Singular integral inequalities with several nonlinearities and integral equations with singular kernels
We deal with an integral inequality with a power nonlinearity in its left-hand side, with n nonlinearities in its right-hand side, and weakly singular kernels. The obtained result is an extention of the Bihari, Henry, Pachpatte and Pinto inequalities and results obtained by the author. Using these...
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Інститут математики НАН України
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| Цитувати: | Singular integral inequalities with several nonlinearities and integral equations with singular kernels / M. Medveď // Нелінійні коливання. — 2008. — Т. 11, № 1. — С. 71-80. — Бібліогр.: 21 назв. — англ. |
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| citation_txt | Singular integral inequalities with several nonlinearities and integral equations with singular kernels / M. Medveď // Нелінійні коливання. — 2008. — Т. 11, № 1. — С. 71-80. — Бібліогр.: 21 назв. — англ. |
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| description | We deal with an integral inequality with a power nonlinearity in its left-hand side, with n nonlinearities in
its right-hand side, and weakly singular kernels. The obtained result is an extention of the Bihari, Henry,
Pachpatte and Pinto inequalities and results obtained by the author. Using these results we prove sufficient
conditions for existence of global solutions of some nonlinear Volterra integral equations with singular
kernels and n nonlinearities.
Розглянуто iнтегральнi нерiвностi зi степеневою нелiнiйнiстю в лiвiй частинi, n нелiнiйностями у правiй частинi та слабкосингулярними ядрами. Отриманий результат є узагальненням
нерiвностей, одержаних Бiхарi, Генрi, Пачпатт та Пiнто, а також результатiв автора. На
основi цих результатiв встановлено достатнi умови iснування глобальних розв’язкiв деяких
нелiнiйних iнтегральних рiвнянь Вольтерри з сингулярними ядрами та n нелiнiйностями.
|
| first_indexed | 2025-12-07T15:23:56Z |
| format | Article |
| fulltext |
UDC 517 . 9
SINGULAR INTEGRAL INEQUALITIES WITH SEVERAL
NONLINEARITIES AND INTEGRAL EQUATIONS
WITH SINGULAR KERNELS*
СИНГУЛЯРНI IНТЕГРАЛЬНI НЕРIВНОСТI З КIЛЬКОМА
НЕЛIНIЙНОСТЯМИ ТА СИНГУЛЯРНИМИ ЯДРАМИ
M. Medveď
Comenius Univ.
Mlinská Dolina, 842 48 Bratislava, Slovakia
e-mail: medved@fmph.uniba.sk
We deal with an integral inequality with a power nonlinearity in its left-hand side, with n nonlinearities in
its right-hand side, and weakly singular kernels. The obtained result is an extention of the Bihari, Henry,
Pachpatte and Pinto inequalities and results obtained by the author. Using these results we prove sufficient
conditions for existence of global solutions of some nonlinear Volterra integral equations with singular
kernels and n nonlinearities.
Розглянуто iнтегральнi нерiвностi зi степеневою нелiнiйнiстю в лiвiй частинi, n нелiнiйностя-
ми у правiй частинi та слабкосингулярними ядрами. Отриманий результат є узагальненням
нерiвностей, одержаних Бiхарi, Генрi, Пачпатт та Пiнто, а також результатiв автора. На
основi цих результатiв встановлено достатнi умови iснування глобальних розв’язкiв деяких
нелiнiйних iнтегральних рiвнянь Вольтерри з сингулярними ядрами та n нелiнiйностями.
1. Introduction. In the paper [1] the integral inequality
u(t) ≤ c +
n∑
i=1
t∫
a
λi(s)ωi(u(s))ds, (1)
has been studied. The aim of this paper is to study a singular version of the inequality (1) and
to generalize the results published in the papers [2 – 5]. A desingularization method developed
in these papers was applied in the papers [3, 5 – 9] to the study of boundary-value problems,
stability problems, and the problem of the existence of global solutions of parabolic differential
equations. It is applied in the paper [10] to the study of integral inequalities of two and more
variables with singular kernels. Using this method a result on the discrete version of an integral
inequality with a singular kernel is also proved in [9]. The method is also used in the papers
[11 – 14].
* This work was supported by the Slovak Grant Agency Vega (Grant N 1/2001/05).
c© M. Medveď, 2008
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 1 71
72 M. MEDVEĎ
2. Main result. First let us consider the inequality
u(t)k ≤ a +
N∑
i=1
t∫
0
(t− s)βi−1λi(s)ωi(u(s))ds+
+
n∑
j=N+1
t∫
0
λj(s)ωj(u(s))ds, t ∈ 〈0, T 〉, (2)
where 0 < βi < 1, i = 1, 2, . . . , n, k ≥ 1, 0 < T < ∞ and a is a positive constant. The
nonsingular version of the integral inequality (2) with one nonlinearity and k = 2 has been
studied by B. G. Pachpatte in the paper [15]. We recall the following notation introduced in [1].
A ⊂ R be a set and ω1, ω2 : A → R \ {0}. If the function ω =
ω2
ω1
is nondecreasing on A, then
we write ω1 ∝ ω2. Obviously the relation ∝ is transitive and reflexive.
Lemma 1. Let α ∈ (0, 1), β > 0. Then there exists a constant C = C(α, β) > 0 such that
t∫
0
(t− s)−αeβsds ≤ ceβt, t ≥ 0 (3)
(see also [4, p. 352, 354, 358]).
Theorem 1. We make the following assumptions:
(H1) the functions ωi : R+ := 〈0,∞) → R, i = 1, 2, . . . , n, are continuous and positive on
(0,∞) such that ωi1 ∝ ωi2 ∝ . . . ∝ ωin for some i1, i2, . . . , in ∈ {1, 2, . . . , n} simultaneously
different,
(H2) the functions u, λi : 〈0, T 〉 → (0,∞), i = 1, 2, . . . , N, are continuous, nonnegative, and
a is a positive constant,
(H3) 0 < βi < 1, i = 1, 2, . . . , n, βi 6= βj for i 6= j.
If ε > 0 and the function u(t) satisfies inequality (1) for t ∈ 〈0, T 〉 then
u(t) ≤
W−1
in
Win(cin−1) +
t∫
0
λ̂in(s)ds
1
kr
, t ∈ 〈0, b〉,
where
Wim(u) =
u∫
um
dσ
[ωim(σ
1
kr )]r
u ≥ um = W−1
im−1
(uim−1), u0 > 0,
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SINGULAR INTEGRAL INEQUALITIES WITH SEVERAL NONLINEARITIES . . . 73
r = r(ε) := q1q2 . . . qN , qi =
1
βi
+ ε, ε > 0,
βi =
1
1 + zi
, zi > 0, i = 1, 2, . . . , N,
W−1
im
is the inverse function of Wim , c0 = (n + 1)r−1ar,
r = r(ε) := q1q2 . . . qN , qi =
1
βi
+ ε, ε > 0, βi =
1
1 + zi
, zi > 0,
cim = W−1
im
Wim(cim−1) +
T∫
0
λ̂im(s)ds
, m = 1, 2, . . . , n,
λ̂i(t) = T q̂i−1(n + 1)r−1dr
i e
−rtλi(t)r, q̂i = q1q2 . . . qi−1qi+1 . . . qN ,
di = c
1
pi
i eT , i = 1, 2, . . . , N, λ̂j(t) = T r−1(n + 1)r−1λj(s)r,
j = N + 1, N + 2, . . . , n, pi =
1 + zi + ε
zi + ε
, c := ci = ci(αi, pi) > 0
satisfies the inequality (3) with α = αipi, αi = 1 − βi, β = βi and the number b ∈ 〈0, T 〉 is the
largest number such that
‖λ̂im‖b :=
b∫
0
λ̂im(s)ds ≤
∞∫
cm−1
dσ
[ωim(σ
1
kr )]r
, m = 1, 2, . . . , n.
This theorem obviously extends the Bihari lemma [16] to n nonlinearities. This is a singular
version of the Pinto inequality [12 ] (Theorem 1) and it is also a generalization of an inequality
proved by J. D. Dauer and N. I. Mahmudov [7] (Theorem 4). They consider the inequality (2)
with N = 1, n = 2, β1 > 0 and in the proof of their result the method of desingularization
developed in [4] is applied.
The following proof of Theorem 1 is based on the method of desingularization of singular
integral inequalities (see [4]), the method applied in the study of integral inequalities with
power nonlinearities in the left-hand sides in the papers [5, 15] and the Pinto’s method [1]
(a generalization of Bihari’s method to integral inequalities with n nonlinearities).
Proof Theorem 1. Let
αi = 1− βi, βi =
1
1 + zi
, zi > 0,
qi =
1
βi
+ ε, pi =
1 + zi + ε
zi + ε
, i = 1, 2, . . . , n.
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 1
74 M. MEDVEĎ
Obviously
1
pi
+
1
qi
= 1. The Hölder inequality and (3) yield
t∫
0
(t− s)βi−1λi(s)ωi(u(s))ds =
=
t∫
0
(t− s)−αies · e−sλi(s)ωi(u(s))ds ≤
≤
t∫
0
(t− s)−αipiepisds
1
pi
t∫
0
e−qisλi(s)qiωi(u(s))qids
1
qi
≤
≤ die
t
t∫
0
νi(s)ωi(u(s))qids
1
qi
,
where
νi(t) = e−qitλi(t)qi , di = c
1
pi
i eT ,
t∫
0
(t− s)−αipiepisds ≤ cie
pit, t ≥ 0, ci = ci(αi, pi) > 0,
see Lemma 1.
From the inequality (2) we obtain
u(t)k ≤ a +
N∑
i=1
di
t∫
0
νi(s)ωi(u(s))qids
1
qi
+
n∑
j=N+1
t∫
0
λj(s)ωj(u(s))ds.
If we define r = q1q2 . . . qN then applying the inequality (A1 + A2 + . . . + An+1)r ≤ (n +
+1)r−1(Ar
1 + Ar
2 + . . . + Ar
n+1), A1, A2, . . . , An+1 ≥ 0, we obtain
u(t)kr ≤ (n + 1)r−1
ar +
N∑
i=1
dr
i
t∫
0
νi(s)ωi(u(s))qids
q̂i
+
+
n∑
j=N+1
t∫
0
λj(s)ωj(u(s))ds
r
, (4)
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SINGULAR INTEGRAL INEQUALITIES WITH SEVERAL NONLINEARITIES . . . 75
where q̂i = q1q2 . . . qi−1qi+1 . . . qN > 1. The Jensen’s inequality in the form
t∫
0
g(s)ds
δ
≤ tδ−1
t∫
0
g(s)δds, δ ≥ 1,
yields
t∫
0
νi(s)ωi(u(s))qids
q̂i
≤ T q̂i−1
t∫
0
νi(s)q̂iωi(u(s)rds =
= T q̂i−1
t∫
0
e−rtλi(s)rds,
t∫
0
λj(s)ωj(u(s))ds
r
≤ T r−1
t∫
0
λj(s)rωj(u(s))rds
and from inequality (4) it follows that
u(t)kr ≤ c0 +
n∑
i=1
t∫
0
λ̂i(s)ωi(u(s))rds,
where
c0 = (n + 1)r−1ar, λ̂i(t) = T q̂i−1(n + 1)r−1dr
i e
−rtλi(t)r, i = 1, 2, . . . , N,
λ̂j(t) = T r−1(n + 1)r−1λj(t)r, j = N + 1, N + 2, . . . , n.
This inequality can also be written as
u(t)kr ≤ c0 +
n∑
m=1
t∫
0
λ̂im(s)ωim(u(s))rds,
where the condition (H1) is satisfied for some i1, i2, . . . , im.
We shall proceed in the sequel by induction with respect to n. For n = 1 the result follows
from [5] (Theorem 2). Let n = 2, i.e., we have the inequality
u(t)kr ≤ γ +
t∫
0
λ̂i1(s)ωi1(u(s))rds +
t∫
0
λ̂i2(s)ωi2(u(s))rds. (5)
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 1
76 M. MEDVEĎ
Without loss of generality one can assume that i1 = 1, i2 = 2, i.e., ω1 ∝ ω2.
Let
z1(t) =
t∫
0
λ̂1(s)ω1(u(s))rds, z2(t) = γ +
t∫
0
λ̂2(s)ω2(u(s))rds, z(t) = z1(t) + z2(t).
Obviously,
u(t)kr ≤ z(t) (6)
and therefore we have
d
dt
(W1(z(t)) =
dz/dt
ω1(z(t))r
≤ λ̂1(t) + λ̂2(t)
[
ω(z(t)
1
kr )
]r
,
where ω =
ω2
ω1
. Hence by integrating over 〈0, T 〉, we obtain
W1(z(t)) ≤ W1(c0) +
t∫
0
λ̂(s)ds +
t∫
0
λ̂(s)
[
ω(z(s)
1
kr )
]r
ds.
Putting v(t) = W1(z(t)) we have
v(t) ≤ W1(c0) +
T∫
0
λ̂1(s)ds +
t∫
0
λ̂2(s)ω(W−1
1 (v(s))rds,
i.e.,
v(t) ≤ W1(c0) +
t∫
0
λ̂2(s)ω̂(v(s))ds,
where ω̂(u) =
(
ω([W−1
1 (u)]
1
kr )
)r
. By the Bihari lemma [16]
v(t) ≤ Ŵ−1
Ŵ (W1(c0)) +
T∫
0
λ̂1(s)ds +
t∫
0
λ̂2(s)ds
,
where
Ŵ (u) =
u∫
u0
dσ
ω̂(σ)
, u ≥ u0 > 0,
and consequently we obtain from (6) that
u(t) ≤ z(t)
1
kr =
(
W−1
1 (v(t)
) 1
kr ≤
W−1
1 ◦ Ŵ−1
Ŵ (c0) +
t∫
0
λ̂2(s)ds
1
kr
. (7)
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SINGULAR INTEGRAL INEQUALITIES WITH SEVERAL NONLINEARITIES . . . 77
Obviously
Ŵ (u) =
u∫
u0
dσ
ω̂(σ)
=
u∫
u0
dσ
ω([W−1
1 (σ)]
1
kr )r
=
=
W−1
1 (u)∫
u1
dτ
[ω2(τ
1
kr )]r
= W2 ◦W−1
1 (u),
where u1 = W−1
1 (u0) and therefore (7) yields
u(t) ≤ W−1
2
W2(c1) +
t∫
0
λ̂2(s)ds
where c1 = W−1
1 (W1(c0) +
∫
λ̂1(s)ds). In the case n > 2 one can proceed by induction in the
same way as in the Pinto’s proof of [1] and we omit this part of the proof.
3. Applications. In this section we apply Theorem 1 to a study of properties of solutions of
the integral equation
Φ(t) = f(t) +
N∑
i=1
t∫
0
(t− s)βi−1Pi(s,Φ(s))ds+
+
n∑
j=N+1
t∫
0
Pj(s,Φ(s))ds, (8)
where Φ, f, Pi, i = 1, 2, . . . , n, are vector-valued functions, and the scalar integral equation of
the form
x(t)k = h(t) +
N∑
i=1
t∫
0
(t− s)βi−1Qi(s, x(s))ds+
+
n∑
j=N+1
t∫
0
Qj(s, x(s))ds, k > 1. (9)
By a solution of (8) and (9), respectively, we mean a continuous solution defined on an
interval 〈0, T ), where 0 < T ≤ ∞. If T = ∞, the solution is called global. A result on the
existence of global solutions of (8) with n = 1 is proved in [5] and it is applied there in the proof
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 1
78 M. MEDVEĎ
of a theorem on the existence of global solutions of some evolution equations. The special case
x(t) = h(t) +
t∫
0
(t− s)β−1w(x(s))ds (10)
of the equation (8), where x, h, w are scalar functions, has been studied in several papers, e.
g. in [9, 17]. In the paper [18] a necessary and sufficient condition for the existence of global
solutions of this equation is proved and this result is used in the paper [19] in the proofs of
some results on the existence of global solutions to a semilinear parabolic evolution equations.
Scalar equations with power nonlinearities in the left-hand sides, i.e., equations of the form (9),
however with n = 1, are studied e. g. in [20]. In this paper the function Q1 = Q is supposed
to be linear, h(t) ≡ 0 and there is a function of t before the integral. The authors have proved
some results on lower and upper estimates of solutions of such equations. Equations of such
type are arising in various applications, e. g., in the nonlinear theory of wave propagation [21].
Theorem 2. Let f ∈ C(R+, RM ), Pi ∈ C(R+ × RM , RM ) with
‖Pi(t, v)‖ ≤ Fi(t)ωi(‖v‖), (t, v) ∈ R+ × RM , i = 1, 2, . . . , n, (11)
where Fi ∈ C(R+, R+) and ωi, i = 1, 2, . . . , n, are functions satisfying the condition (H1) of
Theorem 1. Assume that there exists an ε > 0 such that
∞∫
0
τ r−1dτ
ωi(τ)r
= ∞, i = 1, 2, . . . , n, (12)
where r = r(ε) := q1q2 . . . qN , qi :=
1
βi
+ ε, 0 < βi < 1, i = 1, 2, . . . , N . Then
lim sup
t→T−
‖Φ(t)‖ < ∞
for any solution Φ : 〈0, T ) → RM of the equation (8) with 0 < T < ∞.
Proof. Assume that Φ : 〈0, T ) → Rn is a continuous solution of the equation (8), where
0 < T < ∞ with lim supt→T− ||Φ(t)|| = ∞. The condition (11) yields
‖Φ(t)‖ ≤ ‖f(t)‖+
N∑
i=1
t∫
0
(t− s)βi−1Fi(s)ωi(||Φ(s)||)ds+
+
n∑
j=N+1
t∫
0
Fj(s)ωj(||Φ(s)||)ds
and applying Theorem 1 we obtain
Win(‖Φ(t)‖) ≤ Win(κin) +
t∫
0
F̂in(s)ds,
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SINGULAR INTEGRAL INEQUALITIES WITH SEVERAL NONLINEARITIES . . . 79
where
κ0 = (n + 1)r−1ηr, η = max
t∈〈0,T 〉
‖f(t)‖,
κim = W−1
im
Wim(κim−1) +
T∫
0
F̂im(s)ds
,
Wim(u) =
u∫
um
dσ
[ωim(σ
1
r )]r
, u ≥ um = W−1
im−1
(um−1), u0 = ‖Φ(0)‖r,
F̂im(t) = T q̂im−1(n + 1)r−1dr
imFim(t),
q̂i = q1 . . . qi−1qi+1 . . . qn, dim = cim
1
pim eT , m = 1, 2, . . . , n, cim , pim
are constants from Theorem 1 and the functions Wim , m = 1, 2, . . . , n, are as in Theorem 1.
This yields
‖Φ(t)‖r∫
un
dσ
[ωin(σ
1
r )]r
≤ Win(κin) +
T∫
0
F̂in(s)ds < ∞.
We remark that all the numbers um, m = 1, 2, . . . , n, are well defined because of the divergence
of all integrals from (12). However
lim sup
t→T−
‖Φ(t)‖r∫
un
dσ
[ωin(σ
a
r )]r
= r lim sup
t→T−
‖Φ(t)‖∫
u0
τ r−1dτ
ωin(τ)r
=
= r
∞∫
0
τ r−1dτ
ωin(τ)r
= ∞
and this is a contradiction with (11).
The theorem is proved.
Theorem 3. Let h ∈ C(R+, R), Qi ∈ C(R+, R) with
|Qi(t, w)) ≤ Gi(t)ωi(|w|), (t, w) ∈ R+ × R, i = 1, 2, . . . , n,
where Gi ∈ C(R+, R+) and ωi, i = 1, 2, . . . , n, are functions satisfying the condition (H1) and
the condition
∞∫
0
τkr−1dτ
ωi(τ)kr
= ∞, i = 1, 2, . . . , n,
where r = r(ε) is as in Theorem 2 and k ≥ 1. Then lim supt→T− |Ψ(t)| ≤ ∞ for any solution
Ψ : 〈0, T ) → R of the equation (9) with 0 < T < ∞.
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 1
80 M. MEDVEĎ
This theorem can be proved applying Theorem 1 in the same way as we have proved
Theorem 2.
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Received 09.10.07
ISSN 1562-3076. Нелiнiйнi коливання, 2008, т . 11, N◦ 1
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| id | nasplib_isofts_kiev_ua-123456789-178149 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-3076 |
| language | English |
| last_indexed | 2025-12-07T15:23:56Z |
| publishDate | 2008 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Medveď, M. 2021-02-18T07:08:55Z 2021-02-18T07:08:55Z 2008 Singular integral inequalities with several nonlinearities and integral equations with singular kernels / M. Medveď // Нелінійні коливання. — 2008. — Т. 11, № 1. — С. 71-80. — Бібліогр.: 21 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/178149 517.9 We deal with an integral inequality with a power nonlinearity in its left-hand side, with n nonlinearities in its right-hand side, and weakly singular kernels. The obtained result is an extention of the Bihari, Henry, Pachpatte and Pinto inequalities and results obtained by the author. Using these results we prove sufficient conditions for existence of global solutions of some nonlinear Volterra integral equations with singular kernels and n nonlinearities. Розглянуто iнтегральнi нерiвностi зi степеневою нелiнiйнiстю в лiвiй частинi, n нелiнiйностями у правiй частинi та слабкосингулярними ядрами. Отриманий результат є узагальненням нерiвностей, одержаних Бiхарi, Генрi, Пачпатт та Пiнто, а також результатiв автора. На основi цих результатiв встановлено достатнi умови iснування глобальних розв’язкiв деяких нелiнiйних iнтегральних рiвнянь Вольтерри з сингулярними ядрами та n нелiнiйностями. This work was supported by the Slovak Grant Agency Vega (Grant N 1/2001/05). en Інститут математики НАН України Нелінійні коливання Singular integral inequalities with several nonlinearities and integral equations with singular kernels Сингулярні інтегральні нерівності з кількома нелінійностями та інтегральні рівняння з сингулярними ядрами Сингулярные интегральные неравенства с несколькими нелинейностями и интегральные уравнения с сингулярными ядрами Article published earlier |
| spellingShingle | Singular integral inequalities with several nonlinearities and integral equations with singular kernels Medveď, M. |
| title | Singular integral inequalities with several nonlinearities and integral equations with singular kernels |
| title_alt | Сингулярні інтегральні нерівності з кількома нелінійностями та інтегральні рівняння з сингулярними ядрами Сингулярные интегральные неравенства с несколькими нелинейностями и интегральные уравнения с сингулярными ядрами |
| title_full | Singular integral inequalities with several nonlinearities and integral equations with singular kernels |
| title_fullStr | Singular integral inequalities with several nonlinearities and integral equations with singular kernels |
| title_full_unstemmed | Singular integral inequalities with several nonlinearities and integral equations with singular kernels |
| title_short | Singular integral inequalities with several nonlinearities and integral equations with singular kernels |
| title_sort | singular integral inequalities with several nonlinearities and integral equations with singular kernels |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/178149 |
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