New solvability conditions for a non-local boundary value problem for nonlinear functional differential equations
New efficient conditions are obtained sufficient for the solvability as well as unique solvability of a nonlocal boundary-value problem for nonlinear functional differential equations. Отримано новi ефективнi умови розв’язностi, а також єдиної розв’язностi нелокальних граничних задач для нелiнiйних...
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| Дата: | 2008 |
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| Мова: | Англійська |
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Інститут математики НАН України
2008
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| Цитувати: | New solvability conditions for a non-local boundary value problem for nonlinear functional differential equations / Z. Oplustil // Нелінійні коливання. — 2008. — Т. 11, № 3. — С. 365-386. — Бібліогр.: 12 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859610667085987840 |
|---|---|
| author | Oplustil, Z. |
| author_facet | Oplustil, Z. |
| citation_txt | New solvability conditions for a non-local boundary value problem for nonlinear functional differential equations / Z. Oplustil // Нелінійні коливання. — 2008. — Т. 11, № 3. — С. 365-386. — Бібліогр.: 12 назв. — англ. |
| collection | DSpace DC |
| container_title | Нелінійні коливання |
| description | New efficient conditions are obtained sufficient for the solvability as well as unique solvability of a nonlocal
boundary-value problem for nonlinear functional differential equations.
Отримано новi ефективнi умови розв’язностi, а також єдиної розв’язностi нелокальних граничних задач для нелiнiйних функцiонально-диференцiальних рiвнянь.
|
| first_indexed | 2025-11-28T11:17:51Z |
| format | Article |
| fulltext |
UDC 517.9
NEW SOLVABILITY CONDITIONS FOR A NONLOCAL
BOUNDARY-VALUE PROBLEM FOR NONLINEAR
FUNCTIONAL DIFFERENTIAL EQUATIONS*
ЗНОВI УМОВИ РОЗВ’ЯЗНОСТI НЕЛОКАЛЬНИХ ГРАНИЧНИХ ЗАДАЧ
ДЛЯ НЕЛIНIЙНИХ ФУНКЦIОНАЛЬНО-ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
Z. Opluštil
Inst. Math.
Technická 2, 616 69 Brno, Czech Republic
e-mail: oplustil@fme.vutbr.cz
New efficient conditions are obtained sufficient for the solvability as well as unique solvability of a nonlocal
boundary-value problem for nonlinear functional differential equations.
Отримано новi ефективнi умови розв’язностi, а також єдиної розв’язностi нелокальних гра-
ничних задач для нелiнiйних функцiонально-диференцiальних рiвнянь.
1. Introduction and notation. On the interval [a, b], we consider the functional differential
equation
u′(t) = F (u)(t), (1)
where F : C([a, b]; R) → L([a, b]; R) is a continuous (in general) nonlinear operator. As usual,
by a solution of this equation we understand an absolutely continuous function u : [a, b] → R
satisfying the equality (1) almost everywhere on [a, b].Along with the equation (1), we consider
the nonlocal boundary condition
h(u) = ϕ(u), (2)
where h : C([a, b]; R) → R is a (non-zero) linear bounded functional and ϕ : C([a, b]; R) → R
is a continuous (in general) nonlinear functional.
The question on the solvability of various types of initial and boundary-value problems for
functional differential equations and their systems is a classical topic in the theory of differential
equations (see, e.g., [1 – 11] and references therein). There is a lot of interesting general results
but only a few efficient conditions is known, namely, in the case where the boundary condition
considered is nonlocal. In [12], we studied the question on the unique solvability of the problem
(1), (2) in the linear case, i.e., in the case where the operator F is linear and ϕ ≡ Const. We
found out that it is very useful to consider the boundary condition (2) as a nonlocal perturbation
of the two-point condition
u(a)− λu(b) = ϕ(u) (3)
∗ Published results were acquired using the subsidization of the Ministry of Education, Youth and Sports of the
Czech Republic, research plan MSM 0021630518 “Simulation modelling of mechatronic systems”.
c© Z. Opluštil, 2008
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 3 365
366 Z. OPLUŠTIL
with λ ∈ R. In this paper, the results stated in [3] concerning the problem (1), (3) are gen-
eralized, and new efficient conditions are thus found sufficient for the solvability and unique
solvability of the problem (1), (2).
The following notation is used in the sequel.
1. R is the set of all real numbers, R+ = [0,+∞[.
2. C([a, b]; R) is the Banach space of continuous functions v : [a, b] → R with the norm
‖v‖C = max {|v(t)| : t ∈ [a, b]}.
3. C([a, b]; R+) = {u ∈ C([a, b]; R) : u(t) ≥ 0 for t ∈ [a, b]}.
4. 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.
5. L([a, b]; R+) =
{
p ∈ L([a, b]; R) : p(t) ≥ 0 for almost all t ∈ [a, b]
}
.
6.Lab 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 a.e. t ∈ [a, b] and all v ∈ C([a, b]; R).
7.Pab is the set of operators ` ∈ Lab transforming the setC([a, b]; R+) into the setL([a, b]; R+).
8. Fab is the set of linear bounded functionals h : C([a, b]; R) → R.
9. PFab is the set of functionals h ∈ Fab transforming the set C([a, b]; R+) into the set R+.
10. Bi
hc = {u ∈ C([a, b]; R) : (−1)i+1h(u)sgn
(
(2 − i)u(a) + (i − 1)u(b)
)
≤ c}, where
h ∈ Fab, c ∈ R, i = 1, 2.
11. K([a, b]×A;B), where A,B ⊆ R, is the set of function f : [a, b]×A → B satisfying the
Carathéodory 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 every t ∈ [a, b], and for every r > 0 there exists
qr ∈ L([a, b]; R+) such that
|f(t, x)| ≤ qr(t) for a.e. t ∈ [a, b] and all x ∈ A, |x| ≤ r.
2. Main results. As it was said above, the boundary condition (2) is considered as a non-local
perturbation of the two-point condition (3). Therefore, we assume in the sequel that the linear
functional h appearing in (2) is defined by the formula
h(v) = u(a)− λv(b)− h0(v) + h1(v) for v ∈ C([a, b]; R), (4)
where λ > 0 and h0, h1 ∈ PFab. Moreover, the following assumptions are used:
(H1) F : C([a, b]; R) → L([a, b]; R) is a continuous operator such that the relation
sup
{
|F (v)(·)| : v ∈ C([a, b]; R), ‖v‖C ≤ r
}
∈ L([a, b]; R+)
is satisfied for every r > 0.
(H2) ϕ : C([a, b]; R) → R is a continuous functional such that the condition
sup
{
|ϕ(v)| : v ∈ C([a, b]; R), ‖v‖C ≤ r
}
< +∞
holds for every r > 0.
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NEW SOLVABILITY CONDITIONS FOR A NONLOCAL BOUNDARY-VALUE PROBLEM FOR NONLINEAR . . . 367
Before formulation of the main results we introduce the following notation. Having λ > 0
and h ∈ Fab, we put
α(λ, h) =
(
1− h(1)
)
min
{
1,
1
λ
}
, (5)
β(λ, h) =
(
λ− h(1)
)
min
{
1,
1
λ
}
. (6)
Moreover, for any functional h given by the formula (4), we define the function ω0(· ;h) by
setting
ω0(x;h) =
(
x+ 1
λh0(1)
)(
1− h0(1)
)
1− h0(1)− x
−
( 1
λ
h1(1) +
1− λ
λ
)
if λ ≤ 1,
(
1− λ+ h1(1)
)
x <
(
1− h(1)
)(
1− h0(1)
)
,(
x+ h0(1)
)(
1− h0(1)
)
1− h0(1)− x
−
(
h1(1) + 1− λ
)
if λ ≤ 1,
(
1− λ+ h1(1)
)
x ≥
(
1− h(1)
)(
1− h0(1)
)
,(
x+ λ− 1 + h0(1)
)(
1− h0(1)
)
1− h0(1)− λx
− h1(1)
if λ > 1, λh1(1)x <
(
1− h(1)
)(
1− h0(1)
)
,
(
x+ λ−1
λ + 1
λh0(1)
)(
1− h0(1)
)
1− h0(1)− λx
− 1
λ
h1(1)
if λ > 1, λh1(1)x ≥
(
1− h(1)
)(
1− h0(1)
)
.
In this section, we formulate all the results, the proofs are postponed till Section 5 below.
Theorem 1. Let c ∈ R+, the assumptions (H1) and (H2) be satisfied, and let the functional h
be defined by the formula (4), where λ > 0 and h0, h1 ∈ PFab are such that
h(1) ≥ 0, (7)
h0(1) < 1, h1(1) ≤ λ. (8)
Let, moreover, the condition
ϕ(v)sgn v(a) ≤ c for v ∈ C([a, b]; R) (9)
be fulfilled and there exist
`0, `1 ∈ Pab (10)
such that, on the set B1
hc([a, b]; R), the inequality(
F (v)(t)− `0(v)(t) + `1(v)(t)
)
sgn v(t) ≤ q(t, ‖v‖C) for a.e. t ∈ [a, b] (11)
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 3
368 Z. OPLUŠTIL
holds, where the function q ∈ K([a, b]× R+; R+) satisfies
lim
x→+∞
1
x
b∫
a
q(s, x) ds = 0. (12)
If, in addition,
‖`0(1)‖L < α(λ, h0), (13)
ω0(‖`0(1)‖L;h) < ‖`1(1)‖L < 2
√
α(λ, h0)− ‖`0(1)‖L − h1(1)min
{
1,
1
λ
}
(14)
then the problem (1), (2) has at least one solution.
Theorem 2. Let c ∈ R+, the assumptions (H1) and (H2) be satisfied, and let the functional h
be defined by the formula (4), where λ > 0 and h0, h1 ∈ PFab satisfy the relations (7) and (8).
Let moreover, the condition
ϕ(v)sgn v(b) ≥ −c for v ∈ C([a, b]; R)
be fulfilled and there exist `0, `1 ∈ Pab such that, on the set B2
hc([a, b]; R), the inequality(
F (v)(t)− `0(v)(t) + `1(v)(t)
)
sgn v(t) ≥ −q(t, ‖v‖C) for a.e. t ∈ [a, b]
holds, where the function q ∈ K([a, b]× R+; R+) satisfies (12). If, in addition,
‖`1(1)‖L < β(λ, h1) (15)
and
α(λ, h0)
β(λ, h1)− ‖`1(1)‖L
− 1 < ‖`0(1)‖ <
< 2
√
β(λ, h1)− ‖`0(1)‖L − h0(1)min
{
1,
1
λ
}
(16)
then the problem (1), (2) has at least one solution.
Now we establish theorems concerning the unique solvability of the problem (1), (2).
Theorem 3. Let the assumptions (H1) and (H2) be satisfied and a functional h be defined by
the formula (4), where λ > 0 and h0, h1 ∈ PFab satisfy the relations (7) and (8). Let, moreover,
the condition (
ϕ(v)− ϕ(w)
)
sgn
(
v(a)− w(a)
)
≤ 0 for v, w ∈ C([a, b]; R)
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NEW SOLVABILITY CONDITIONS FOR A NONLOCAL BOUNDARY-VALUE PROBLEM FOR NONLINEAR . . . 369
be fulfilled and there exist `0, `1 ∈ Pab such that, on the set B1
hc([a, b]; R) with c = |ϕ(0)|, the
inequality(
F (v)(t)− F (w)(t)− `0(v − w)(t) + `1(v − w)(t)
)
sgn
(
v(t)− w(t)
)
≤ 0 for a.e. t ∈ [a, b]
holds. If, in addition, the conditions (13) and (14) are fulfilled then the problem (1), (2) is
uniquely solvable.
Theorem 4. Let the assumptions (H1) and (H2) be satisfied and the functional h be defined by
the formula (4), where λ > 0 and h0, h1 ∈ PFab satisfy the relations (7) and (8). Let, moreover,
the condition (
ϕ(v)− ϕ(w)
)
sgn
(
v(b)− w(b)
)
≥ 0 for v, w ∈ C([a, b]; R)
be fulfilled and there exist `0, `1 ∈ Pab such that, on the set B2
hc([a, b]; R) with c = |ϕ(0)|, the
inequality(
F (v)(t)− F (w)(t)− `0(v − w)(t) + `1(v − w)(t)
)
sgn
(
v(t)− w(t)
)
≥ 0 for a.e. t ∈ [a, b]
holds. If, in addition, the conditions (15) and (16) are fulfilled then the problem (1), (2) is
uniquely solvable.
Remark 1. Let the functional h be defined by the formula (4), where λ > 0 and h0, h1 ∈
∈ PFab. Define the operator ψ : L([a, b]; R) → L([a, b]; R) by setting
ψ(z)(t) df= z(a+ b− t), t ∈ [a, b],
for an arbitrary z ∈ L([a, b]; R). Let ω be the restriction of ψ to the space C([a, b]; R), and
F̂ (z)(t) df= −ψ
(
F (ω(z))
)
(t) for a.e. t ∈ [a, b] and all z ∈ C([a, b]; R),
ĥ(z) df= z(a)− 1
λ
z(b) +
1
λ
h0
(
ω(z)
)
− 1
λ
h1
(
ω(z)
)
for z ∈ C([a, b]; R),
ϕ̂(z) df= − 1
λ
ϕ(ω(z)) for z ∈ C([a, b]; R).
It is not difficult to verify that if u is a solution to the problem (1), (2) then the function v df= ω(u)
is a solution to the problem
v′(t) = F̂ (v)(t), ĥ(v) = ϕ̂(v), (17)
and vice versa, if v is a solution to the problem (17) then the function u df= ω(v) is a solution to
the problem (1), (2).
Therefore, using the above transformation, we can immediately derive conditions for the
solvability and unique solvability of the problem (1), (2) in the case where h(1) ≤ 0 (we do not
formulate them here in detail).
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370 Z. OPLUŠTIL
3. An example. As an example, on the interval [0, 1], we consider the integro-differential
equation
u′(t) = d cos(2πt)
1∫
0
u(τ(s))√
s
ds− g1(t)u(t)eu
2(ω(t)) + g2(t)|u(t)|ν (18)
subjected to the nonlocal boundary condition
u(0) =
1
2
u(1) + k
1∫
0
sin(2πs)u(s) ds− u(0)eu(1/4) + arctg u
(
1
2
)
, (19)
where d, k ∈ R+, ν ∈ [0, 1[, g1, g2 ∈ L([0, 1]; R), and τ, ω : [0, 1] → [0, 1] are measurable
functions.
Theorem 1 yields the following corollary.
Corollary 1. Let the function g1 be nonnegative on [0, 1] and the numbers d and k satisfy
k ≤ π
2
, d <
π − k
2
, (20)
and
(d+ k)(π − k)
π − k − 2d
−
(
k +
π
2
)
< d < 2
√
π(π − k − 2d)− k
2
. (21)
Then the problem (18), (19) has at least one solution.
4. Auxiliary propositions. The main results are proved using a lemma on a priory estimate
stated in [6] by Kiguradze and Půža. This lemma can be formulated as follows.
Lemma 1 ([6], Corollary 2). Let there exist a positive number ρ and an operator ` ∈ Lab
such that homogeneous problem
u′(t) = `(u)(t), h(u) = 0 (22)
has only the trivial solution, and, for every δ ∈ ]0, 1[, an arbitrary function u ∈ C̃([a, b]; R)
satisfying the relations
u′(t) = `(u)(t) + δ[F (u)(t)− `(u)(t)] for a.e. t ∈ [a, b], h(u) = δϕ(u) (23)
admits the estimate
‖u‖C ≤ ρ. (24)
Then the problem (1), (2) has at least one solution.
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NEW SOLVABILITY CONDITIONS FOR A NONLOCAL BOUNDARY-VALUE PROBLEM FOR NONLINEAR . . . 371
Definition 1. Let i ∈ {1, 2}, h ∈ Fab. We say that an operator ` ∈ Lab belongs to the set
Ui(h), if there exists r > 0 such that, for arbitrary q∗ ∈ L([a, b]; R+) and c ∈ R+, every function
u ∈ C̃([a, b]; R) satisfying the inequalities
(−1)i+1h(u)sgn
(
(2− i)u(a) + (i− 1)u(b)
)
≤ c, (25)
(−1)i+1
(
u′(t)− `(u)(t)
)
sgnu(t) ≤ q∗(t) for a.e. t ∈ [a, b] (26)
admits the estimate
‖u‖C ≤ r(c+ ‖q∗‖L). (27)
Lemma 2. Let i ∈ {1, 2}, c ∈ R+, the assumptions (H1) and (H2) be satisfied, and
(−1)i+1ϕ(v)sgn
(
(2− i)v(a) + (i− 1)v(b)
)
≤ c for v ∈ C([a, b]; R). (28)
Let, moreover, there exist ` ∈ Ui(h) such that, on the set Bi
hc([a, b]; R), the inequality
(−1)i+1
(
F (v)(t)− `(v)(t)
)
sgn v(t) ≤ q(t, ‖v‖C) for a.e. t ∈ [a, b] (29)
is fulfilled. Then the problem (1), (2) has at least one solution.
Proof. First note that, due to the condition ` ∈ Ui(h), the homogeneous problem (22) has
only the trivial solution.
Let r be the number appearing in Definition 1. According to (12), there exists ρ > 2rc such
that
1
x
b∫
a
q(s, x) <
1
2r
for x > ρ.
Now assume that a function u ∈ C̃([a, b]; R) satisfies (23) for some δ ∈ ]0, 1[. Then, according
to (28), u satisfies inequality (25), i.e., u ∈ Bi
hc([a, b]; R). By (29), we obtain that inequality (26)
is fulfilled with q∗ ≡ q(· , ‖u‖C). Hence, by virtue of the condition ` ∈ Ui(λ) and the definition
of the number ρ, we get the estimate (24).
Since ρ depends neither on u nor on δ, it follows from Lemma 1 that the problem (1), (2)
has at least one solution.
Lemma 3. Let i ∈ {1, 2}, the assumptions (H1) and (H2) be satisfied, and let the relation
(−1)i+1
(
ϕ(u1)− ϕ(u2)
)
sgn
(
(2− i)(u1(a)− u2(a)) + (i− 1)(u1(b)− u2(b))
)
≤ 0 (30)
hold for every u1, u2 ∈ C([a, b]; R). Let, moreover, there exist ` ∈ Ui(λ) such that, on the set
Bi
hc([a, b]; R) with c = |ϕ(0)|, the inequality
(−1)i+1
(
F (u1)(t)− F (u2)(t)− `(u1 − u2)(t)
)
sgn
(
u1(t)− u2(t)
)
≤ 0 (31)
is fulfilled for a.e. t ∈ [a, b]. Then the problem (1), (2) is uniquely solvable.
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372 Z. OPLUŠTIL
Proof. It follows from the condition (30) that the inequality (28) is fulfilled, where c =
= |ϕ(0)|. Using (31), we get that, on the set Bi
hc([a, b]; R), the inequality (29) holds, where q ≡
≡ |F (0)|. Consequently, all the assumptions of Lemma 2 are fulfilled, and thus the problem (1),
(2) has at least one solution. It remains to show that problem (1), (2) has at most one solution.
Let u1, u2 be arbitrary solutions of the problem (1), (2). Put u(t) = u1(t) − u2(t) for t ∈
∈ [a, b]. Then, by virtue of (30) and (31), we get u1, u2 ∈ Bi
hc([a, b]; R) and
(−1)i+1h(u)sgn
(
(2− i)u(a) + (i− 1)u(b)
)
≤ 0,
(−1)i+1
(
u′(t)− `(u)(t)
)
sgnu(t) ≤ 0 for a.e. t ∈ [a, b].
The last relations, together with the assumption ` ∈ Ui(λ), result in u ≡ 0. Consequently,
u1 ≡ u2.
Lemma 4. Let the functional h be defined by the formula (4), where λ > 0 and h0, h1 ∈
∈ PFab satisfy the conditions (7) and (8). Let, moreover, the operator ` admit the representation
` = `0−`1, where `0 and `1 are such that the conditions (10), (13), and (14) hold. Then ` belongs
to the set U1(h).
Proof. Let c ∈ R+, q∗ ∈ L([a, b]; R+), and u ∈ C̃([a, b]; R) satisfy (25) and (26) with i = 1.
We shall show that (27) holds, where r depends only on ‖`0(1)‖L, ‖`1(1)‖L, λ, h0(1), and h1(1).
It is clear that
u′(t) = `0(u)(t)− `1(u)(t) + q̃(t) for a.e. t ∈ [a, b], (32)
where
q̃(t) = u′(t)− `(u)(t) for a.e. t ∈ [a, b]. (33)
From (25) and (26) we get(
u(a)− λu(b)− h0(u) + h1(u)
)
sgnu(a) ≤ c (34)
and
q̃(t)sgnu(t) ≤ q∗(t) for a.e. t ∈ [a, b]. (35)
First suppose that u does not change its sign. Put
M = max {|u(t)| : t ∈ [a, b]}, m = min{|u(t)| : t ∈ [a, b]} (36)
and choose t1, t2 ∈ [a, b] such that t1 6= t2 and
|u(t1)| = M, |u(t2)| = m. (37)
It is clear that M ≥ 0, m ≥ 0, and either
t1 < t2 (38)
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NEW SOLVABILITY CONDITIONS FOR A NONLOCAL BOUNDARY-VALUE PROBLEM FOR NONLINEAR . . . 373
or
t1 > t2. (39)
Moreover, according to (10), (35), and (36), from (32) we obtain
|u(t)|′ ≤ M `0(1)(t)−m`1(1)(t) + q∗(t) for a.e. t ∈ [a, b]. (40)
If u(a) = 0 then m = 0 and the integration of (40) from a to t1, on account of (37), yields
M ≤ M
t1∫
a
`0(1)(s) ds+
t1∫
a
q∗(s) ds.
By virtue of (13), it follows from the last inequality that
‖u‖C = M ≤ r0(‖q∗‖+ c),
where
r0 = [1− ‖`0(1)‖L]−1.
Consequently, the estimate (27) holds with r = r0.
If u(a) 6= 0 then, according to (34), we obtain
|u(a)| ≤ λ|u(b)|+ h0(|u|)− h1(|u|) + c,
and thus
|u(a)| − |u(b)| ≤ h0(|u|)− h1(|u|) + |u(b)|(λ− 1) + c (41)
and
|u(a)| − |u(b)| ≤ 1
λ
h0(|u|)−
1
λ
h1(|u|) + |u(a)|λ− 1
λ
+
c
λ
. (42)
Let first (38) hold. Then the integration of (40) from a to t1 and from t2 to b results in
M − |u(a)| ≤ M
t1∫
a
`0(1)(s) ds+
t1∫
a
q∗(s) ds,
|u(b)| −m ≤ M
b∫
t2
`0(1)(s) ds+
b∫
t2
q∗(s) ds.
Summing two last inequalities, we get
M −m+ |u(b)| − |u(a)| ≤ M
∫
J
`0(1)(s) ds+
∫
J
q∗(s) ds, (43)
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374 Z. OPLUŠTIL
where J = [a, t1] ∪ [t2, b].
Let λ ≤ 1. Taking into account (10) and (41), we obtain from (43) that
M −m− h0(|u|) + h1(|u|)− |u(b)|(λ− 1)− c ≤ M
b∫
a
`0(1)(s) ds+
b∫
a
q∗(s) ds.
From the last inequality, according to (36), we get
M(1− h0(1))−m(λ− h1(1))− c ≤ M‖`0(1)‖L + ‖q∗‖L.
Due to (7), the last inequality implies
(M −m)(1− h0(1)) ≤ M‖`0(1)‖L + ‖q∗‖L + c. (44)
Let λ > 1. From (43), in view of (7), (42), and (36), it follows that
1
λ
(M −m)(1− h0(1))− c ≤ M‖`0(1)‖L + ‖q∗‖L,
which, together with (44), yields
min
{
1,
1
λ
}
(M −m)(1− h0(1)) ≤ M‖`0(1)‖L + ‖q∗‖L + c.
Now suppose that (39) is fulfilled. Then the integration of (40) from t2 to t1, on account of
(10) and (36), yields
min
{
1,
1
λ
}
(M −m)(1− h0(1))− c ≤ M −m ≤ M‖`0(1)‖L + ‖q∗‖L.
Therefore, in both cases (38) and (39), the inequality
min
{
1,
1
λ
}
(M −m)(1− h0(1)) ≤ M‖`0(1)‖L + ‖q∗‖L + c (45)
holds.
On the other hand, the integration of (40) from a to b, yields
|u(b)| − |u(a)| ≤ M‖`0(1)‖L −m‖`1(1)‖L + ‖q∗‖L. (46)
Now we shall divide the discussion into the following four cases:
Case (a): λ ≤ 1 and (1− λ+ h1(1))‖`0(1)‖L < (1− h(1))(1− h0(1)). Using (42) and (36) in
(46), we get
m ≤ M(h0(1) + λ‖`0(1)‖L) + c+ λ‖q∗‖L
λ‖`1(1)‖L + h1(1)− λ+ 1
.
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On the other hand, (45) implies
M(1− h0(1)− ‖`0(1)‖L) ≤ m(1− h0(1)) + c+ ‖q∗‖L. (47)
Hence, on account of the first inequality in (14), we have
‖u‖C = M ≤ r1(c+ ‖q∗‖L)(λ‖`1(1)‖L + h1(1)− h0(1)− λ+ 2),
where
r1 =
[
(1− h0(1)− ‖`0(1)‖L)(λ‖`1(1)‖L + h1(1)− λ+ 1)×
× (−(1− h0(1))(h0(1) + λ‖`0(1)‖L))
]−1
> 0,
and thus the estimate (27) holds, where r is defined by
r = r1(λ‖`1(1)‖L + h1(1)− h0(1)− λ+ 2).
Case (b): λ ≤ 1 and (1−λ+h1(1))‖`0(1)‖L ≥ (1−h(1))(1−h0(1)). As above, (45) implies
(47). If we use the estimate (41) in (46), according to (36), we obtain
m ≤ M(h0(1) + ‖`0(1)‖L) + c+ ‖q∗‖L
‖`1(1)‖L + h1(1)− λ+ 1
.
From the last inequality, the first inequality in (14), and (47) we get
‖u‖C = M ≤ r2(c+ ‖q∗‖L)(‖`1(1)‖L + h1(1)− h0(1)− λ+ 2),
where
r2 =
[
(1− h0(1)− ‖`0(1)‖L)(‖`1(1)‖L + h1(1)− λ+ 1)×
× (−(1− h0(1))(h0(1) + ‖`0(1)‖L))
]−1
> 0,
and thus the estimate (27) holds, where r is defined by
r = r2(λ‖`0(1)‖L + h1(1)− h0(1)− λ+ 2).
Case (c): λ > 1 and λh1(1)‖`0(1)‖L < (1−h(1))(1−h0(1)). From (46), in view of (41), and
(36), it follows that
m ≤ M(h0(1) + ‖`0(1)‖L + λ− 1) + c+ ‖q∗‖L
‖`1(1)‖L + h1(1)
.
On the other hand from the (45) we have
M(1− h0(1)− λ‖`0(1)‖L) ≤ m(1− h0(1)) + λ(c+ ‖q∗‖L). (48)
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Hence, on account of (14), the relation
‖u‖C = M ≤ r3(c+ ‖q∗‖L)(λ‖`1(1)‖L + λh1(1) + 1− h0(1)),
is satisfied, where
r3 =
[
(1− h0(1))(‖`1(1)‖L + h1(1)− h0(1)− λ+ 1− ‖`0(1)‖L)×
× (−λ‖`0(1)‖L(h1(1) + ‖`1(1)‖L))
]−1
> 0,
and thus the estimate (27) holds, where r is defined by
r = r3(λ‖`1(1)‖L + λh1(1) + 1− h0(1)).
Case (d): λ > 1 and λh1(1)‖`0(1)‖L ≥ (1 − h(1))(1 − h0(1)). As above, (45) yields (48). If
we use estimate (42) in (46), according to (36), we obtain
m ≤ M(h0(1) + λ‖`1(1)‖L + λ− 1) + c+ λ‖q∗‖L
λ‖`1(1)‖L + h1(1)
.
From the last inequality and (48), in view of (14), it follows that
‖u‖C = M ≤ r4λ(c+ ‖q∗‖L)(λ‖`1(1)‖L + h1(1) + 1− h0(1)),
where
r4 =
[
(1− h0(1)− λ‖`0(1)‖L)(λ‖`1(1)‖L + h1(1))×
× (−(λ‖`0(1)‖L + λ− 1 + h0(1))(1− h0(1)))
]−1
> 0,
and thus the estimate (27) holds, where r is defined by
r = r4λ(λ‖`1(1)‖L + h1(1) + 1− h0(1)).
Therefore, in all cases (a) – (d), the estimate (27) holds.
Now suppose that u changes its sign. Put
M = max{u(t) : t ∈ [a, b]}, m = −min{u(t) : t ∈ [a, b]} (49)
and choose tM , tm ∈ [a, b] such that
u(tM ) = M, u(tm) = −m. (50)
Obviously, M > 0, m > 0, and either
tm < tM (51)
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or
tm > tM . (52)
First suppose that (51) holds. It is clear that there exists α2 ∈ ]tm, tM [ such that
u(t) > 0 for α2 < t ≤ tM , u(α2) = 0. (53)
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. (54)
Put
α3 =
{
b, if u(b) ≥ 0,
inf{t ∈ ]tM , b] : u(s) < 0 for t ≤ s ≤ b}, if u(b) < 0.
It is clear that if α3 < b then
u(t) < 0 for α3 < t ≤ b, u(α3) = 0. (55)
The integration of (32) from α1 to tm, from α2 to tM , and from α3 to b, in view of (10), (35),
(49), (50), and (53) – (55), yields
u(α1) +m ≤ m
tm∫
α1
`0(1)(s) ds+M
tm∫
α1
`1(1)(s) ds+
tm∫
α1
q∗(s) ds, (56)
M ≤ M
tM∫
α2
`0(1)(s) ds+m
tM∫
α2
`0(1)(s) ds+
tM∫
α2
q∗(s) ds, (57)
u(α3)− u(b) ≤ m
b∫
α3
`0(1)(s) ds+M
b∫
α3
`1(1)(s) ds+
b∫
α3
q∗(s) ds. (58)
If u(b) ≥ 0 or u(a) ≥ 0 then, according to (34), (49), (54), and the assumption λ > 0, we
obtain
u(α1) ≥ −c−mh0(1)−Mh1(1),
and thus from (56) we find
−c−mh0(1)−Mh1(1) +m ≤
≤ max {1, λ}
m tm∫
α1
`0(1)(s) ds+M
tm∫
α1
`1(1)(s) ds+
tm∫
α1
q∗(s) ds
.
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378 Z. OPLUŠTIL
Hence,
min
{
1,
1
λ
}(
−c−mh0(1)−Mh1(1) +m
)
≤
≤ m
∫
J
`0(1)(s) ds+M
∫
J
`1(1)(s) ds+
∫
J
q∗(s) ds, (59)
where J = [α1, tm] ∪ [α3, b].
Let u(b) < 0 and u(a) < 0.Multiplying both sides of (58) by λ and taking (55) into account,
we get
−λu(b) ≤ λ
m b∫
α3
`0(1)(s) ds+M
b∫
α3
`1(1)(s) ds+
b∫
α3
q∗(s) ds
.
Summing the last inequality and (56), according to (34), (49), and (54), we can verify that the
inequality (59) is fulfilled, where J = [α1, tm] ∪ [α3, b].
From (57) and (59) we get
M(1− C1) ≤ mA1 + ‖q∗‖L + c, (60)
m(α(λ, h0)−D1) ≤ M
(
B1 + h1(1)min
{
1,
1
λ
})
+ ‖q∗‖L + c, (61)
where
A1 =
tM∫
α2
`1(1)(s) ds, B1 =
∫
J
`1(1)(s) ds
and
C1 =
tM∫
α2
`0(1)(s) ds, D1 =
∫
J
`0(1)(s) ds.
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Due to (13), it is clear that C1 < 1 and D1 < α(λ, h0). Consequently, (60) and (61) imply
0 < M(1− C1)(α(λ, h0)−D1) ≤
≤ A1
[
M
(
B1 + min
{
1,
1
λ
}
h1(1)
)
+ ‖q∗‖L + c
]
+ (‖q∗‖L + c)(α(λ, h0)−D1) ≤
≤ MA1
(
B1 + min
{
1,
1
λ
}
h1(1)
)
+ (‖q∗‖L + c)(1 + ‖`1(1)‖L),
(62)
0 < m(α(λ, h0)−D1)(1− C1) ≤
≤
(
B1 + min
{
1,
1
λ
}
h1(1)
)
(mA1 + ‖q∗‖L + c) + c+ ‖q∗‖L ≤
≤ mA1
(
B1 + min
{
1,
1
λ
}
h1(1)
)
+ (‖q∗‖L + c)(1 + h1(1) + ‖`1(1)‖L).
Obviously,
(1− C1)(α(λ, h0)−D1) ≥ α(λ, h0)− (C1 +D1) ≥ α(λ, h0)− ‖`0(1)‖L > 0
and
4A1
(
B1 + min
{
1,
1
λ
}
h1(1)
)
≤
(
A1 +B1 + min
{
1,
1
λ
}
h1(1)
)2
≤
≤
(
‖`1(1)‖L + min
{
1,
1
λ
}
h1(1)
)2
.
By the last inequalities and the second inequality in (14), from (62) we get
M ≤ r5(‖`1(1)‖L + 1)(‖q∗‖L + c),
m ≤ r5(‖`1(1)‖L + 1 + h1(1))(‖q∗‖L + c),
where
r5 =
(
α(λ, h0)− ‖`0(1)‖L −
(‖`1(1)‖L + min{1, 1
λ}h1(1))2
4
)−1
.
Consequently, the estimate (27) holds, where r is given by
r = r5(‖`1(1)‖L + 1 + h1(1)).
If (52) holds, the validity of the estimate (27) can be proved analogously.
The lemma is proved.
Lemma 5. Let the functional h be defined by the formula (4), where λ > 0 and h0, h1 ∈
∈ PFab satisfy the conditions (7) and (8). Let, moreover, the operator ` admit the representation
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380 Z. OPLUŠTIL
` = `0−`1, where `0 and `1 are such that the conditions (10), (15), and (16) hold. Then ` belongs
to the set U2(h).
Proof. Let c ∈ R+, q
∗ ∈ L([a, b]; R+), and u ∈ C̃([a, b]; R) satisfy (25) and (26) with i = 2,
i.e.,
h(u)sgnu(b) ≥ −c
and (
u′(t)− `(u)(t)
)
sgnu(t) ≥ −q∗(t) for a.e. t ∈ [a, b].
We shall show that (27) holds, where r depends only on ‖`0(1)‖L, ‖`1(1)‖L λ, h0(1), and h1(1).
Obviously, u satisfies (32), where q̃ is defined by (33). Clearly,
−q̃(t) sgnu(t) ≤ q∗(t) for a.e. t ∈ [a, b] (63)
and
(−u(a) + λu(b) + h0(u)− h1(u)) sgnu(b) ≤ c. (64)
First suppose that u does not change its sign. Define numbers M and m by (36) and choose
t1, t2 ∈ [a, b] such that t1 6= t2 and (37) is fulfilled. Obviously, M ≥ 0, m ≥ 0, and either (38)
or (39) holds. Moreover, according to (10), (36), and (63), from (32) we get
−|u(t)|′ ≤ M `1(1)(t)−m`0(1)(t) + q∗(t) for a.e. t ∈ [a, b]. (65)
If u(b) = 0 then m = 0 and the integration of (32) from t1 to b, on account of (37), yields
M ≤ M
b∫
t1
`1(1)(s) ds+
b∫
t1
q∗(s) ds.
From the last inequality, in view of (15), it follows that
‖u‖C = M ≤ r0(‖q∗‖+ c),
where
r0 = [1− ‖`1(1)‖L]−1 .
Consequently, the estimate (27) holds with r = r0.
If u(b) 6= 0 then, according to (64), we obtain
λ|u(b)| − |u(a)| ≤ h1(|u|)− h0(|u|) + c, (66)
and thus
|u(b)| − |u(a)| ≤ h1(|u|)− h0(|u|) + |u(b)|(1− λ) + c (67)
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and
λ (|u(b)| − |u(a)|) ≤ h1(|u|)− h0(|u|) + |u(a)|(1− λ) + c. (68)
Let first (39) hold. The integration of (65) from a to t2 and from t1 to b, in view of (10) and
(37), results in
|u(a)| −m ≤ M
t2∫
a
`1(1)(s) ds+
t2∫
a
q∗(s) ds,
and
M − |u(b)| ≤ M
b∫
t1
`1(1)(s) ds+
b∫
t1
q∗(s) ds. (69)
Multiplying both sides of (69) by λ and summing the last inequalities, in view of (10), (36), and
(66), we get
M (λ− h1(1)−max{1, λ}‖`1(1)‖L) ≤ m(1− h0(1)) + c+ max{1, λ}‖q∗‖L.
Hence, by virtue of (5) and (6), we obtain
M(β(λ, h1)− ‖`1(1)‖L) ≤ mα(λ, h0) + c+ ‖q∗‖L. (70)
If (38) is fulfilled then the integration of (65) from t1 to t2, on account of (5), (6), and (7),
yields
Mβ(λ, h1)−mα(λ, h0) ≤ (M −m)(λ− h1(1))min
{
1,
1
λ
}
≤
≤ M −m ≤ M‖`1(1)‖L + ‖q∗‖L.
Therefore, in both cases (38) and (39), the inequality (70) holds.
On the other hand, the integration of (65) from a to b implies
|u(a)| − |u(b)| ≤ M‖`1(1)‖L −m‖`1(1)‖L + ‖q∗‖L. (71)
Hence, using (36), (67), and (68) in (71) we get
m ≤ M(1− β(λ, h1) + ‖`1(1)‖L) + ‖q∗‖L + c
1 + ‖`0(1)‖L − α(λ, h0)
.
From the last inequality and (70) we obtain
M(β(λ, h1)− ‖`1(1)‖L) ≤
≤ M(1− β(λ, h1) + ‖`1(1)‖L) + ‖q∗‖L + c
1 + ‖`0(1)‖L − α(λ, h0)
α(λ, h0) + ‖q∗‖L + c.
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Thus, on account of (15) and the first inequality in (16), we have
‖u‖C = M ≤ r1 (c+ ‖q∗‖L)(1 + ‖`0(1)‖L),
where
r1 = [(β(λ, h1)− ‖`1(1)‖L)(1 + ‖`0(1)‖L)− α(λ, h0)]
−1 > 0.
Therefore, the estimate (27) holds, where r is defined by
r = r1(1 + ‖`0(1)‖L).
Now suppose that u changes its sign. Define numbersM andm by (49) and choose tM , tm ∈
∈ [a, b] such that (50) is fulfilled. Obviously, M > 0, m > 0, and either (51) or (52) holds.
First suppose that (51) is fulfilled. It is clear that there exists α1 ∈ ]tm, tM [ such that
u(t) < 0 for tm ≤ t < α1, u(α1) = 0. (72)
Let
α2 = sup{t ∈ [a, b] : u(s) > 0 for tM ≤ s ≤ t}.
Obviously,
u(t) > 0 for tM ≤ t < α2 and u(α2) = 0 if α2 < b. (73)
Put
α3 =
{
a, if u(a) ≤ 0,
sup{t ∈ [a, tm] : u(s) > 0 for a ≤ s ≤ t}, if u(a) > 0.
It is clear that if α3 > a then
u(t) > 0 for a ≤ t < α3, u(α3) = 0. (74)
The integration of (32) from tm to α1, from tM to α2, and from a to α3, in view of (10), (49),
(63) and (72) – (74), yields
m ≤ M
α1∫
tm
`0(1)(s) ds+m
α1∫
tm
`1(1)(s) ds+
α1∫
tm
q∗(s) ds, (75)
M − u(α2) ≤ M
α2∫
tM
`1(1)(s) ds+m
α2∫
tM
`0(1)(s) ds+
α2∫
tM
q∗(s) ds, (76)
u(a)− u(α3) ≤ M
α3∫
a
`1(1)(s) ds+m
α3∫
a
`0(1)(s) ds+
α3∫
a
q∗(s) ds. (77)
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If u(b) ≤ 0 or u(a) ≤ 0 then, according to (49), (64), and (73), we obtain
λu(α2) ≤ mh0(1) +Mh1(1) + c. (78)
Multiplying both sides of (76) by λ and taking into account (10) and (78), we get
λM − c−mh0(1)−Mh1(1) ≤
≤ λ
M α2∫
tM
`1(1)(s) ds+m
α2∫
tM
`0(1)(s) ds+
α2∫
tM
q∗(s) ds
, (79)
and thus
min
{
1,
1
λ
}
(λM − c−mh0(1)−Mh1(1)) ≤
≤ M
∫
I
`1(1)(s) ds+m
∫
I
`0(1)(s) ds+
∫
I
q∗(s) ds, (80)
where I = [a, α3] ∪ [tM , α2].
If u(b) > 0 and u(a) > 0 then multiplying both sides of (76) by λ we get
λM − λu(α2) ≤ λ
M α2∫
tM
`1(1)(s) ds+m
α2∫
tM
`0(1)(s) ds+
α2∫
tM
q∗(s) ds
.
Summing the last inequality and (77), according to (49), (64), (73), and (74), we obtain that the
inequality (80) with I = [a, α3] ∪ [tM , α2] holds.
From (75) and (80) we get
m(1−A1) ≤ MC1 + ‖q∗‖L + c, (81)
M(β(λ, h1)−B1) ≤ m
(
min
{
1,
1
λ
}
h0(1) +D1
)
+ ‖q∗‖L + c, (82)
where
A1 =
α1∫
tm
`1(1)(s) ds, B1 =
∫
I
`1(1)(s) ds,
and
C1 =
α1∫
tm
`0(1)(s) ds, D1 =
∫
I
`0(1)(s) ds.
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384 Z. OPLUŠTIL
Due to (15), it is clear that A1 < 1 and B1 < β(λ, h1). Consequently, (81) and (82) imply
0 < m(1−A1)(β(λ, h1)−B1) ≤
≤ mC1
(
min
{
1,
1
λ
}
h0(1) +D1
)
+ (‖q∗‖L + c)(1 + ‖`0(1)‖L),
(83)
0 < M(1−A1)(β(λ, h1)−B1) ≤
≤ MC1
(
min
{
1,
1
λ
}
h0(1) +D1
)
+ (‖q∗‖L + c)(2 + ‖`0(1)‖L).
Obviously,
(1−A1)(β(λ, h1)−B1) ≥ β(λ, h1)−A1β(λ, h1)−B1 ≥
≥ β(λ, h1)− (A1 +B1) ≥ β(λ, h1)− ‖`1(1)‖L > 0
and
4C1
(
min
{
1,
1
λ
}
h0(1) +D1
)
≤
(
min
{
1,
1
λ
}
h0(1) + C1 +D1
)2
≤
≤
(
min
{
1,
1
λ
}
h0(1) + ‖`0(1)‖L
)2
.
Hence, from the second inequality in (16) and (83) we obtain
M ≤ r2(‖`0(1)‖L + 2)(c+ ‖q∗‖L),
m ≤ r2(‖`0(1)‖L + 1)(c+ ‖q∗‖L),
where
r2 =
[
β(λ, h1)− ‖`1(1)‖L −
(
‖`0(1)‖L + min
{
1, 1
λ
}
h0(1)
)2
4
]−1
.
Consequently, the estimate (27) holds, where r is defined by
r = r2(‖`0(1)‖L + 2)(c+ ‖q∗‖L).
If (52) holds, the validity of the estimate (27) can be proved analogously.
The lemma is proved.
5. Proofs. Proof of Theorems 1 and 3. The validity of theorems follows from Lemmas 2, 3,
and 4.
Proof of Theorem 2 and 4. The assertion of theorems can be derived from Lemmas 2, 3,
and 5.
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 3
NEW SOLVABILITY CONDITIONS FOR A NONLOCAL BOUNDARY-VALUE PROBLEM FOR NONLINEAR . . . 385
Proof of Corollary 1. It is clear that (18), (19) is a particular case of (1), (2) in which a = 0,
b = 1, and F, h, and ϕ are defined by the formulae
F (z)(t) df= d cos(2πt)
1∫
0
z(τ(s))√
s
ds− g1(t)z(t)ez
2(ω(t)) + g2(t)|z(t)|ν
for a.e. t ∈ [0, 1] and all z ∈ C([0, 1]; R) and
h(z) df= z(0)− 1
2
z(1)− k
1∫
0
sin(2πs)z(s) ds, ϕ(z) df= −z(0)ez(1/4) + arctan z(1/2)
for z ∈ C([0, 1]; R), respectively. Moreover, the above-defined functional h admits the repre-
sentation (4), where λ = 1/2 and
h0(z)
df= k
1∫
0
max
{
sin(2πs), 0
}
z(s) ds, h1(z)
df= k
1∫
0
max
{
− sin(2πs), 0
}
z(s) ds
for z ∈ C([0, 1]; R).
Now we put
`0(z)(t)
df= d max
{
cos(2πt), 0
} 1∫
0
z(τ(s))√
s
ds,
`1(z)(t)
df= d max
{
− cos(2πt), 0
} 1∫
0
z(τ(s))√
s
ds
for a.e. t ∈ [0, 1] and all z ∈ C([0, 1]; R).
It is easy to verify that `0, `1 ∈ Pab, h0, h1 ∈ PFab, and
‖`0(1)‖L = ‖`1(1)‖L =
2d
π
, h0(1) = h1(1) =
k
π
.
Therefore, in view of the assumptions (20) and (21), the conditions (7), (8), (13), and (14) of
Theorem 1 are fulfilled. On the other hand, operators F and ϕ satisfy the assumptions (H1)
and (H2), the relation (9) with c = π/2 holds, and the inequality (11) is satisfied on the set
C([0, 1]; R), where q(t, x) = |g2(t)|xν for a.e. t ∈ [0, 1] and all x ∈ R+.
Applying Theorem 1, we establish solvability of the problem (18), (19).
1. Azbelev N. V., Maksimov V. P., Rakhmatullina L. F. Introduction to the theory of functional differential equa-
tions (in Russian). — Moscow: Nauka, 1991.
2. Hakl R., Kiguradze I., Půža B. Upper and lower solutions of boundary-value problems for functional differ-
ential equations and theorems on functional differential inequalities // Georg. Math. — 2000. — 7, №. 3. —
P. 489 – 512.
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 3
386 Z. OPLUŠTIL
3. Hakl R., Lomtatidze A., Šremr J. On a periodic type boundary-value problem for first order nonlinear func-
tional differential equations // Nonlinear Anal. — 2002. — 51. — P. 425 – 447.
4. Hakl R., Lomtatidze A., Šremr J. Some boundary-value problems for first order scalar functional differential
equations // Folia Fac. Sci. Natur. Univ. Masar. Brunensis, Brno. — 2002.
5. Hale J. Theory of functional differential equations. — New York etc.: Springer, 1977.
6. Kiguradze I., Půža B. On boundary-value problems for functional differential equations // Mem. Different.
Equat. Math. Phys. — 1997. — 12. — P. 106 – 113.
7. Kiguradze I., Sokhadze Z. On the global solvability of the Cauchy problem for singular functional differential
equations // Georg. Math. J. — 1997. — 4, № 4. — P. 355 – 373.
8. Kiguradze I., Sokhadze Z. On the uniqueness of a solution to the Cauchy problem for functional differential
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10. Kolmanovskii V., Myshkis A. Introduction to the theory and applications of functional differential equations.
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Received 19.11.07
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 3
|
| id | nasplib_isofts_kiev_ua-123456789-178203 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-3076 |
| language | English |
| last_indexed | 2025-11-28T11:17:51Z |
| publishDate | 2008 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Oplustil, Z. 2021-02-18T08:15:55Z 2021-02-18T08:15:55Z 2008 New solvability conditions for a non-local boundary value problem for nonlinear functional differential equations / Z. Oplustil // Нелінійні коливання. — 2008. — Т. 11, № 3. — С. 365-386. — Бібліогр.: 12 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/178203 517.9 New efficient conditions are obtained sufficient for the solvability as well as unique solvability of a nonlocal boundary-value problem for nonlinear functional differential equations. Отримано новi ефективнi умови розв’язностi, а також єдиної розв’язностi нелокальних граничних задач для нелiнiйних функцiонально-диференцiальних рiвнянь. Published results were acquired using the subsidization of the Ministry of Education, Youth and Sports of the Czech Republic, research plan MSM 0021630518 “Simulation modelling of mechatronic systems”. en Інститут математики НАН України Нелінійні коливання New solvability conditions for a non-local boundary value problem for nonlinear functional differential equations Новi умови розв’язностi нелокальних граничних задач для нелiнiйних функцiонально-диференцiальних рiвнянь Article published earlier |
| spellingShingle | New solvability conditions for a non-local boundary value problem for nonlinear functional differential equations Oplustil, Z. |
| title | New solvability conditions for a non-local boundary value problem for nonlinear functional differential equations |
| title_alt | Новi умови розв’язностi нелокальних граничних задач для нелiнiйних функцiонально-диференцiальних рiвнянь |
| title_full | New solvability conditions for a non-local boundary value problem for nonlinear functional differential equations |
| title_fullStr | New solvability conditions for a non-local boundary value problem for nonlinear functional differential equations |
| title_full_unstemmed | New solvability conditions for a non-local boundary value problem for nonlinear functional differential equations |
| title_short | New solvability conditions for a non-local boundary value problem for nonlinear functional differential equations |
| title_sort | new solvability conditions for a non-local boundary value problem for nonlinear functional differential equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/178203 |
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