On an antiperiodic type boundary-value problem for first order nonlinear functional differential equations of non-Volterra's type
Nonimprovable, in a certain sense, sufficient conditions are established for the solvability and unique solvability of the boundary-value problem u'(t) = F(u)(t), u(a) + λu(b) = h(u), where F : C([a, b]; R) → L([a, b]; R) is a continuous operator satisfying the Caratheodory conditions, ` h :...
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irk-123456789-1769502021-02-10T01:26:28Z On an antiperiodic type boundary-value problem for first order nonlinear functional differential equations of non-Volterra's type Hakl, R. Lomtatidze, A. Šremr, J. Nonimprovable, in a certain sense, sufficient conditions are established for the solvability and unique solvability of the boundary-value problem u'(t) = F(u)(t), u(a) + λu(b) = h(u), where F : C([a, b]; R) → L([a, b]; R) is a continuous operator satisfying the Caratheodory conditions, ` h : C([a, b]; R) → R is a continuous functional, and λ ∈ R₊. Отримано неполiпшуванi у певному сенсi достатнi умови для iснування розв’язкiв або єдиного розв’язку граничної задачi u'(t) = F(u)(t), u(a) + λu(b) = h(u), де F : C([a, b]; R) → L([a, b]; R) — неперервний оператор, що задовольняє умови Каратеодорi, h : C([a, b]; R) → R — неперервний функцiонал i λ ∈ R₊. 2003 Article On an antiperiodic type boundary-value problem for first order nonlinear functional differential equations of non-Volterra's type / R. Hakl, A. Lomtatidze, J. Šremr // Нелінійні коливання. — 2003. — Т. 6, № 4. — С. 550-573. — Бібліогр.: 27 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/176950 517.948 en Нелінійні коливання Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Nonimprovable, in a certain sense, sufficient conditions are established for the solvability and unique
solvability of the boundary-value problem u'(t) = F(u)(t), u(a) + λu(b) = h(u),
where F : C([a, b]; R) → L([a, b]; R) is a continuous operator satisfying the Caratheodory conditions, `
h : C([a, b]; R) → R is a continuous functional, and λ ∈ R₊. |
| format |
Article |
| author |
Hakl, R. Lomtatidze, A. Šremr, J. |
| spellingShingle |
Hakl, R. Lomtatidze, A. Šremr, J. On an antiperiodic type boundary-value problem for first order nonlinear functional differential equations of non-Volterra's type Нелінійні коливання |
| author_facet |
Hakl, R. Lomtatidze, A. Šremr, J. |
| author_sort |
Hakl, R. |
| title |
On an antiperiodic type boundary-value problem for first order nonlinear functional differential equations of non-Volterra's type |
| title_short |
On an antiperiodic type boundary-value problem for first order nonlinear functional differential equations of non-Volterra's type |
| title_full |
On an antiperiodic type boundary-value problem for first order nonlinear functional differential equations of non-Volterra's type |
| title_fullStr |
On an antiperiodic type boundary-value problem for first order nonlinear functional differential equations of non-Volterra's type |
| title_full_unstemmed |
On an antiperiodic type boundary-value problem for first order nonlinear functional differential equations of non-Volterra's type |
| title_sort |
on an antiperiodic type boundary-value problem for first order nonlinear functional differential equations of non-volterra's type |
| publisher |
Інститут математики НАН України |
| publishDate |
2003 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/176950 |
| citation_txt |
On an antiperiodic type boundary-value problem for first order nonlinear functional differential equations of non-Volterra's type / R. Hakl, A. Lomtatidze, J. Šremr // Нелінійні коливання. — 2003. — Т. 6, № 4. — С. 550-573. — Бібліогр.: 27 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT haklr onanantiperiodictypeboundaryvalueproblemforfirstordernonlinearfunctionaldifferentialequationsofnonvolterrastype AT lomtatidzea onanantiperiodictypeboundaryvalueproblemforfirstordernonlinearfunctionaldifferentialequationsofnonvolterrastype AT sremrj onanantiperiodictypeboundaryvalueproblemforfirstordernonlinearfunctionaldifferentialequationsofnonvolterrastype |
| first_indexed |
2025-07-15T14:54:15Z |
| last_indexed |
2025-07-15T14:54:15Z |
| _version_ |
1837725128695218176 |
| fulltext |
UDC 517.948
ON AN ANTIPERIODIC TYPE BOUNDARY-VALUE PROBLEM FOR FIRST
ORDER NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS
OF NON-VOLTERRA’S TYPE
ПРО ГРАНИЧНI ЗАДАЧI АНТИПЕРIОДИЧНОГО ТИПУ ДЛЯ
НЕЛIНIЙНИХ ФУНКЦIОНАЛЬНО-ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
НЕВОЛЬТЕРРIВСЬКОГО ТИПУ
R. Hakl
Math. Inst. Czech Acad. Sci.
Žižkova 22, 616 62 Brno, Czech Republic
e-mail: hakl@ipm.cz
A. Lomtatidze
Masaryk Univ.
Janáčkovo nám. 2a, 662 95 Brno, Czech Republic
e-mail: bacho@math.muni.cz
J. Šremr
Math. Inst. Czech Acad. Sci.
Žižkova 22, 616 62 Brno, Czech Republic
e-mail: sremr@ipm.cz
Nonimprovable, in a certain sense, sufficient conditions are established for the solvability and unique
solvability of the boundary-value problem
u′(t) = F (u)(t), u(a) + λu(b) = h(u),
where F : C([a, b];R) → L([a, b];R) is a continuous operator satisfying the Carathèodory conditions,
h : C([a, b];R) → R is a continuous functional, and λ ∈ R+.
Отримано неполiпшуванi у певному сенсi достатнi умови для iснування розв’язкiв або єдиного
розв’язку граничної задачi
u′(t) = F (u)(t), u(a) + λu(b) = h(u),
де F : C([a, b];R) → L([a, b];R) — неперервний оператор, що задовольняє умови Каратеодорi,
h : C([a, b];R) → R — неперервний функцiонал i λ ∈ R+.
Introduction. The following notation is used throughout.
R is the set of all real numbers, R+ = [0,+∞[.
C([a, b];R) is the Banach space of continuous functions u : [a, b] → R with the norm
‖u‖C = max{|u(t)| : a ≤ t ≤ b}.
C([a, b];R+) = {u ∈ C([a, b];R) : u(t) ≥ 0 for t ∈ [a, b]}.
C̃([a, b];R) is the set of absolutely continuous functions u : [a, b] → R.
c© R. Hakl, A. Lomtatidze, J. Šremr, 2003
550 ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 4
ON AN ANTIPERIODIC TYPE BOUNDARY-VALUE PROBLEM FOR FIRST ORDER NONLINEAR FUNCTIONAL . . . 551
Bi
λc([a, b];R) = {u ∈ C([a, b];R) : (u(a)+λu(b)) sgn((2− i)u(a)+(i−1)u(b)) ≤ c}, where
c ∈ R, i = 1, 2.
L([a, b];R) is the Banach space of Lebesgue integrable functions p : [a, b] → R with the
norm ‖p‖L =
b∫
a
|p(s)|ds.
L([a, b];R+) = {p ∈ L([a, b];R) : p(t) ≥ 0 for almost all t ∈ [a, b]}.
Mab is the set of measurable functions τ : [a, b] → [a, b].
L̃ab is the set of linear operators ` : C([a, b];R) → L([a, b];R) for which there is a function
η ∈ L([a, b];R+) such that
|`(v)(t)| ≤ η(t)‖v‖C for t ∈ [a, b], v ∈ C([a, b];R).
Pab is the set of linear operators ` ∈ L̃ab transforming the set C([a, b];R+) into the set
L([a, b];R+).
Kab is the set of continuous operatorsF : C([a, b];R) → L([a, b];R) satisfying the Carathèo-
dory conditions, i.e., for every r > 0 there exists qr ∈ L([a, b];R+) such that
|F (v)(t)| ≤ qr(t) for t ∈ [a, b], ‖v‖C ≤ r.
K([a, b]×A;B), where A ⊆ R2, B ⊆ R, is the set of functions f : [a, b]×A → B satisfying
the Carathèodory conditions, i.e., f(·, x) : [a, b] → B is a measurable function for all x ∈ A,
f(t, ·) : A → B is a continuous function for almost all t ∈ [a, b], and for every r > 0 there
exists qr ∈ L([a, b];R+) such that
|f(t, x)| ≤ qr(t) for t ∈ [a, b], x ∈ A, ‖x‖ ≤ r.
[x]+ =
1
2
(|x|+ x), [x]− =
1
2
(|x| − x).
By a solution of the equation
u′(t) = F (u)(t), (0.1)
where F ∈ Kab, we understand a function u ∈ C̃([a, b];R) satisfying the equation (0.1) almost
everywhere in [a, b].
Consider the problem on the existence and uniqueness of a solution of (0.1) satisfying the
boundary condition
u(a) + λu(b) = h(u), (0.2)
where λ ∈ R+ and h : C([a, b];R) → R is a continuous functional.
The general boundary-value problems for functional differential equations have been studi-
ed very intensively. There are a lot of interesting general results (see, e.g., [1 – 27] and the
references therein), but still only a few effective criteria for the solvability of special boundary-
value problems for functional differential equations are known even in the linear case. In the
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 4
552 R. HAKL, A. LOMTATIDZE, J. ŠREMR
present paper, we try to fill to some extent the existing gap. More precisely, in Section 1 there are
established nonimprovable effective sufficient conditions for the solvability and unique solvabi-
lity of the problem (0.1), (0.2). Sections 2, 3 and 4 are devoted respectively to the auxiliary
propositions, the proofs of the main results and the examples verifying their optimality.
All results will be concretized for the differential equation with deviating arguments of the
form
u′(t) = p(t)u(τ(t))− g(t)u(µ(t)) + f(t, u(t), u(ν(t))), (0.3)
where p, g ∈ L([a, b];R+), τ, µ, ν ∈ Mab, and f ∈ K([a, b]×R2;R).
The special case of the discussed boundary-value problem is the Cauchy problem (for λ = 0
and h ≡ Const). In this case, the below theorems coincide with the results obtained in [5]. The
periodic type boundary-value problem (i.e. the case λ < 0) for the linear equation and for the
nonlinear one is studied respectively in [14] and [15].
From the general theory of linear boudary-value problems for functional differential equati-
ons we need the following well-known result (see, e.g., [3, 19, 27]).
Theorem 0.1. Let ` ∈ L̃ab. Then the problem
u′(t) = `(u)(t) + q0(t), u(a) + λu(b) = c 0, (0.4)
where q0 ∈ L([a, b];R), c 0 ∈ R, is uniquely solvable if and only if the corresponding homogeneous
problem
u′(t) = `(u)(t), (0.10)
u(a) + λu(b) = 0 (0.20)
has only the trivial solution.
Remark 0.1. From the Riesz – Schauder theory it follows that if ` ∈ L̃ab and the problem
(0.10), (0.20) has a nontrivial solution, then there exist q0 ∈ L([a, b];R) and c 0 ∈ R such that
the problem (0.4) has no solution.
1. Main results. Throughout the paper we assume that q ∈ K([a, b]×R+;R+) is nondecrea-
sing in the second argument, and satisfies
lim
x→+∞
1
x
b∫
a
q(s, x)ds = 0. (1.1)
Theorem 1.1. Let λ ∈ ]0, 1], c ∈ R+,
h(v) sgn v(a) ≤ c for v ∈ C([a, b];R), (1.2)
and let there exist
`0, `1 ∈ Pab (1.3)
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ON AN ANTIPERIODIC TYPE BOUNDARY-VALUE PROBLEM FOR FIRST ORDER NONLINEAR FUNCTIONAL . . . 553
such that on the set B1
λc([a, b];R) the inequality
[F (v)(t)− `0(v)(t) + `1(v)(t)] sgn v(t) ≤ q(t, ‖v‖C) for t ∈ [a, b] (1.4)
holds. If, moreover,
‖`0(1)‖L < 1, ‖`1(1)‖L < α(λ), (1.5)
where
α(λ) =
−λ+ 2
√
1− ‖`0(1)‖L for ‖`0(1)‖L < 1− λ2;
1
λ
(1− ‖`0(1)‖L) for ‖`0(1)‖L ≥ 1− λ2,
(1.6)
then the problem (0.1), (0.2) has at least one solution.
Remark 1.1. Theorem 1.1 is nonimprovable in a certain sense. More precisely, the second
inequality in (1.5) cannot be replaced by
‖`1(1)‖L < (1 + ε)α(λ)
no matter how small ε > 0 would be (see Examples 4.1 – 4.3).
Theorem 1.2. Let λ ∈ ]0, 1], c ∈ R+,
h(v) sgn v(b) ≤ c for v ∈ C([a, b];R), (1.7)
and let there exist `0, `1 ∈ Pab such that on the set B2
λc([a, b];R) the inequality
[F (v)(t)− `0(v)(t) + `1(v)(t)] sgn v(t) ≥ −q(t, ‖v‖C) for t ∈ [a, b] (1.8)
holds. If, moreover,
‖`0(1)‖L + λ‖`1(1)‖L < λ, (1.9)
then the problem (0.1), (0.2) has at least one solution.
Remark 1.2. Theorem 1.2 is nonimprovable in a certain sense. More precisely, the inequality
(1.9) cannot be replaced by
‖`0(1)‖L + λ‖`1(1)‖L < λ+ ε
no matter how small ε > 0 would be (see Examples 4.4 and 4.5).
Remark 1.3. Let λ ∈ [1,+∞[. Define an operator ψ : L([a, b];R) → L([a, b];R) by
ψ(w)(t) df= w(a+ b− t) for t ∈ [a, b].
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 4
554 R. HAKL, A. LOMTATIDZE, J. ŠREMR
Let ϕ be a restriction of ψ to the space C([a, b];R). Put ϑ =
1
λ
, and
F̂ (w)(t) df= −ψ(F (ϕ(w)))(t), ĥ(w) df= ϑh(ϕ(w)).
It is clear that if u is a solution of the problem (0.1), (0.2), then the function v
df= ϕ(u) is a
solution of the problem
v′(t) = F̂ (v)(t), v(a) + ϑv(b) = ĥ(v), (1.10)
and vice versa, if v is a solution of the problem (1.10), then the function u df= ϕ(v) is a solution
of the problem (0.1), (0.2).
Therefore, the following theorems immediately follow from Theorems 1.1 and 1.2.
Theorem 1.3. Let λ ∈ [1,+∞[, c ∈ R+, the condition (1.7) be fulfilled, and let there exist
`0, `1 ∈ Pab such that on the set B2
λc([a, b];R) the inequality (1.8) holds. If, moreover,
‖`1(1)‖L < 1, ‖`0(1)‖L < β(λ), (1.11)
where
β(λ) =
− 1
λ
+ 2
√
1− ‖`1(1)‖L for ‖`1(1)‖L < 1− 1
λ2
;
λ (1− ‖`1(1)‖L) for ‖`1(1)‖L ≥ 1− 1
λ2
,
(1.12)
then the problem (0.1), (0.2) has at least one solution.
Theorem 1.4. Let λ ∈ [1,+∞[, c ∈ R+, the condition (1.2) be fulfilled, and let there exist
`0, `1 ∈ Pab such that on the set B1
λc([a, b];R) the inequality (1.4) holds. If, moreover,
λ‖`1(1)‖L + ‖`0(1)‖L < 1, (1.13)
then the problem (0.1), (0.2) has at least one solution.
Remark 1.4. On account of Remarks 1.1 – 1.3, it is clear that Theorems 1.3 and 1.4 are also
nonimprovable.
Next we establish theorems on the unique solvability of the problem (0.1), (0.2).
Theorem 1.5. Let λ ∈ ]0, 1],
[h(v)− h(w)] sgn(v(a)− w(a)) ≤ 0 for v, w ∈ C([a, b];R), (1.14)
and let there exist `0, `1 ∈ Pab such that on the set B1
λc([a, b];R), where c = |h(0)|, the inequality
[F (v)(t)− F (w)(t)− `0(v − w)(t) + `1(v − w)(t)] sgn(v(t)− w(t)) ≤ 0 (1.15)
holds. Let, moreover, (1.5) be fulfilled, where α(λ) is defined by (1.6). Then the problem (0.1),
(0.2) is uniquely solvable.
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ON AN ANTIPERIODIC TYPE BOUNDARY-VALUE PROBLEM FOR FIRST ORDER NONLINEAR FUNCTIONAL . . . 555
Theorem 1.6. Let λ ∈ ]0, 1],
[h(v)− h(w)] sgn(v(b)− w(b)) ≤ 0 for v, w ∈ C([a, b];R), (1.16)
and let there exist `0, `1 ∈ Pab such that on the set B2
λc([a, b];R), where c = |h(0)|, the inequality
[F (v)(t)− F (w)(t)− `0(v − w)(t) + `1(v − w)(t)] sgn(v(t)− w(t)) ≥ 0 (1.17)
holds. Let, moreover, (1.9) be fulfilled. Then the problem (0.1), (0.2) is uniquely solvable.
According to Remark 1.3, Theorems 1.5 and 1.6 imply the following results.
Theorem 1.7. Let λ ∈ [1,+∞[, the condition (1.16) be satisfied, and let there exist `0, `1 ∈ Pab
such that on the set B2
λc([a, b];R), where c = |h(0)|, the inequality (1.17) holds. Let, moreover,
(1.11) be fulfilled, where β(λ) is defined by (1.12). Then the problem (0.1), (0.2) is uniquely
solvable.
Theorem 1.8. Let λ ∈ [1,+∞[, the condition (1.14) be satisfied, and let there exist `0, `1 ∈ Pab
such that on the set B1
λc([a, b];R), where c = |h(0)|, the inequality (1.15) holds. Let, moreover,
(1.13) be fulfilled. Then the problem (0.1), (0.2) is uniquely solvable.
Remark 1.5. Theorems 1.5 – 1.8 are nonimprovable in a certain sense (see Examples
4.1 – 4.5).
For the equation of the type (0.3), from Theorems 1.1 – 1.8 we get the following assertions.
Corollary 1.1. Let λ ∈ ]0, 1], c ∈ R+, the condition (1.2) be fulfilled, and let
f(t, x, y) sgnx ≤ q(t) for t ∈ [a, b], x, y ∈ R, (1.18)
where q ∈ L([a, b];R+). If, moreover,
b∫
a
p(s)ds < 1,
b∫
a
g(s)ds < γ(λ), (1.19)
where
γ(λ) =
−λ+ 2
√√√√√1−
b∫
a
p(s)ds for
b∫
a
p(s)ds < 1− λ2;
1
λ
1−
b∫
a
p(s)ds
for
b∫
a
p(s)ds ≥ 1− λ2,
(1.20)
then the problem (0.3), (0.2) has at least one solution.
Corollary 1.2. Let λ ∈ ]0, 1], c ∈ R+, the condition (1.7) be fulfilled, and let
f(t, x, y) sgnx ≥ −q(t) for t ∈ [a, b], x, y ∈ R, (1.21)
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 4
556 R. HAKL, A. LOMTATIDZE, J. ŠREMR
where q ∈ L([a, b];R+). If, moreover,
λ
b∫
a
g(s)ds+
b∫
a
p(s)ds < λ, (1.22)
then the problem (0.3), (0.2) has at least one solution.
Corollary 1.3. Let λ ∈ [1,+∞[, c ∈ R+, the conditions (1.7) and (1.21) be fulfilled, and let
b∫
a
g(s)ds < 1,
b∫
a
p(s)ds < δ(λ), (1.23)
where
δ(λ) =
− 1
λ
+ 2
√√√√√1−
b∫
a
g(s)ds for
b∫
a
g(s)ds < 1− 1
λ2
;
λ
1−
b∫
a
g(s)ds
for
b∫
a
g(s)ds ≥ 1− 1
λ2
.
(1.24)
Then the problem (0.3), (0.2) has at least one solution.
Corollary 1.4. Let λ ∈ [1,+∞[, c ∈ R+, the conditions (1.2) and (1.18) be fulfilled, and let
λ
b∫
a
g(s)ds+
b∫
a
p(s)ds < 1. (1.25)
Then the problem (0.3), (0.2) has at least one solution.
Corollary 1.5. Let λ ∈ ]0, 1], the condition (1.14) be fulfilled and let
[f(t, x1, y1)− f(t, x2, y2)] sgn(x1 − x2) ≤ 0
for t ∈ [a, b], x1, x2, y1, y2 ∈ R. (1.26)
If, moreover, (1.19) holds, where γ(λ) is defined by (1.20), then the problem (0.3), (0.2) is uniquely
solvable.
Corollary 1.6. Let λ ∈ ]0, 1], the conditions (1.16),
[f(t, x1, y1)− f(t, x2, y2)] sgn(x1 − x2) ≥ 0
for t ∈ [a, b], x1, x2, y1, y2 ∈ R, (1.27)
and (1.22) hold. Then the problem (0.3), (0.2) is uniquely solvable.
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 4
ON AN ANTIPERIODIC TYPE BOUNDARY-VALUE PROBLEM FOR FIRST ORDER NONLINEAR FUNCTIONAL . . . 557
Corollary 1.7. Let λ ∈ [1,+∞[, and the conditions (1.16) and (1.27) be fulfilled. Let, moreover,
(1.23) hold, where δ(λ) is defined by (1.24). Then the problem (0.3), (0.2) is uniquely solvable.
Corollary 1.8. Let λ ∈ [1,+∞[, and the conditions (1.14), (1.25), and (1.26) hold. Then the
problem (0.3), (0.2) is uniquely solvable.
2. Auxiliary propositions. First we formulate the result from [22] (Theorem 1) in a suitable
for us form.
Lemma 2.1. Let there exist a positive number ρ and an operator ` ∈ L̃ab such that the
homogeneous problem (0.10), (0.20) has only the trivial solution, and let for every δ ∈ ]0, 1[ and
for an arbitrary function u ∈ C̃([a, b];R) satisfying
u′(t) = `(u)(t) + δ[F (u)(t)− `(u)(t)], u(a) + λu(b) = δh(u), (2.1)
the estimate
‖u‖C ≤ ρ (2.2)
hold. Then the problem (0.1), (0.2) has at least one solution.
Definition 2.1. We say that the operator ` ∈ L̃ab belongs to the set Ui(λ), i ∈ {1, 2}, if there
exists a positive number r such that for any q∗ ∈ L([a, b];R+) and c ∈ R+, every function
u ∈ C̃([a, b];R), satisfying the inequalities
[u(a) + λu(b)] sgn ((2− i)u(a) + (i− 1)u(b)) ≤ c, (2.3)
(−1)i+1[u′(t)− `(u)(t)] sgnu(t) ≤ q∗(t) for t ∈ [a, b], (2.4)
admits the estimate
‖u‖C ≤ r (c+ ‖q∗‖L) . (2.5)
Lemma 2.2. Let i ∈ {1, 2}, c ∈ R+,
h(v) sgn((2− i)v(a) + (i− 1)v(b)) ≤ c for v ∈ C([a, b];R), (2.6)
and let there exist ` ∈ Ui(λ) such that on the set Bi
λc([a, b];R) the inequality
(−1)i+1[F (v)(t)− `(v)(t)] sgn v(t) ≤ q(t, ‖v‖C) for t ∈ [a, b] (2.7)
is fulfilled. Then the problem (0.1), (0.2) has at least one solution.
Proof. First note that due to the condition ` ∈ Ui(λ), the homogeneous problem (0.10),
(0.20) has only the trivial solution.
Let r be the number appearing in Definition 2.1. According to (1.1) there exists ρ > 2rc
such that
1
x
b∫
a
q(s, x)ds <
1
2r
for x > ρ.
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558 R. HAKL, A. LOMTATIDZE, J. ŠREMR
Now assume that a function u ∈ C̃([a, b];R) satisfies (2.1) for some δ ∈ ]0, 1[. Then, accor-
ding to (2.6), u satisfies the inequality (2.3), i.e., u ∈ Bi
λc([a, b];R). By (2.1) and (2.7) we obtain
that the inequality (2.4) is fulfilled for q∗(t) = q(t, ‖u‖C). Hence, by the condition ` ∈ Ui(λ)
and the definition of the number ρ, we get the estimate (2.2).
Since ρ depends neither on u nor on δ, from Lemma 2.1 it follows that the problem (0.1),
(0.2) has at least one solution.
The lemma is proved.
Lemma 2.3. Let i ∈ {1, 2},
[h(u1)− h(u2)] sgn((2− i)(u1(a)− u2(a)) + (i− 1)(u1(b)− u2(a))) ≤ 0
for u1, u2 ∈ C([a, b];R), (2.8)
and let there exist ` ∈ Ui(λ) such that on the set Bi
λc([a, b];R), where c = |h(0)|, the inequality
(−1)i+1[F (u1)(t)− F (u2)(t)− `(u1 − u2)(t)] sgn(u1(t)− u2(t)) ≤ 0 (2.9)
holds. Then the problem (0.1), (0.2) is uniquely solvable.
Proof. From (2.8) it follows that the condition (2.6) is fulfilled, where c = |h(0)|. By (2.9),
on the set Bi
λc([a, b];R) the inequality (2.7) holds, where q ≡ |F (0)|. Consequently, all the
assumptions of Lemma 2.2 are fulfilled and this guarantees that the problem (0.1), (0.2) has at
least one solution. It remains to show that the problem (0.1), (0.2) has at most one solution.
Let u1, u2 be arbitrary solutions of the problem (0.1), (0.2). Put u(t) = u1(t) − u2(t) for
t ∈ [a, b]. Then by (2.8) and (2.9) we get
[u(a) + λu(b)] sgn((2− i)u(a) + (i− 1)u(b)) ≤ 0,
(−1)i+1[u′(t)− `(u)(t)] sgnu(t) ≤ 0 for t ∈ [a, b].
This, together with the condition ` ∈ Ui(λ), results in u ≡ 0. Consequently, u1 ≡ u2.
The lemma is proved.
Lemma 2.4. Let λ ∈ ]0, 1], the operator ` admit the representation ` = `0 − `1 with `0 and
`1 satisfying the conditions (1.3) and (1.5), where α is defined by (1.6). Then ` belongs to the set
U1(λ).
Proof. Suppose q∗ ∈ L([a, b];R+), c ∈ R+ and u ∈ C̃([a, b];R) satisfies (2.3) and (2.4) for
i = 1. We show that (2.5) holds, where
r =
‖`1(1)‖L + 1 + λ
1− ‖`0(1)‖L −
1
4
(‖`1(1)‖L + λ)2
if ‖`0(1)‖L < 1− λ2;
‖`1(1)‖L + 1 + λ
1− ‖`0(1)‖L − λ‖`1(1)‖L
if ‖`0(1)‖L ≥ 1− λ2.
(2.10)
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ON AN ANTIPERIODIC TYPE BOUNDARY-VALUE PROBLEM FOR FIRST ORDER NONLINEAR FUNCTIONAL . . . 559
It is clear that
u′(t) = `0(u)(t)− `1(u)(t) + q̃(t), (2.11)
where
q̃(t) = u′(t)− `(u)(t) for t ∈ [a, b]. (2.12)
Obviously,
q̃(t) sgnu(t) ≤ q∗(t) for t ∈ [a, b], (2.13)
and
[u(a) + λu(b)] sgnu(a) ≤ c. (2.14)
First suppose that u does not change its sign. According to (2.14) and the assumption λ ∈
∈ ]0, 1], we obtain
|u(a)| ≤ c. (2.15)
Choose t0 ∈ [a, b] such that
|u(t0)| = ‖u‖C . (2.16)
Due to (1.3) and (2.13), (2.11) implies
|u(t)|′ ≤ ‖u‖C `0(1)(t) + q∗(t) for t ∈ [a, b]. (2.17)
The integration of (2.17) from a to t0, on account of (1.3), (2.15) and (2.16), results in
‖u‖C − c ≤ ‖u‖C − |u(a)| ≤ ‖u‖C
t0∫
a
`0(1)(s)ds+
t0∫
a
q∗(s)ds ≤ ‖u‖C‖`0(1)‖L + ‖q∗‖L.
Thus
‖u‖C (1− ‖`0(1)‖L) ≤ c+ ‖q∗‖L
and, consequently, the estimate (2.5) holds.
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560 R. HAKL, A. LOMTATIDZE, J. ŠREMR
Now suppose that u changes its sign. Put
M = max{u(t) : t ∈ [a, b]}, m = −min{u(t) : t ∈ [a, b]} (2.18)
and choose tM , tm ∈ [a, b] such that
u(tM ) = M, u(tm) = −m. (2.19)
Obviously, M > 0, m > 0, and either
tm < tM , (2.20)
or
tm > tM . (2.21)
First suppose that (2.20) is fulfilled. It is clear that there exists α2 ∈ ]tm, tM [ such that
u(t) > 0 for α2 < t ≤ tM , u(α2) = 0. (2.22)
Let
α1 = inf{t ∈ [a, tm] : u(s) < 0 for t ≤ s ≤ tm}.
Obviously,
u(t) < 0 for α1 < t ≤ tm and u(α1) = 0 if α1 > a. (2.23)
From (2.14), (2.23) and the assumption λ ∈ ]0, 1] it follows that
u(α1) ≥ −λ[u(b)]+ − c ≥ −λM − c. (2.24)
The integration of (2.11) from α1 to tm and from α2 to tM , in view of (1.3), (2.13), (2.18), (2.19),
(2.22), (2.23) and (2.24), yields
m− λM − c ≤ m+ u(α1) ≤ M
tm∫
α1
`1(1)(s)ds+m
tm∫
α1
`0(1)(s)ds+
tm∫
α1
q∗(s)ds,
M ≤ M
tM∫
α2
`0(1)(s)ds+m
tM∫
α2
`1(1)(s)ds+
tM∫
α2
q∗(s)ds.
From the last two inequalities we obtain
m(1− C1) ≤ M(A1 + λ) + ‖q∗‖L + c, M(1−D1) ≤ mB1 + ‖q∗‖L, (2.25)
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ON AN ANTIPERIODIC TYPE BOUNDARY-VALUE PROBLEM FOR FIRST ORDER NONLINEAR FUNCTIONAL . . . 561
where
A1 =
tm∫
α1
`1(1)(s)ds, B1 =
tM∫
α2
`1(1)(s)ds,
C1 =
tm∫
α1
`0(1)(s)ds, D1 =
tM∫
α2
`0(1)(s)ds.
Due to the first inequality in (1.5), C1 < 1, D1 < 1. Consequently, (2.25) implies
0 < m(1− C1)(1−D1) ≤ (A1 + λ)(mB1 + ‖q∗‖L) + ‖q∗‖L + c ≤
≤ m(A1 + λ)B1 + (‖q∗‖L + c)(‖`1(1)‖L + 1 + λ),
(2.26)
0 < M(1− C1)(1−D1) ≤ B1(M(A1 + λ) + ‖q∗‖L + c) + ‖q∗‖L ≤
≤ M(A1 + λ)B1 + (‖q∗‖L + c)(‖`1(1)‖L + 1 + λ).
Obviously,
(1− C1)(1−D1) ≥ 1− (C1 +D1) ≥ 1− ‖`0(1)‖L > 0. (2.27)
If ‖`0(1)‖L ≥ 1 − λ2, then, according to (1.6) and the second inequality in (1.5), we obtain
‖`1(1)‖L < λ. Hence, B1 < λ and
(A1 + λ)B1 = A1B1 + λB1 ≤ λ(A1 +B1) ≤ λ‖`1(1)‖L.
By the last inequality and (2.27), from (2.26) we get
m ≤ r0(‖`1(1)‖L + 1 + λ)(c+ ‖q∗‖L),
(2.28)
M ≤ r0(‖`1(1)‖L + 1 + λ)(c+ ‖q∗‖L),
where
r0 = (1− ‖`0(1)‖L − λ‖`1(1)‖L)−1. (2.29)
Therefore, the estimate (2.5) holds.
If ‖`0(1)‖L < 1− λ2, then by the inequalities
4(A1 + λ)B1 ≤ (A1 +B1 + λ)2 ≤ (‖`1(1)‖L + λ)2
and (2.27), (2.26) implies
m ≤ r1(‖`1(1)‖L + 1 + λ)(c+ ‖q∗‖L),
(2.30)
M ≤ r1(‖`1(1)‖L + 1 + λ)(c+ ‖q∗‖L),
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562 R. HAKL, A. LOMTATIDZE, J. ŠREMR
where
r1 =
[
1− ‖`0(1)‖L −
1
4
(‖`1(1)‖L + λ)2
]−1
. (2.31)
Therefore, the estimate (2.5) is valid.
Now suppose that (2.21) is satisfied. Obviously there exists α4 ∈ ]tm, tM [ such that
u(t) < 0 for α4 < t ≤ tm, u(α4) = 0. (2.32)
Let
α3 = inf{t ∈ [a, tM ] : u(s) > 0 for t ≤ s ≤ tM}.
Obviously,
u(t) > 0 for α3 < t ≤ tM and u(α3) = 0 if α3 > a. (2.33)
From (2.14), (2.33) and the assumption λ ∈ ]0, 1] we get
u(α3) ≤ λ[u(b)]− + c ≤ λm+ c. (2.34)
The integration of (2.11) from α3 to tM and from α4 to tm, in view of (1.3), (2.13), (2.18), (2.19),
(2.32), (2.33) and (2.34), results in
M − λm− c ≤ M − u(α3) ≤ M
tM∫
α3
`0(1)(s)ds+m
tM∫
α3
`1(1)(s)ds+
tM∫
α3
q∗(s)ds,
m ≤ M
tm∫
α4
`1(1)(s)ds+m
tm∫
α4
`0(1)(s)ds+
tm∫
α4
q∗(s)ds.
From the last two inequalities we obtain
M(1− C2) ≤ m(A2 + λ) + ‖q∗‖L + c, m(1−D2) ≤ MB2 + ‖q∗‖L, (2.35)
where
A2 =
tM∫
α3
`1(1)(s)ds, B2 =
tm∫
α4
`1(1)(s)ds,
C2 =
tM∫
α3
`0(1)(s)ds, D2 =
tm∫
α4
`0(1)(s)ds.
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ON AN ANTIPERIODIC TYPE BOUNDARY-VALUE PROBLEM FOR FIRST ORDER NONLINEAR FUNCTIONAL . . . 563
Due to the first inequality in (1.5), C2 < 1, D2 < 1. Consequently, (2.35) implies
0 < M(1− C2)(1−D2) ≤ (A2 + λ)(MB2 + ‖q∗‖L) + ‖q∗‖L + c ≤
≤ M(A2 + λ)B2 + (‖q∗‖L + c)(‖`1(1)‖L + 1 + λ),
(2.36)
0 < m(1− C2)(1−D2) ≤ B2(m(A2 + λ) + ‖q∗‖L + c) + ‖q∗‖L ≤
≤ m(A2 + λ)B2 + (‖q∗‖L + c)(‖`1(1)‖L + 1 + λ).
Obviously,
(1− C2)(1−D2) ≥ 1− (C2 +D2) ≥ 1− ‖`0(1)‖L > 0. (2.37)
If ‖`0(1)‖L ≥ 1 − λ2, then according to (1.6) and the second inequality in (1.5), we obtain
‖`1(1)‖L < λ. Hence, B2 < λ and
(A2 + λ)B2 = A2B2 + λB2 ≤ λ(A2 +B2) ≤ λ‖`1(1)‖L.
By the last inequality and (2.37), (2.36) implies (2.28), where r0 is defined by (2.29). Therefore,
the estimate (2.5) is valid.
If ‖`0(1)‖L < 1− λ2, then by the inequalities
4(A2 + λ)B2 ≤ (A2 +B2 + λ)2 ≤ (‖`1(1)‖L + λ)2
and (2.37), (2.36) implies (2.30), where r1 is defined by (2.31). Therefore, the estimate (2.5)
holds.
The lemma is proved.
Lemma 2.5. Let λ ∈ ]0, 1], the operator ` admit the representation ` = `0 − `1, where `0 and
`1 satisfy the conditions (1.3) and (1.9). Then ` belongs to the set U2(λ).
Proof. Let q∗ ∈ L([a, b];R+), c ∈ R+ and u ∈ C̃([a, b];R) satisfy (2.3) and (2.4) for i = 2.
We show that (2.5) holds, where
r =
λ‖`0(1)‖L + 1 + λ
λ− λ‖`1(1)‖L − ‖`0(1)‖L
. (2.38)
Obviously, u satisfies (2.11), where q̃ is defined by (2.12). Clearly,
−q̃(t) sgnu(t) ≤ q∗(t) for t ∈ [a, b], (2.39)
and
[u(a) + λu(b)] sgnu(b) ≤ c. (2.40)
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564 R. HAKL, A. LOMTATIDZE, J. ŠREMR
First suppose that u does not change its sign. According to (2.40) and the assumption λ ∈
∈ ]0, 1], we obtain
|u(b)| ≤ c
λ
. (2.41)
Choose t0 ∈ [a, b] such that (2.16) holds. Due to (1.3) and (2.41), (2.11) implies
−|u(t)|′ ≤ ‖u‖C `1(1)(t) + q∗(t) for t ∈ [a, b]. (2.42)
The integration of (2.42) from t0 to b, on account of (1.3), (2.41) and (2.16), results in
‖u‖C −
c
λ
≤ ‖u‖C − |u(b)| ≤ ‖u‖C
b∫
t0
`1(1)(s)ds+
b∫
t0
q∗(s)ds ≤
≤ ‖u‖C‖`1(1)‖L + ‖q∗‖L.
Thus
‖u‖C (1− ‖`1(1)‖L) ≤ c+ ‖q∗‖L
λ
,
and, consequently, the estimate (2.5) holds.
Now suppose that u changes its sign. Define numbers M and m by (2.18) and choose
tM , tm ∈ [a, b] such that (2.19) is fulfilled. Obviously, M > 0, m > 0, and either (2.20) or
(2.21) is valid.
First suppose that (2.21) holds. It is clear that there exists α1 ∈ ]tM , tm[ such that
u(t) > 0 for tM ≤ t < α1, u(α1) = 0. (2.43)
Let
α2 = sup{t ∈ [tm, b] : u(s) < 0 for tm ≤ s ≤ t}.
Obviously,
u(t) < 0 for tm ≤ t < α2 and u(α2) = 0 if α2 < b. (2.44)
From (2.40), (2.44) and the assumption λ ∈ ]0, 1] we obtain
u(α2) ≥ − 1
λ
[u(a)]+ −
c
λ
≥ −M
λ
− c
λ
. (2.45)
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ON AN ANTIPERIODIC TYPE BOUNDARY-VALUE PROBLEM FOR FIRST ORDER NONLINEAR FUNCTIONAL . . . 565
The integration of (2.11) from tM to α1 and from tm to α2, in view of (1.3), (2.18), (2.19), (2.39),
(2.43), (2.44) and (2.45), implies
M ≤ M
α1∫
tM
`1(1)(s)ds+m
α1∫
tM
`0(1)(s)ds+
α1∫
tM
q∗(s)ds,
m− M
λ
− c
λ
≤ m+ u(α2) ≤ M
α2∫
tm
`0(1)(s)ds+m
α2∫
tm
`1(1)(s)ds+
α2∫
tm
q∗(s)ds.
From the last two inequalities we get
M(1−A1) ≤ mC1 + ‖q∗‖L, m(1−B1) ≤ M
(
D1 +
1
λ
)
+ ‖q∗‖L +
c
λ
, (2.46)
where
A1 =
α1∫
tM
`1(1)(s)ds, B1 =
α2∫
tm
`1(1)(s)ds,
C1 =
α1∫
tM
`0(1)(s)ds, D1 =
α2∫
tm
`0(1)(s)ds.
Due to (1.9), A1 < 1, B1 < 1. Consequently, (2.46) implies
0 < M(1−A1)(1−B1) ≤ C1
(
M
(
D1 +
1
λ
)
+ ‖q∗‖L +
c
λ
)
+ ‖q∗‖L ≤
≤ MC1
(
D1 +
1
λ
)
+
(
‖`0(1)‖L + 1 +
1
λ
)
(‖q∗‖L + c) ,
(2.47)
0 < m(1−A1)(1−B1) ≤
(
D1 +
1
λ
)
(mC1 + ‖q∗‖L) + ‖q∗‖L +
c
λ
≤
≤ mC1
(
D1 +
1
λ
)
+
(
‖`0(1)‖L + 1 +
1
λ
)
(‖q∗‖L + c) .
Obviously,
(1−A1)(1−B1) ≥ 1− (A1 +B1) ≥ 1− ‖`1(1)‖L > 0. (2.48)
According to (1.9) and the assumption λ ∈ ]0, 1], we obtain ‖`0(1)‖L <
1
λ
. Hence, C1 <
1
λ
and
C1
(
D1 +
1
λ
)
= C1D1 +
1
λ
C1 ≤
1
λ
(C1 +D1) ≤ 1
λ
‖`0(1)‖L.
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566 R. HAKL, A. LOMTATIDZE, J. ŠREMR
By the last inequality, (2.48) and the assumption λ ∈ ]0, 1], from (2.47) we get
M ≤ r0 (λ‖`0(1)‖L + 1 + λ) (c+ ‖q∗‖L),
(2.49)
m ≤ r0 (λ‖`0(1)‖L + 1 + λ) (c+ ‖q∗‖L),
where
r0 = (λ− λ‖`1(1)‖L − ‖`0(1)‖L)−1. (2.50)
Therefore, the estimate (2.5) holds.
Now suppose that (2.20) is valid. Obviously there exists α3 ∈ ]tm, tM [ such that
u(t) < 0 for tm ≤ t < α3, u(α3) = 0. (2.51)
Let
α4 = sup{t ∈ [tM , b] : u(s) > 0 for tM ≤ s ≤ t}.
It is clear that
u(t) > 0 for tM ≤ t < α4 and u(α4) = 0 if α4 < b. (2.52)
From (2.40), (2.52), and the assumption λ ∈ ]0, 1] it follows that
u(α4) ≤ 1
λ
[u(a)]− +
c
λ
≤ m
λ
+
c
λ
. (2.53)
The integration of (2.11) from tm to α3 and from tM to α4, in view of (1.3), (2.18), (2.19), (2.39),
(2.51), (2.52) and (2.53), yields
m ≤ M
α3∫
tm
`0(1)(s)ds+m
α3∫
tm
`1(1)(s)ds+
α3∫
tm
q∗(s)ds,
M − m
λ
− c
λ
≤ M − u(α4) ≤ M
α4∫
tM
`1(1)(s)ds+m
α4∫
tM
`0(1)(s)ds+
α4∫
tM
q∗(s)ds.
From the last two inequalities we get
m(1−A2) ≤ MC2 + ‖q∗‖L, M(1−B2) ≤ m
(
D2 +
1
λ
)
+ ‖q∗‖L +
c
λ
, (2.54)
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ON AN ANTIPERIODIC TYPE BOUNDARY-VALUE PROBLEM FOR FIRST ORDER NONLINEAR FUNCTIONAL . . . 567
where
A2 =
α3∫
tm
`1(1)(s)ds, B2 =
α4∫
tM
`1(1)(s)ds,
C2 =
α3∫
tm
`0(1)(s)ds, D2 =
α4∫
tM
`0(1)(s)ds.
Due to (1.9), A2 < 1, B2 < 1. Consequently, (2.54) implies
0 < m(1−A2)(1−B2) ≤ C2
(
m
(
D2 +
1
λ
)
+ ‖q∗‖L +
c
λ
)
+ ‖q∗‖L ≤
≤ mC2
(
D2 +
1
λ
)
+
(
‖`0(1)‖L + 1 +
1
λ
)
(‖q∗‖L + c) ,
(2.55)
0 < M(1−A2)(1−B2) ≤
(
D2 +
1
λ
)
(MC2 + ‖q∗‖L) + ‖q∗‖L +
c
λ
≤
≤ MC2
(
D2 +
1
λ
)
+
(
‖`0(1)‖L + 1 +
1
λ
)
(‖q∗‖L + c) .
Obviously,
(1−A2)(1−B2) ≥ 1− (A2 +B2) ≥ 1− ‖`1(1)‖L > 0. (2.56)
According to (1.9) and the assumption λ ∈ ]0, 1], we obtain ‖`0(1)‖L <
1
λ
. Hence, C2 <
1
λ
and
C2
(
D2 +
1
λ
)
= C2D2 +
1
λ
C2 ≤
1
λ
(C2 +D2) ≤ 1
λ
‖`0(1)‖L.
By the last inequality and (2.56), (2.55) implies (2.49), where r0 is defined by (2.50). Therefore,
the estimate (2.5) is valid.
The lemma is proved.
3. Proofs of the main results. Theorem 1.1 follows from Lemmas 2.2 and 2.4, Theorem 1.2
follows from Lemmas 2.2 and 2.5, Theorem 1.5 follows from Lemmas 2.3 and 2.4, and Theo-
rem 1.6 follows from Lemmas 2.3 and 2.5.
Proof of Corollary 1.1. Obviously, the conditions (1.18) and (1.19), with γ defined by (1.20),
yield the conditions (1.4) and (1.5) with α defined by (1.6), where
F (v)(t) df= p(t)v(τ(t))− g(t)v(µ(t)) + f(t, v(t), v(ν(t))),
(3.1)
`0(v)(t) df= p(t)v(τ(t)), `1(v)(t) df= g(t)v(µ(t)).
Consequently, all the assumptions of Theorem 1.1 are fulfilled.
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 4
568 R. HAKL, A. LOMTATIDZE, J. ŠREMR
Proof of Corollary 1.5. Obviously, the conditions (1.26) and (1.19), with γ defined by (1.20),
yield the conditions (1.15) and (1.5) with α defined by (1.6), where F , `0 and `1 are defined by
(3.1). Consequently, all the assumptions of Theorem 1.5 are fulfilled.
Corollaries 1.2 – 1.4 and 1.6 – 1.8 can be proved analogously.
4. On remarks 1.1 and 1.2. On Remark 1.1. Let λ ∈ ]0, 1] (for the case λ = 0, see [5]).
Denote by G the set of pairs (x, y) ∈ R+ ×R+ such that either
x < 1− λ2, y < 2
√
1− x− λ,
or
1− λ2 ≤ x < 1, y <
1− x
λ
.
According to Theorem 1.1, if (1.2) is fulfilled and there exist `0, `1 ∈ Pab such that (‖`0(1)‖L,
‖`1(1)‖L) ∈ G, and on the set B1
λc([a, b];R) the inequality (1.4) holds, then the problem (0.1),
(0.2) is solvable.
Below we give examples which show that for any pair (x0, y0) 6∈ G, x0 ≥ 0, y0 ≥ 0, there
exist functions p0 ∈ L([a, b];R), −p1 ∈ L([a, b];R+), and τ ∈ Mab such that
b∫
a
[p0(s)]+ds = x0,
b∫
a
[p0(s)]−ds = y0, (4.1)
and the problem
u′(t) = p0(t)u(τ(t)) + p1(t)u(t), u(a) + λu(b) = 0 (4.2)
has a nontrivial solution. Then by Remark 0.1, there exist q0 ∈ L([a, b];R) and c 0 ∈ R such
that the problem (0.1), (0.2) with
F (v)(t) df= p0(t)v(τ(t)) + p1(t)v(t) + q0(t), h(v) df= c 0 (4.3)
has no solution, while the conditions (1.2) and (1.4) are fulfilled with `0(v)(t) df= [p0(t)]+v(τ(t)),
`1(v)(t) df= [p0(t)]−v(τ(t)), q ≡ |q0| and c = |c 0|.
It is clear that if x0, y0 ∈ R+ and (x0, y0) 6∈ G, then (x0, y0) belongs to at least one of the
following sets
G1 = {(x, y) ∈ R+ ×R+ : 1 < x, 0 ≤ y} ,
G2 =
{
(x, y) ∈ R+ ×R+ : 1− λ2 ≤ x ≤ 1,
1− x
λ
< y
}
,
G3 =
{
(x, y) ∈ R+ ×R+ : 0 ≤ x < 1− λ2, 2
√
1− x− λ < y
}
.
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ON AN ANTIPERIODIC TYPE BOUNDARY-VALUE PROBLEM FOR FIRST ORDER NONLINEAR FUNCTIONAL . . . 569
Example 4.1. Let (x0, y0) ∈ G1, and ε > 0 be such that x0 − ε ≥ 1, λ − ε > 0. Put a = 0,
b = 4, t0 = 3 +
ε
1 + ε
,
p0(t) =
0 for t ∈ [0, 1[;
−y0 for t ∈ [1, 2[;
x0 − 1− ε for t ∈ [2, 3[;
1 + ε for t ∈ [3, 4],
p1(t) =
−
λ− ε
λ− (λ− ε)t
for t ∈ [0, 1[;
0 for t ∈ [1, 4],
τ(t) =
t0 for t ∈ [0, 3[;
4 for t ∈ [3, 4].
Then (4.1) holds, and the problem (4.2) has the nontrivial solution
u(t) =
−(λ− ε)t+ λ for t ∈ [0, 1[;
ε for t ∈ [1, 3[;
−(1 + ε)(t− 3) + ε for t ∈ [3, 4].
Example 4.2. Let (x0, y0) ∈ G2, and ε > 0 be such that
1− x0 + ε
λ
≤ y0, λ − ε > 0. Put
a = 0, b = 4, t0 = 2 +
ε
1− x0 + ε
,
p0(t) =
0 for t ∈ [0, 1[;
−y0 +
1− x0 + ε
λ
for t ∈ [1, 2[;
−1− x0 + ε
λ
for t ∈ [2, 3[;
x0 for t ∈ [3, 4],
p1(t) =
− λ− ε
λ− (λ− ε)t
for t ∈ [0, 1[;
0 for t ∈ [1, 4],
τ(t) =
t0 for t ∈ [0, 2[;
0 for t ∈ [2, 3[;
4 for t ∈ [3, 4].
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570 R. HAKL, A. LOMTATIDZE, J. ŠREMR
Then (4.1) holds, and the problem (4.2) has the nontrivial solution
u(t) =
−(λ− ε)t+ λ for t ∈ [0, 1[;
ε for t ∈ [1, 2[;
−(1− x0 + ε)(t− 2) + ε for t ∈ [2, 3[;
−x0(t− 3)− (1− x0) for t ∈ [3, 4].
Example 4.3. Let (x0, y0) ∈ G3, and ε > 0 be such that y0 ≥ 2
√
1− x0 − λ + ε,
ε < 1−
√
1− x0. Put a = 0, b = 5,
p0(t) =
−
√
1− x0 + λ for t ∈ [0, 1[;
0 for t ∈ [1, 3−
√
1− x0 − ε[;
−1 for t ∈ [3−
√
1− x0 − ε, 3[;
−y0 + 2
√
1− x0 − λ+ ε for t ∈ [3, 4[;
x0 for t ∈ [4, 5],
p1(t) =
0 for t ∈ [0, 1[∪[3−
√
1− x0 − ε, 5];
− 1− x0
(1− x0)(1− t) +
√
1− x0
for t ∈ [1, 2[;
−
√
1− x0√
1− x0(3− t)− (1− x0)
for t ∈ [2, 3−
√
1− x0 − ε[,
τ(t) =
5 for t ∈ [0, 1[;
1 for t ∈ [1, 3[;
3−
√
1− x0 for t ∈ [3, 4[;
5 for t ∈ [4, 5].
Then (4.1) holds, and the problem (4.2) has the nontrivial solution
u(t) =
(
√
1− x0 − λ)t+ λ for t ∈ [0, 1[;
(1− x0)(1− t) +
√
1− x0 for t ∈ [1, 2[;
√
1− x0(3− t)− (1− x0) for t ∈ [2, 3[;
−(1− x0) for t ∈ [3, 4[;
x0(5− t)− 1 for t ∈ [4, 5].
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ON AN ANTIPERIODIC TYPE BOUNDARY-VALUE PROBLEM FOR FIRST ORDER NONLINEAR FUNCTIONAL . . . 571
On Remark 1.2. Let λ ∈ ]0, 1]. Denote by H the set of pairs (x, y) ∈ R+ ×R+ such that
x+ λy < λ.
By Theorem 1.2, if (1.7) is fulfilled and there exist `0, `1 ∈ Pab and q ∈ L([a, b];R+) such
that (‖`0(1)‖L, ‖`1(1)‖L) ∈ H , and on the set B2
λc([a, b];R) the inequality (1.8) holds, then the
problem (0.1), (0.2) is solvable.
Below we give examples which show that for any pair (x0, y0) 6∈ H , x0 ≥ 0, y0 ≥ 0, there
exist functions p0 ∈ L([a, b];R), p1 ∈ L([a, b];R+), and τ ∈ Mab such that (4.1) is fulfilled and
the problem (4.2) has a nontrivial solution. Then by Remark 0.1, there exist q0 ∈ L([a, b];R) and
c 0 ∈ R such that the problem (0.1), (0.2), where F and h are defined by (4.3), has no solution,
while the conditions (1.7) and (1.8) are fulfilled with `0(v)(t) df= [p0(t)]+v(τ(t)), `1(v)(t) df=
df= [p0(t)]−v(τ(t)), q ≡ |q0| and c = |c 0|.
It is clear that if x0, y0 ∈ R+ and (x0, y0) 6∈ H , then (x0, y0) belongs to at least one of the
following sets
H1 = {(x, y) ∈ R+ ×R+ : λ < x, 0 ≤ y} ,
H2 =
{
(x, y) ∈ R+ ×R+ : 0 ≤ x ≤ λ, −x
λ
+ 1 < y
}
.
Example 4.4. Let (x0, y0) ∈ H1, and ε > 0 be such that x0 − λ ≥ ε, 1 − ε > 0. Put a = 0,
b = 4, t0 =
λ
λ+ ε
,
p0(t) =
λ+ ε for t ∈ [0, 1[;
−y0 for t ∈ [1, 2[;
x0 − λ− ε for t ∈ [2, 3[;
0 for t ∈ [3, 4],
p1(t) =
0 for t ∈ [0, 3[;
1− ε
(1− ε)(t− 4) + 1
for t ∈ [3, 4],
τ(t) =
4 for t ∈ [0, 1[;
t0 for t ∈ [1, 4].
Then (4.1) holds, and the problem (4.2) has the nontrivial solution
u(t) =
(λ+ ε)t− λ for t ∈ [0, 1[;
ε for t ∈ [1, 3[;
(1− ε)(t− 4) + 1 for t ∈ [3, 4].
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 4
572 R. HAKL, A. LOMTATIDZE, J. ŠREMR
Example 4.5. Let (x0, y0) ∈ H2, and ε > 0 be such that
λ− x0 + ε
λ
≤ y0, 1 − ε > 0. Put
a = 0, b = 4, t0 = 2− ε
λ− x0 + ε
,
p0(t) =
x0 for t ∈ [0, 1[;
−λ− x0 + ε
λ
for t ∈ [1, 2[;
−y0 +
λ− x0 + ε
λ
for t ∈ [2, 3[;
0 for t ∈ [3, 4],
p1(t) =
0 for t ∈ [0, 3[;
1− ε
1− (1− ε)(4− t)
for t ∈ [3, 4],
τ(t) =
4 for t ∈ [0, 1[;
0 for t ∈ [1, 2[;
t0 for t ∈ [2, 4].
Then (4.1) holds, and the problem (4.2) has the nontrivial solution
u(t) =
−x0t+ λ for t ∈ [0, 1[;
(λ− x0 + ε)(2− t)− ε for t ∈ [1, 2[;
−ε for t ∈ [2, 3[;
(1− ε)(4− t)− 1 for t ∈ [3, 4].
Acknowledgement. For the first author this work was supported by the Grant
№ 201/99/0295 of the Grant Agency of the Czech Republic and for the second author by the RI
No. J07/98:143100001.
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Received 08.09.2003
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