Control of hyperbolic equations
The new analogs of Wendroff’s type inequalities for discontinuous functions are considered. The impulse influence on the behaviour of the solutions of hyperbolic equations with nonlinearities of the Lipschitz and H¨older types is investigated. Розглянуто новi аналоги нерiвностей Вендрофа для розри...
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| Cite this: | Control of hyperbolic equations / S.D. Borysenko, M. Ciarletta, G. Iovane, A.M. Piccirillo, V. Zampoli // Доповiдi Нацiональної академiї наук України. — 2013. — № 4. — С. 43–52. — Бібліогр.: 15 назв. — англ. |
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| author | Borysenko, S.D. Ciarletta, M. Iovane, G. Piccirillo, A.M. Zampoli, V. |
| author_facet | Borysenko, S.D. Ciarletta, M. Iovane, G. Piccirillo, A.M. Zampoli, V. |
| citation_txt | Control of hyperbolic equations / S.D. Borysenko, M. Ciarletta, G. Iovane, A.M. Piccirillo, V. Zampoli // Доповiдi Нацiональної академiї наук України. — 2013. — № 4. — С. 43–52. — Бібліогр.: 15 назв. — англ. |
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| description | The new analogs of Wendroff’s type inequalities for discontinuous functions are considered. The
impulse influence on the behaviour of the solutions of hyperbolic equations with nonlinearities
of the Lipschitz and H¨older types is investigated.
Розглянуто новi аналоги нерiвностей Вендрофа для розривних функцiй. Дослiджено вплив
iмпульсного збурення на поведiнку розв’язкiв гiперболiчних рiвнянь з нелiнiйностями як
лiпшицевого так i гельдерового характеру.
Рассмотрены новые аналоги неравенств Вендрофа для разрывных функций. Исследовано воздействие импульсных возмущений на поведение решений гиперболических уравнений с нелинейностями как липшицевого так и гельдерового характера.
|
| first_indexed | 2025-12-07T16:25:31Z |
| format | Article |
| fulltext |
UDC 517.9
S. D. Borysenko, M. Ciarletta, G. Iovane, A.M. Piccirillo, V. Zampoli
Control of hyperbolic equations
(Presented by Corresponding Member of the NAS of Ukraine S. I. Lyashko)
The new analogs of Wendroff’s type inequalities for discontinuous functions are considered. The
impulse influence on the behaviour of the solutions of hyperbolic equations with nonlinearities
of the Lipschitz and Hölder types is investigated.
Introduction. In the present article we found new analogs of the Wendroff inequality for dis-
continuous functions with finite jumps on some curves Γj ⊂ R
2
+ and discontinuities of the Lipschi-
tz and non-Lipschitz types. New conditions of boundedness for solutions of impulsive nonlinear
hyperbolic equations are obtained.
Our paper is devoted to a generalization of results [1–15], and it is based on new analogs
of a Wendroff type inequality.
We consider some set D∗ ⊂ R
2, where D∗ = D \ Γ, D =
⋃
j
Dj , j = 1, 2 . . .; Γ =
⋃
j
Γj,
Γj = {(x, y) : ϕj(x, y) = 0, j = 1, 2 . . .}, Γk ∩ Γk+1 = ∅, k = 1, 2, . . .;
ϕj(x, y) are real-valued continuously differentiable functions such that grad ϕj(x, y) > 0, for
all j = 1, 2 . . .;
D1 = {(x, y) : x > 0, y > 0, ϕ1(x, y) < 0};
Dk = {(x, y) : x > 0, y > 0, ϕk−1(x, y) > 0, ϕk(x, y) < 0,∀ k > 2, k ∈ N};
Gp = {(u, v) : (x, y) ∈ Dp, 0 6 u 6 x, 0 6 v 6 y, p ∈ N}; µϕn
is the Lebesgue–Stieltjes
measure concentrated on the curves {Γn}.
Let us consider a real-valued nonnegative, discontinuous, nondecreasing function u(x, y)
in D∗, which has finite jumps on the curves {Γj}.
Previous results. Lipschitz type discontinuities.
Proposition 1. Let us suppose that a function u(x1, x2) satisfies the following integro-sum
inequality in D∗:
u(x1, x2) 6 q(x1, x2) + g(x1, x2)
∫∫
Gn
ψ(τ, s)W [u(τ, s)] dτds +
+
n−1
∑
j=1
∫
Γj∩Gn
βj(x, y)u(x1, x2)dµϕj
, (1)
where q(x1, x2) is positive and nondecreasing, g(x1, x2) > 1, βj(x1, x2) > 0, ψ(τ, s) > 0; the
function W belongs to the class Φ1 of functions such that:
1. W (σ1σ2) 6W (σ1)W (σ2) ∀σ1, σ2 > 0;
2. W : [0,∞[→ [0,∞[, W (0) = 0;
3. W is nondecreasing.
© S.D. Borysenko, M. Ciarletta, G. Iovane, A.M. Piccirillo, V. Zampoli, 2013
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2013, №4 43
Moreover, u(x1, x2) is a nonnegative discontinuous function, which has finite jumps on the
curves {Γj}, j = 1, 2, . . ..
Then for arbitrary {0 < x1 < ∞, 0 < x2 < ∞}, the following estimate is fulfilled:
u(x1, x2) 6 q(x1, x2)g(x1, x2)Ψ
−1
i
{
∫∫
Di
ψ(τ, s)
q(τ, s)
W [q(τ, s)g(τ, s)] dτds
}
, (2)
∀x ∈ Di :
∫∫
Di
ψ(τ, s)
q(τ, s)
W [q(τ, s)g(τ, s)] dτds ∈ Dom(Ψ−1
i ),
Ψ0(V )
def
=
V
∫
1
dσ
W (σ)
, Ψi(V )
def
=
V
∫
Ci
dσ
W (σ)
, i = 1, 2,
where
V = (V1, V2), σ = (σ1, σ2)
and
Ci =
(
1 +
∫
Γi∩Gn
βj(x1, x2)g(x1, x2)duϕi
)
Ψ−1
i
{
∫∫
Gi+1\Gi
ψ(τ, s)
q(τ, s)
W [q(τ, s)g(τ, s)] dτds
}
.
Proposition 2. Let us suppose that a nonnegative discontinuous function u(x1, x2), which
has finite jumps on the curves {Γj}, satisfies inequality (1), where the functions W belongs to
the class Φ1 of functions such that:
1. W : [0,∞[→ [0,∞[ is continuous and nondecreasing;
2. ∀ t > 0, u > 0, t−1(W (u)) 6W (t−1u);
3. W (0) = 0.
If all functions q, g, ψ, βj satisfy the conditions of Proposition 1, then, for arbitrary {0 6
6 x1 6 x∗1, 0 6 x2 6 x∗2} the following inequality is justified:
u(x1, x2) 6 q(x1, x2)g(x1, x2)Ψ
−1
i
{
∫∫
Di
ψ(τ, s)g(τ, s) dτds
}
, i = 0, 1, . . . ,
where
Ψ0(V ) =
V
∫
1
dσ
W (σ)
, Ψi(V ) =
V
∫
Ci
dσ
W (σ)
, i = 1, 2, . . . ,
Ci =
(
1 +
∫
Γi∩Gn
βj(x1, x2)g(x1, x2) dµϕi
)
Ψ
−1
i−1
{
∫∫
Di
ψ(τ, s)g(τ, s) dτds
}
,
44 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2013, №4
(x∗1, x
∗
2) = x∗ = sup
x
{
x :
∫∫
Gi+1\Gi
ψ(τ, s)g(τ, s)dτds ∈ Dom(Ψ
−1
i (V )), i = 1, 2, . . .
}
.
Non-Lipschitz type discontinuities. Let us consider the following inequality:
u(x1, x2) 6 ϕ(x1, x2) +
∫∫
Gn
f(σ1, σ2)u
α(σ1, σ2) dσ1dσ2 +
+
n−1
∑
j=1
∫
Γj
⋂
Gn
βj(x1, x2)u
m(x1, x2) dµϕj
. (3)
Proposition 3. If the function u(x1, x2) satisfies inequality (3) with f > 0, βj > 0, α = 1,
m > 0, then the following estimates are valid:
u(x1, x2) 6 ϕ(x1, x2)
∞
∏
j=1
∫
Γj∩Gj+1
ϕm−1(x1, x2)βj(x1, x2) dµϕj
×
× exp
[ x1
∫
0
x2
∫
0
f(σ1, σ2) dσ1dσ2
]
, if 0 < m 6 1, (4)
u(x1, x2) 6 ϕ(x1, x2)
∞
∏
j=1
(
1 +
∫
Γj∩Gj+1
ϕm−1(x1, x2)βj(x1, x2) dµϕj
)
×
× exp
[
m
x1
∫
0
x2
∫
0
f(σ1, σ2) dσ1, dσ2
]
, if m > 1. (5)
Proposition 4. If the function u(x1, x2) satisfies inequality (3) with α = m > 0, m 6= 1 and
the conditions of the above theorem are valid, then the following estimates hold:
u(x1, x2) 6 ϕ(x1, x2)
∞
∏
j=1
(
1 +
∫
Γj∩Gj+1
βj(x1, x2)ϕ
m−1(x1, x2) dµϕj
)
×
×
[
1 + (1−m)
x1
∫
0
x2
∫
0
ϕm−1(σ1, σ2)f(σ1, σ2) dσ1dσ2
]1/(1−m)
for 0 < m < 1; (6)
u(x1, x2) 6 ϕ(x1, x2)
∞
∏
j=1
(
1 +m
∫
Γj∩Gj+1
βj(x1, x2)ϕ
m−1(x1, x2) dµϕj
)
×
×
[
1− (m− 1)
[
∞
∏
j=1
(
1 +m
∫
Γj∩Gj+1
βj(x1, x2)ϕ
m−1(x1, x2) dµϕj
)]m−1
×
×
x1
∫
0
x2
∫
0
ϕm−1(σ1, σ2)f(σ1, σ2) dσ1dσ2
]1/(1−m)
for m > 1 (7)
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2013, №4 45
such that
x1
∫
0
x2
∫
0
ϕm−1(σ1, σ2)f(σ1, σ2) dσ1dσ2 6
1
m
,
∞
∏
j=1
(
1 +m
∫
Γj∩Gj+1
βj(x1, x2)ϕ
m−1(x1, x2) dµϕj
)
<
(
1 +
1
m− 1
)1/(1−m)
.
(8)
Proposition 5. Let us suppose that a function u(x1, x2) satisfies the inequality
u(x1, x2) 6 ϕ(x1, x2) + g(x1, x2)
∫∫
Gn
ψ(τ, s)um(τ, s) dτds +
+
n−1
∑
j=1
∫
Γj∩Gn
βj(x1, x2)u
n(x1, x2) dµϕj
, (9)
where ϕ is a positive and nondecreasing function, g(x1, x2) > 1, βj(x1, x2) > 0, ψ(τ, s) > 0; the
function u(x1, x2) is nonnegative and has finite jumps on the curves Γj , j = 1, 2, . . .; m,n > 0.
With these conditions, the following estimates take place:
u(x1, x2) 6 ϕ(x1, x2)g(x1, x2)(Ψ
−1
i
∫∫
Di
ψ(τ, s)ϕm−1(τ, s)gm(τ, s)dτds) (10)
∀ (x1, x2) ∈ Di :
∫∫
Di
ψ(τ, s)g(τ, s) dτds ∈ Dom(Ψ−1
i ),
Ψi(V ) =
V
∫
Ci
σ−mdσ, C0 = 1,
Ci =
(
1 +
∫
Γi∩Gi+1
βi(x1, x2)g
n(x1, x2)ϕ
n−1(x1, x2)dµϕi
)
×
×Ψ−1
i−1
(
∫∫
Gi+1\Gi
ψ(τ, s)ϕm−1(τ, s)gm(τ, s) dτds
)
, i = 1, 2, . . . ,
where m = 1;
Ψ−1
i (V ) = Ci expV, i = 1, 2, . . . ;
if 0 < m < 1 ∀ (x1, x2) ∈ Di:
Ψ−1
i (V ) = (Ci + (1−m)V )
1
1−m , i = 1, 2, . . . ;
if m > 1,
Ψ−1
i (V ) = [Ci − (m− 1)V ]−
1
m−1 , i = 1, 2, . . . ,
46 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2013, №4
∀ (x1, x2) ∈ Di :
∫∫
Di
ψ(τ, s)ϕm−1(τ, s)gm(τ, s) dτds <
Ci
m− 1
.
Applications. Lipschitz type discontinuities. Let us suppose that the evolution of
some real processes may be described by hyperbolic partial differential equations with impulse
perturbations concentrated on the surfaces
∂2u(x1, x2)
∂x1∂x2
= H(x, u(x)), (x1, x2) ∈ Γi,
u(x1, 0) = φ1(x1),
u(0, x2) = φ2(x2),
φ1(0) = φ2(0),
∆u
∣
∣
(x1,x2)∈Γi
=
∫
Γi∩Gn
βi(x1, x2)u(x1, x2)dµφi
.
(11)
Here ∆u
∣
∣
(x1,x2)∈Γi
are the characteristic values of finite jumps u(x)(x = (x1, x2)), when the
solution of (11) meets the hypersurfaces Γi : u(x)
⋂
Γi.
We investigate equation (11) in the domain D∗ ⊂ R
2
+, which was described in Introduction.
Denote, by φ(x1, x2) the boundary conditions in (11). Every solution of (11), satisfying the
boundary conditions, is also a solution of the Volterra integro-sum equation:
u(x1, x2) = φ(x1, x2) +
∫∫
Gn
H(τ, s, u(τ, s)) dτds +
n−1
∑
j=1
∫
Γj∩Gn
βj(x1, x2)u(x1, x2) dµϕj
. (12)
Let us suppose that
|H(τ, s, u(τ, s))| 6 ψ(τ, s)W [|u(τ, s)|], (13)
where ψ(τ, s) > 0, W (σ) ∈ Φ1.
By using the result of Proposition 1, we obtain the following statement:
Proposition 6. If H(x, u(x)) in (11) satisfies condition (13), then, for all solutions of equa-
tion (13), the following inequality is valid for all x1 > 0, x2 > 0
|u(x1, x2)| 6 |φ(x1, x2)|Ψ
−1
i
{
∫∫
Di
ψ(τ, s)
|φ(τ, s)|
W [|φ(τ, s)|] dτds
}
(14)
∀x ∈ Di :
∫∫
Di
ψ(τ, s)
|φ(τ, s)|
W [‖φ(τ, s)|] dτds ∈ Dom(Ψ−1
i ),
where
Ψ0(V1) =
V1
∫
1
dσ
W (σ)
, Ψi(V1) =
V1
∫
Ci
dσ1
W (σ1)
i = 1, 2, . . .
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2013, №4 47
Ci =
(
1 +
∫
Γi∩Gi+1
‖βi(x1, x2)| dµφi
)
Ψ−1
i−1
(
∫∫
Gi+1\Gi
ψ(τ, s)
‖φ(τ, s)|
W [‖φ(τ, s)|] dτds
)
.
By using the result of Proposition 2, we obtain:
Proposition 7. If the function H satisfies (13), where W belongs to the class of functions
Φ1 : W ∈ Φ1, then all solutions of equation (11) satisfy such estimate:
‖u(x1, x2)| 6 ‖φ(x1, x2)|Ψ
−1
j
(
∫∫
Dj
ψ(τ, s)dτds
)
, ∀ j = 0, 1, . . . ,
where
Ψ0(V ) =
V
∫
1
dσ
W (σ)
, Ψi(V ) =
V1
∫
C1
dσ
W (σ)
, i = 1, 2, . . . ,
Ci =
(
1 +
∫
Γi∩Gi+1
‖βi(x1, x2)| dµφi
)
Ψ
−1
i−1
(
∫∫
Gi+1\Gi
ψ(τ, s) dτds
)
,
∀x : 0 < x < x∗ : x∗ = sup
x
{
x :
∫∫
Gi+1\Gi
ψ(τ, s) dτds ∈ Dom(Ψ
−1
i ), i = 1, 2, . . .
}
.
From Proposition 3, the next result follows:
Proposition 8. Let us suppose that the following conditions take place:
A) ‖H(x1, x2, u(x1, x2))| 6 f(x1, x2)‖u(x1, x2)|
α, α = const > 0, where f is a continuous
nonnegative function in R
2
+.
B) ∃M = const > 0: ‖φ(x1, x2)| 6 M . Then, for solutions of equations (11), the following
estimates take place:
1. ‖u(x1, x2)| 6M
∞
∏
j=1
(
1 +
∫
Γi∩Gj+1
‖βj(x1, x2)|dµφi
)
exp
[ x1
∫
0
x2
∫
0
f(σ1, σ2) dσ1dσ2
]
,
if α = 1;
2. ‖u(x1, x2)| 6M
∞
∏
j=1
(
1 +
∫
Γi∩Gj+1
‖βj(x1, x2)|dµφi
)
×
×
[
1 + (1− α)Mα−1
x1
∫
0
x2
∫
0
f(σ1, σ2)dσ1dσ2
]1/(1−α)
, if 0 < α < 1;
3. ‖u(x1, x2)| 6M
∞
∏
j=1
(
1 +
∫
Γi∩Gj+1
‖βj(x1, x2)|dµφi
){
1 + (α− 1)Mα−1 ×
48 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2013, №4
×
[
∞
∏
j=1
(
1 +
∫
Γi∩Gj+1
‖βj(x1, x2)| dµφi
)]α−1 x1
∫
0
x2
∫
0
f(σ1, σ2)dσ1dσ2
}−1/(α−1)
for α > 1 and arbitrary (x1, x2) ∈ D∗ such that
x1
∫
0
x2
∫
0
f(σ1, σ2)dσ1dσ2 <
{
(α− 1)Mα−1
[
∞
∏
j=1
(
1 +
∫
Γi∩Gj+1
‖βj(x1, x2)| dµφi
)]α−1}−1
.
From Proposition 8, we obtain the following statement:
Proposition 9. Let us consider the following conditions for equation (11):
1. ‖H(x1, x2, u(x1, x2))| 6 ψ(x1, x2)‖u(x1, x2)|
α;
2. ∃M = const > 0 : ‖ϕ(x1, x2)| 6 M ;
3. ∃ ξ, η:
∞
∏
j=1
(
1 +
∫
Γi∩Gj+1
‖βj(x1, x2)|dµϕi
)
6 ξ <∞;
x1
∫
0
x2
∫
0
ψ(σ1, σ2) dσ1dσ2 6 η <∞.
Then all solutions u(x1, x2) of equation (11) are bounded for 0 < α 6 1. If additionally
∞
∏
j=1
(
1 +
∫
Γi∩Gj+1
‖βj(x1, x2)|dµϕi
)
<
M1−α
(α− 1)η
,
all solutions of equation (11) are bounded also for α > 1.
Hölder type discontinuities. Let us consider such problem:
∂2u(x1, x2)
∂x1∂x2
= F (x, u(x)), x = (x1, x2) ∈ Γi,
u(x1, 0) = ψ1(x1),
u(0, x2) = ψ2(x2), (15)
ψ1(0) = ψ2(0),
∆u|x∈Γi
=
∫
Γi∩Gn
βi(x)u
m(x)dµφi
, m > 0.
In (15) we suppose that the boundary conditions ψ(x1, x2) are bounded, i. e.
|ψ(x1, x2)| 6M = const <∞,
and F (x, u) satisfies the estimate:
|F (x, u)| 6 f(x1, x2)|u(x1, x2)|
α, (16)
with f > 0, α = const > 0.
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2013, №4 49
By using Propositions 3 and estimates A)–D), we obtain the following statement:
Proposition 10. Let us suppose that, for problem (15), the assumptions in Introduction about
curves Γi, domains Bk, Gk and functions ϕk are valid. Moreover, let F satisfy inequality (16).
I. Then the following estimates take place:
A′) α = 1, m 6 1 ⇒
⇒ |u(x1, x2)| 6M
∞
∏
j=1
(
1 +Mm−1
∫
Γj∩Gj+1
|β∗j (x1, x2)|dµϕj
)
exp
[ x1
∫
0
x2
∫
0
f(σ1, σ2) dσ1dσ2
]
,
B′) α = 1, m > 1 ⇒
⇒ |u(x1, x2)| 6M
∞
∏
j=1
(
1 +Mm−1
∫
Γj∩Gj+1
|β∗j (x1, x2)| dµϕj
)
exp
[
m
x1
∫
0
x2
∫
0
f(σ1, σ2) dσ1dσ2
]
,
C ′) 0 < α = m < 1 ⇒ |u(x1, x2)| 6M
∞
∏
j=1
(
1 +Mm−1
∫
Γj∩Gj+1
|β∗j (x1, x2)|dµϕj
)
×
×
[
1 + (1−m)Mm−1
x1
∫
0
x2
∫
0
f(σ1, σ2)dσ1dσ2
]1/(1−m)
,
D′) α = m > 1 ⇒ |u(x1, x2)| 6M
∞
∏
j=1
(
1 +mMm−1
∫
Γj∩Gj+1
|β
(
j∗)(x1, x2)| dµϕj
)
×
×
[
1− (m− 1)Mm−1
[
∞
∏
j=1
(
1 +mMm−1
∫
Γj∩Gj+1
|β∗j (x1, x2)| dµϕj
)]m−1
×
×
x1
∫
0
x2
∫
0
f(σ1, σ2) dσ1dσ2
]−1/(m−1)
,
for all x1, x2 > 0:
x1
∫
0
x2
∫
0
f(σ1, σ2) dσ1dσ2 6
1
mMm−1
, (17)
∞
∏
j=1
(
1 +mMm−1
∫
Γj∩Gj+1
|β∗j (x1, x2)| dµϕj
)
<
(
1 +
1
m− 1
)1/(1−m)
. (18)
II. All solutions u(x1, x2) of (15) are bounded in cases A′)–C′) only if the values
∞
∏
j=1
(
1 +
∫
Γj∩Gj+1
|β∗j | dµϕj
)
,
x1
∫
0
x2
∫
0
f(σ1, σ2) dσ1dσ2
50 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2013, №4
are bounded. Referring to case D′), (17), (18) guarantee conditions of boundedness for all solutions
of (15).
1. Borysenko S.D., Piccirillo A.M. Impulsive integral inequalities: applications. – Caserta: Lunaset, 2011. –
150 p.
2. Borysenko S.D. On some generalizations of the Bellman–Bihari result for integro-functional inequalities
for discontinuous functions and their applications // The nonlinear Analysis and Applications 2009. –
Materials of the Int. sci. conf. (April 2–4, 2009, Kyiv). – Kyiv: NTUU “KPI”, 2009. – P. 83.
3. Borysenko S.D., Piccirillo A.M. Impulsive differential models: stability. – Caserta: Lunaset, 2013. – 130 p.
4. Gallo A., Piccirillo A.M. New Wendroff type inequalities for discontinuous functions and its applications //
Nonlinear Studies. – 2012. – 19, No 1. – P. 1–11.
5. Borysenko S.D., Ciarletta M., Iovane G. Integro-sum inequalities and motion stability of systems with
impulse perturbations // Nonlinear Analysis. – 62. – P. 417–428.
6. Borysenko S.D., Iovane G. Integro-sum inequalities and qualitative analysis of dynamical systems with
perturbations. – S. Severino: Elda., 2006. – 180 p.
7. Borysenko S.D., Iovane G. About some integral inequalities of Wendroff type for discontinuous functions //
Nonlinear Analysis. – 2007. – 66. – P. 2190–2203.
8. Borysenko S.D., Iovane G. Differential models: dynamical systems with perturbations. – Rome: Aracne,
2009. – 280 p.
9. Borysenko S.D., Iovane G., Giordano P. Investigations of the properties of motion for essential nonlinear
systems perturbed by impulses on some hypersufaces // Nonlinear Analysis. – 2005. – 62. – P. 345–363.
10. Piccirillo A.M., Gallo A. About new analogies of Gronwall–Bellman–Bihari type inequalities for disconti-
nuous functions and estimated solution for impulsive differential systems // Ibid. – 2007. – 67, No 5. –
P. 1550–1559.
11. Gallo A., Piccirillo A.M. About some new generalizations of Bellman–Bihari results for integro-functional
inequalities for discontinuous functions and their applications // Ibid. – 2009. – 71. – P. 2276–2288.
12. Iovane G. Some new integral inequalities of Bellman–Bihari type with delay for discontinuous functions //
Ibid. – 2007. – 66. – P. 498–508.
13. Iovane G., Borysenko S.D. Boundedness, stability, practical stability of motion of impulsive systems //
Proc. DE&CAS. – Brest, 2005. – P. 15–21.
14. Mitropolsky Yu.A., Borysenko S.D., Toscano S. Investigations of the properties of solutions of impulsive
differential systems in the linear approximation // Reports of the NAS of Ukraine. – 2007. – No 7. –
P. 36–42.
15. Mitropolsky Yu.A., Iovane G., Borysenko S.D. About a generalization of Bellman–Bihari type inequalities
for discontinuous functions and their applications // Nonlinear Analysis. – 2007. – 66. – P. 2140–2165.
Received 12.06.2012NTU of Ukraine “Kiev Polytechnical Institute”
Second University of Naples, Italy
University of Salerno, Italy
С.Д. Борисенко, М. Чарлетта, Ж. Iоване, А. М. Пiчiрiлло, В. Замполi
Iмпульснi управлiння гiперболiчними рiвняннями
Розглянуто новi аналоги нерiвностей Вендрофа для розривних функцiй. Дослiджено вплив
iмпульсного збурення на поведiнку розв’язкiв гiперболiчних рiвнянь з нелiнiйностями як
лiпшицевого так i гельдерового характеру.
ISSN 1025-6415 Доповiдi Нацiональної академiї наук України, 2013, №4 51
С.Д. Борисенко, М. Чарлетта, Ж. Иоване, А.М. Пичирилло, В. Замполи
Импульсное управление гиперболическими уравнениями
Рассмотрены новые аналоги неравенств Вендрофа для разрывных функций. Исследовано воз-
действие импульсных возмущений на поведение решений гиперболических уравнений с не-
линейностями как липшицевого так и гельдерового характера.
52 ISSN 1025-6415 Reports of the National Academy of Sciences of Ukraine, 2013, №4
|
| id | nasplib_isofts_kiev_ua-123456789-85636 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1025-6415 |
| language | English |
| last_indexed | 2025-12-07T16:25:31Z |
| publishDate | 2013 |
| publisher | Видавничий дім "Академперіодика" НАН України |
| record_format | dspace |
| spelling | Borysenko, S.D. Ciarletta, M. Iovane, G. Piccirillo, A.M. Zampoli, V. 2015-08-11T13:12:13Z 2015-08-11T13:12:13Z 2013 Control of hyperbolic equations / S.D. Borysenko, M. Ciarletta, G. Iovane, A.M. Piccirillo, V. Zampoli // Доповiдi Нацiональної академiї наук України. — 2013. — № 4. — С. 43–52. — Бібліогр.: 15 назв. — англ. 1025-6415 https://nasplib.isofts.kiev.ua/handle/123456789/85636 517.9 The new analogs of Wendroff’s type inequalities for discontinuous functions are considered. The impulse influence on the behaviour of the solutions of hyperbolic equations with nonlinearities of the Lipschitz and H¨older types is investigated. Розглянуто новi аналоги нерiвностей Вендрофа для розривних функцiй. Дослiджено вплив iмпульсного збурення на поведiнку розв’язкiв гiперболiчних рiвнянь з нелiнiйностями як лiпшицевого так i гельдерового характеру. Рассмотрены новые аналоги неравенств Вендрофа для разрывных функций. Исследовано воздействие импульсных возмущений на поведение решений гиперболических уравнений с нелинейностями как липшицевого так и гельдерового характера. en Видавничий дім "Академперіодика" НАН України Доповіді НАН України Інформатика та кібернетика Control of hyperbolic equations Iмпульснi управлiння гiперболiчними рiвняннями Импульсное управление гиперболическими уравнениями Article published earlier |
| spellingShingle | Control of hyperbolic equations Borysenko, S.D. Ciarletta, M. Iovane, G. Piccirillo, A.M. Zampoli, V. Інформатика та кібернетика |
| title | Control of hyperbolic equations |
| title_alt | Iмпульснi управлiння гiперболiчними рiвняннями Импульсное управление гиперболическими уравнениями |
| title_full | Control of hyperbolic equations |
| title_fullStr | Control of hyperbolic equations |
| title_full_unstemmed | Control of hyperbolic equations |
| title_short | Control of hyperbolic equations |
| title_sort | control of hyperbolic equations |
| topic | Інформатика та кібернетика |
| topic_facet | Інформатика та кібернетика |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/85636 |
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