Existence of positive solutions for a coupled system of nonlinear fractional differential equations
We study the following nonlinear boundary-value problems for fractional differential equations $$D^{\alpha} u(t) = f(t, v(t),D^{\beta - 1}v(t)), t > 0,\\ D^{\beta} v(t) = g(t, u(t),D^{\alpha - 1}u(t)), t > 0,\\ u > 0,\; v > 0 \in (0,\infty), \lim_{t\rightarrow 0+} u(t) =...
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2019
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507163676377088 |
|---|---|
| author | Ghanmi, A. Ганмі, А. |
| author_facet | Ghanmi, A. Ганмі, А. |
| author_sort | Ghanmi, A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T08:54:16Z |
| description | We study the following nonlinear boundary-value problems for fractional differential equations
$$D^{\alpha} u(t) = f(t, v(t),D^{\beta - 1}v(t)), t > 0,\\
D^{\beta} v(t) = g(t, u(t),D^{\alpha - 1}u(t)), t > 0,\\
u > 0,\; v > 0 \in (0,\infty),
\lim_{t\rightarrow 0+}
u(t) = \lim_{t\rightarrow 0+}
v(t) = 0,$$
where $1 < \alpha \leq 2$ and $1 < \beta \leq 2$. Under certain conditions on $f$ and $g$, the existence of positive solutions is obtained by
applying the Schauder fixed-point theorem. |
| first_indexed | 2026-03-24T02:04:57Z |
| format | Article |
| fulltext |
UDC 517.9
A. Ghanmi, S. Horrigue (Univ. Jeddah, KSA; Univ. Tunis El Manar, Tunisia)
EXISTENCE OF POSITIVE SOLUTIONS FOR A COUPLED SYSTEM
OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS
ПРО IСНУВАННЯ ДОДАТНИХ РОЗВ’ЯЗКIВ ЗВ’ЯЗАНИХ СИСТЕМ
НЕЛIНIЙНИХ ДРОБОВО-ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
We study the following nonlinear boundary-value problems for fractional differential equations
D\alpha u(t) = f(t, v(t), D\beta - 1v(t)), t > 0,
D\beta v(t) = g(t, u(t), D\alpha - 1u(t)), t > 0,
u > 0 and v > 0 in (0,\infty ),
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow 0+
u(t) = \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow 0+
v(t) = 0,
where 1 < \alpha \leq 2 and 1 < \beta \leq 2. Under certain conditions on f and g, the existence of positive solutions is obtained by
applying the Schauder fixed-point theorem.
Вивчаються нелiнiйнi граничнi задачi для дробово-диференцiальних рiвнянь
D\alpha u(t) = f(t, v(t), D\beta - 1v(t)), t > 0,
D\beta v(t) = g(t, u(t), D\alpha - 1u(t)), t > 0,
u > 0 i v > 0 в (0,\infty ),
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow 0+
u(t) = \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow 0+
v(t) = 0,
де 1 < \alpha \leq 2 та 1 < \beta \leq 2. За деяких умов, накладених на f i g, iснування додатних розв’язкiв встановлюється
за допомогою теореми Шаудера про нерухому точку.
1. Introduction. Fractional differential equations are gaining much importance and attention since
they can be applied in various fields of science and engineering. Many phenomena in viscoelasticity,
electrochemistry, control, porous media, electromagnetic, etc., can be modeled by fractional diffe-
rential equations. They also serve as an excellent tool for the description of hereditary properties
of various materials and processes. We refer the reader to [8, 11, 12, 20] and references therein for
details.
Recently, many authors have investigated sufficient conditions for the existence of solutions for
the following coupled systems of nonlinear fractional differential equations with different boundary
conditions on finite domain
D\alpha u(t) = f(t, v(t)),
D\beta v(t) = g(t, u(t)),
and more generally,
D\alpha u(t) = f(t, v(t), D\mu v(t)),
c\bigcirc A. GHANMI, S. HORRIGUE, 2019
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1 37
38 A. GHANMI, S. HORRIGUE
D\beta v(t) = g(t, u(t), D\nu u(t)),
where D\alpha is the standard Riemann – Liouville deravitive of order \alpha , see, for example, [2, 4, 6, 9,
10, 16 – 19]. However, to the best of our knowledge few papers consider the existence of solutions
of fractional differential equations on the half-line. Maagli in [13], studied the existence of solutions
for differential equations involving the Riemann – Liouville fractional derivative on the half-line
\BbbR + := (0,\infty )
D\alpha u(t) = f(t, u(t)) in (0,\infty ),
u > 0 in (0,\infty ),
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow 0
u(t) = 0,
where 1 < \alpha \leq 2 and f is a Borel measurable function in (0,\infty )\times (0,\infty ).
Maagli and Dhifli [14] considered the following boundary-value problem for fractional differen-
tial equations:
D\alpha u(t) = f
\bigl(
t, u(t), D\alpha - 1u(t)
\bigr)
in (0,\infty ),
u > 0 in (0,\infty ),
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow 0
u(t) = 0,
where 1 < \alpha \leq 2 and f is a Borel measurable function in (0,\infty )\times (0,\infty )\times (0,\infty ) satisfying some
appropriate conditions.
Our aim in this paper is to extend the above results to the coupled system of nonlinear fractional
differential equations on an unbounded domain
D\alpha u(t) + f
\bigl(
t, v(t), D\beta - 1v(t)
\bigr)
= 0, t > 0,
D\beta v(t) + g
\bigl(
t, u(t), D\alpha - 1u(t)
\bigr)
= 0, t > 0,
u > 0 and v > 0 in (0,\infty ),
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow 0+
u(t) = \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow 0+
v(t) = 0,
(1.1)
where 1 < \alpha \leq 2, 1 < \beta \leq 2, f and g are Borel measurable functions in \BbbR + \times \BbbR + \times \BbbR + satisfying
the following assumptions:
(H1) f and g are continuous with respect to the second and third variable.
(H2) There exist nonnegative measurable functions h1, h2, k1, and k2 on \BbbR + \times \BbbR + \times \BbbR + such
that
(i) for all x, y, z \in \BbbR + we have
| f(x, y, z)| \leq h(x, y, z),
| g(x, y, z)| \leq k(x, y, z),
where h(x, y, z) := yh1(x, y, z) + zh2(x, y, z) and k(x, y, z) := yk1(x, y, z) + zk2(x, y, z);
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
EXISTENCE OF POSITIVE SOLUTIONS FOR A COUPLED SYSTEM . . . 39
(ii) for j = 1, 2, the functions hj and kj are nondecreasing with respect to the second and the
third variables and satisfying for all x \in \BbbR +
\mathrm{l}\mathrm{i}\mathrm{m}
(y,z)\rightarrow (0,0)
hj(x, y, z) = \mathrm{l}\mathrm{i}\mathrm{m}
(y,z)\rightarrow (0,0)
kj(x, y, z) = 0 for j = 1, 2;
(iii) the integrals
\int \infty
0
h(t, \omega \beta (t), 1) dt and
\int \infty
0
k(t, \omega \alpha (t), 1) dt converge, where \omega \nu =
t\nu
\Gamma (\nu )
for 1 < \nu \leq 2.
Our main result is the following.
Theorem 1.1. Assume (H1) and (H2). Then problem (1.1) has infinitely many solutions. More
precisely, there exists a number b > 0 such that for each c \in (0, b], problem (1.1) has a continuous
solution (u, v) satisfying
u(t) = c\omega \alpha (t) +
\infty \int
0
\Biggl(
1 -
\biggl( \Bigl(
1 - s
t
\Bigr) +\biggr) \alpha - 1
\Biggr)
f(s, v(s), D\beta - 1v(s)) ds,
v(t) = c\omega \beta (t) +
\infty \int
0
\Biggl(
1 -
\biggl( \Bigl(
1 - s
t
\Bigr) +\biggr) \beta - 1
\Biggr)
g(s, u(s), D\alpha - 1u(s)) ds
and
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
u(t)
\omega \alpha (t)
= \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
D\alpha - 1u(t) = c,
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
v(t)
\omega \beta (t)
= \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
D\beta - 1v(t) = c,
where, for every x \in \BbbR , x+ = \mathrm{m}\mathrm{a}\mathrm{x}(x, 0).
This paper is organized as follows. In Section 2, some facts and results about fractional calculus
are given. We prove the main result in Section 3. Finally, we conclude this paper by considering an
example in Section 4.
2. Preliminaries. In this section, we introduce some necessary definitions and results which
will used throughout this paper.
Definition 2.1. The Riemann – Liouville fractional integral of order \theta > 0 of any function u :
\BbbR + \rightarrow \BbbR is defined by
I\theta u(t) =
1
\Gamma (\theta )
t\int
0
(t - s)\theta - 1u(s) ds
provided the right-hand side is point-wise defined on \BbbR +.
Definition 2.2. The Riemann – Liouville fractional derivative of order \theta > 0 of a continuous
function u : (0,\infty ) - \rightarrow \BbbR is given by
D\theta u(t) =
1
\Gamma (n - \theta )
\biggl(
d
dt
\biggr) n
t\int
0
u(s)
(t - s)\theta - n+1
ds,
where \Gamma is the Gamma function and n = [\theta ] + 1, provided that the right-hand side is point-wise
defined.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
40 A. GHANMI, S. HORRIGUE
Remark 2.1. The following properties are well known (see [12, 15]):
(i) D\theta I\theta u(t) = u(t), a.e. in \BbbR +, \theta > 0, u \in L1
loc([0,\infty )).
(ii) I\delta I\theta u(t) = I\theta +\delta u(t), a.e. in \BbbR +, \theta + \delta \geq 1, u \in L1
loc([0,\infty )).
(iii) Let \theta > 0, then D\theta u(t) = 0 if and only if u(t) =
\sum n
j=1
cjt
\theta - j , where n = - [ - \theta ] the
smallest integer greater than or equal to \theta and (c1, . . . , cn) \in \BbbR n.
(iv) Let 1 < \theta \leq 2, and t \geq 0, then we have
I\theta - 1(1)(t) = \omega \theta (t) =
t\theta - 1
\Gamma (\theta )
.
In the sequel, we denote by C([0,\infty ]) the set of continuous functions u on \BbbR + such that
\mathrm{l}\mathrm{i}\mathrm{m}t\rightarrow 0+ u(t) and \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
u(t) exist. It is easy to see that C([0,\infty ]) is a Banach space with the norm
\| u\| \infty = \mathrm{s}\mathrm{u}\mathrm{p}t\geq 0 | u(t)| . For 1 < \theta \leq 2, we define
E\theta =
\Bigl\{
u \in C([0,\infty ]) : D\theta - 1(\omega \theta u) \in C([0,\infty ])
\Bigr\}
(2.1)
endowed with the norm \| u\| \theta = \| D\theta - 1(\omega \theta u)\| \infty . Then, it’s easy to see that the map
(E\theta , \| .\| \theta ) - \rightarrow
\bigl(
C([0,\infty ]), \| .\| \infty
\bigr)
,
u \mapsto - \rightarrow D\theta - 1\omega \theta (u)
is an isometry. It follows that (E\theta , \| .\| \theta ) is a Banach space.
Let E = E\alpha \times E\beta endowed with the norm
\| (u, v)\| = \mathrm{m}\mathrm{a}\mathrm{x}
\bigl(
\| u\| \alpha , \| u\| \beta
\bigr)
,
then (E, \| .\| ) is a Banach space.
Next, we quote some results in the following lemmas that will be used later.
Lemma 2.1 (see [7]). Let 1 < \theta \leq 2 and let f be a function in C([0,\infty )) such that f(0) = 0
and D\theta - 1f belongs to C([0,\infty )). Then, for t \geq 0, we have
I\theta - 1D\theta - 1f(t) = f(t).
Lemma 2.2 (see [14]). Let m1,m2 \in \BbbR and u \in C([0,\infty )) such that D\theta - 1(\omega \theta u) \in C([0,\infty ))
and m1 \leq D\theta - 1(\omega \theta u)(t) \leq m2 for all t \geq 0. Then, for each t \geq 0,
m1 \leq u(t) \leq m2.
In particular, \| u\| \infty \leq \| D\theta - 1(\omega \theta u)\| \infty and E\theta \subset C([0,\infty ]).
Let \scrF \mu =
\bigl\{
u \in E\mu : 0 \leq D\mu - 1(\omega \mu u) \leq 1
\bigr\}
. Then we have the following result.
Lemma 2.3 (Assume (H2)). Then the family of functions\left\{
t\int
0
\Bigl(
1 - s
t
\Bigr) \alpha - 1
f
\bigl(
s, \omega \beta (s)v(s), D
\alpha - 1(\omega \beta v)(s)
\bigr)
ds, v \in \scrF \beta
\right\}
and \left\{
t\int
0
\Bigl(
1 - s
t
\Bigr) \beta - 1
g
\Bigl(
s, \omega \alpha (s)u(s), D
\beta - 1(\omega \alpha u)(s)
\Bigr)
ds, u \in \scrF \alpha
\right\}
are relatively compact in C([0,\infty ]).
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
EXISTENCE OF POSITIVE SOLUTIONS FOR A COUPLED SYSTEM . . . 41
Proof. The proof is very similar to and based on the technique used in the proofs of [13]
(Lemma 1.5); hence we omit it.
3. Proof of Theorem 1.1. Let BC([0,\infty )) be the Banach space of all bounded continuous
real-valued functions on [0,\infty ), endowed with the sup-norm \| .\| \infty . In order to prove Theorem 1.1,
we need the following compactness criterion for a subset of BC([0,\infty )), which is a consequence of
the well-known Arzela – Ascoli theorem. This compactness criterion is an adaptation of a lemma due
to Avramescu [3]. In order to formulate this criterion, we note that a set U of real-valued functions
defined on [0,\infty ) is said to be equiconvergent at \infty if all the functions in U are convergent in \BbbR at
the point \infty and, in addition, for each \epsilon > 0, there exists T = T (\epsilon ) > 0 such that, for any function
\psi \in U, we have | \psi (t) - \mathrm{l}\mathrm{i}\mathrm{m}s\rightarrow \infty \psi (s)| < \epsilon for t > T.
Theorem 3.1 (see [3]). Let U be an equicontinuous and uniformly bounded subset of the Banach
space BC([0,\infty )). If U is equiconvergent at \infty , it is also relatively compact.
In the sequel, for x, y, z \in \BbbR +, we denote
F (x, y, z) = \omega \beta (x)h1(x, y, z) + h2(x, y, z) and G(x, y, z) = \omega \alpha (x)k1(x, y, z) + k2(x, y, z).
It follows from (H2) and Lebesgue’s theorem that
\mathrm{l}\mathrm{i}\mathrm{m}
s\rightarrow 0
\infty \int
0
F (t, s\omega \beta (t), s)dt = 0 and \mathrm{l}\mathrm{i}\mathrm{m}
s\rightarrow 0
\infty \int
0
G(t, s\omega \alpha (t), s)dt = 0.
Hence, we can fix a number 0 < \lambda < 1 such that
\mathrm{m}\mathrm{a}\mathrm{x}
\left( \infty \int
0
F (t, \lambda \omega \beta (t), \lambda ) dt,
\infty \int
0
G(t, \lambda \omega \alpha (t), \lambda ) dt
\right) \leq 1
3
.
Let b =
2\lambda
3
and c \in (0, b]. To apply a fixed point argument, set
\Lambda := \Lambda \alpha \times \Lambda \beta ,
where
\Lambda \alpha =
\biggl\{
u \in E\alpha :
c
2
\leq D\alpha - 1(\omega \alpha u) \leq
3c
2
\biggr\}
and \Lambda \beta =
\biggl\{
v \in E\beta :
c
2
\leq D\beta - 1(\omega \beta v) \leq
3c
2
\biggr\}
.
Then \Lambda is a nonempty closed bounded and convex set in E. Now, we define the operator T on \Lambda by
T (u, v) := (A1v,A2u),
where, for a given t > 0,
A1v(t) = c+
\infty \int
0
\Biggl(
1 -
\biggl( \Bigl(
1 - s
t
\Bigr) +\biggr) \alpha - 1
\Biggr)
f
\bigl(
s, \omega \beta (s)v(s), D
\alpha - 1(\omega \beta v)(s)
\bigr)
ds, v \in \Lambda \beta ,
and
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
42 A. GHANMI, S. HORRIGUE
A2u(t) = c+
\infty \int
0
\Biggl(
1 -
\biggl( \Bigl(
1 - s
t
\Bigr) +\biggr) \beta - 1
\Biggr)
f
\Bigl(
s, \omega \alpha (s)u(s), D
\beta - 1(\omega \alpha u)(s)
\Bigr)
ds, u \in \Lambda \alpha .
First, we shall prove that the operator T maps \Lambda into itself. Let v \in \Lambda \beta . Using Lemma 2.3, we
deduce that the function A1v is in C([0,\infty ]). On the other hand, for t \geq 0, we obtain
\omega \alpha (t)A1v = \omega \alpha (t)
\left( c+ \infty \int
0
f
\bigl(
s, \omega \beta (s)v(s), D
\alpha - 1(\omega \beta v)(s)
\bigr)
ds
\right) - I\alpha (f(., \omega \beta v,D
\alpha - 1(\omega \beta v)))(t).
Hence, applying D\alpha - 1 on both sides of this equality, we conclude that, for each t \geq 0,
D\alpha - 1(\omega \alpha A1v)(t) = c+
\infty \int
t
f
\bigl(
s, \omega \beta (s)v(s), D
\alpha - 1(\omega \beta v)(s)
\bigr)
ds. (3.1)
This implies that D\alpha - 1(\omega \alpha A1v) is in C([0,\infty ]) and A1\Lambda \alpha \subset E\alpha . Furthermore, for v \in \Lambda \beta and
t \geq 0, we have
| D\alpha - 1(\omega \alpha A1v)(t) - c| \leq
\infty \int
0
\bigm| \bigm| f \bigl( s, \omega \beta (s)v(s), D
\alpha - 1(\omega \beta v)(s)
\bigm| \bigm| \bigr) ds \leq
\leq
\infty \int
0
h
\bigl(
s, \omega \beta (s)v(s), D
\alpha - 1(\omega \beta v)(s)
\bigr)
ds \leq
\leq
\infty \int
0
h
\biggl(
s,
3c
2
\omega \alpha (s),
3c
2
\biggr)
ds =
=
3c
2
\infty \int
0
F
\biggl(
s,
3c
2
\omega \beta (s),
3c
2
\biggr)
ds \leq
\leq 3c
2
\infty \int
0
F (s, \lambda \omega \beta (s), \lambda ) ds \leq
c
2
.
It follows that for each t \geq 0
c
2
\leq D\alpha - 1(\omega \alpha A1v)(t) \leq
3c
2
.
So, since from Lemma 2.3 A1\Lambda \alpha \subset C([0,\infty ]), we conclude that \Lambda \alpha is invariant under A1. Similarly
we prove that \Lambda \beta is invariant under A2 and hence \Lambda is invariant under T.
Next, we prove that T\Lambda is relatively compact in (E, \| \cdot \| ). For any v \in \Lambda \beta and t > 0, we have
d
dt
D\alpha - 1(\omega \alpha A1v)(t) = - f
\bigl(
x, \omega \beta (t)v(t), D
\beta - 1(\omega \beta v)(t)
\bigr)
a.e. in \BbbR +.
Since
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
EXISTENCE OF POSITIVE SOLUTIONS FOR A COUPLED SYSTEM . . . 43\bigm| \bigm| f(t, \omega \beta (t)v(t), D
\beta - 1(\omega \beta v)(t))
\bigm| \bigm| \leq h
\bigl(
t, \omega \beta (t)v(t), D
\beta - 1(\omega \beta v)(t)
\bigr)
\leq h(t, \omega \beta (t), 1) (3.2)
and
\infty \int
0
h(x, \omega \beta (t), 1) dt <\infty , (3.3)
it follows that the family
\bigl\{
D\alpha - 1(\omega \alpha A1v), v \in \Lambda \beta
\bigr\}
is equicontinuous on [0,\infty ]. Moreover,\bigl\{
D\alpha - 1(\omega \alpha A1v), v \in \Lambda \beta
\bigr\}
is uniformly bounded. Thus, by Theorem 3.1, to prove that
\bigl\{
D\alpha - 1(\omega \alpha u),
u \in \Lambda \beta
\bigr\}
is relatively compact, it suffice to prove that all elements of
\bigl\{
D\alpha - 1(\omega \alpha v), v \in \Lambda \beta
\bigr\}
are
equiconvergent at infinity. Endeed, since for all v \in \Lambda \beta , D
\alpha - 1(\omega \alpha A1v) \subset C([0,\infty ]), it follows that
\mathrm{l}\mathrm{i}\mathrm{m}t - \rightarrow \infty D\alpha - 1(\omega \alpha v)(t) exists. On the other hand, it follows from (3.2), (3.3) and the dominated
convergence theorem that
\mathrm{l}\mathrm{i}\mathrm{m}
t - \rightarrow \infty
\infty \int
t
f
\bigl(
s, \omega \beta (s)v(s), D
\alpha - 1(\omega \beta v)(s)
\bigr)
ds = 0.
So, using (3.1), we obtain
\mathrm{l}\mathrm{i}\mathrm{m}
t - \rightarrow \infty
\bigm| \bigm| \bigm| D\beta - 1(\omega \alpha v)(t) - \mathrm{l}\mathrm{i}\mathrm{m}
t - \rightarrow \infty
D\beta - 1(\omega \alpha v)(t)
\bigm| \bigm| \bigm| =
= \mathrm{l}\mathrm{i}\mathrm{m}
t - \rightarrow \infty
\infty \int
t
f
\bigl(
s, \omega \beta (s)v(s), D
\alpha - 1(\omega \beta v)(s)
\bigr)
ds = 0.
That is
\bigl\{
D\beta - 1(\omega \alpha u), u \in \Lambda \beta
\bigr\}
is relatively compact in (C([0,\infty ]), \| \cdot \| \infty ). This implies that A1\Lambda \alpha
is relatively compact in (E\alpha , \| \cdot \| \alpha ).
Similar process can be repeated to prove that A2\Lambda \beta is relatively compact in (E\beta , \| \cdot \| \beta ). Thus
T\Lambda is relatively compact in (E, \| \cdot \| ).
Now, we prove the continuity of T in \Lambda . Let (vk) be a sequence in \Lambda \beta such that
\| vk - v\| \beta =
\bigm\| \bigm\| D\beta - 1(\omega \beta vk) - D\beta - 1(\omega \beta v)
\bigm\| \bigm\|
\infty \rightarrow 0 as k \rightarrow \infty .
Then, by Lemma 2.2, \| vk - v\| \infty \rightarrow 0 as k \rightarrow \infty , and, for any t \in [0,\infty ], we have\bigm| \bigm| D\alpha - 1(\omega \alpha A1vk)(t) - D\alpha - 1(\omega \alpha A1v)(t)
\bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \int
t
\bigl[
f(s, \omega \alpha (s)vk(s), D
\beta - 1(\omega \beta vk)(s)) - f(s, \omega \beta (s)v(s), D
\beta - 1(\omega \beta v)(s))
\bigr]
ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\infty \int
0
\bigm| \bigm| \bigm| f(s, \omega \beta (s)vk(s), D
\alpha - 1(\omega \beta vk)(s)) - f(s, \omega \beta (s)v(s), D
\beta - 1(\omega \beta v)(s))
\bigm| \bigm| \bigm| ds.
Since\bigm| \bigm| f(s, \omega \beta (s)vk(s), D
\alpha - 1(\omega \beta vk)(s)) - f(s, \omega \beta (s)v(s), D
\alpha - 1(\omega \beta v)(s))
\bigm| \bigm| \leq 2h(s, \omega \beta (s), 1),
and, by (\mathrm{H}1) and Lebesgue’s theorem, we get
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
44 A. GHANMI, S. HORRIGUE\bigm\| \bigm\| A1vk - A1v
\bigm\| \bigm\|
\alpha
=
\bigm\| \bigm\| D\alpha - 1(\omega \alpha A1vk) - D\alpha - 1(\omega \alpha A1v)
\bigm\| \bigm\|
\infty \rightarrow 0 as k \rightarrow \infty .
Hence, A1 is continuous in \Lambda \alpha . In a similar way, A2 is continuous in \Lambda \beta and so T is continuous in
\Lambda . Therefore, by Schauder fixed point theorem there exists (x, y) \in \Lambda such that T (x, y) = (x, y).
That is, for t > 0,
x(t) = c+
\infty \int
0
\Biggl(
1 -
\biggl( \Bigl(
1 - s
t
\Bigr) +\biggr) \alpha - 1
\Biggr)
f(s, \omega \beta (s)y(s), D
\beta - 1(\omega \beta y)(s)) ds
and
y(t) = c+
\infty \int
0
\Biggl(
1 -
\biggl( \Bigl(
1 - s
t
\Bigr) +\biggr) \beta - 1
\Biggr)
g(s, \omega \alpha (s)x(s), D
\alpha - 1(\omega \alpha x)(s)) ds.
We put u(t) = \omega \alpha (t)x(t) and v(t) = \omega \beta (t)y(t). Then for any t > 0, we have
u(t) = c\omega \alpha (t) + \omega \alpha (t)
\infty \int
0
\Biggl(
1 -
\biggl( \Bigl(
1 - s
t
\Bigr) +\biggr) \alpha - 1
\Biggr)
f(s, v(s), D\beta - 1(v)(s)) ds (3.4)
and
v(t) = c\omega \beta (t) + \omega \beta (t)
\infty \int
0
\Biggl(
1 -
\biggl( \Bigl(
1 - s
t
\Bigr) +\biggr) \beta - 1
\Biggr)
g(s, u(s), D\alpha - 1(u)(s)) ds. (3.5)
Moreover, for t > 0, we obtain
c
2
\omega \alpha (t) \leq u(t) \leq 3c
2
\omega \alpha (t),
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
u(t)
\omega \alpha (t)
= \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
D\alpha - 1u(t) = c
and
c
2
\omega \beta (t) \leq v(t) \leq 3c
2
\omega \beta (t),
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
v(t)
\omega \beta (t)
= \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
D\beta - 1v(t) = c.
It remains to show that u is a solution of problem (1.1). Indeed, applying D\alpha on both sides of
(3.4) and D\beta on both sides of (3.5), we obtain by Remark 2.1
D\alpha u(x) = - f(x, u,D\beta - 1u) a.e. in \BbbR +
and
D\beta v(x) = - g(x, u,D\alpha - 1u) a.e. in \BbbR +.
Theorem 1.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
EXISTENCE OF POSITIVE SOLUTIONS FOR A COUPLED SYSTEM . . . 45
Example 3.1. Let p1, p2, q1, q2 \geq 0 such that \mathrm{m}\mathrm{a}\mathrm{x}(p1, q1) > 1, \mathrm{m}\mathrm{a}\mathrm{x}(p2, q2) > 1 and let k, h
be measurable functions satisfying
\infty \int
0
t(\alpha - 1)p1
\bigm| \bigm| k(t)\bigm| \bigm| dt <\infty
and
\infty \int
0
t(\beta - 1)p2
\bigm| \bigm| h(t)\bigm| \bigm| dt <\infty .
Then, there exists a constant b > 0 such that for each c \in (0, b], the problem
D\alpha u+ k(x)vp1(D\beta - 1v)q1 = 0, u > 0, in \BbbR +,
D\beta v + h(x)up2(D\alpha - 1u)q2 = 0, v > 0, in \BbbR +,
\mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow 0+
u(x) = \mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow 0+
v(x) = 0,
has a continuous solution (u, v) in \BbbR + satisfying
\mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow 0+
u(x)
\omega \alpha (x)
= \mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow \infty
D\alpha - 1u(x) = c
and
\mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow 0+
v(x)
\omega \beta (x)
= \mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow \infty
D\beta - 1v(x) = c.
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Received 21.04.16,
after revision — 06.09.16
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
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| id | umjimathkievua-article-1417 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:04:57Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/0b/8eb476c3ffc112729dbc2f7c538d070b.pdf |
| spelling | umjimathkievua-article-14172019-12-05T08:54:16Z Existence of positive solutions for a coupled system of nonlinear fractional differential equations Про iснування додатних розв’язкiв зв’язаних систем нелiнiйних дробово-диференцiальних рiвнянь Ghanmi, A. Ганмі, А. We study the following nonlinear boundary-value problems for fractional differential equations $$D^{\alpha} u(t) = f(t, v(t),D^{\beta - 1}v(t)), t > 0,\\ D^{\beta} v(t) = g(t, u(t),D^{\alpha - 1}u(t)), t > 0,\\ u > 0,\; v > 0 \in (0,\infty), \lim_{t\rightarrow 0+} u(t) = \lim_{t\rightarrow 0+} v(t) = 0,$$ where $1 < \alpha \leq 2$ and $1 < \beta \leq 2$. Under certain conditions on $f$ and $g$, the existence of positive solutions is obtained by applying the Schauder fixed-point theorem. Вивчаються нелiнiйнi граничнi задачi для дробово-диференцiальних рiвнянь $$D^{\alpha} u(t) = f(t, v(t),D^{\beta - 1}v(t)), t > 0,\\ D^{\beta} v(t) = g(t, u(t),D^{\alpha - 1}u(t)), t > 0,\\ u > 0,\; v > 0 \in (0,\infty), \lim_{t\rightarrow 0+} u(t) = \lim_{t\rightarrow 0+} v(t) = 0,$$ де $1 < \alpha \leq 2$ та $1 < \beta \leq 2$. За деяких умов, накладених на $f$ i $g$, iснування додатних розв’язкiв встановлюється за допомогою теореми Шаудера про нерухому точку. Institute of Mathematics, NAS of Ukraine 2019-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1417 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 1 (2019); 37-46 Український математичний журнал; Том 71 № 1 (2019); 37-46 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1417/401 Copyright (c) 2019 Ghanmi A. |
| spellingShingle | Ghanmi, A. Ганмі, А. Existence of positive solutions for a coupled system of nonlinear fractional differential equations |
| title | Existence of positive solutions for a coupled system of nonlinear
fractional differential equations |
| title_alt | Про iснування додатних розв’язкiв зв’язаних систем
нелiнiйних дробово-диференцiальних рiвнянь |
| title_full | Existence of positive solutions for a coupled system of nonlinear
fractional differential equations |
| title_fullStr | Existence of positive solutions for a coupled system of nonlinear
fractional differential equations |
| title_full_unstemmed | Existence of positive solutions for a coupled system of nonlinear
fractional differential equations |
| title_short | Existence of positive solutions for a coupled system of nonlinear
fractional differential equations |
| title_sort | existence of positive solutions for a coupled system of nonlinear
fractional differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1417 |
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