On the solvability of nonlinear ordinary differential equation in grand Lebesgue spaces
UDC 517.9We study the relationship between the second-order nonlinear ordinary differential equations and the Hardy inequality in grand Lebesgue spaces. In particular, we give a characterization of the Hardy inequality by using nonlinear ordinary differential equations in grand Lebesgue spaces.
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Institute of Mathematics, NAS of Ukraine
2022
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512274485084160 |
|---|---|
| author | Bandaliyev, R. A. Safarova, K. H. Bandaliyev, R. A. Safarova, K. H. |
| author_facet | Bandaliyev, R. A. Safarova, K. H. Bandaliyev, R. A. Safarova, K. H. |
| author_sort | Bandaliyev, R. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-11-27T13:39:22Z |
| description | UDC 517.9We study the relationship between the second-order nonlinear ordinary differential equations and the Hardy inequality in grand Lebesgue spaces. In particular, we give a characterization of the Hardy inequality by using nonlinear ordinary differential equations in grand Lebesgue spaces. |
| doi_str_mv | 10.37863/umzh.v74i8.6146 |
| first_indexed | 2026-03-24T03:26:11Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i8.6146
UDC 517.9
R. A. Bandaliyev1, K. H. Safarova (Inst. Math. and Mech. Nat. Acad. Sci. Azerbaijan and Azerbaijan Univ.
Architecture and Construction, Baku)
ON THE SOLVABILITY OF NONLINEAR ORDINARY DIFFERENTIAL
EQUATION IN GRAND LEBESGUE SPACES
ПРО РОЗВ’ЯЗНIСТЬ НЕЛIНIЙНИХ ЗВИЧАЙНИХ ДИФЕРЕНЦIАЛЬНИХ
РIВНЯНЬ У ВЕЛИКИХ ПРОСТОРАХ ЛЕБЕГА
We study the relationship between the second-order nonlinear ordinary differential equations and the Hardy inequality
in grand Lebesgue spaces. In particular, we give a characterization of the Hardy inequality by using nonlinear ordinary
differential equations in grand Lebesgue spaces.
Вивчається зв’язок мiж нелiнiйними звичайними диференцiальними рiвняннями другого порядку та нерiвнiстю
Гардi у великих просторах Лебега. Зокрема, дано характеристику нерiвностi Гардi нелiнiйними звичайними дифе-
ренцiальними рiвняннями у великих просторах Лебега.
1. Introduction. It is well-known that in 1925 G. H. Hardy [15] proved the integral inequality using
the calculus of variations, which states that if f \in Lp is a nonnegative function on (0,\infty ), then\left( \infty \int
0
\left( 1
x
x\int
0
f(t)dt
\right) p
dx
\right)
1
p
\leq p
p - 1
\left( \infty \int
0
fp(x)dx
\right) 1
p
, p > 1. (1.1)
The constant
p
p - 1
in (1.1) is the best possible (see also [16]). Also, inequality (1.1) holds in
any finite interval [a, b], 0 \leq a < b < \infty . The prehistory of the classical Hardy inequality has been
described in [23]. Some important steps in the further development of what today is called Hardy type
inequalities are described in [24]. A systematic investigation of the generalized Hardy inequality with
weights that started in [3]. Namely, in [3] two-weight Hardy inequality in its equivalent differential
form \left( \infty \int
0
fp(x)\omega (x) dx
\right) 1
p
\leq C
\left( \infty \int
0
\bigl(
f \prime (x)
\bigr) p
v(x)dx
\right) 1
p
, f(0) = f(+0) = 0, (1.2)
was connected with the Euler – Lagrange differential equation. It should be mentioned that in [4]
Hardy inequality was studied not only with the case p > 1, but also with p < 0 and even with
0 < p < 1. Beesack’s approach was extended to a class of inequalities containing the Hardy ine-
quality (1.2) as a special case (see, e.g., [4] or [30]). In particular, a necessary and sufficient
condition on weight functions for validity (1.2) was obtained in [31] and [32]. The study of the case
1 Corresponding author, e-mail: bandaliyevr@gmail.com.
c\bigcirc R. A. BANDALIYEV, K. H. SAFAROVA, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8 1011
1012 R. A. BANDALIYEV, K. H. SAFAROVA
with different parameters p and q was started in [5] and developed in [22, 24, 25]. In the case p \not = q
the other type criterion on weight functions for validity (1.1) was obtained in [14] and [27]. Namely,
in [14] and [27] the inequality (1.2) was connected with nonlinear ordinary differential equation
in weighted Lebesgue spaces. Similar problems for two-dimensional Hardy operator in weighted
Lebesgue spaces with mixed norm is studied in [1]. Moreover, the Hardy inequality has numerous
applications in the spectral theory of operators, in the theory of integral equations, in the theory of
function spaces etc. (see, e.g., [6, 7, 24 – 26]).
In this paper we study similar problem in grand Lebesgue spaces. Namely, we give a new
characterization of Hardy inequality by nonlinear ordinary differential equation in grand Lebesgue
spaces. The main contribution in this paper is the characterization of best possible constant in Hardy
inequality by specially introduced quantity.
Grand Lebesgue spaces proved to be useful in application to partial differential equations (see,
e.g., [11, 13, 18, 19, 28, 29]). In particular, in the theory of PDE’s, it turned out that these are the
right spaces in which some nonlinear equations have to be considered (see, e.g., [8, 10, 12, 34]).
We note that the boundedness of classical Hardy operator in grand and small Lebesgue spaces
was first proved in [9]. Later, the characterization of boundedness of the Hardy type operators
between weighted grand Lebesgue spaces was studied in [20] (see, e.g., [21]). Similar results for
one-dimensional and multidimensional Hardy operators in grand Lebesgue spaces on unbounded
domains were proved in [33]. Recently, the boundedness of Hausdorff operator in grand Lebesgue
spaces was obtained in [2].
This paper is organized as follows. Section 2 contains some preliminaries along with the standard
ingredients used in the proofs. The main results are stated and proved in Section 3. Namely, in
Section 3, we establish necessary condition and sufficient condition on the best possible constant in
Hardy inequality on grand Lebesgue spaces.
2. Preliminaries. Let 1 < p < \infty and p\prime =
p
p - 1
. In 1992 T. Iwaniec and C. Sbordone [17], in
their studies related with the integrability properties of the Jacobian in a bounded open set \Omega \subset \BbbR n,
introduced a new type of function spaces Lp)(\Omega ) called grand Lebesgue spaces. Namely, the grand
Lebesgue space is defined as the space of the Lebesgue measurable functions f on \Omega such that
\| f\| Lp)(\Omega ) = \mathrm{s}\mathrm{u}\mathrm{p}
0<\varepsilon <p - 1
\left( \varepsilon
| \Omega |
\int
\Omega
| f(x)| p - \varepsilon dx
\right) 1
p - \varepsilon
< \infty ,
where | \Omega | is the Lebesgue measure of \Omega . Throughout this paper we assume that all functions are
Lebesgue measurable. Let n = 1 and let \Omega = (0, 1). Then the norm in grand Lebesgue space has
the form
\| f\| Lp)(0,1) = \| f\| p) = \mathrm{s}\mathrm{u}\mathrm{p}
0<\varepsilon <p - 1
\left( \varepsilon
1\int
0
| f(x)| p - \varepsilon dx
\right)
1
p - \varepsilon
=
= \mathrm{s}\mathrm{u}\mathrm{p}
0<\varepsilon <p - 1
\varepsilon
1
p - \varepsilon \| f\| p - \varepsilon < \infty .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
ON THE SOLVABILITY OF NONLINEAR ORDINARY DIFFERENTIAL EQUATION . . . 1013
We denote by C1(0, 1) the space of continuously differentiable functions on (0, 1). The set of all
absolutely continuous functions on (0, 1) is denoted by AC(0, 1).
Let 1 < a < p be a fixed number and 0 < \varepsilon < p - a. Suppose that \lambda is a positive measurable
function defined on (0, p - a) such that \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{i}\mathrm{n}\mathrm{f}0<\varepsilon <p - a \lambda (\varepsilon ) > 0. Let us consider the nonlinear
differential equation
\lambda (\varepsilon )
d
dt
\bigl(
[y\prime (t)]p - \varepsilon - 1
\bigr)
+ t\varepsilon - p
\bigl[
y(t)
\bigr] p - \varepsilon - 1
= 0, (2.1)
where
y(t) > 0, y\prime (t) > 0, 0 < t < 1, y\prime \in AC(0, 1). (2.2)
We say that y is a solution of the problem (2.1), (2.2), if y satisfies the differential equation (2.1)
almost everywhere on (0, 1) and the condition (2.2). We denote y(0) = \mathrm{l}\mathrm{i}\mathrm{m}t\rightarrow +0 y(t) and let
A(p, \lambda ) = \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}0<\varepsilon <p - a
\bigl(
\lambda (\varepsilon )
\bigr) 1
p - \varepsilon .
We denote by Lp),a(0, 1) the grand Lebesgue space the set of all measurable functions with the
finite norm
\| u\| Lp),a(0,1) = \| u\| p),a = \mathrm{s}\mathrm{u}\mathrm{p}
0<\varepsilon <p - a
\varepsilon
1
p - \varepsilon \| u\| p - \varepsilon .
It is obvious that Lp),1(0, 1) = Lp)(0, 1) and Lp)(0, 1) \lhook \rightarrow Lp),a(0, 1).
First we prove the following theorem.
Theorem 2.1. Let a < p < \infty and \lambda be a positive measurable function defined by (2.1) and
A(p, \lambda ) < \infty . Suppose that u is an absolutely continuous function on (0, 1) satisfies condition
u(0) = u(+0) = 0. If the problem (2.1), (2.2) has a solution u, then\bigm\| \bigm\| \bigm\| u
x
\bigm\| \bigm\| \bigm\|
p),a
\leq A(p, \lambda )
\bigm\| \bigm\| u\prime \bigm\| \bigm\|
p),a
.
Proof. It is well-known that for any absolutely continuous function the representation
u(x) = u(0) +
x\int
0
u\prime (t)dt
holds. Since u(0) = 0, it follows that
u(x) =
x\int
0
u\prime (t)dt.
Let a function y be a solution of problem (2.1), (2.2). Suppose that 0 < \varepsilon < p - a be any
number. Then, using Hölder inequality with exponents p - \varepsilon and (p - \varepsilon )\prime , we have
\bigm| \bigm| u(x)\bigm| \bigm| \leq x\int
0
\bigm| \bigm| u\prime (t)\bigm| \bigm| dt = x\int
0
\bigm| \bigm| u\prime (t)\bigm| \bigm| \bigl[ y\prime (t)\bigr] - 1
(p - \varepsilon )\prime
\bigl[
y\prime (t)
\bigr] 1
(p - \varepsilon )\prime dt \leq
\leq
\left( x\int
0
y\prime (t)dt
\right) 1
(p - \varepsilon )\prime
\left( x\int
0
\bigm| \bigm| u\prime (t)\bigm| \bigm| p - \varepsilon \bigl[
y\prime (t)
\bigr] - p - \varepsilon
(p - \varepsilon )\prime dt
\right) 1
p - \varepsilon
=
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
1014 R. A. BANDALIYEV, K. H. SAFAROVA
= (y(x) - y(0))
1
(p - \varepsilon )\prime
\left( x\int
0
\bigm| \bigm| u\prime (t)\bigm| \bigm| p - \varepsilon \bigl[
y\prime (t)
\bigr] \varepsilon - p+1
dt
\right) 1
p - \varepsilon
\leq
\leq [y(x)]
1
(p - \varepsilon )\prime
\left( x\int
0
\bigm| \bigm| u\prime (t)\bigm| \bigm| p - \varepsilon \bigl[
y\prime (t)
\bigr] \varepsilon - p+1
dt
\right) 1
p - \varepsilon
=
=
\biggl[
- xp - \varepsilon \lambda (\varepsilon )
d
dx
\Bigl( \bigl[
y\prime (x)
\bigr] p - \varepsilon - 1
\Bigr) \biggr] 1
p - \varepsilon
\left( x\int
0
\bigm| \bigm| u\prime (t)\bigm| \bigm| p - \varepsilon \bigl[
y\prime (t)
\bigr] \varepsilon - p+1
dt
\right) 1
p - \varepsilon
=
= x(\lambda (\varepsilon ))
1
p - \varepsilon
\left( x\int
0
\bigm| \bigm| u\prime (t)\bigm| \bigm| p - \varepsilon \bigl[
y\prime (t)
\bigr] \varepsilon - p+1
\biggl(
- d
dx
\Bigl( \bigl[
y\prime (x)
\bigr] p - \varepsilon - 1
\Bigr) \biggr)
dt
\right) 1
p - \varepsilon
.
Hence, we have
| u(x)|
x
\leq (\lambda (\varepsilon ))
1
p - \varepsilon
\left( x\int
0
\bigm| \bigm| u\prime (t)\bigm| \bigm| p - \varepsilon \bigl[
y\prime (t)
\bigr] \varepsilon - p+1
\biggl(
- d
dx
\bigl(
[y\prime (x)]p - \varepsilon - 1
\bigr) \biggr)
dt
\right) 1
p - \varepsilon
.
Thus, one has
\left( 1\int
0
\biggl(
| u(x)|
x
\biggr) p - \varepsilon
dx
\right)
1
p - \varepsilon
\leq
\leq A(p, \lambda )
\left( 1\int
0
\left( x\int
0
\bigm| \bigm| u\prime (t)\bigm| \bigm| p - \varepsilon \bigl[
y\prime (t)
\bigr] \varepsilon - p+1
\biggl(
- d
dx
\Bigl( \bigl[
y\prime (x)
\bigr] p - \varepsilon - 1
\Bigr) \biggr)
dt
\right) dx
\right)
1
p - \varepsilon
=
= A(p, \lambda )
\left( 1\int
0
1\int
0
\bigm| \bigm| u\prime (t)\bigm| \bigm| p - \varepsilon \bigl[
y\prime (t)
\bigr] \varepsilon - p+1
\biggl(
- d
dx
\Bigl( \bigl[
y\prime (x)
\bigr] p - \varepsilon - 1
\Bigr) \biggr)
\chi (0,x)(t)dt dx
\right)
1
p - \varepsilon
=
= A(p, \lambda )
\left( 1\int
0
\bigm| \bigm| u\prime (t)\bigm| \bigm| p - \varepsilon \bigl[
y\prime (t)
\bigr] \varepsilon - p+1
\left( 1\int
t
\biggl(
- d
dx
\Bigl( \bigl[
y\prime (x)
\bigr] p - \varepsilon - 1
\Bigr) \biggr)
dx
\right) dt
\right)
1
p - \varepsilon
=
= A(p, \lambda )
\left( 1\int
0
\bigm| \bigm| u\prime (t)\bigm| \bigm| p - \varepsilon \bigl[
y\prime (t)
\bigr] \varepsilon - p+1
\Bigl( \bigl[
y\prime (t)
\bigr] p - \varepsilon - 1 -
\bigl[
y\prime (1)
\bigr] p - \varepsilon - 1
\Bigr)
dt
\right)
1
p - \varepsilon
. (2.3)
From equation (2.1), it follows that y\prime \prime < 0. Therefore y\prime is a decreasing function on (0, 1). Thus,
(2.3) implies that
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
ON THE SOLVABILITY OF NONLINEAR ORDINARY DIFFERENTIAL EQUATION . . . 1015
\left( 1\int
0
\biggl(
| u(x)|
x
\biggr) p - \varepsilon
dx
\right)
1
p - \varepsilon
\leq A(p, \lambda )
\left( 1\int
0
\bigm| \bigm| u\prime (x)\bigm| \bigm| p - \varepsilon
dx
\right)
1
p - \varepsilon
. (2.4)
Multiply both side of (2.4) by \varepsilon
1
p - \varepsilon and passing to supremum over all \varepsilon \in (0, p - a), we complete
the proof of Theorem 2.1.
Let us set
M(\varepsilon ) =
1
p - \varepsilon - 1
\mathrm{i}\mathrm{n}\mathrm{f}
g
\mathrm{s}\mathrm{u}\mathrm{p}
0<x<1
1
g(x) - x
x\int
0
(g(t))p - \varepsilon t\varepsilon - p dt, 0 < \varepsilon < p - a, (2.5)
where the infimum is taken over the class of all measurable functions g such that g(x) > x for
0 < x < 1.
Remark 2.1. Let 0 < x < 1 and g(x) = 2x or g(x) = x(1+ex). Then \mathrm{s}\mathrm{u}\mathrm{p}0<\varepsilon <p - aM(\varepsilon ) < \infty .
The following lemma establishes a connection between problem (2.1), (2.2) and M(\varepsilon ).
Lemma 2.1. Let \lambda and M be two functions defined on (0, p - a). Then the following statements
are equivalent:
(i) if \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{i}\mathrm{n}\mathrm{f}0<\varepsilon <p - a \lambda (\varepsilon ) > 0 and the problem (2.1), (2.2) has a solution with an absolutely
continuous first derivative, then \lambda (\varepsilon ) \geq M(\varepsilon ) for all 0 < \varepsilon < p - 1;
(ii) if \mathrm{s}\mathrm{u}\mathrm{p}0<\varepsilon <p - aM(\varepsilon ) < \infty , then the problem (2.1), (2.2) has a solution for every \lambda (\varepsilon ) >
> M(\varepsilon ).
Proof. Suppose that (i) holds. Let y be a solution of (2.1) – (2.3). Let us take \omega =
y
y\prime
. Then \omega
is positive solution of the nonlinear differential equation
\omega \prime =
1
(p - \varepsilon - 1)\lambda (\varepsilon )
t\varepsilon - p \omega p - \varepsilon + 1. (2.6)
By (2.6), we have
\omega (x) \geq
x\int
0
\omega \prime (t)dt =
1
(p - \varepsilon - 1)\lambda (\varepsilon )
x\int
0
t\varepsilon - p
\bigl(
\omega (t)
\bigr) p - \varepsilon
dt+ x. (2.7)
This implies that \omega (x) \geq x. By (2.7), one has
\lambda (\varepsilon ) \geq 1
p - \varepsilon - 1
\mathrm{i}\mathrm{n}\mathrm{f}
g
\mathrm{s}\mathrm{u}\mathrm{p}
0<x<1
1
g(x) - x
x\int
0
t\varepsilon - p
\bigl(
g(t)
\bigr) p - \varepsilon
dt. (2.8)
Therefore, by (2.8) and (2.5), we conclude that \lambda (\varepsilon ) \geq M(\varepsilon ) for all 0 < \varepsilon < p - 1. This completes
the proof of (i).
Let us assume that (ii) holds. Let us fix \lambda (\varepsilon ) > M(\varepsilon ). By the definition of M(\varepsilon ) there exists a
measurable functions g such that
g(x) \geq 1
(p - \varepsilon - 1)\lambda (\varepsilon )
x\int
0
t\varepsilon - p(g(t))p - \varepsilon dt+ x.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
1016 R. A. BANDALIYEV, K. H. SAFAROVA
We define a sequence of functions \omega n(x) by setting
\omega 0(x) = g(x), \omega n+1(x) =
1
(p - \varepsilon - 1)\lambda (\varepsilon )
x\int
0
t\varepsilon - p(\omega n(t))
p - \varepsilon dt+ x, n = 0, 1, 2, . . . .
It is obvious that \omega 0(x) \geq \omega 1(x). Let \omega n - 1(x) \geq \omega n(x). So, one has
\omega n(x) - \omega n+1(x) =
1
(p - \varepsilon - 1)\lambda (\varepsilon )
x\int
0
t\varepsilon - p
\bigl[
(\omega n - 1(t))
p - \varepsilon - (\omega n(t))
p - \varepsilon
\bigr]
dt \geq 0.
This implies that
\bigl\{
\omega n(x)
\bigr\} \infty
n=0
is a nonincreasing by n on x \in (0, 1). Since \omega n(x) \geq 0, this implies
that a sequence
\bigl\{
\omega n(x)
\bigr\} \infty
n=0
is converges. Let \omega (x) = \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \omega n(x) for a.e. x \in (0, 1). By
the Levi monotone convergence theorem, it follows that \omega is a nonnegative solution of the integral
equation
\omega (x) =
1
(p - \varepsilon - 1)\lambda (\varepsilon )
x\int
0
t\varepsilon - p
\bigl(
\omega (t)
\bigr) p - \varepsilon
dt+ x.
So, \omega is an absolutely continuous function and satisfies the differential equation
\omega \prime (x) =
1
(p - \varepsilon - 1)\lambda (\varepsilon )
x\varepsilon - p(\omega (x))p - \varepsilon + 1.
Therefore, for any fixed number a \in (0, 1) the function
y(x) = Ce
\int x
a
dt
\omega (t)
dt
, C = y(a),
satisfies problem (2.1), (2.2).
The lemma is proved.
3. Main results. In this section, we proved the solvability of problem (2.1), (2.2) in grand
Lebesgue space Lp), a(0, 1).
We need the following theorem.
Theorem 3.1. Let 1 < a \leq p < \infty , M be a positive function defined on (0, p - a) by (2.5)
and A(p,M) < \infty . Suppose that u is an absolutely continuous function on (0, 1) satisfies condition
u(0) = 0. Let C > 0 be the best constant such that\bigm\| \bigm\| \bigm\| u
x
\bigm\| \bigm\| \bigm\|
p),a
\leq C\| u\prime \| p),a. (3.1)
Then
1 \leq C \leq A(p,M) \leq a
a - 1
. (3.2)
Proof. Let us suppose that (3.1) holds and we choose the test function as u(x) = x, 0 < x < 1.
Then u\prime (x) = 1 and
\| 1\| p),a = \mathrm{s}\mathrm{u}\mathrm{p}
0<\varepsilon <p - a
\varepsilon
1
p - \varepsilon = (p - a)
1
a .
On the other hand, one has
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
ON THE SOLVABILITY OF NONLINEAR ORDINARY DIFFERENTIAL EQUATION . . . 1017\bigm\| \bigm\| \bigm\| u
x
\bigm\| \bigm\| \bigm\|
p),a
= \| 1\| p),a.
Hence C \geq 1.
Now we show that C \leq A(p,M). Let u\prime (x) = f(x). Since u(0) = 0, it follows that
u(x) =
x\int
0
f(t)dt.
We set Hf(x) =
1
x
\int x
0
f(t)dt. Obviously,
C = \mathrm{s}\mathrm{u}\mathrm{p}
\| f\| p),a=1
\| Hf\| p),a.
Assume the contrary. Let C > A(p,M). Then there exists a number \mu > 0 such that C > \mu >
> A(p,M). So \mu >
\bigl(
M(\varepsilon )
\bigr) 1
p - \varepsilon for all 0 < \varepsilon < p - 1. This implies that \mu p - \varepsilon > M(\varepsilon ). We choose
\lambda (\varepsilon ) = \mu p - \varepsilon . Since A(p,M) < \infty , by Lemma 2.1, problem (2.1), (2.2) has a solution for every
\lambda (\varepsilon ) > M(\varepsilon ). Therefore, by Theorem 2.1, we have\bigm\| \bigm\| \bigm\| u
x
\bigm\| \bigm\| \bigm\|
p),a
\leq \mu \| u\prime \| p),a.
Hence C is not the best possible constant in (3.1). This contradiction completes the proof.
Finally we show that A(p,M) \leq a
a - 1
. By the definition of M(\varepsilon ) for every function g satisfying
condition g(x) > x, we have
M(\varepsilon ) \leq 1
p - \varepsilon - 1
\mathrm{s}\mathrm{u}\mathrm{p}
0<x<1
1
g(x) - x
x\int
0
t\varepsilon - p
\bigl(
g(t)
\bigr) p - \varepsilon
dt.
We choose g\varepsilon (x) = (p - \varepsilon )\prime x. It is obvious that g\varepsilon (x) > x. So, one has
M(\varepsilon ) \leq 1
p - \varepsilon - 1
\bigl(
(p - \varepsilon )\prime
\bigr) p - \varepsilon
(p - \varepsilon )\prime - 1
=
\bigl(
(p - \varepsilon )\prime
\bigr) p - \varepsilon
.
Hence
\bigl(
M(\varepsilon )
\bigr) 1
p - \varepsilon \leq (p - \varepsilon )\prime and passing to supremum over all \varepsilon \in (0, p - a), we get
A(p,M) \leq \mathrm{s}\mathrm{u}\mathrm{p}
0<\varepsilon <p - a
p - \varepsilon
p - \varepsilon - 1
=
a
a - 1
.
Theorem 3.1 is proved.
Now we proved our main theorem.
Theorem 3.2. Let 1 < a \leq p < \infty and \varepsilon \in (0, p - a). Suppose that u is an absolutely
continuous function on (0, 1) satisfies condition u(0) = 0. Then, for the solvability of problem (2.1),
(2.2), it is necessary and sufficient that there exists a constant C0 > 0 such that the inequality\bigm\| \bigm\| \bigm\| u
x
\bigm\| \bigm\| \bigm\|
p),a
\leq C0 \| u\prime \| p),a (3.3)
holds.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
1018 R. A. BANDALIYEV, K. H. SAFAROVA
Proof. The sufficiency part of the Theorem 3.2 follows from Theorem 2.1. On the other hand,
the inequality (3.3) holds with constant C0 = A(p, \lambda ). We shall prove only the necessity part. Let u
be an absolutely continuous function satisfying condition u(0) = 0 and let the inequality (3.3) holds.
Then C \leq C0 < \infty , where C is the constant in (3.1). By (3.2) for all \varepsilon \in (0, p - a), we get that
M(\varepsilon ) \leq
\biggl(
a
a - 1
\biggr) p - \varepsilon
\leq
\biggl(
a
a - 1
\biggr) p
\leq
\biggl(
a
a - 1
\biggr) p
C < \infty .
So A(p,M) \leq
\biggl(
a
a - 1
\biggr) p
C < \infty . Then, by Lemma 2.1, problem (2.1), (2.2) has a solution for any
\lambda (\varepsilon ) > M(\varepsilon ).
Theorem 3.2 is proved.
Example 3.1. Let 1 < a \leq p < \infty , 0 < \alpha < 1, and \lambda (\varepsilon ) =
\alpha \varepsilon - p+1
(1 - \alpha ) (p - \varepsilon - 1)
. Then
y(t) = t\alpha is the solution of problem (2.1), (2.2). It is easy to see that A(p, \lambda ) < \infty . Thus, by
Theorem 3.2, there exists a constant C0 > 0 such that (3.3) holds.
References
1. R. A. Bandaliyev, Connection of two nonlinear differential equations with a two-dimensional Hardy operator in
weighted Lebesgue spaces with mixed norm, Electron. J. Different. Equat., 2016, 1 – 10 (2016).
2. R. A. Bandaliyev, P. Górka, Hausdorff operator in Lebesgue spaces, Math. Inequal. Appl., 22, 657 – 676 (2019).
3. P. R. Beesack, Hardy’s inequality and its extensions, Pacific J. Math., 11, 39 – 61 (1961).
4. P. R. Beesack, Integral inequalities involving a function and its derivatives, Amer. Math. Monthly, 78, 705 – 741
(1971).
5. J. S. Bradley, The Hardy inequalities with mixed norms, Canad. Math. Bull., 21, 405 – 408 (1978).
6. P. Drabek, A. Kufner, Note on spectra of quasilinear equations and the Hardy inequality, Nonlinear Analysis and
Applications: to V. Lakshmikantham on his 80th birthday, vol. 1, Kluwer Acad. Publ., Dordrecht (2003), p. 505 – 512.
7. D. E. Edmunds, W. D. Evans, Hardy operators, function spaces and embeddings, Springer-Verlag, Berlin (2004).
8. A. Fiorenza, M. R. Formica, A. Gogatishvili, On grand and small Lebesgue and Sobolev spaces and some applications
to PDE’s, Different. Integral Equat., 10, № 1, 21 – 46 (2018).
9. A. Fiorenza, B. Gupta, P. Jain, The maximal theorem in weighted grand Lebesgue spaces, Stud. Math., 188, № 2,
123 – 133 (2008).
10. A. Fiorenza, J. M. Rakotoson, Compactness, interpolation inequalities for small Lebesgue – Sobolev spaces and
applications, Cal. Var. Partial Different. Equat., 25, № 2, 187 – 203 (2006).
11. A. Fiorenza, C. Sbordone, Existence and uniqueness results for solutions of nonlinear equations with right-hand side
in L1 , Stud. Math., 127, № 3, 223 – 231 (1998).
12. L. Greco, A remark on the equality det, Df = \mathrm{D}\mathrm{e}\mathrm{t}Df , Different, Integral Equat., 6, 1089 – 1100 (1993).
13. L. Greco, T. Iwaniec, C. Sbordone, Inverting the p-harmonic operator, Manuscripta Math., 92, 249 – 258 (1997).
14. P. Gurka, Generalized Hardy’s inequality, Čas. pro Pêstoánı́ Mat., 109, 194 – 203 (1984).
15. G. H. Hardy, Notes on some points in the integral calculus, LX. An inequality between integrals, Messenger Math.,
54, 150 – 156 (1925).
16. G. H. Hardy, J. E. Littlewood, G. Pòlya, Inequalities, Cambridge Univ. Press (1934).
17. T. Iwaniec, C. Sbordone, On the integrability of the Jacobian under minimal hypotheses, Arch. Ration. Mech. and
Anal., 119, 129 – 143 (1992).
18. T. Iwaniec, C. Sbordone, Weak minima of variational integrals, J. reine und angew. Math., 454, 143 – 161 (1994).
19. T. Iwaniec, C. Sbordone, Riesz transforms and elliptic pde’s with VMO coefficients, J. Anal. Math., 74, 183 – 212
(1998).
20. P. Jain, M. Singh, A. P. Singh, Hardy type operators on grand Lebesgue spaces for non-increasing functions, Trans.
A. Razmadze Math. Inst., 270, 34 – 46 (2016).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 8
ON THE SOLVABILITY OF NONLINEAR ORDINARY DIFFERENTIAL EQUATION . . . 1019
21. P. Jain, M. Singh, A. P. Singh, Hardy-type integral inequalities for quasi-monotone functions, Georgian Math. J., 24,
№ 4, 523 – 533 (2016).
22. V. M. Kokilashvili, On Hardy’s inequality in weighted spaces, Bull. Akad. Nauk Geor. USSR, 96, 37 – 40 (1979).
23. A. Kufner, L. Maligranda, L. E. Persson, The Hardy inequality-about its history and some related results, Research
report, Luleå Univ. Technology, Sweden (2005).
24. A. Kufner, L.-E. Persson, Weighted inequalities of Hardy type, World Sci. Publ. Co, New Jersey etc. (2003).
25. V. G. Maz’ya, Sobolev spaces, Springer-Verlag, Berlin (1985).
26. B. Muckenhoupt, Hardy’s inequalities with weights, Stud. Math., 44, 31 – 38 (1972).
27. S. H. Saker, R. R. Mahmoud, A connection between weighted Hardy’s inequality and half-linear dynamic equations,
Adv. Different. Equat. 129, 1 – 15 (2019).
28. C. Sbordone, Grand Sobolev spaces and their applications to variational problems, Matematiche, 55, № 2, 335 – 347
(1996).
29. C. Sbordone, Nonlinear elliptic equations with right-hand side in nonstandard spaces, Rend. Semin. Mat. Fis.
Modena, Supp., 46, 361 – 368 (1998).
30. D. T. Shum, On a class of new inequalities, Trans. Amer. Math. Soc., 204, 299 – 341 (1975).
31. G. Talenti, Osservazione sopra una classe di disuguaglianze, Rend. Semin. Mat. Fiz. Milano, 39, 171 – 185 (1969).
32. G. Tomaselli, A class of inequalities, Boll. Unione Mat. Ital., 2, 622 – 631 (1969).
33. S. M. Umarkhadzhiev, On one-dimensional and multidimensional Hardy operators in grand Lebesgue spaces on sets
which may have infinite measure, Azerb. J. Math., 7, № 2, 132 – 152 (2017).
34. S. M. Umarkhadzhiev, On elliptic homogeneous differential operators in grand spaces, Russian Math., 64, 57 – 65
(2020).
Received 30.05.20
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| id | umjimathkievua-article-6146 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| language | English |
| last_indexed | 2026-03-24T03:26:11Z |
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| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-61462022-11-27T13:39:22Z On the solvability of nonlinear ordinary differential equation in grand Lebesgue spaces On the solvability of nonlinear ordinary differential equation in grand Lebesgue spaces Bandaliyev, R. A. Safarova, K. H. Bandaliyev, R. A. Safarova, K. H. Hardy inequality, nonlinear ordinary differential equation, grand Lebesgue spaces, absolutely continuous functions UDC 517.9We study the relationship between the second-order nonlinear ordinary differential equations and the Hardy inequality in grand Lebesgue spaces. In particular, we give a characterization of the Hardy inequality by using nonlinear ordinary differential equations in grand Lebesgue spaces. УДК 517.9 Про розв’язнiсть нелiнiйних звичайних диференцiальнихрiвнянь у великих просторах Лебега Вивчається зв’язок мiж нелiнiйними звичайними диференцiальними рiвняннями другого порядку та нерiвнiстю Гардi у великих просторах Лебега. Зокрема, дано характеристику нерiвностi Гардi нелiнiйними звичайними диференцiальними рiвняннями у великих просторах Лебега. Institute of Mathematics, NAS of Ukraine 2022-10-04 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6146 10.37863/umzh.v74i8.6146 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 8 (2022); 1011 - 1019 Український математичний журнал; Том 74 № 8 (2022); 1011 - 1019 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6146/9283 Copyright (c) 2022 Rovshan Bandaliyev |
| spellingShingle | Bandaliyev, R. A. Safarova, K. H. Bandaliyev, R. A. Safarova, K. H. On the solvability of nonlinear ordinary differential equation in grand Lebesgue spaces |
| title | On the solvability of nonlinear ordinary differential equation in grand Lebesgue spaces |
| title_alt | On the solvability of nonlinear ordinary differential equation in grand Lebesgue spaces |
| title_full | On the solvability of nonlinear ordinary differential equation in grand Lebesgue spaces |
| title_fullStr | On the solvability of nonlinear ordinary differential equation in grand Lebesgue spaces |
| title_full_unstemmed | On the solvability of nonlinear ordinary differential equation in grand Lebesgue spaces |
| title_short | On the solvability of nonlinear ordinary differential equation in grand Lebesgue spaces |
| title_sort | on the solvability of nonlinear ordinary differential equation in grand lebesgue spaces |
| topic_facet | Hardy inequality nonlinear ordinary differential equation grand Lebesgue spaces absolutely continuous functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6146 |
| work_keys_str_mv | AT bandaliyevra onthesolvabilityofnonlinearordinarydifferentialequationingrandlebesguespaces AT safarovakh onthesolvabilityofnonlinearordinarydifferentialequationingrandlebesguespaces AT bandaliyevra onthesolvabilityofnonlinearordinarydifferentialequationingrandlebesguespaces AT safarovakh onthesolvabilityofnonlinearordinarydifferentialequationingrandlebesguespaces |