Nonpositive solutions to a certain functional differential inequality
In the paper, efficient conditions are found guaranteeing that every solution to the problem u'(t) ≥ ι(u)(t), u(a) ≥ h(u) is nonpositive, where ι : C([a, b]; R) → L([a, b]; R) and h : C([a, b]; R) → R are linear bounded operators. The results obtained are very useful for the investigation of...
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| Cite this: | Nonpositive solutions to a certain functional differential inequality / A. Lomtatidze, Z. Opluštil, J. Šremr // Нелінійні коливання. — 2009. — Т. 12, № 4. — С. 461-494. — Бібліогр.: 18 назв. — англ. |
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irk-123456789-1784172021-02-20T01:25:51Z Nonpositive solutions to a certain functional differential inequality Lomtatidze, A. Opluštil, Z. Šremr, J. In the paper, efficient conditions are found guaranteeing that every solution to the problem u'(t) ≥ ι(u)(t), u(a) ≥ h(u) is nonpositive, where ι : C([a, b]; R) → L([a, b]; R) and h : C([a, b]; R) → R are linear bounded operators. The results obtained are very useful for the investigation of the question on solvability and unique solvability of the nonlocal boundary-value problems for the first order functional differential equations in both linear and nonlinear cases. Знайдено ефективнi умови для того, щоб кожен розв’язок задачi u'(t) ≥ ι(u)(t), u(a) ≥ h(u), де ι : C([a, b]; R) → L([a, b]; R) h : C([a, b]; R) → R — лiнiйнi обмеженi оператори, був недодатним. Отриманi результати є корисними для вивчення задачi розв’язностi та iснування єдиного розв’язку нелокальних граничних задач для функцiонально-диференцiальних рiвнянь першого порядку як в лiнiйному, так i в нелiнiйному випадках. 2009 Article Nonpositive solutions to a certain functional differential inequality / A. Lomtatidze, Z. Opluštil, J. Šremr // Нелінійні коливання. — 2009. — Т. 12, № 4. — С. 461-494. — Бібліогр.: 18 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/178417 517.9 en Нелінійні коливання Інститут математики НАН України |
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In the paper, efficient conditions are found guaranteeing that every solution to the problem
u'(t) ≥ ι(u)(t), u(a) ≥ h(u)
is nonpositive, where ι : C([a, b]; R) → L([a, b]; R) and h : C([a, b]; R) → R are linear bounded operators. The results obtained are very useful for the investigation of the question on solvability and unique
solvability of the nonlocal boundary-value problems for the first order functional differential equations in
both linear and nonlinear cases. |
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Article |
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Lomtatidze, A. Opluštil, Z. Šremr, J. |
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Lomtatidze, A. Opluštil, Z. Šremr, J. Nonpositive solutions to a certain functional differential inequality Нелінійні коливання |
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Lomtatidze, A. Opluštil, Z. Šremr, J. |
| author_sort |
Lomtatidze, A. |
| title |
Nonpositive solutions to a certain functional differential inequality |
| title_short |
Nonpositive solutions to a certain functional differential inequality |
| title_full |
Nonpositive solutions to a certain functional differential inequality |
| title_fullStr |
Nonpositive solutions to a certain functional differential inequality |
| title_full_unstemmed |
Nonpositive solutions to a certain functional differential inequality |
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nonpositive solutions to a certain functional differential inequality |
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Інститут математики НАН України |
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2009 |
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http://dspace.nbuv.gov.ua/handle/123456789/178417 |
| citation_txt |
Nonpositive solutions to a certain functional differential inequality / A. Lomtatidze, Z. Opluštil, J. Šremr // Нелінійні коливання. — 2009. — Т. 12, № 4. — С. 461-494. — Бібліогр.: 18 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT lomtatidzea nonpositivesolutionstoacertainfunctionaldifferentialinequality AT oplustilz nonpositivesolutionstoacertainfunctionaldifferentialinequality AT sremrj nonpositivesolutionstoacertainfunctionaldifferentialinequality |
| first_indexed |
2025-07-15T16:53:37Z |
| last_indexed |
2025-07-15T16:53:37Z |
| _version_ |
1837732639015960576 |
| fulltext |
UDC 517 . 9
NONPOSITIVE SOLUTIONS TO A CERTAIN
FUNCTIONAL DIFFERENTIAL INEQUALITY
НЕДОДАТНI РОЗВ’ЯЗКИ ДЕЯКИХ
ФУНКЦIОНАЛЬНО-ДИФЕРЕНЦIАЛЬНИХ НЕРIВНОСТЕЙ
A. Lomtatidze
Masaryk Univ.
Kotlářská 2, 611 37 Brno, Czech Republic
e-mail: bacho@math.muni.cz
Z. Opluštil
Inst. Math.
Technická 2, 616 69 Brno, Czech Republic
e-mail: oplustil@fme.vutbr.cz
J. Šremr
Inst. Math. Acad. Sci. Czech Republic
Žižkova 22, 616 62 Brno, Czech Republic
e-mail: sremr@ipm.cz
In the paper, efficient conditions are found guaranteeing that every solution to the problem
u′(t) ≥ `(u)(t), u(a) ≥ h(u)
is nonpositive, where ` : C([a, b]; R) → L([a, b]; R) and h : C([a, b]; R) → R are linear bounded oper-
ators. The results obtained are very useful for the investigation of the question on solvability and unique
solvability of the nonlocal boundary-value problems for the first order functional differential equations in
both linear and nonlinear cases.
Знайдено ефективнi умови для того, щоб кожен розв’язок задачi
u′(t) ≥ `(u)(t), u(a) ≥ h(u),
де ` : C([a, b]; R) → L([a, b]; R) h : C([a, b]; R) → R — лiнiйнi обмеженi оператори, був недодат-
ним. Отриманi результати є корисними для вивчення задачi розв’язностi та iснування єдиного
розв’язку нелокальних граничних задач для функцiонально-диференцiальних рiвнянь першого
порядку як в лiнiйному, так i в нелiнiйному випадках.
1. Introduction and notation. On the interval [a, b], we consider the functional differential
inequality
u′(t) ≥ `(u)(t), (1.1)
where ` : C([a, b]; R) → L([a, b]; R) is a linear bounded operator. By a solution to inequality
(1.1) we understand an absolutely continuous function u : [a, b] → R satisfying inequality (1.1)
almost everywhere on the interval [a, b].
c© A. Lomtatidze, Z. Opluštil, J. Šremr, 2009
ISSN 1562-3076. Нелiнiйнi коливання, 2009, т . 12, N◦ 4 461
462 A. LOMTATIDZE, Z. OPLUŠTIL, J. ŠREMR
Theorems on differential inequalities play a very important role in the theory of differen-
tial equations. For example, well-known Gronwall’s inequality is also a corollary of a certain
theorem on differential inequalities. Various types of differential inequalities are studied in the
literature (see, e.g., [1, 3 – 6, 8, 9, 11, 13, 15 – 17]). In the present paper, effective sufficient con-
ditions are found guaranteeing that every solution to inequality (1.1) satisfying the condition
u(a) ≥ h(u) (1.2)
with a linear bounded functional h : C([a, b]; R) → R is nonpositive on the interval [a, b]. State-
ments obtained here can be used in the investigation of the question on solvability and unique
solvability of the nonlocal boundary-value problems for functional differential equations in
both linear and nonlinear cases.
In order to simplify the formulation of the main results we introduce the following defini-
tion.
Definition 1.1. Let h ∈ Fab. An operator ` ∈ Lab is said to belong to the set Ṽ −
ab (h) (resp.
Ṽ +
ab (h)) if every solution to the problem (1.1), (1.2) is nonpositive (resp. nonnegative).
As it was mentioned above, the aim of the paper is to find conditions guaranteeing the
inclusions ` ∈ Ṽ −
ab (h) and ` ∈ Ṽ +
ab (h) to hold. In the case where the functional h is given by the
formula
h(v) df= λv(b) for v ∈ C([a, b]; R)
with λ ≥ 0, the sets Ṽ +
ab (h) and Ṽ −
ab (h) are described in detail (see [7, 8]). In [14], the case where
h ∈ PFab is considered. However, a general case of h has not been studied yet.
We shall suppose throughout the paper that the functional h ∈ Fab is defined by the formula
h(v) df= λv(b) + h0(v)− h1(v) for v ∈ C([a, b]; R), (1.3)
where λ > 0 and h0, h1 ∈ PFab. There is no lost of generality in assuming this, because an
arbitrary linear bounded functional can be represented in this form.
The following notation is used in the sequel:
(1) N is the set of all natural numbers, R is the set of all real numbers, R+ = [0,+∞[. If
x ∈ R then we put
[x]+ =
|x|+ x
2
, [x]− =
|x| − x
2
.
(2) C([a, b]; R) is the Banach space of continuous functions v : [a, b] → R endowed with
the norm ‖v‖C = max{|v(t)| : t ∈ [a, b]}.
(3) C̃([a, b];D), where D ⊆ R, is the set of absolutely continuous functions v : [a, b] → D.
(4) L([a, b]; R) is the Banach space of Lebesgue integrable functions p : [a, b] → R endowed
with the norm ‖p‖L =
∫ b
a |p(s)|ds.
(5) L([a, b];D) = {p ∈ L([a, b]; R) : p : [a, b] → D}, where D ⊆ R.
(6) C([a, b];D) = {v ∈ C([a, b]; R) : v : [a, b] → D}, where D ⊆ R.
(7) Lab is the set of linear bounded operators ` : C([a, b]; R) → L([a, b]; R), Pab is the set of
operators ` ∈ Lab mapping the set C([a, b]; R+) into the set L([a, b]; R+).
(8) Fab is the set of linear bounded functionals h : C([a, b]; R) → R, PFab is the set of
functionals h ∈ Fab mapping the set C([a, b]; R+) into the set R+.
ISSN 1562-3076. Нелiнiйнi коливання, 2009, т . 12, N◦ 4
NONPOSITIVE SOLUTIONS TO A CERTAIN FUNCTIONAL DIFFERENTIAL INEQUALITY 463
Definition 1.2. Let t0 ∈ [a, b]. We say that ` ∈ Lab is a t0 Volterra operator if, for arbitrary
a1 ∈ [a, t0], b1 ∈ [t0, b], a1 6= b1, and v ∈ C([a, b]; R) with the property
v(t) = 0 for t ∈ [a1, b1],
the relation
`(v)(t) = 0 for a.e. t ∈ [a1, b1]
holds.
2. Preliminary remarks. Recall that we suppose ` ∈ Lab and h ∈ Fab. The following two
assumptions are natural:
(A) If h(1) = 1 then the operator ` is supposed to be nontrivial in the sense that the
condition `(1) 6≡ 0 holds.
(B) h̃ 6≡ 0, where the functional h̃ is defined by the formula
h̃(v) = h(v)− v(a) for v ∈ C([a, b]; R).
Remark 2.1. It follows from Definition 1.1 that if ` ∈ Ṽ −
ab (h) (resp. ` ∈ Ṽ +
ab (h)) then the
homogeneous problem
u′(t) = `(u)(t), u(a) = h(u) (2.1)
has only the trivial solution. Therefore, the inclusion ` ∈ Ṽ −
ab (h) (resp. ` ∈ Ṽ +
ab (h)) guarantees
the unique solvability of the problem
u′(t) = `(u)(t) + q(t), u(a) = h(u) + c (2.2)
for every q ∈ L([a, b]; R) and c ∈ R. This fact follows from the Fredholm property of problem
(2.2) (see, e.g., [2, 10]; in the case, where the operator ` is strongly bounded, see also [1, 12,
18]). Moreover, under the condition ` ∈ Ṽ −
ab (h) (resp. ` ∈ Ṽ +
ab (h)), the unique solution to the
problem (2.2) is nonpositive (resp. nonnegative) whenever q ∈ L([a, b]; R+) and c ∈ R+.
Remark 2.2. It is easy to verify that the condition (−Pab) ∩ Ṽ −
ab (h) 6= ∅ implies
h(1) > 1. (2.3)
Indeed, if ` ∈ (−Pab) ∩ Ṽ −
ab (h) and h(1) ≤ 1, then the function u ≡ 1 is a positive solution to
the problem (1.1), (1.2), which contradicts the inclusion ` ∈ Ṽ −
ab (h).
On the other hand if, together with (2.3), the inequality h0(1) ≤ 1 holds then the zero
operator belongs to the set Ṽ −
ab (h). Indeed, let u ∈ C̃([a, b]; R) satisfy (1.2) and
u′(t) ≥ 0 for a.e. t ∈ [a, b].
Then it is clear that
u(a) ≤ u(t) ≤ u(b) for t ∈ [a, b]. (2.4)
By virtue of condition (2.4) and the assumptions h0, h1 ∈ PFab, it follows from (1.2) that
u(a) ≥ λu(b) + h0(u)− h1(u) ≥ u(a)h0(1) + (λ− h1(1)) u(b).
ISSN 1562-3076. Нелiнiйнi коливання, 2009, т . 12, N◦ 4
464 A. LOMTATIDZE, Z. OPLUŠTIL, J. ŠREMR
Taking now condition (2.4) and the assumption h0(1) ≤ 1 into account, we get
(λ− h1(1))u(b) ≤ (1− h0(1))u(a) ≤ (1− h0(1))u(b),
and thus
(h(1)− 1) u(b) ≤ 0.
The last inequality and (2.3) result in u(b) ≤ 0. Hence, condition (2.4) guarantees u(t) ≤ 0 for
t ∈ [a, b], and thus 0 ∈ Ṽ −
ab (h).
We have shown that condition (2.3) is necessary for the validity of the relation (−Pab) ∩
∩Ṽ −
ab (h) 6= ∅ and conditions (2.3) and h0(1) ≤ 1 are sufficient for the inclusion 0 ∈ Ṽ −
ab (h) to
hold.
Definition 2.1. An operator ` ∈ Lab is said to belong to the set Sab(a) (resp. Sab(b)) if every
solution u to inequality (1.1), which satisfies u(a) ≥ 0 (resp. u(b) ≤ 0), is nonnegative (resp.
nonpositive).
Remark 2.3. The sets Sab(a) and Sab(b) are investigated in [6].
3. Auxiliary statements. In this section, auxiliary statements are given. More precisely, pro-
perties of the sets U−
ab and Ũ+
ab(h) are studied that are very useful in the investigation of the
validity of the desired inclusion ` ∈ Ṽ −
ab (h).
3.1. Formulation of results. We first formulate all the results, the proofs are given in the next
section.
Definition 3.1. Let h ∈ Fab. An operator ` ∈ Lab is said to belong to the set U−
ab, if the
problem (1.1), (1.2) has no nontrivial nonnegative solution.
Remark 3.1. It follows immediately from Definitions 1.1 and 3.1 that Ṽ −
ab (h) ⊆ U−
ab(h).
Since the set U−
ab(h) is wider than Ṽ −
ab (h), conditions for the inclusion ` ∈ U−
ab can be derived
relatively easy. In Theorem 3.1 (Theorem 3.2), the case ` ∈ Pab (−` ∈ Pab) is considered,
whereas Theorems 3.3 and 3.4 concern the case where ` = `0 − `1 with `0, `1 ∈ Pab.
Theorem 3.1. Let ` ∈ Pab and
h1(1) < λ. (3.1)
Let, moreover, there exist a function γ ∈ C̃([a, b]; R+) satisfying
γ′(t) ≤ `(γ)(t) for a.e. t ∈ [a, b], (3.2)
γ(a) < h(γ). (3.3)
Then ` ∈ U−
ab(h).
Remark 3.2. If ` ∈ Pab, h(1) ≥ 1, and h1(1) < λ then the operator ` belongs to the set
U−
ab(h) without any additional assumptions. Indeed, since the operator ` is supposed to be
nontrivial in the case where h(1) = 1, the function
γ(t) = 1 +
t∫
a
`(1)(s)ds for t ∈ [a, b]
ISSN 1562-3076. Нелiнiйнi коливання, 2009, т . 12, N◦ 4
NONPOSITIVE SOLUTIONS TO A CERTAIN FUNCTIONAL DIFFERENTIAL INEQUALITY 465
satisfies the conditions (3.2) and (3.3).
Theorem 3.2. Let −` ∈ Pab and
h(1) > 1, h0(1) ≤ 1. (3.4)
Then ` ∈ U−
ab(h) if and only if there exists a function γ ∈ C̃([a, b]; ]0,+∞[) satisfying the condi-
tions (3.2) and (3.3).
Theorem 3.3. Let ` = `0 − `1, where `0, `1 ∈ Pab, and
h(1) ≤ 1, h1(1) < λ. (3.5)
If, moreover,
b∫
a
`1(1)(s) ds < (λ− h1(1))min
{
1,
1
λ
}
(3.6)
and
b∫
a
`0(1)(s) ds >
(1− h0(1))min
{
1, 1
λ
}
(λ− h1(1))min
{
1, 1
λ
}
−
∫ b
a `1(1)(s) ds
− 1, (3.7)
then ` ∈ U−
ab(h).
Theorem 3.4. Let ` = `0 − `1, where `0, `1 ∈ Pab, and
h(1) > 1, h1(1) < λ. (3.8)
Let, moreover, the inequality (3.6) hold and
b∫
a
`0(1)(s) ds > ω
b∫
a
`1(1)(s) ds
, (3.9)
where
ω(y) =
(y + h1(1))
(
1− 1
λh1(1)
)
1− 1
λh1(1)− y
− (h0(1) + λ− 1)
if λ ≥ 1, y <
(h(1)− 1)
(
1− 1
λh1(1)
)
λ− 1 + h0(1)
,(
y + 1
λh1(1)
) (
1− 1
λh1(1)
)
1− 1
λh1(1)− y
−
(
1
λ
h0(1) +
λ− 1
λ
)
if λ ≥ 1, y ≥
(h(1)− 1)
(
1− 1
λh1(1)
)
λ− 1 + h0(1)
,(
y + 1−λ
λ + 1
λh1(1)
)
(λ− h1(1))
λ− h1(1)− y
− 1
λ
h0(1)
if λ < 1, y <
(h(1)− 1) (λ− h1(1))
h0(1)
,
(y + 1− λ + h1(1)) (λ− h1(1))
λ− h1(1)− y
− h0(1)
if λ < 1, y ≥ (h(1)− 1) (λ− h1(1))
h0(1)
.
(3.10)
ISSN 1562-3076. Нелiнiйнi коливання, 2009, т . 12, N◦ 4
466 A. LOMTATIDZE, Z. OPLUŠTIL, J. ŠREMR
Then ` ∈ U−
ab(h).
Now we introduce the following definition.
Definition 3.2. Let h ∈ Fab. An operator ` ∈ Lab is said to belong to the set Ũ+
ab(h) if there is
no nonpositive solution u to inequality (1.1) satisfying the condition
u(a) > h(u). (3.11)
Remark 3.3. It is clear that Ũ+
ab(0) = Lab and Ṽ +
ab (h) ⊆ Ũ+
ab(h).
Theorem 1. Let ` ∈ Pab and h ∈ PFab be such that h(1) ≤ 1. If there exists a function
γ ∈ C̃([a, b]; ]0,+∞[) satisfying the conditions
γ′(t) ≥ `(γ)(t) for a.e. t ∈ [a, b], (3.12)
γ(a) ≥ h(γ), (3.13)
then ` ∈ Ũ+
ab(h).
3.2. Proofs. We first recall a result established in [6].
Lemma 3.1 ([6], Theorem 1.1). Let ` ∈ Pab. Then ` ∈ Sab(a) if and only if there exists
a function γ ∈ C̃([a, b]; ]0,+∞[) satisfying condition (3.12).
Proof of Theorem 3.1. Let u be a nonnegative solution to the problem (1.1), (1.2). We shall
show that u ≡ 0. Since ` ∈ Pab and u is a nonnegative function, it follows from (1.1) that
0 ≤ u(a) ≤ u(t) ≤ u(b) for t ∈ [a, b]. (3.14)
Suppose that u(b) > 0. Then condition (1.2), in view of (3.1), (3.14), and the assumptions
h0, h1 ∈ PFab, results in
u(a) ≥ λu(b) + h0(u)− h1(u) ≥ (λ− h1(1))u(b) > 0.
Consequently, the relation (3.14) implies
u(t) > 0 for t ∈ [a, b]. (3.15)
Put
v(t) = ru(t)− γ(t) for t ∈ [a, b],
where
r = max
{
γ(t)
u(t)
: t ∈ [a, b]
}
.
According to (3.3), (3.15), and the assumption γ ∈ C̃([a, b]; R+), we get
r > 0. (3.16)
ISSN 1562-3076. Нелiнiйнi коливання, 2009, т . 12, N◦ 4
NONPOSITIVE SOLUTIONS TO A CERTAIN FUNCTIONAL DIFFERENTIAL INEQUALITY 467
It is obvious that
v(t) ≥ 0 for t ∈ [a, b] (3.17)
and there exists t0 ∈ [a, b] such that
v(t0) = 0. (3.18)
Taking now (1.1), (3.2), (3.16), (3.17), and the assumption ` ∈ Pab into account, we obtain
v′(t) ≥ `(v)(t) ≥ 0 for a.e. t ∈ [a, b]. (3.19)
Therefore, relation (3.19), on account of (3.17) and (3.18), yields
0 = v(a) ≤ v(t) ≤ v(b) for t ∈ [a, b]. (3.20)
However, using (1.2), (3.1), (3.3), (3.16), (3.20), and the assumptions h0, h1 ∈ PFab, we get
0 = v(a) > λv(b) + h0(v)− h1(v) ≥ (λ− h1(1)) v(b) ≥ 0,
which is a contradiction.
The contradiction obtained proves that u(b) ≤ 0. However, relation (3.14) then implies
u ≡ 0, and thus ` ∈ U−
ab(h).
The theorem is proved.
Proof of Theorem 3.2. First suppose that there exists a function γ ∈ C̃([a, b]; ]0 + ∞[)
satisfying relations (3.2) and (3.3). Let u be a nonnegative solution to the problem (1.1), (1.2).
We shall show that u ≡ 0. Suppose that, on the contrary, there exists t∗ ∈ [a, b] such that
u(t∗) > 0. (3.21)
Put
v(t) = rγ(t)− u(t) for t ∈ [a, b],
where
r = max
{
u(t)
γ(t)
: t ∈ [a, b]
}
.
According to (3.21), inequality (3.16) holds. It is clear that condition (3.17) is satisfied and
there exists t0 ∈ [a, b] such that (3.18) is true. Taking now (1.1), (3.2), (3.16), (3.17), and the
assumption −` ∈ Pab into account, we obtain
v′(t) ≤ `(v)(t) ≤ 0 for a.e. t ∈ [a, b]. (3.22)
Therefore, on account of (3.17) and (3.18), the relation (3.22) yields
0 = v(b) ≤ v(t) ≤ v(a) for t ∈ [a, b]. (3.23)
However, using (1.2), (3.3), (3.4), (3.16), (3.23), and the assumptions h0, h1 ∈ PFab, we get
0 = λv(b) = rλγ(b)− λu(b) > v(a)− h0(v) + h1(v) ≥ v(a) (1− h0(1)) ≥ 0,
ISSN 1562-3076. Нелiнiйнi коливання, 2009, т . 12, N◦ 4
468 A. LOMTATIDZE, Z. OPLUŠTIL, J. ŠREMR
a contradiction. The contradiction obtained proves that u ≡ 0, and thus ` ∈ U−
ab(h).
Now suppose that ` ∈ U−
ab(h). We first show that the homogeneous problem (2.1) has only
the trivial solution. Let u be a solution to problem (2.1). Using Remark 2.2, we have 0 ∈ Ṽ −
ab (h).
Therefore, according to Remark 2.1, the problem
α′(t) = `([u]−)(t), (3.24)
α(a) = h(α) (3.25)
has a unique solution α and the relation
α(t) ≥ 0 for t ∈ [a, b] (3.26)
holds. From (1.1), (1.2), (3.24), (3.25), and the assumption −` ∈ Pab, we get the relations
v′(t) = `([u]+)(t) ≤ 0 for a.e. t ∈ [a, b], v(a) = h(v),
where
v(t) = u(t) + α(t) for t ∈ [a, b]. (3.27)
Consequently, using the inclusion 0 ∈ Ṽ −
ab (h), we obtain v(t) ≥ 0 for t ∈ [a, b], and thus
−u(t) ≤ α(t) for t ∈ [a, b]. (3.28)
Taking now relation (3.26) into account, inequality (3.28) implies
[u(t)]− ≤ α(t) for t ∈ [a, b].
Therefore, in view of the assumption −` ∈ Pab, equation (3.24) yields
α′(t) ≥ `(α)(t) for a.e. t ∈ [a, b]. (3.29)
Consequently, α is a nonnegative function satisfying the conditions (3.25) and (3.29). Hence,
the assumption ` ∈ U−
ab(h) implies α ≡ 0, and thus relation (3.28) yields
u(t) ≥ 0 for t ∈ [a, b]. (3.30)
Since−u is also solution to the homogeneous problem (2.1), according to the above-proved we
have −u(t) ≥ 0 for t ∈ [a, b]. Consequently, u ≡ 0, i.e., the homogeneous problem (2.1) has
only the trivial solution. By virtue of the Fredholm property of the problem (2.2) (see, e.g., [2,
10]), the problem
γ′(t) = `(γ)(t), γ(a) = h(γ) + 1− h(1) (3.31)
has a unique solution γ. Setting
γ̄(t) = γ(t)− 1 for t ∈ [a, b],
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NONPOSITIVE SOLUTIONS TO A CERTAIN FUNCTIONAL DIFFERENTIAL INEQUALITY 469
we get from (3.31) the relations
γ̄′(t) ≤ `(γ̄)(t) for a.e. t ∈ [a, b], γ̄(a) = h(γ̄).
Now, analogously as above one can show that γ̄(t) ≥ 0 for t ∈ [a, b]. Therefore, in view of the
assumption h(1) > 1, it follows from (3.31) that γ is a positive function satisfying inequalities
(3.2) and (3.3).
The theorem is proved.
Proof of Theorem 3.3. Let u be a nonnegative solution to the problem (1.1), (1.2). We shall
show that u ≡ 0. Suppose that, on the contrary, u 6≡ 0. Put
x0 =
b∫
a
`0(1)(s) ds, y0 =
b∫
a
`1(1)(s) ds, (3.32)
M = max {u(t) : t ∈ [a, b]} , m = min {u(t) : t ∈ [a, b]} , (3.33)
and choose tM , tm ∈ [a, b] such that
u(tM ) = M, u(tm) = m. (3.34)
Obviously,
M > 0, m ≥ 0, (3.35)
and either
tm < tM (3.36)
or
tm ≥ tM . (3.37)
First suppose that (3.36) holds. The integrations of (1.1) from a to tm and from tM to b, in
view of (3.33) – (3.35) and the assumptions `0, `1 ∈ Pab, yield
u(a)−m ≤
tm∫
a
`1(u)(s) ds−
tm∫
a
`0(u)(s) ds ≤ M
tm∫
a
`1(1)(s) ds, (3.38)
M − u(b) ≤
b∫
tM
`1(u)(s) ds−
b∫
tM
`0(u)(s) ds ≤ M
b∫
tM
`1(1)(s) ds. (3.39)
Moreover, on account of (3.33) and the assumptions h0, h1 ∈ PFab, condition (1.2) implies
u(a)− λu(b) ≥ h0(u)− h1(u) ≥ mh0(1)−Mh1(1). (3.40)
We get from (3.38) – (3.40) the inequality
M (λ− h1(1))−m (1− h0(1)) ≤ M
tm∫
a
`1(1)(s) ds + λ
b∫
tM
`1(1)(s) ds
,
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470 A. LOMTATIDZE, Z. OPLUŠTIL, J. ŠREMR
i.e.,
M
(
(λ− h1(1))min
{
1,
1
λ
}
− y0
)
≤ m (1− h0(1))min
{
1,
1
λ
}
. (3.41)
Now suppose that (3.37) holds. The integration of (1.1) from tM to tm, in view of (3.33) –
(3.35) and the assumptions `0, `1 ∈ Pab, results in
M −m ≤
tm∫
tM
`1(u)(s) ds−
tm∫
tM
`0(u)(s) ds ≤ M
tm∫
tM
`1(1)(s) ds. (3.42)
It is not difficult to verify that, by virtue of (3.5) and (3.42), inequality (3.41) is true.
We have proved that, in both cases (3.36) and (3.37), inequality (3.41) is satisfied. On the
other hand, integration of (1.1) from a to b, in view of (3.32), (3.33), and the assumptions `0,
`1 ∈ Pab, yields
u(a)− u(b) ≤
b∫
a
`1(u)(s) ds−
b∫
a
`0(u)(s) ds ≤ My0 −mx0,
i.e.,
mx0 ≤ My0 + u(b)− u(a). (3.43)
Moreover, condition (1.2) implies
u(b)− u(a) ≤ u(b) (1− λ)− h0(u) + h1(u), (3.44)
u(b)− u(a) ≤ u(a)
(
1
λ
− 1
)
− 1
λ
h0(u) +
1
λ
h1(u). (3.45)
First suppose that λ ≤ 1. Inequalities (3.43) and (4.44), together with (3.33) and the as-
sumptions h0, h1 ∈ PFab, result in
mx0 ≤ My0 + M (1− λ)−mh0(1) + Mh1(1). (3.46)
Hence, by virtue of (3.6), (3.32), and (3.35), we get from (3.41) and (3.46) the relation m > 0
and the inequality
(λ− h1(1)− y0) (x0 + h0(1)) ≤ (y0 + 1− λ + h1(1)) (1− h0(1)) ,
which, in view of (3.6) and (3.32), contradicts (3.7).
Now suppose that λ > 1. The inequalities (3.43) and (3.45), together with (3.33) and the
assumptions h0, h1 ∈ PFab, imply
mx0 ≤ My0 −m
λ− 1
λ
− 1
λ
mh0(1) +
1
λ
Mh1(1). (3.47)
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NONPOSITIVE SOLUTIONS TO A CERTAIN FUNCTIONAL DIFFERENTIAL INEQUALITY 471
Hence, by virtue of (3.6), (3.32), and (3.35), we get from (3.41) and (3.47) the relation m > 0
and the inequality(
1− 1
λ
h1(1)− y0
) (
x0 +
λ− 1
λ
+
1
λ
h0(1)
)
≤
(
y0 +
1
λ
h1(1)
)
1− h0(1)
λ
,
which, in view of (3.6) and (3.32), contradicts (3.7).
The contradictions obtained prove the relation u ≡ 0, and thus ` ∈ U−
ab(h).
The theorem is proved.
Proof of Theorem 3.4. Let u be a nonnegative solution to the problem (1.1), (1.2). We shall
show that u ≡ 0. Suppose that, on the contrary, u 6≡ 0. Define the numbers x0, y0 and M, m
by formulae (3.32) and (3.33), respectively, and choose tM , tm ∈ [a, b] such that relations (3.34)
hold. Obviously, condition (3.35) is true and either the relation (3.36) or (3.37) is satisfied.
First suppose that (3.36) holds. Analogously to the proof of Theorem 3.3, the validity of
inequality (3.41) can be proved. Consequently, in view of (2.3) and (3.35), we get
M
(
(λ− h1(1))min
{
1,
1
λ
}
− y0
)
≤ m (λ− h1(1))min
{
1,
1
λ
}
. (3.48)
Now suppose that (3.37) holds. Analogously to the proof of Theorem 3.3, it can be shown
that (3.42) is satisfied. Consequently, it is not difficult to verify that, by virtue of (3.1) and (3.42),
inequality (3.48) is true.
We have proved that, in both cases (3.36) and (3.37), inequality (3.48) is satisfied. On the
other hand, analogously to the proof of Theorem 3.3, inequalities (3.43) – (3.45) can be derived.
First suppose that
λ ≥ 1, y0 <
(h(1)− 1)
(
1− 1
λh1(1)
)
λ− 1 + h0(1)
.
Relations (3.43) and (3.44), together with (3.33) and the assumptions h0, h1 ∈ PFab, result in
mx0 ≤ My0 −m(λ− 1)−mh0(1) + Mh1(1). (3.49)
Hence, by virtue of (3.6), (3.32), and (3.35), we get, from (3.48) and (3.49), the relation m > 0
and the inequality(
1− 1
λ
h1(1)− y0
)
(x0 + λ− 1 + h0(1)) ≤ (y0 + h1(1))
(
1− 1
λ
h1(1)
)
,
which, in view of (3.6) and (3.32), contradicts (3.9) with ω given by (3.10).
Now suppose that
λ ≥ 1, y0 ≥
(h(1)− 1)
(
1− 1
λh1(1)
)
λ− 1 + h0(1)
.
The inequalities (3.43) and (3.45), together with (3.33) and the assumptions h0, h1 ∈ PFab,
result in
mx0 ≤ My0 −m
λ− 1
λ
− 1
λ
mh0(1) +
1
λ
Mh1(1). (3.50)
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472 A. LOMTATIDZE, Z. OPLUŠTIL, J. ŠREMR
Hence, by virtue of (3.6), (3.32), and (3.35), we get, from (3.48) and (3.50), the relation m > 0
and the inequality(
1− 1
λ
h1(1)− y0
) (
x0 +
λ− 1
λ
+
1
λ
h0(1)
)
≤
(
y0 +
1
λ
h1(1)
)
λ− h1(1)
λ
,
which, in view of (3.6) and (3.32), contradicts (3.9) with ω given by (3.10).
Now suppose that
λ < 1, y0 <
(h(1)− 1) (λ− h1(1))
h0(1)
.
The inequalities (3.43) and (3.45), together with (3.33) and the assumptions h0, h1 ∈ PFab,
result in
mx0 ≤ My0 + M
1− λ
λ
− 1
λ
mh0(1) +
1
λ
Mh1(1). (3.51)
Hence, by virtue of (3.6), (3.32), and (3.35), we get, from (3.48) and (3.51), the relation m > 0
and the inequality
(λ− h1(1)− y0)
(
x0 +
1
λ
h0(1)
)
≤
(
y0 +
1− λ
λ
+
1
λ
h1(1)
)
(λ− h1(1)) ,
which, in view of (3.6) and (3.32), contradicts (3.9) with ω given by (3.10).
Finally suppose that
λ < 1, y0 ≥
(h(1)− 1) (λ− h1(1))
h0(1)
.
The inequalities (3.43) and (3.44), together with (3.33) and the assumptions h0, h1 ∈ PFab,
result in
mx0 ≤ My0 + M (1− λ)−mh0(1) + Mh1(1). (3.52)
Hence, by virtue of (3.6), (3.32), and (3.35), we get, from (3.48) and (3.52), the relation m > 0
and the inequality
(λ− h1(1)− y0) (x0 + h0(1)) ≤ (y0 + 1− λ + h1(1)) (λ− h1(1)) ,
which, in view of (3.6) and (3.32), contradicts (3.9) with ω given by (3.10).
The contradictions obtained prove the relation u ≡ 0 and thus ` ∈ U−
ab(h).
The theorem is proved.
Proof of Theorem 3.5. By virtue of the inequality (3.12) and the assumption ` ∈ Pab,
Lemma 3.1 guarantees that ` ∈ Sab(a).
Let u be a nonpositive solution to the problem (1.1), (3.11). It is not difficult to verify that
u(a) < 0. (3.53)
Indeed, if u(a) = 0 then inequality (1.1), in view of the inclusion ` ∈ Sab(a), yields u(t) ≥ 0
for t ∈ [a, b]. Hence we get u ≡ 0, which contradicts relation (3.11).
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NONPOSITIVE SOLUTIONS TO A CERTAIN FUNCTIONAL DIFFERENTIAL INEQUALITY 473
Put
w(t) = γ(a)u(t)− u(a)γ(t) for t ∈ [a, b].
We immediately obtain, from (1.1), (3.12), (3.53), and the assumption γ(a) > 0, the relations
w′(t) ≥ `(w)(t) for a.e. t ∈ [a, b], (3.54)
w(a) = 0. (3.55)
Therefore, the inclusion ` ∈ Sab(a) implies
w(t) ≥ 0 for t ∈ [a, b]. (3.56)
On the other hand, it follows from (3.11), (3.13), (3.53), (3.56), and the assumptions γ(a) > 0
and h ∈ PFab that
w(a) > h(w) ≥ 0,
which contradicts relation (3.55).
The contradiction obtained proves that there is no nonpositive solution to the problem (1.1),
(3.11), and thus ` ∈ Ũ+
ab(h).
The theorem is proved.
4. Main results. In this sections, we give main results of the paper, which are efficient con-
ditions under which the operator ` belongs to the set Ṽ −
ab (h). The results are formulated in
Sections 4.1 – 4.3, their proofs are presented in Section 4.5.
We first give a rather theoretical statement.
Proposition 4.1. Let h ∈ Fab. Then ` ∈ Ṽ −
ab (h) if and only if ` ∈ U−
ab(h) and there exists
¯̀ ∈ Pab such that ` + ¯̀ ∈ Ṽ −
ab (h).
Now we present a general result.
Theorem 4.1. Let ` ∈ Sab(b) ∩ Ũ+
ab(h0). Then ` ∈ Ṽ −
ab (h) if and only if there exists a function
γ ∈ C̃([a, b]; R+) satisfying the conditions (3.2) and (3.3).
4.1. The case ` ∈ Pab. The following statements can be proved in the case where ` ∈ Pab.
Theorem 4.2. Let ` ∈ Pab ∩ Ũ+
ab(h0) be a b-Volterra operator and condition (3.4) hold. Then
` ∈ Ṽ −
ab (h) if and only if ` ∈ Sab(b).
Corollary 4.1. Let ` ∈ Pab be a b-Volterra operator and condition (3.4) be fulfilled. If, more-
over, there exists a function γ ∈ C̃([a, b]; ]0,+∞[) such that the conditions (3.12) and
γ(a) ≥ h0(γ) (4.1)
hold, then ` ∈ Ṽ −
ab (h).
Corollary 4.2. Let ` ∈ Pab be a b-Volterra operator and
h(1) > 1, h0(1) < 1. (4.2)
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474 A. LOMTATIDZE, Z. OPLUŠTIL, J. ŠREMR
Assume that
h0(ϕ1) > 0 (4.3)
and there exist m, k ∈ N such that m > k and
%m(t) ≤ %k(t) for t ∈ [a, b], (4.4)
where %1 ≡ 1 and
%i+1(t)
df=
h0(ϕi)
1− h0(1)
+ ϕi(t) for t ∈ [a, b], i ∈ N, (4.5)
ϕi(t)
df=
t∫
a
`(%i)(s)ds for t ∈ [a, b], i ∈ N. (4.6)
Then ` ∈ Ṽ −
ab (h).
Remark 4.1. It follows from Corollary 4.2 (for k = 1 and m = 2) that if ` ∈ Pab is a
b-Volterra operator, condition (4.2) is fulfilled, and relation (4.3) holds with ϕ1 given by formula
(4.6), then ` ∈ Ṽ −
ab (h) provided that
b∫
a
`(1)(s)ds ≤ 1− h0(1).
Corollary 4.3. Let ` ∈ Pab be a b-Volterra operator and condition (4.2) be fulfilled. Then the
operator ` belongs to the set Ṽ −
ab (h) provided that ` ∈ Ṽ +
ab (h0).
Remark 4.2. Recall that efficient conditions guaranteeing the validity of the inclusion ` ∈
∈ Ṽ +
ab (h0) are stated in [14].
4.2. The case −` ∈ Pab. The following statements can be proved in the case where −` ∈
∈ Pab.
Theorem 4.3. Let −` ∈ Pab and condition (3.4) be fulfilled. Then ` ∈ Ṽ −
ab (h) if and only if
` ∈ U−
ab(h).
Corollary 4.4. Let −` ∈ Pab and condition (3.4) be fulfilled. Assume that at least one of the
following conditions is satisfied:
(a) there exist m, k ∈ N and a constant δ ∈ [0, 1[ such that m > k and
%m(t) ≤ δ%k(t) for t ∈ [a, b], (4.7)
where %1 ≡ 1, %i+1 ≡ ϑ(%i) for i ∈ N, and
ϑ(v)(t) df=
h̃(v)
h(1)− 1
− z(v)(a)
h(1)− 1
− z(v)(t) for t ∈ [a, b], v ∈ C([a, b]; R), (4.8)
h̃(v) df= h(z(v)), z(v)(t) df=
b∫
t
`(v)(s) ds for t ∈ [a, b], v ∈ C([a, b]; R); (4.9)
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NONPOSITIVE SOLUTIONS TO A CERTAIN FUNCTIONAL DIFFERENTIAL INEQUALITY 475
(b) there exists ¯̀ ∈ Pab such that
h(z0) > z0(a), (4.10)
z0(a) (1− h(z1)) + h(z0)z1(a) < h(z0), (4.11)
and the inequality
`(1)(t)ϑ(v)(t)− `(ϑ(v))(t) ≤ ¯̀(v)(t) for a.e. t ∈ [a, b] (4.12)
holds on the set {v ∈ C([a, b]; R+) : v(a) = h(v)}, where the operator ϑ is defined by formulae
(4.8) and (4.9),
z0(t) = exp
b∫
t
|`(1)(s)|ds
for t ∈ [a, b], (4.13)
z1(t) =
b∫
t
¯̀(1)(s) exp
s∫
t
|`(1)(ξ)| dξ
ds for t ∈ [a, b]. (4.14)
Then ` ∈ Ṽ −
ab (h).
Remark 4.3. Let −` ∈ Pab and the condition (3.4) be fulfilled. Then it follows from Corol-
lary 4.4(a) (for k = 1 and m = 2) that ` ∈ Ṽ −
ab (h) provided
b∫
a
|`(1)(s)| ds < 1− 1 + h1(1)
λ + h0(1)
.
Moreover, it follows from Corollary 4.4(b) (with ¯̀ ≡ 0) that ` ∈ Ṽ −
ab (h) provided that ` is a
b-Volterra operator and the condition (4.10)
z0(a) < h(z0)
holds, where the function z0 is given by formula (4.13).
4.3. The case ` = `0 − `1 with `0, `1 ∈ Pab. The following statements can be proved
in the case where the operator is regular, i.e., admits the representation ` = `0 − `1 with
`0, `1 ∈ Pab.
Theorem 4.4. Let ` = `0 − `1, where `0, `1 ∈ Pab,
h1(1) < λ, h0(1) ≤ 1, (4.15)
and
b∫
a
`0(1)(s)ds ≤ (1− h0(1))min
{
1,
1
λ
}
. (4.16)
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476 A. LOMTATIDZE, Z. OPLUŠTIL, J. ŠREMR
Then ` ∈ Ṽ −
ab (h) if and only if ` ∈ U−
ab(h).
Theorem 4.5. Let ` = `0 − `1, where `0, `1 ∈ Pab, h ∈ Fab, and condition (2.3) hold. If
`0 ∈ Ṽ −
ab (h), −`1 ∈ Ṽ −
ab (h), (4.17)
then ` ∈ Ṽ −
ab (h).
4.4. Further remarks. Introduce the operator ϕ : C([a, b]; R) → C([a, b]; R) by setting
ϕ(w)(t) df= w(a + b− t) for t ∈ [a, b], w ∈ C([a, b]; R).
Let
̂̀(w)(t) df= −` (ϕ(w)) (a + b− t) for a.e. t ∈ [a, b] and all w ∈ C([a, b]; R),
ĥ(w) df=
1
λ
v(b)− 1
λ
h0 (ϕ(w)) +
1
λ
h1 (ϕ(w)) for w ∈ C([a, b]; R).
It is clear that if u is a solution to the problem (1.1), (1.2) then the function v
df= −ϕ(u) is
a solution to the problem
v′(t) ≥ ̂̀(v)(t), v(a) ≥ ĥ(v), (4.18)
and vice versa, if v is a solution to the problem (4.18) then the function u
df= −ϕ(v) is a solution
to the problem (1.1), (1.2).
Consequently, the relation
` ∈ Ṽ +
ab (h) ⇔ ̂̀∈ Ṽ −
ab
(
ĥ
)
holds.
Therefore, efficient conditions guaranteeing the validity of the inclusion ` ∈ Ṽ +
ab (h) can be
immediately derived from the results stated in Sections 4.1 – 4.3. For example, Corollary 4.1 of
Section 4.1 immediately yields the following.
Corollary 4.5. Let −` ∈ Pab be an a-Volterra operator and
h(1) < 1, h1(1) ≤ λ.
If, moreover, there exists a function γ ∈ C̃([a, b]; ]0,+∞[) such that the conditions (3.2) and
λγ(b) ≥ h1(γ)
hold then ` ∈ Ṽ +
ab (h).
4.5. Proofs. To prove statements formulated in Sections 4 and 4.5 we need the following
lemmas.
Lemma 4.1. Let h ∈ Fab and ` ∈ U−
ab(h). Then ` + ¯̀ ∈ U−
ab(h) for every ¯̀ ∈ Pab.
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NONPOSITIVE SOLUTIONS TO A CERTAIN FUNCTIONAL DIFFERENTIAL INEQUALITY 477
Proof. It follows immediately from Definition 3.1.
Lemma 1. ([6], Theorem 1.6). Let ` ∈ Pab be a b-Volterra operator and there exist a function
γ ∈ C̃([a, b]; R+) satisfying the conditions (3.12) and
γ(t) > 0 for t ∈ ]a, b].
Then ` ∈ Sab(b).
Proof of Proposition 4.1. First suppose that ` ∈ Ṽ −
ab (h). Then, according to Remark 3.1, we
have ` ∈ U−
ab(h). Moreover, it is clear that the inclusion ` + ¯̀ ∈ Ṽ −
ab (h) is true with ¯̀≡ 0.
Now suppose that ` ∈ U−
ab(h) and there exists an operator ¯̀ ∈ Pab such that ` + ¯̀ ∈ Ṽ −
ab (h).
Let u be a solution to the problem (1.1), (1.2). We shall show that the function u is nonpositive.
According to the assumption ` + ¯̀ ∈ Ṽ −
ab (h) and Remark 2.1, the problem
α′(t) =
(
` + ¯̀)
(α)(t)− ¯̀([u]+)(t), (4.19)
α(a) = h(α) (4.20)
has a unique solution α and the relation
α(t) ≥ 0 for t ∈ [a, b] (4.21)
holds. From (1.1), (1.2), (4.19), (4.20), and the assumption ¯̀ ∈ Pab, we get the relations
v′(t) ≥ (` + ¯̀)(v)(t) for a.e. t ∈ [a, b], v(a) ≥ h(v),
where
v(t) = u(t)− α(t) for t ∈ [a, b]. (4.22)
Consequently, using the inclusion ` + ¯̀ ∈ Ṽ −
ab (h), we obtain v(t) ≤ 0 for t ∈ [a, b], and thus
u(t) ≤ α(t) for t ∈ [a, b]. (4.23)
Taking now relation (4.21) into account, inequality (4.23) implies
[u(t)]+ ≤ α(t) for t ∈ [a, b].
Therefore, in view of the assumption ¯̀ ∈ Pab, equation (4.19) yields
α′(t) ≥ (` + ¯̀)(α)(t)− ¯̀(α)(t) = `(α)(t) for a.e. t ∈ [a, b]. (4.24)
Consequently, α is a nonnegative function satisfying the conditions (4.20) and (4.24). Hence,
the assumption ` ∈ U−
ab(h) implies α ≡ 0, and thus relation (4.23) yields
u(t) ≤ 0 for t ∈ [a, b]. (4.25)
Therefore, the inclusion ` ∈ Ṽ −
ab (h) is true.
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478 A. LOMTATIDZE, Z. OPLUŠTIL, J. ŠREMR
The proposition is proved.
Proof of Theorem 4.1. First suppose that ` ∈ Ṽ −
ab (h). According to Remark 2.1, the problem
γ′(t) = `(γ)(t), γ(a) = h(γ)− 1 (4.26)
has a unique solution γ and, moreover, the relation
γ(t) ≥ 0 for t ∈ [a, b] (4.27)
holds. Obviously, the function γ satisfies the conditions (3.2) and (3.3).
Now suppose that there exists a function γ ∈ C̃([a, b]; R+) satisfying the conditions (3.2)
and (3.3). We shall show that ` ∈ Ṽ −
ab (h). Let u be a solution to the problem (1.1), (1.2). It is
clear that either
u(b) > 0 (4.28)
or
u(b) ≤ 0. (4.29)
Assume that condition (4.28) holds. Put
w(t) = γ(b)u(t)− u(b)γ(t) for t ∈ [a, b].
We get, from (1.1), (3.2), and (4.28), the relations
w′(t) ≥ `(w)(t) for a.e. t ∈ [a, b], (4.30)
w(b) = 0. (4.31)
Therefore, the assumption ` ∈ Sab(b) yields
w(t) ≤ 0 for t ∈ [a, b]. (4.32)
On the other hand, it follows from (1.2), (3.3), (4.28), (4.31), (4.32), and the assumption h1 ∈
∈ PFab that
w(a) > λw(b) + h0(w)− h1(w) ≥ h0(w).
Consequently, the function w is a nonpositive solution to the problem
w′(t) ≥ `(w)(t), w(a) > h0(w),
which contradicts the assumption ` ∈ Ũ+
ab(h0).
The contradiction obtained proves that u satisfies condition (4.29). Taking now (1.1) and
(4.29) into account, the assumption ` ∈ Sab(b) implies relation (4.25), and thus ` ∈ Ṽ −
ab (h).
The theorem is proved.
Proof of Theorem 4.2. First suppose that ` ∈ Sab(b). It is clear that, in view of (2.3) and the
assumption ` ∈ Pab, the function γ ≡ 1 satisfies the conditions (3.2) and (3.3). Hence, by virtue
of Theorem 4.1, we get ` ∈ Ṽ −
ab (h).
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NONPOSITIVE SOLUTIONS TO A CERTAIN FUNCTIONAL DIFFERENTIAL INEQUALITY 479
Now assume that ` ∈ Ṽ −
ab (h). Suppose that, on the contrary, ` 6∈ Sab(b). Then there exists a
solution u to the inequality (1.1) satisfying the relations u(b) = c and
u(t0) > 0, (4.33)
where c ≤ 0 and t0 ∈]a, b[. According to the assumption ` ∈ Ṽ −
ab (h) and Remark 2.1, the
problem
u′
0(t) = `(u0)(t), (4.34)
u0(a) = h(u0)− 1 (4.35)
has a unique solution u0 and, moreover, the relation
u0(t) ≥ 0 for t ∈ [a, b] (4.36)
holds. It is not difficult to verify that
u0(b) > 0. (4.37)
Indeed, suppose that (4.37) does not hold. Then, in view of (4.36), we find u0(b) = 0. Hence,
by virtue of (4.36) and the assumption h1 ∈ PFab, the condition (4.35) implies
u0(a) = λu0(b) + h0(u0)− h1(u0)− 1 < h0(u0),
which, together with (4.34) and (4.36), contradicts the assumption ` ∈ Ũ+
ab(h0). The contradic-
tion obtained proves the validity of relation (4.37).
Since ` 6∈ Sab(b), if follows from Lemma 4.2, on account of (4.34), (4.36), (4.37), and the
assumption ` ∈ Pab, that there exists a0 ∈ ]a, b[ such that
u0(t) = 0 for t ∈ [a, a0], (4.38)
u0(t) > 0 for t ∈ ]a0, b]. (4.39)
Denote by ˜̀ the restriction of the operator ` to the space C([a0, b]; R). By virtue of the condi-
tions (4.34) and (4.39), we get
u′
0(t) = ˜̀(u0)(t) for a.e. t ∈ [a0, b], u0(t) > 0 for t ∈ ]a0, b],
and thus Lemma 4.2 guarantees validity of the inclusion ˜̀∈ Sa0b(b). It follows from inequality
(1.1) and condition (4.34) that
w′(t) ≥ ˜̀(w)(t) for a.e. t ∈ [a0, b], w(b) = 0, (4.40)
where
w(t) = u(t)− c
u0(b)
u0(t) for t ∈ [a0, b].
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480 A. LOMTATIDZE, Z. OPLUŠTIL, J. ŠREMR
Since ˜̀∈ Sa0b(b), relations (4.40) result in w(t) ≤ 0 for t ∈ [a0, b], i.e.,
u(t) ≤ c
u0(b)
u0(t) for t ∈ [a0, b].
From the latter inequality, (4.33), and (4.39) we get
a < t0 < a0. (4.41)
Now we put
v(t) = u(t) + (u(a)− h(u))u0(t) for t ∈ [a, b]. (4.42)
It is clear that
v′(t) ≥ `(v)(t) for a.e. t ∈ [a, b], v(a) = h(v).
Consequently, by virtue of the assumption ` ∈ Ṽ −
ab (h), the inequality v(t) ≤ 0 holds for t ∈
∈ [a, b]. Finally, in view of (4.38) and (4.41), relation (4.42) yields
0 ≥ v(t0) = u(t0) + (u(a)− h(u))u0(t0) = u(t0),
which contradicts the inequality (4.33).
The contradiction obtained proves the validity of the inclusion ` ∈ Sab(b).
The theorem is proved.
Proof of Corollary 4.1. According to Lemma 4.2, inequality (3.12) yields ` ∈ Sab(b). On
the other hand, by virtue of the conditions (3.12), (3.4), and (4.1), using Theorem 3.5 we get
` ∈ Ũ+
ab(h0). Consequently, the assertion of the corollary follows from Theorem 4.2.
Proof of Corollary 4.2. Put
γ(t) =
m∑
j=k+1
%j(t) for t ∈ [a, b].
In view of condition (4.3), where the function ϕ1 is given by the formula (4.6), we get γ ∈
∈ C̃([a, b]; ]0,+∞[ ). On the other hand, by virtue of the relations (4.4) – (4.6) and the assump-
tion ` ∈ Pab, it is clear that the function γ satisfies the conditions (3.12) and (4.1). Consequently,
the assumptions of Corollary 4.1 are satisfied.
Proof of Corollary 4.3. According to the assumption ` ∈ Pab ∩ Ṽ +
ab (h0), Theorem 2.1 in [14]
guarantees that there exists a function γ ∈ C̃([a, b]; ]0,+∞[) satisfying the conditions (3.12)
and
γ(a) > h0(γ).
Consequently, the assumptions of Corollary 4.1 are satisfied.
Proof of Theorem 4.3. The validity of the theorem follows immediately from Proposition 4.1
(with ¯̀≡ −`) and Remark 2.2.
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NONPOSITIVE SOLUTIONS TO A CERTAIN FUNCTIONAL DIFFERENTIAL INEQUALITY 481
Proof of Corollary 4.4. (a) It is not difficult to verify that the function γ, defined by the
formula
γ(t) =
m∑
j=1
%j(t)− δ
k∑
j=1
%j(t) for t ∈ [a, b],
is positive and satisfies the conditions (3.2) and (3.3). Consequently, the assertion of the corol-
lary follows from Theorems 3.2 and 4.3.
(b) According to relations (4.10) and (4.11), there exists ε > 0 such that
γ0 (ε− h(z1)) z0(a) + γ0h(z0)z1(a) ≤ 1, (4.43)
where γ0 = (h(z0)− z0(a))−1 . Put
γ(t) = γ0
(ε− h(z1)) exp
b∫
t
|`(1)(s)| ds
+
+ exp
b∫
a
|`(1)(s)| ds
t∫
a
¯̀(1)(s) exp
s∫
t
|`(1)(ξ)| dξ
ds+
+ h(z0)
b∫
t
¯̀(1)(s) exp
s∫
t
|`(1)(ξ)| dξ
ds
for t ∈ [a, b],
where the functions z0 and z1 are defined by the formulae (4.13) and (4.14), respectively. It is
not difficult to verify that γ is a solution to the problem
γ′(t) = `(1)(t)γ(t)− ¯̀(1)(t), (4.44)
γ(a) = h(γ)− ε. (4.45)
Using the inequalities (3.4) and (4.10), and the assumptions h0, h1 ∈ PFab, it follows from the
definition of the function γ that γ(b) > 0, and thus the relation γ(t) > 0 holds for t ∈ [a, b].
Since −`, ¯̀ ∈ Pab, equality (4.44) implies γ(t) ≤ γ(a) for t ∈ [a, b]. Taking now inequality
(4.43) into account, the conditions (4.44) and (4.45) result in
γ′(t) ≤ `(1)(t)γ(t)− ¯̀(γ)(t) for a.e. t ∈ [a, b], γ(a) < h(γ).
Consequently, Theorem 4.3 guarantees validity of inclusion
˜̀∈ Ṽ −
ab (h), (4.46)
where
˜̀(v)(t) df= `(1)(t)v(t)− ¯̀(v)(t) for a.e. t ∈ [a, b] and all v ∈ C([a, b]; R).
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482 A. LOMTATIDZE, Z. OPLUŠTIL, J. ŠREMR
Since−` ∈ Pab, in order to prove the inclusion ` ∈ Ṽ −
ab (h) it is sufficient to show that ` ∈ U−
ab(h)
(see Theorem 4.3). Hence, let u be a nonnegative solution to the problem (1.1), (1.2). We shall
show that u ≡ 0. Put
w(t) = ϑ(v)(t) for t ∈ [a, b], (4.47)
where the operator ϑ is defined by the formulae (4.8) and (4.6), and
v(t) = u(t) +
u(a)− h(u)
h(1)− 1
for t ∈ [a, b].
Obviously,
v(t) ≥ u(t) for t ∈ [a, b]
and
v′(t) ≥ `(v)(t) for a.e. t ∈ [a, b], v(a) = h(v), (4.48)
w′(t) = `(v)(t) for a.e. t ∈ [a, b], w(a) = h(w). (4.49)
It follows from (4.48) and (4.49) that
y′(t) ≥ 0 for a.e. t ∈ [a, b], y(a) = h(y),
where y(t) = v(t) − w(t) for t ∈ [a, b]. By virtue of Remark 2.2, we have 0 ∈ Ṽ −
ab (h). Conse-
quently, y(t) ≤ 0 for t ∈ [a, b], i.e.,
0 ≤ u(t) ≤ v(t) ≤ w(t) for t ∈ [a, b]. (4.50)
On the other hand, using (4.12), (4.47) – (4.50), and the assumptions −`, ¯̀ ∈ Pab, we get
w′(t) = `(v)(t) ≥ `(1)(t)w(t) + `(w)(t)− `(1)(t)w(t) =
= `(1)(t)w(t) + `(ϑ(v))(t)− `(1)(t)ϑ(v)(t) ≥ `(1)(t)w(t)− ¯̀(v)(t) ≥
≥ `(1)(t)w(t)− ¯̀(w)(t) = ˜̀(w)(t) for a.e. t ∈ [a, b].
Taking now (4.46) and (4.49) into account, we find w(t) ≤ 0 for t ∈ [a, b]. Hence, the relation
(4.50) implies u ≡ 0, and thus ` ∈ Ṽ −
ab (h).
The corollary is proved.
Proof of Theorem 4.4. Assume that ` ∈ U−
ab(h). Since `1 ∈ Pab, Lemma 4.1 guarantees that
`0 = ` + `1 ∈ U−
ab(h).
We shall show that `0 ∈ Ṽ −
ab (h). Assume that, on the contrary, there exists a solution u to
the inequality
u′(t) ≥ `0(u)(t) (4.51)
satisfying condition (1.2), which is not nonpositive on the interval [a, b]. Then, in view of the
above-proved inclusion `0 ∈ U−
ab(h), it is clear that u assumes both positive and negative values,
i.e.,
M > 0, m > 0, (4.52)
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NONPOSITIVE SOLUTIONS TO A CERTAIN FUNCTIONAL DIFFERENTIAL INEQUALITY 483
where
M = max {u(t) : t ∈ [a, b]} , m = −min {u(t) : t ∈ [a, b]} . (4.53)
Now we choose tM , tm ∈ [a, b] such that
u(tM ) = M, u(tm) = −m. (4.54)
Obviously, either
tM < tm (4.55)
or
tM > tm. (4.56)
If (4.55) holds then the integration of (4.51) from tM to tm, in view of (4.52), (4.53) and the
assumption `0 ∈ Pab, results in
M + m ≤ −
tm∫
tM
`0(u)(s) ds ≤ m
b∫
a
`0(1)(s) ds.
Hence, by virtue of (4.16) and the second inequality in (4.52), we get M ≤ 0, which contradicts
the first inequality in (4.52).
If (4.56) holds then the integrations of (4.51) from a to tm and from tM to b, in view of
(4.52), (4.53) and the assumption `0 ∈ Pab, yield
u(a) + m ≤ −
tm∫
a
`0(u)(s) ds ≤ m
tm∫
a
`0(1)(s) ds, (4.57)
M − u(b) ≤ −
b∫
tM
`0(u)(s) ds ≤ m
b∫
tM
`0(1)(s) ds. (4.58)
On the other hand, the condition (1.2), on account of (4.53) and the assumptions h0, h1 ∈ PFab,
implies
u(a)− λu(b) ≥ h0(u)− h1(u) ≥ −mh0(1)−Mh1(1). (4.59)
Now we get, from (4.57) – (4.59), the inequality
M (λ− h1(1)) + m (1− h0(1)) ≤ m
tm∫
a
`0(1)(s) ds + λ
b∫
tM
`0(1)(s) ds
,
i.e.,
M (λ− h1(1))min
{
1,
1
λ
}
+ m (1− h0(1))min
{
1,
1
λ
}
≤ m
b∫
a
`0(1)(s) ds.
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484 A. LOMTATIDZE, Z. OPLUŠTIL, J. ŠREMR
Hence, by virtue of (3.1), (4.16), and the second inequality in (4.52), we find M ≤ 0, which
contradicts the first inequality in (4.52).
The contradictions obtained prove the validity of the inclusion `0 ∈ Ṽ −
ab (h).
Now we put ¯̀ ≡ `1. Since ` ∈ U−
ab(h) and ` + ¯̀ = `0 ∈ Ṽ −
ab (h), Proposition 4.1 yields
` ∈ Ṽ −
ab (h).
The validity of the converse implication follows immediately from Remark 3.1.
The theorem is proved.
Proof of Theorem 4.5. It is easy to verify that ` ∈ U−
ab(h). Indeed, the assumption −`1 ∈
∈ Ṽ −
ab (h) yields −`1 ∈ U−
ab(h) (see Remark 3.1), and thus, in view of Lemma 4.1, we get ` =
= −`1 + `0 ∈ U−
ab(h).
Now we put ¯̀ ≡ `1. Then it is clear that ¯̀ ∈ Pab and ` + ¯̀ = `0 ∈ Ṽ −
ab (h). Consequently,
Proposition 4.1 yields ` ∈ Ṽ −
ab (h).
The theorem is proved.
5. Differential inequalities with argument deviations. In this section, we give some corollar-
ies of the main results for operators with argument deviations. More precisely, efficient criteria
are proved below for validity of the inclusion ` ∈ Ṽ −
ab (h) in the case where the operator ` is
given by one of the following formulae:
`(v)(t) df= p(t)v(τ(t)) for a.e. t ∈ [a, b] and all v ∈ C([a, b]; R), (5.1)
`(v)(t) df= −g(t)v(µ(t)) for a.e. t ∈ [a, b] and all v ∈ C([a, b]; R), (5.2)
and
`(v)(t) df= p(t)v(τ(t))− g(t)v(µ(t)) for a.e. t ∈ [a, b] and all v ∈ C([a, b]; R). (5.3)
Here we suppose that p, g ∈ L([a, b]; R+) and τ, µ : [a, b] → [a, b] are measurable functions.
Throughout this section, the following notation is used:
µ∗ = ess inf {µ(t) : t ∈ [a, b]} , τ∗ = ess sup {τ(t) : t ∈ [a, b]} , (5.4)
and
α(t) = exp
b∫
t
g(s) ds
, β(t) = exp
t∫
a
p(s) ds
for t ∈ [a, b]. (5.5)
We first formulate all the results, their proofs are given later, in Section 5.1 below.
Theorem 5.1. Let condition (3.1) be fulfilled and
h(1) < 1. (5.6)
Assume that
0 <
b∫
a
p(s) ds ≤ (1− h0(1))min
{
1,
1
λ
}
(5.7)
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NONPOSITIVE SOLUTIONS TO A CERTAIN FUNCTIONAL DIFFERENTIAL INEQUALITY 485
and
ess inf
τ(t)∫
t
p(s) ds : t ∈ [a, b]
> η∗, (5.8)
where
η∗ = inf
{
1
x
ln
xβx(τ∗)
βx(τ∗) + (h(βx)− 1) (1− h(1))−1 : x > 0, h(βx) > 1
}
. (5.9)
Then the operator ` given by formula (5.1) belongs to the set Ṽ −
ab (h).
Corollary 5.1. Let the inequalities (3.1) and (5.6) be fulfilled. Assume that condition (5.7) is
satisfied and
ess inf
τ(t)∫
t
p(s) ds : t ∈ [a, b]
> ξ∗, (5.10)
where
ξ∗ = inf
{
||p||L
y
ln
yey (1− h(1))
||p||L (ey − 1) (1− h0(1))
: y > ln
1− h0(1)
λ− h1(1)
}
. (5.11)
Then the operator ` defined by formula (5.1) belongs to the set Ṽ −
ab (h).
Theorem 5.2. Let the conditions (3.1) and (5.6) be fulfilled. Assume that τ(t) ≥ t for a.e.
t ∈ [a, b],
b∫
a
p(s)ds > ln
1− h0(1)
λ− h1(1)
, (5.12)
and at least one of the following conditions is satisfied:
(a) h0(z0) > 0 and
max
{
h0(z1) + (1− h0(1)) z1(t)
h0(z0) + (1− h0(1)) z0(t)
: t ∈ [a, b]
}
< 1− h0(z0)
1− h0(1)
, (5.13)
where
z0(t) =
t∫
a
p(s) ds for t ∈ [a, b], (5.14)
z1(t) =
t∫
a
p(s)
τ(s)∫
a
p(ξ)dξ
ds for t ∈ [a, b]; (5.15)
(b)
h0(β) < 1, (5.16)
h0(γ0)
1− h0(β)
β(b) + γ0(b) < 1, (5.17)
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486 A. LOMTATIDZE, Z. OPLUŠTIL, J. ŠREMR
where
γ0(t) =
t∫
a
p(s)
τ(s)∫
s
p(ξ)dξ
exp
t∫
s
p(η)dη
ds for t ∈ [a, b]; (5.18)
(c) h0(1) 6= 0
ess sup
τ(t)∫
t
p(s)ds : t ∈ [a, b]
< κ∗, (5.19)
where
κ∗ = sup
{
‖p‖L
x
ln
xex(1− h0(1))
‖p‖L(ex − 1)
: 0 < x < ln
1
h0(1)
}
. (5.20)
Then the operator ` given by the formula (5.1) belongs to the set Ṽ −
ab (h).
Theorem 5.3. Let the conditions (2.3) and (4.2) be fulfilled. Assume that τ(t) ≥ t for a.e.
t ∈ [a, b] and at least one of the following conditions is satisfied:
(a) the inequality (5.13) holds, where the functions z0 and z1 are defined by the formulae
(5.14) and (5.15), respectively;
(b) the inequalities (5.16) and (5.17) hold, where the function γ0 is given by the formula
(5.18);
(c) h0(1) 6= 0 and the condition (5.19) holds, where the number κ∗ is defined by the formula
(5.20).
Then the operator ` given by the formula (5.1) belongs to the set Ṽ −
ab (h).
Remark 5.1. If h0(z0) > 0, where z0 is defined by formula (5.14), then the strict inequality
(5.13) in Theorem 5.3(a) can be weakened. More precisely, the following assertion is true.
Theorem 5.4. Let the conditions (2.3) and (4.2) be fulfilled. Assume that τ(t) ≥ t for a.e.
t ∈ [a, b],
h0(z0) > 0,
and
max
{
h0(z1) + (1− h0(1)) z1(t)
h0(z0) + (1− h0(1)) z0(t)
: t ∈ [a, b]
}
≤ 1− h0(z0)
1− h0(1)
, (5.21)
where the functions z0 and z1 are defined by the formulae (5.14) and (5.15), respectively. Then
the operator ` given by the formula (5.1) belongs to the set Ṽ −
ab (h).
Theorem 5.5. Let the conditions (2.3) and (3.4) be fulfilled. Assume that
ess sup
t∫
µ(t)
g(s) ds : t ∈ [a, b]
< ω∗, (5.22)
where
ω∗ = sup
{
1
x
ln
xαx(µ∗)
αx(µ∗)− f(x)
: x > 0, ĥ(αx) > αx(a)
}
,
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NONPOSITIVE SOLUTIONS TO A CERTAIN FUNCTIONAL DIFFERENTIAL INEQUALITY 487
f(x) df=
ĥ(αx)− αx(a)
h(1)− 1
for x > 0, (5.23)
ĥ(v) df= min {h(1), h(v)} for v ∈ C([a, b]; R).
Then the operator ` given by formula (5.2) belongs to the set Ṽ −
ab (h).
Corollary 5.2. Let the conditions (2.3) and (3.4) be fulfilled. Assume that g 6≡ 0 and
ess sup
t∫
µ(t)
g(s)ds : t ∈ [a, b]
< ξ∗,
where
ξ∗ = sup
{
||g||L
y
ln
yey (h(1)− 1)
||g||L(ey − 1) (λ + h0(1))
: 0 < y < ln
λ + h0(1)
1 + h1(1)
}
. (5.24)
Then the operator ` defined by the formula (5.2) belongs to the set Ṽ −
ab (h).
Theorem 5.6. Let the conditions (2.3) and (4.2) be fulfilled. Assume that g 6≡ 0 and
max
{
z1(a)− h(z1) + (h(1)− 1) z1(t)
z0(a)− h(z0) + (h(1)− 1) z0(t)
: t ∈ [a, b]
}
< 1− z0(a)− h(z0)
h(1)− 1
, (5.25)
where
z0(t) =
b∫
t
g(s) ds for t ∈ [a, b],
z1(t) =
b∫
t
g(s)
b∫
µ(s)
g(ξ) dξ
ds for t ∈ [a, b].
Then the operator ` given by formula (5.2) belongs to the set Ṽ −
ab (h).
Theorem 5.7. Let the conditions (2.3) and (3.4) be fulfilled. Assume that the inequalities
(4.10) and (4.11) are satisfied, where
z0(t) = exp
b∫
t
g(s) ds
for t ∈ [a, b],
(5.26)
z1(t) =
b∫
t
g(s)σ(s)
s∫
µ(s)
g(ξ) dξ
exp
s∫
t
g(η) dη
ds for t ∈ [a, b],
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488 A. LOMTATIDZE, Z. OPLUŠTIL, J. ŠREMR
and
σ(t) =
1
2
(1 + sgn (t− µ(t))) for a.e. t ∈ [a, b]. (5.27)
Then the operator ` given by formula (5.2) belongs to the set Ṽ −
ab (h).
Theorem 5.8. Let the conditions (3.1) and (3.5) be fulfilled. If
b∫
a
g(s) ds < (λ− h1(1))min
{
1,
1
λ
}
(5.28)
and
(1− h0(1))min
{
1, 1
λ
}
(λ− h1(1))min
{
1, 1
λ
}
−
∫ b
a g(s) ds
− 1 <
∫ b
a
p(s) ds ≤ (1− h0(1))min
{
1,
1
λ
}
,
then the operator ` given by formula (5.3) belongs to the set Ṽ −
ab (h).
Theorem 5.9. Let the conditions (2.3) and (3.4) be fulfilled. Assume that the inequality (5.28)
is satisfied and
ω
b∫
a
g(s) ds
<
b∫
a
p(s) ds ≤ (1− h0(1))min
{
1,
1
λ
}
,
where the function ω is defined by formula (3.10). Then the operator ` given by formula (5.3)
belongs to the set Ṽ −
ab (h).
Corollary 5.3. Let the conditions (3.1) and (3.4) be fulfilled. Assume that either
h(1) ≤ 1,
1− h0(1)
λ− h1(1)
− 1 <
b∫
a
p(s) ds ≤ (1− h0(1))min
{
1,
1
λ
}
or
h(1) > 1,
b∫
a
p(s) ds ≤ (1− h0(1))min
{
1,
1
λ
}
.
Then the operator ` given by formula (5.1) belongs to the set Ṽ −
ab (h).
Theorem 5.10. Let the conditions (2.3) and (3.4) be fulfilled. Assume that the functions p, τ
satisfy condition (5.7) or the assumptions of Theorems 5.3 or 5.4, whereas the functions g, µ fulfil
the assumptions of Theorems 5.5, 5.6 or 5.7. Then the operator ` given by formula (5.3) belongs
to the set Ṽ −
ab (h).
5.1. Proofs. We give the following lemmas before we prove statements formulated above.
Lemma 5.1. Let the functional h be defined by formula (1.3), where λ > 0 and h0, h1 ∈ PFab
are such that the conditions (3.1) and (5.6) are fulfilled. Let, moreover, the operator ` be defined
by the formula (5.1), p 6≡ 0, and condition (5.8) be satisfied, where the number η∗ is defined by
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NONPOSITIVE SOLUTIONS TO A CERTAIN FUNCTIONAL DIFFERENTIAL INEQUALITY 489
formula (5.9). Then there exists a function γ ∈ C̃([a, b]; ]0,+∞[) satisfying the inequalities (3.2)
and (3.3).
Proof. According to (5.8) with η∗ given by (5.9), there exist x0 > 0 and ε > 0 such that
h(βx0) ≥ 1 + ε (5.29)
and the relation
τ(t)∫
t
p(s) ds ≥ 1
x0
ln
x0β
x0(τ∗)
βx0(τ∗) + (h(βx0)− 1− ε) (1− h(1))−1 for a.e. t ∈ [a, b] (5.30)
holds. Put
δ =
h(βx0)− 1− ε
1− h(1)
. (5.31)
By virtue of the conditions (5.6) and (5.29), we get δ ≥ 0. Hence, relation (5.30) yields
e
x0
τ(t)R
t
p(s) ds
≥ x0β
x0(τ∗)
βx0(τ∗) + δ
≥ x0β
x0(τ(t))
βx0(τ(t)) + δ
for a.e. t ∈ [a, b].
Consequently, we have
x0e
x0
tR
a
p(s) ds
≤ e
x0
τ(t)R
a
p(s) ds
+ δ for a.e. t ∈ [a, b]. (5.32)
Now we put
γ(t) = e
x0
tR
a
p(s) ds
+ δ for t ∈ [a, b].
It is clear that γ(t) > 0 for t ∈ [a, b] and, using condition (5.32), we get
`(γ)(t) = p(t)
e
x0
τ(t)R
a
p(s) ds
+ δ
≥ x0p(t)e
x0
tR
a
p(s) ds
= γ′(t) for a.e. t ∈ [a, b],
i.e., inequality (3.2) holds. On the other hand, in view of equality (5.31) and the assumption
h(1) < 1, inequality (3.3) is satisfied.
The lemma is proved.
Lemma 5.2. Let the operator ` be defined by formula (5.2), h ∈ Fab satisfies condition (2.3),
and let inequality (5.22) be fulfilled, where the number ω∗ is defined by formulae (5.23). Then
there exists a function γ ∈ C̃([a, b]; ]0,+∞[) satisfying the inequalities (3.2) and (3.3).
Proof. According to (5.22) with ω∗ given by (5.23), there exist x0 > 0 and ε > 0 such that
ĥ(αx0) ≥ αx0(a) + ε (5.33)
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490 A. LOMTATIDZE, Z. OPLUŠTIL, J. ŠREMR
and the inequality
t∫
µ(t)
g(s)ds ≤ 1
x0
ln
x0α
x0(µ∗)
αx0(µ∗)−
(
ĥ(αx0)− αx0(a)− ε
)
(h(1)− 1)−1
(5.34)
holds for a.e. t ∈ [a, b]. Put
δ =
ĥ(αx0)− αx0(a)− ε
h(1)− 1
. (5.35)
By virtue of the conditions (2.3), (5.23), and (5.33), we get δ ∈ [0, 1[. Hence, relation (5.34)
yields
e
x0
tR
µ(t)
g(s) ds
≤ x0α
x0(µ∗)
αx0(µ∗)− δ
≤ x0α
x0(µ(t))
αx0(µ(t))− δ
for a.e. t ∈ [a, b].
Consequently, we have
e
x0
bR
µ(t)
g(s) ds
− δ ≤ x0e
x0
bR
t
g(s) ds
for a.e. t ∈ [a, b]. (5.36)
Now we put
γ(t) = e
x0
bR
t
g(s) ds
− δ for t ∈ [a, b].
It is clear that γ(t) > 0 for t ∈ [a, b] and, using condition (5.36), we get
`(γ)(t) = −g(t)
e
x0
bR
µ(t)
g(s) ds
− δ
≥ −x0g(t)e
x0
bR
t
g(s) ds
= γ′(t) for a.e. t ∈ [a, b],
i.e., inequality (3.2) holds. On the other hand, in view of (5.23), (5.35), and the assumption
h(1) > 1, the inequality (3.3) is satisfied.
The lemma is proved.
Now we are in position to prove Theorems 5.1 – 5.10.
Proof of Theorem 5.1. Let the operator ` be defined by formula (5.1). It is clear that ` ∈ Pab
and condition (5.7) implies validity of relation (4.16) with `0 ≡ `. According to Lemma 5.1,
there exists a function γ ∈ C̃([a, b]; ]0,+∞[) satisfying the conditions (3.2) and (3.3), which
guarantees validity of the inclusion ` ∈ U−
ab(h) (see Theorem 3.1). Consequently, in view of
Theorem 4.4 (with `0 ≡ ` and `1 ≡ 0), we get ` ∈ Ṽ −
ab (h).
The theorem is proved.
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NONPOSITIVE SOLUTIONS TO A CERTAIN FUNCTIONAL DIFFERENTIAL INEQUALITY 491
Proof of Corollary 5.1. It is not difficult to verify that
xβx(τ∗)
βx(τ∗) + (h(βx)− 1) (1− h(1))−1 ≤
≤ xβx(b)
βx(b) +
(
(λ− h1(1)) ex||p||L + h0(1)− 1
)
(1− h(1))−1 =
=
xex||p||L (1− h(1))(
ex||p||L − 1
)
(1− h0(1))
for every x > 0 such that (λ− h1(1)) ex||p||L > 1−h0(1). Therefore, the relation η∗ ≤ ξ∗ holds,
where η∗ and ξ∗ are defined by the formulae (5.9) and (5.11), respectively. Consequently, the
assertion of the corollary follows immediately from Theorem 5.1.
Proof of Theorem 5.2. Let the operator ` be defined by formula (5.1). It is clear that ` ∈ Pab
and ` is a b-Volterra operator. According to Theorems 4.1 and 4.2, and Corollary 4.2 in [14], we
conclude that each of the conditions (a) – (c) guarantees validity of the inclusion ` ∈ Ṽ +
ab (h0).
Moreover, by virtue of Theorem 2.1 in [14], there exists a function γ ∈
∈ C̃([a, b]; ]0,+∞[) satisfying inequality (3.12). Therefore, Lemma 4.2 guarantees that ` ∈
∈ Sab(b). Furthermore, the above-proved inclusion ` ∈ Ṽ +
ab (h0) yields ` ∈ Ũ+
ab(h0) (see Re-
mark 3.3).
On the other hand, since we suppose that τ(t) ≥ t for a. e. t ∈ [a, b], condition (5.12) implies
validity of condition (5.10), where ξ∗ is defined by formula (5.11). Therefore, analogously to the
proof of Corollary 5.1 it can be shown that relation (5.8) is satisfied with η∗ given by formula
(5.9), and thus, according to Lemma 5.1, there exists a function γ ∈ C̃([a, b]; ]0,+∞[ ) satisfying
the conditions (3.2) and (3.3).
Consequently, by virtue of Theorem 4.1, we get ` ∈ Ṽ −
ab (h).
The theorem is proved.
Proof of Theorem 5.3. Let the operator ` be defined by formula (5.1). It is clear that ` ∈ Pab
and ` is a b-Volterra operator. According to Theorems 4.1 and 4.2, and Corollary 4.2 in [14], we
conclude that each of the conditions (a) – (c) guarantees validity of the inclusion ` ∈ Ṽ +
ab (h0).
Therefore, the assumptions of Corollary 4.3 are satisfied.
The theorem is proved.
Proof of Theorem 5.4. Let the operator ` be defined by formula (5.1). It is clear that ` ∈ Pab
and ` is a b-Volterra operator. Using condition (5.21), it is not difficult to verify that
%3(t) ≤ %2(t) for t ∈ [a, b],
where the functions %2 and %3 are defined by the formulae (4.5) and (4.6). Consequently, the
assumptions of Corollary 4.2 are satisfied with k = 2 and m = 3.
The theorem is proved.
Proof of Theorem 5.5. The assertion of the theorem follows immediately from Lemma 5.2
and Theorem 4.3.
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492 A. LOMTATIDZE, Z. OPLUŠTIL, J. ŠREMR
Proof of Corollary 5.2. It is not difficult to verify that
xαx(µ∗)
αx(µ∗)−
(
ĥ(αx)− αx(a)
)
(h(1)− 1)−1
≥
≥ xαx(a)
αx(a)− (λ + h0(1)− (1 + h1(1))αx(a)) (h(1)− 1)−1 =
=
xex||g||L (h(1)− 1)(
ex||g||L − 1
)
(λ + h0(1))
for every x > 0 such that λ+h0(1) > (1 + h1(1)) ex||g||L . Therefore, the relation ξ∗ ≤ ω∗ holds,
where ω∗ and ξ∗ are defined by the formulae (5.23) and (5.24), respectively. Consequently,
validity of the corollary follows immediately from Theorem 5.5.
Proof of Theorem 5.6. Let the operator ` be defined by formula (5.2). It is clear that −` ∈
∈ Pab. According to condition (5.25), there exists δ ∈ [0, 1[ such that the inequality
z1(a)− h(z1)
h(1)− 1
+ z1(t) ≤
(
δ − z0(a)− h(z0)
h(1)− 1
) (
z0(a)− h(z0)
h(1)− 1
+ z0(t)
)
holds for t ∈ [a, b]. However, it means that
%3(t) ≤ δ%2(t) for t ∈ [a, b],
where the functions %2 and %3 are defined in Corollary 4.4 (a). Consequently, the assumptions
of Corollary 4.4(a) are satisfied with k = 2 and m = 3.
The theorem is proved.
Proof of Theorem 5.7. Let the operators ` and ¯̀be defined by formulae (5.2) and
¯̀(v)(t) df= g(t)σ(t)
t∫
µ(t)
g(s)v(µ(s))ds
for a.e. t ∈ [a, b], all v ∈ C([a, b]; R),
respectively, where the function σ is given by formula (5.27). It is clear that −` ∈ Pab, ¯̀ ∈ Pab,
and
`(1)(t)ϑ(v)(t)− `(ϑ(v))(t) = g(t)
t∫
µ(t)
g(s)v(µ(s))ds ≤
≤ ¯̀(v)(t) for a.e. t ∈ [a, b] and all v ∈ C([a, b]; R+),
where the operator ϑ is defined by formulae (4.8) and (4.9), and thus condition (4.12) holds on
the set C([a, b]; R+). Therefore, the assumptions of Corollary 4.4(b) are satisfied.
The theorem is proved.
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NONPOSITIVE SOLUTIONS TO A CERTAIN FUNCTIONAL DIFFERENTIAL INEQUALITY 493
Proof of Theorem 5.8. Let the operator ` be defined by formula (5.3),
`0(v)(t) df= p(t)v(τ(t)) for a.e. t ∈ [a, b] and all v ∈ C([a, b]; R), (5.37)
and
`1(v)(t) df= g(t)v(µ(t)) for a.e. t ∈ [a, b] and all v ∈ C([a, b]; R). (5.38)
It is clear that `0, `1 ∈ Pab and ` = `0 − `1. Therefore, validity of the theorem follows from
Theorems 3.3 and 4.4.
Proof of Theorem 5.9. Let the operators `, `0, and `1 be defined by formulae (5.3), (5.37),
and (5.38), respectively. It is clear that `0, `1 ∈ Pab and ` = `0 − `1. Therefore, the assertion of
the theorem follows from Theorems 3.4 and 4.4.
Proof of Corollary 5.3. Validity of the corollary follows immediately from Theorems 5.8
and 5.9 with g ≡ 0.
Proof of Theorem 5.10. Let the operators `, `0, and `1 be defined by the formulae (5.3),
(5.37), and (5.38), respectively. It is clear that `0, `1 ∈ Pab and ` = `0 − `1. Therefore, the as-
sertion of the theorem follows immediately from Theorem 4.5, Theorems 5.3 – 5.7, and Corol-
lary 5.3.
Acknowledgement. For the first author, the research was supported by the Ministry of Ed-
ucation of the Czech Republic under the project MSM0021622409. For the second author, the
published results were acquired using the subsidization of the Ministry of Education of the
Czech Republic, Research Plan 2E08017 ,,Procedures and Methods to Increase Number of Re-
searchers” . For the third author, the research was supported by the Grant Agency of the Czech
Republic, Grant No. 201/06/0254, and by the Academy of Sciences of the Czech Republic, In-
stitutional Research Plan No. AV0Z10190503.
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Received 07.10.08
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