for two-dimensional systems of linear functional differential equations with monotone operator |
| title_full_unstemmed |
On the Cauchy problem for two-dimensional systems of linear functional differential equations with monotone operator |
| title_sort |
on the cauchy problem for two-dimensional systems of linear functional differential equations with monotone operator |
| publisher |
Інститут математики НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/177215 |
| citation_txt |
On the Cauchy problem for two-dimensional systems of linear functional differential equations with monotone operator / J. Šremr, R. Hakl // Нелінійні коливання. — 2007. — Т. 10, № 4. — С. 560-573. — Бібліогр.: 19 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT sremrj onthecauchyproblemfortwodimensionalsystemsoflinearfunctionaldifferentialequationswithmonotoneoperator AT haklr onthecauchyproblemfortwodimensionalsystemsoflinearfunctionaldifferentialequationswithmonotoneoperator |
| first_indexed |
2025-07-15T15:15:04Z |
| last_indexed |
2025-07-15T15:15:04Z |
| _version_ |
1837726439097499648 |
| fulltext |
UDC 517 . 9
ON THE CAUCHY PROBLEM FOR TWO-DIMENSIONAL SYSTEMS
OF LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS
WITH MONOTONE OPERATORS*
ПРО ЗАДАЧУ КОШI ДЛЯ ДВОВИМIРНИХ СИСТЕМ ЛIНIЙНИХ
ФУНКЦIОНАЛЬНО-ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
З МОНОТОННИМИ ОПЕРАТОРАМИ
J. Šremr, R. Hakl
Math. Inst. Acad. Sci. Czech Republic
Žižkova 22, CZ-61662 Brno, Czech Republic
e-mail: sremr@ipm.cz
hakl@ipm.cz
We establish new efficient conditions sufficient for the unique solvability of the Cauchy problem for two-
dimensional systems of linear functional differential equations with monotone operators.
Знайдено новi ефективнi умови, що є достатнiми для iснування єдиного розв’язку задачi Кошi
для двовимiрних систем лiнiйних функцiонально-диференцiальних рiвнянь з монотонними опе-
раторами.
1. Introduction and rotation. On the interval [a, b], we consider two-dimensional differential
system
u′i(t) = σi1 `i1(u1)(t) + σi2 `i2(u2)(t) + qi(t), i = 1, 2, (1.1)
with the initial conditions
u1(a) = c1, u2(a) = c2 , (1.2)
where `ik : C([a, b]; R) → L([a, b]; R) are linear nondecreasing operators, σik ∈
∈ {−1, 1}, qi ∈ L([a, b]; R), and ci ∈ R, i, k = 1, 2. By a solution of the problem (1.1), (1.2)
we understand an absolutely continuous vector function u = (u1, u2)T : [a, b] → R2 satisfying
(1.1) almost everywhere on [a, b] and verifying also the initial conditions (1.2).
The problem of solvability of the Cauchy problem for linear functional differential equa-
tions and their systems has been studied by many authors (see, e.g., [1 – 6] and references
therein). There are a lot of interesting results but only a few efficient conditions is known
at present. Furthermore, most of them are available for the one-dimensional case only or for
systems with the so-called Volterra operators (see, e.g., [2, 3, 5, 7 – 9]). Let us mention that
the efficient conditions guaranteeing the unique solvability of the initial value problem for n-
dimensional systems of linear functional differential equations are given, e.g., in [4, 10 – 13].
In this paper, we establish new efficient condition sufficient for the unique solvability of the
problem (1.1), (1.2) with σ11 = 1 and σ22 = 1. The cases where σ11σ22 = −1 and σ11 = σ22 =
= −1 are studied in [14] and [15], respectively.
∗ For the first author, the research was supported by the Grant Agency of the Czech Republic, No. 201/06/0254,
and for the second author by the Grant Agency of the Czech Republic, No. 201/04/P183. The research was also
supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.
c© J. Šremr, R. Hakl, 2007
560 ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 4
ON THE CAUCHY PROBLEM FOR TWO-DIMENSIONAL SYSTEMS OF LINEAR FUNCTIONAL . . . 561
The integral conditions given in Theorems 2.1 and 2.2 are optimal in a certain sense which
is shown by counter-examples constructed in the last part of the paper.
The following notation is used throughout the paper:
(1) R is the set of all real numbers, R+ = [0,+∞[ ;
(2) C([a, b]; R) is the Banach space of continuous functions u : [a, b] → R equipped with
the norm
‖u‖C = max
{
|u(t)| : t ∈ [a, b]
}
;
(3) L([a, b]; R) is the Banach space of Lebesgue integrable functions h : [a, b] → R equipped
with the norm
‖h‖L =
b∫
a
|h(s)|ds;
(4) L
(
[a, b]; R+
)
=
{
h ∈ L([a, b]; R) : h(t) ≥ 0 for a.a. t ∈ [a, b]
}
;
(5) an operator ` : C([a, b]; R) → L([a, b]; R) is said to be nondecreasing if the inequality
`(u1)(t) ≤ `(u2)(t) for a.a. t ∈ [a, b]
holds for every functions u1, u2 ∈ C([a, b]; R) such that
u1(t) ≤ u2(t) for t ∈ [a, b];
(6) Pab is the set of linear nondecreasing operators ` : C([a, b]; R) → L([a, b]; R).
In what follows, the equalities and inequalities with integrable functions are understood to
hold almost everywhere.
2. Main results. In this section, we present the main results of the paper. The proofs are
given later, in Section 3. Theorems formulated below contain the efficient conditions sufficient
for the unique solvability of the problem (1.1), (1.2) with σ11 = 1 and σ22 = 1. Recall that
the operators `ik are supposed to be linear and nondecreasing, i.e., such that `ik ∈ Pab for
i, k = 1, 2.
Put
Aik =
b∫
a
`ik(1)(s)ds for i, k = 1, 2. (2.1)
At first, we consider the case where σ12σ21 > 0.
Theorem 2.1. Let σ11 = 1, σ22 = 1, and σ12σ21 > 0. Let, moreover,
A11 < 1, A22 < 1, (2.2)
and
A12 A21 < (1−A11)(1−A22), (2.3)
where the numbers Aik, i, k = 1, 2, are defined by (2.1). Then the problem (1.1), (1.2) has
a unique solution.
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562 J. ŠREMR, R. HAKL
Fig. 2.1
Remark 2.1. Neither one of the strict inequalities (2.2) and (2.3) can be replaced by the
nonstrict one (see Examples 4.1 and 4.2).
Remark 2.2. Let H1 be the set of triplets (x, y, z) ∈ R3
+ satisfying
x < 1, y < 1, z < (1− x)(1− y)
(see Fig. 2.1). According to Theorem 2.1, the problem (1.1), (1.2) is uniquely solvable if `ik ∈
∈ Pab, i, k = 1, 2, are such that b∫
a
`11(1)(s)ds ,
b∫
a
`22(1)(s)ds ,
b∫
a
`12(1)(s)ds
b∫
a
`21(1)(s)ds
∈ H1 .
Remark 2.3. It should be noted that Theorem 2.1 can be derived as a consequence of Corol-
lary 1.3.1 given in [4]. However, we shall prove this theorem using the technique common for
both theorems given in this paper.
Remark 2.4. It follows from Corollary 3.2 of [16] that if σ11 = 1, σ22 = 1, σ12σ21 > 0, and
A11 + A12 < 1, A21 + A22 < 1, (2.4)
where the numbers Aik, i, k = 1, 2, are defined by (2.1), then the problem (1.1), (1.2) has
a unique solution (u1, u2)T . Moreover, this solution satisfies
u1(t) ≥ 0, σ12u2(t) ≥ 0 for t ∈ [a, b]
provided that c1 ≥ 0, σ12c2 ≥ 0, and
q1(t) ≥ 0, σ12q2(t) ≥ 0 for t ∈ [a, b].
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ON THE CAUCHY PROBLEM FOR TWO-DIMENSIONAL SYSTEMS OF LINEAR FUNCTIONAL . . . 563
Fig. 2.2
On the other hand, if the assumption (2.4) is weakened to the assumptions (2.2), (2.3) then
the problem (1.1), (1.2) has still a unique solution but no information about the sign of this
solution is guaranteed in general.
Now we consider the case where σ12σ21 < 0.
Theorem 2.2. Let σ11 = 1, σ22 = 1, and σ12σ21 < 0. Let, moreover, the condition (2.2) be
satisfied and
A12A21 < 4
√
(1−A11)(1−A22) +
(√
1−A11 +
√
1−A22
)2
, (2.5)
where the numbers Aik, i, k = 1, 2, are defined by (2.1). Then the problem (1.1), (1.2) has
a unique solution.
Remark 2.5. The strict inequalities (2.2) in Theorem 2.2 cannot be replaced by the non-
strict ones (see Example 4.1). Furthermore, the strict inequality (2.5) cannot be replaced by
the nonstrict one provided A11 = A22 (see Example 4.3).
Remark 2.6. Let H2 be the set of triplets (x, y, z) ∈ R3
+ satisfying
x < 1, y < 1, z < 4
√
(1− x)(1− y) +
(
√
1− x +
√
1− y
)2
(see Fig. 2.2). According to Theorem 2.2, the problem (1.1), (1.2) is uniquely solvable if `ik ∈
∈ Pab, i, k = 1, 2, are such that
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564 J. ŠREMR, R. HAKL
b∫
a
`11(1)(s)ds ,
b∫
a
`22(1)(s)ds ,
b∫
a
`12(1)(s)ds
b∫
a
`21(1)(s)ds
∈ H2 .
At last, we give consequences of Theorems 2.1 and 2.2 for the system with argument devia-
tions,
u′1(t) = h11(t)u1
(
τ11(t)
)
+ σ1h12(t)u2
(
τ12(t)
)
+ q1(t),
(2.6)
u′2(t) = σ2h21(t)u1
(
τ21(t)
)
+ h22(t)u2
(
τ22(t)
)
+ q2(t),
where hik ∈ L
(
[a, b]; R+
)
, τik : [a, b] → [a, b] are measurable functions, σi ∈ {−1, 1}, and
qi ∈ L
(
[a, b]; R
)
, i, k = 1, 2.
Corollary 2.1. Let σ1σ2 > 0 and let the conditions (2.2) and (2.3) be fulfilled, where
Aik =
b∫
a
hik(s)ds for i, k = 1, 2. (2.7)
Then the problem (2.6), (1.2) has a unique solution.
Corollary 2.2. Let σ1σ2 < 0 and let the conditions (2.2) and (2.5) be fulfilled, where the
numbers Aik, i, k = 1, 2, are defined by (2.7). Then the problem (2.6), (1.2) has a unique solution.
3. Proofs of the main results. In this section, we shall prove the statements formulated
above. Recall that the numbers Aik, i, k = 1, 2, are defined by (2.1).
It is well-known from the general theory of boundary-value problems for functional differ-
ential equations (see, e.g., [4, 11, 17, 18]) that the following lemma is true.
Lemma 3.1. The problem (1.1), (1.2) is uniquely solvable if and only if the corresponding
homogeneous problem
u′i(t) = σi1 `i1(u1)(t) + σi2 `i2(u2)(t), i = 1, 2, (3.1)
u1(a) = 0, u2(a) = 0, (3.2)
has only the trivial solution.
In order to simplify the discussion in the proofs, we formulate the following obvious lemma.
Lemma 3.2. (u1, u2)T is a solution of the problem (3.1), (3.2) if and only if (u1,−u2)T is
a solution of the problem
v′i(t) = (−1)i−1σi1 `i1(v1)(t) + (−1)iσi2 `i2(v2)(t), i = 1, 2, (3.3)
v1(a) = 0, v2(a) = 0 . (3.4)
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ON THE CAUCHY PROBLEM FOR TWO-DIMENSIONAL SYSTEMS OF LINEAR FUNCTIONAL . . . 565
Lemma 3.3 ([19], Remark 1.1). Let ` ∈ Pab be such that
b∫
a
`(1)(s)ds < 1.
Then every absolutely continuous function u : [a, b] → R such that
u′(t) ≥ `(u)(t) for t ∈ [a, b], u(a) ≥ 0,
satisfies u(t) ≥ 0 for t ∈ [a, b].
Now we are in a position to prove the main results.
Proof of Theorem 2.1. According to Lemmas 3.1 and 3.2, in order to prove the theorem it
is sufficient to show that the system
u′i(t) = `i1(u1)(t) + `i2(u2)(t), i = 1, 2, (3.5)
has only the trivial solution satisfying (3.2).
Suppose that, on the contrary, (u1, u2)T is a nontrivial solution of the problem (3.5), (3.2).
If the inequality
ui(t) ≥ 0 for t ∈ [a, b] (3.6)
holds for some i ∈ {1, 2} then, by virtue of (2.2), the assumption `3−i i ∈ Pab, and Lemma 3.3,
we get
u3−i(t) ≥ 0 for t ∈ [a, b]. (3.7)
Consequently, the functions u1 and u2 satisfy one of the following alternatives.
(a) Both functions u1 and u2 do not change their signs. Then, without loss of generality, we
can assume that (3.6) holds for i = 1, 2.
(b) Both functions u1 and u2 change their signs.
Put
Mi = max
{
ui(t) : t ∈ [a, b]
}
, i = 1, 2, (3.8)
and choose αi ∈ [a, b], i = 1, 2, such that
ui(αi) = Mi for i = 1, 2. (3.9)
Obviously, in both cases (a) and (b), we have
M1 ≥ 0, M2 ≥ 0, M1 + M2 > 0. (3.10)
The integration of (3.5) from a to αi, in view of (3.8) – (3.10), and the assumptions `i1, `i2 ∈ Pab,
yield
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566 J. ŠREMR, R. HAKL
Mi =
αi∫
a
`i1(u1)(s)ds +
αi∫
a
`i2(u2)(s)ds ≤
≤ M1
αi∫
a
`i1(1)(s)ds + M2
αi∫
a
`i2(1)(s)ds ≤
≤ M1Ai1 + M2Ai2, i = 1, 2. (3.11)
By virtue of (2.2) and (3.10), we get from (3.11) that
0 ≤ Mi
(
1−Aii
)
≤ M3−iAi 3−i, i = 1, 2. (3.12)
Using (2.2) and (3.10) once again, (3.12) implies M1 > 0, M2 > 0, and
(1−A11)(1−A22) ≤ A12A21,
which contradicts (2.3).
The contradiction obtained proves that the problem (3.5), (3.2) has only the trivial solution.
Proof of Theorem 2.2. According to Lemmas 3.1 and 3.2, in order to prove the theorem it
is sufficient to show that the system
u′1(t) = `11(u1)(t) + `12(u2)(t), (3.13)
u′2(t) = −`21(u1)(t) + `22(u2)(t) (3.14)
has only the trivial solution satisfying (3.2).
Suppose that, on the contrary, (u1, u2)T is a nontrivial solution of the problem (3.13), (3.14),
(3.2). It is clear that u1 and u2 satisfy one of the following.
(a) One of the functions u1 and u2 is of a constant sign. According to Lemma 3.2, we can
assume without loss of generality that u1(t) ≥ 0 for t ∈ [a, b].
(b) Both functions u1 and u2 change their signs.
Case (a): u1(t) ≥ 0 for t ∈ [a, b]. In view of (2.2) and the assumption `21 ∈ Pab, Lemma 3.3
yields u2(t) ≤ 0 for t ∈ [a, b]. Now, by virtue of (2.2) and the assumption `12 ∈ Pab, Lemma 3.3
again implies u1(t) ≤ 0 for t ∈ [a, b]. Consequently, u1 ≡ 0 and Lemma 3.3 once again results
in u2 ≡ 0, which is a contradiction.
Case (b): u1 and u2 change their signs. For i = 1, 2, we put
Mi = max
{
ui(t) : t ∈ [a, b]
}
, mi = −min
{
ui(t) : t ∈ [a, b]
}
. (3.15)
Choose αi, βi ∈ [a, b], i = 1, 2, such that the equalities
u1(α1) = M1 , u1(β1) = −m1 (3.16)
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ON THE CAUCHY PROBLEM FOR TWO-DIMENSIONAL SYSTEMS OF LINEAR FUNCTIONAL . . . 567
and
u2(α2) = M2 , u2(β2) = −m2 (3.17)
are satisfied. Obviously,
Mi > 0, mi > 0 for i = 1, 2. (3.18)
Furthermore, for i, k = 1, 2, we denote
Bik =
min{αi,βi}∫
a
`ik(1)(s)ds, Dik =
max{αi,βi}∫
min{αi,βi}
`ik(1)(s)ds. (3.19)
It is clear that
Bik + Dik ≤ Aik for i, k = 1, 2. (3.20)
According to Lemma 3.2, we can assume without loss of generality that α1 < β1 and α2 < β2.
The integrations of (3.13) from a to α1 and from α1 to β1, in view of (3.15), (3.16), (3.19), and
the assumptions `11, `12 ∈ Pab, result in
M1 =
α1∫
a
`11(u1)(s)ds +
α1∫
a
`12(u2)(s)ds ≤
≤ M1
α1∫
a
`11(1)(s)ds + M2
α1∫
a
`12(1)(s)ds = M1B11 + M2B12
and
M1 + m1 = −
β1∫
α1
`11(u1)(s)ds−
β1∫
α1
`12(u2)(s)ds ≤
≤ m1
β1∫
α1
`11(1)(s)ds + m2
β1∫
α1
`12(1)(s)ds = m1D11 + m2D12 .
The last relations, by virtue of (2.2) and (3.18), imply
0 <
M1
M2
(1−B11) +
m1
m2
(1−D11) +
M1
m2
≤ B12 + D12 ≤ A12 . (3.21)
On the other hand, the integrations of (3.14) from a to α2 and from α2 to β2, using (3.15),
ISSN 1562-3076. Нелiнiйнi коливання, 2007, т . 10, N◦ 4
568 J. ŠREMR, R. HAKL
(3.17), (3.19), and the assumptions `21, `22 ∈ Pab, give
M2 = −
α2∫
a
`21(u1)(s)ds +
α2∫
a
`22(u2)(s)ds ≤
≤ m1
α2∫
a
`21(1)(s)ds + M2
α2∫
a
`22(1)(s)ds = m1B21 + M2B22
and
M2 + m2 =
β2∫
α2
`21(u1)(s)ds−
β2∫
α2
`22(u2)(s)ds ≤
≤ M1
β2∫
α2
`21(1)(s)ds + m2
β2∫
α2
`22(1)(s)ds = M1D21 + m2D22 .
The last relations, by virtue of (2.2) and (3.18), yield
0 <
M2
m1
(1−B22) +
m2
M1
(1−D22) +
M2
M1
≤ B21 + D21 ≤ A21 . (3.22)
Now, it follows from (3.21) and (3.22) that
A12A21 ≥
M1
m1
(1−B11)(1−B22) +
m2
M2
(1−B11)(1−D22) + 1−B11+
+
M2
m2
(1−D11)(1−B22) +
m1
M1
(1−D11)(1−D22) +
m1M2
m2M1
(1−D11)+
+
M2M1
m1m2
(1−B22) + 1−D22 +
M2
m2
. (3.23)
Using the relation
x + y ≥ 2
√
xy for x ≥ 0, y ≥ 0,
it is easy to verify that
M1
m1
(1−B11)(1−B22) +
m1
M1
(1−D11)(1−D22) ≥
≥ 2
√
(1−B11)(1−B22)(1−D11)(1−D22) ≥
≥ 2
√
(1−B11 −D11)(1−B22 −D22) ≥ 2
√
(1−A11)(1−A22) ,
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ON THE CAUCHY PROBLEM FOR TWO-DIMENSIONAL SYSTEMS OF LINEAR FUNCTIONAL . . . 569
m1M2
m2M1
(1−D11) +
M2M1
m1m2
(1−B22) ≥ 2
M2
m2
√
(1−D11)(1−B22) , (3.24)
M2
m2
(1−D11)(1−B22) + 2
M2
m2
√
(1−D11)(1−B22) +
M2
m2
=
=
M2
m2
(√
(1−D11)(1−B22) + 1
)2
,
and
m2
M2
(1−B11)(1−D22) +
M2
m2
(√
(1−D11)(1−B22) + 1
)2
≥
≥ 2
√
(1−B11)(1−D22)
(√
(1−D11)(1−B22) + 1
)
≥
≥ 2
√
(1−B11 −D11)(1−B22 −D22) + 2
√
(1−B11)(1−D22) ≥
≥ 2
√
(1−A11)(1−A22) + 2
√
(1−B11)(1−D22) . (3.25)
Therefore, by virtue of (3.24), (3.25), (3.23) implies
A12A21 ≥
≥ 4
√
(1−A11)(1−A22) + 1−B11 + 2
√
(1−B11)(1−D22) + 1−D22 ≥
≥ 4
√
(1−A11)(1−A22) +
(√
1−A11 +
√
1−A22
)2
,
which contradicts (2.5).
The contradictions obtained in (a) and (b) prove that the problem (3.13), (3.14), (3.2) has
only the trivial solution.
Proof of Corollary 2.1. The validity of the corollary follows immediately from Theorem 2.1.
Proof of Corollary 2.2. The validity of the corollary follows immediately from Theorem 2.2.
4. Counter-examples. In this part, the counter-examples are constructed verifying that the
results obtained above are optimal in a certain sense.
Example 4.1. Let σik ∈ {−1, 1}, hik ∈ L
(
[a, b]; R+
)
, i, k = 1, 2, be such that
σ11 = 1,
b∫
a
h11(s)ds ≥ 1.
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570 J. ŠREMR, R. HAKL
It is clear that there exists t0 ∈ ]a, b] such that
t0∫
a
h11(s)ds = 1.
Let the operators `ik ∈ Pab, i, k = 1, 2, be defined by
`ik(v)(t) df= hik(t)v
(
τik(t)
)
for t ∈ [a, b], v ∈ C([a, b]; R), (4.1)
where τ11(t) = t0, τ12(t) = a, τ21(t) = a, and τ22(t) = a for t ∈ [a, b]. Put
u(t) =
t∫
a
h11(s)ds for t ∈ [a, b].
It is easy to verify that (u, 0)T is a nontrivial solution of the problem (1.1), (1.2) with qi ≡ 0 and
ci = 0, i = 1, 2.
An analogous example can be constructed for the case where
σ22 = 1,
b∫
a
h22(s)ds ≥ 1.
This example shows that the constant 1 in the right-hand side of the inequalities in (2.2) is
optimal and cannot be weakened.
Example 4.2. Let σik = 1 for i, k = 1, 2 and let hik ∈ L
(
[a, b]; R+
)
, i, k = 1, 2, be such that
b∫
a
h11(s)ds < 1,
b∫
a
h22(s)ds < 1, (4.2)
and
b∫
a
h12(s)ds
b∫
a
h21(s)ds ≥
1−
b∫
a
h11(s)ds
1−
b∫
a
h22(s)ds
.
It is clear that there exists t0 ∈ ]a, b] such that
t0∫
a
h12(s)ds
t0∫
a
h21(s)ds =
1−
t0∫
a
h11(s)ds
1−
t0∫
a
h22(s)ds
.
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ON THE CAUCHY PROBLEM FOR TWO-DIMENSIONAL SYSTEMS OF LINEAR FUNCTIONAL . . . 571
Let the operators `ik ∈ Pab, i, k = 1, 2, be defined by (4.1), where τik(t) = t0 for t ∈ [a, b],
i, j = 1, 2. Put
u1(t) =
t∫
a
h11(s)ds +
1−
t0∫
a
h11(s)ds
t0∫
a
h12(s)ds
t∫
a
h12(s)ds for t ∈ [a, b],
u2(t) =
t∫
a
h21(s)ds +
t0∫
a
h21(s)ds
1−
t0∫
a
h22(s)ds
t∫
a
h22(s)ds for t ∈ [a, b].
It is easy to verify that (u1, u2)T is a nontrivial solution of the problem (1.1), (1.2) with qi ≡ 0
and ci = 0, i = 1, 2.
This example shows that the strict inequality (2.3) in Theorem 2.1 cannot be replaced by the
nonstrict one.
Example 4.3. Let σ11 = 1, σ12 = 1, σ21 = −1, and σ22 = 1. Let α ∈ [0, 1[ and h12, h21 ∈
∈ L
(
[a, b]; R+
)
be such that
b∫
a
h12(s)ds
b∫
a
h21(s)ds ≥ 8(1− α).
It is clear that there exist t0 ∈ ]a, b] and t1, t2 ∈ ]a, t0[ such that
t0∫
a
h12(s)ds
t0∫
a
h21(s)ds = 8(1− α)
and
t1∫
a
h12(s)ds =
1
4
t0∫
a
h12(s)ds,
t2∫
a
h21(s)ds =
1
2
t0∫
a
h21(s)ds.
Furthermore, we choose h11, h22 ∈ L
(
[a, b]; R+
)
with the properties
h11(t) = 0 for t ∈ [a, t1] ∪ [t0, b], h22(t) = 0 for t ∈ [t2, b],
and
b∫
a
h11(s)ds =
b∫
a
h22(s)ds = α.
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572 J. ŠREMR, R. HAKL
Let the operators `ik ∈ Pab, i, k = 1, 2, be defined by (4.1), where τ11(t) = t0, τ22(t) = t2 for
t ∈ [a, b], and
τ12(t) =
{
t0 for t ∈ [a, t1[,
t2 for t ∈ [t1, b],
τ21(t) =
{
t1 for t ∈ [a, t2[,
t0 for t ∈ [t2, b].
Put
u1(t) =
t0∫
t2
h21(s)ds
t∫
a
h12(s)ds for t ∈ [a, t1[,
1− α− 2
t∫
t1
h11(s)ds−
t0∫
t2
h21(s)ds
t∫
t1
h12(s)ds for t ∈ [t1, b],
u2(t) =
−(1− α)
t∫
a
h21(s)ds−
t0∫
t2
h21(s)ds
t∫
a
h22(s)ds for t ∈ [a, t2[,
−
t0∫
t2
h21(s)ds + 2
t∫
t2
h21(s)ds for t ∈ [t2, b].
It is easy to verify that (u1, u2)T is a nontrivial solution of the problem (1.1), (1.2) with qi ≡ 0
and ci = 0, i = 1, 2.
This example shows that the strict inequality (2.5) in Theorem 2.2 cannot be replaced by the
nonstrict one provided A11 = A22.
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Received 08.03.06
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