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ABSTRACT
This paper deals with existence, uniqueness and global behaviour of a positive solution for the fractional boundary value problem in
with the condition
where
,
, p>1,
, the differential operator is taken in the Riemann–Liouville sense and
are non-negative and continuous functions that may are singular at x=0 or x=1 and satisfies some appropriate conditions.
1. Introduction
In this paper, we consider the following nonlinear boundary value problem of fractional differential equation with p-Laplacian
(1)
(1) where
and for
,
,
and
. The operator
(and
) is the Riemann–Liouville derivative of order α (of order β). The functions ψ and a are positive and continuous in
that may are singular at x=0 or x=1 and satisfying some conditions detailed below. Then, we will adress the question of existence, uniqueness and exact asymptotic behaviour of positive solutions to problem (Equation1
(1)
(1) ).
We believe that there are few papers dedicated to the study of fractional differential equations with p-Laplacian operator, see for instance [Citation1–11]. These types of problems arise in applied field such as turbulent flow of a gas in a porous medium, biophysics, plasma physics and material science.
This work is motivated by recent advances in the study of p-Laplacian fractional differential equations involving singular or sublinear nonlinearities with different boundary conditions. Namely, in [Citation12], Liu considered the fractional differential equation
(2)
(2) where
,
and f is a non-negative function on
allowed to be singular at t=0. The author proved the existence of positive solution with fractional non-local integral boundary conditions.
Recently, in [Citation13], Mâagli et al. considered the following problem
(3)
(3) where
,
and the function a is required to satisfy some assumptions related to
, the set of all Karamata functions L defined on
, by
for some
where c>0 and z is a continuous function on
with
To describe the result of [Citation13] in more details, we need some notations.
For two non-negative functions f and g defined on a set S, the notation
,
means that there exists c>0 such that
, for all
.
.
In [Citation13], Mâagli et al. studied problem (Equation3(3)
(3) ) where a verifies the assumption
:
satisfying for each
(4)
(4) where
,
and
such that
(5)
(5) Based on the Schauder fixed-point theorem, the authors showed in [Citation13] the following result.
Theorem 1.1
Assume that a satisfies . Then problem (Equation3
(3)
(3) ) has a unique positive solution
satisfying for
,
(6)
(6) where
defined on
by
(7)
(7)
The main goal of this paper is to improve and extend the above results on the boundary behaviour of solutions to problem (Equation1(1)
(1) ). More precisely, we consider the following assumption:
a and ψ are non-negative functions in
satisfying for each
(8)
(8) where
,
,
and
such that
(9)
(9)
Theorem 1.2
Assume , then problem (Equation1
(1)
(1) ) has a unique positive solution
satisfying for
,
(10)
(10)
The rest of the paper is as follows. In Section 2, we state some already known results on functions in and some definitions and lemmas from fractional calculus theory. In Section 3, we present some necessary conditions to existence result and we prove our main result.
2. Preliminaries: Karamata properties and fractional calculus
In what follows, we are quoting without proof some fundamental properties of functions belonging to the class collected from [Citation14, Citation15]. We recall that a function L defined on
belongs to the class
, if
(11)
(11) for some
c>0 and z is a continuous function on
with
Proposition 2.1
A function L is in
if and only if L is a positive function in
such that
(12)
(12)
Let
and
. Then we have
(13)
(13)
Let
and
Then we have
(14)
(14)
As a standard example of functions belonging to the class (see [Citation15]), we give
Example 2.2
Let and
. Let
and w be a sufficiently large positif real number such that the function
(15)
(15) is defined and positive on
, where
. Then we have
.
Lemma 2.3
Let L be a function in
. Then we have
(16)
(16) In particular,
(17)
(17)
If
converges, then
(18)
(18) In particular,
(19)
(19)
Applying Karamata's theorem, we get the following.
Lemma 2.4
Let and L be a function in
defined on
for some
. We have
If
, then
converges and
(20)
(20)
If
, then
diverges and
(21)
(21)
Next, we give some definitions and fundamental facts of fractional calculus theory, which can be found in [Citation16, Citation17].
Definition 2.5
Let , the Riemann–Liouville fractional integral of order γ of a measurable function
is given by
(22)
(22) provided that the right-hand side is pointwise defined on
.
Here Γ is the Euler Gamma function.
Definition 2.6
The Riemann–Liouville derivative of order of a measurable function
is given by
provided that the right-hand side is pointwise defined on
.
Here , where
means the integer part of the number γ.
Lemma 2.7
Let , n be the smallest integer greater than or equal to α and
. Then, we have
for
and
for a.e.
.
if and only if
, where
.
Since our approach is based on potential theory, we recall in the following some basic tools. For , we denote by
the Green's function for the boundary value problem (Equation3
(3)
(3) ). From [Citation13], we have
(23)
(23) where
.
Proposition 2.8
see [Citation13]
Let and f be a non-negative measurable function on
. Then we have
For
,
(24)
(24)
For
,
if and only if
.
If the map
is continuous and integrable on
, then
is the unique solution in
of the boundary value problem
(25)
(25)
Lemma 2.9
[Citation13]
Let and let for
(26)
(26) with
and
. Assume that
(27)
(27) Then we have for
,
(28)
(28)
Remark 2.1
We need to verify condition and
in Lemma 6, only if
and
. This is due to Lemma 2.
3. Proof of main result
We begin this section by announcing and proving some propositions and lemmas, that will play a crucial role in the proof of our main result.
Let p>1 and let . Then
is the function inverse of
. We say easily that
is increasing monotone and multiplicative on
.
Lemma 3.1
Let . Let f be a non-negative function in
such that
and
are continuous and integrable on
, then the boundary value problem
(29)
(29) has a unique positive solution in
given by
(30)
(30)
Proof.
Since is continuous and integrable on
, we deduce by Proposition 2
that for
, we have
(31)
(31) Thus
(32)
(32) In addition, using the fact that
is continuous and integrable on
, we conclude again by Proposition 2
that problem (Equation29
(29)
(29) ) has a unique solution u in
given by
(33)
(33)
Here, below we provide a crucial property concerning continuity.
Lemma 3.2
Let ϕ and ψ be tow non-negative functions in such that
(34)
(34) Then the family
(35)
(35) is relatively compact in
.
Proof.
Let such that
for all
.
Let , we have
Using Proposition 2
, we obtain that
(36)
(36) Thus
is uniformly bounded. Now, let us prove that
is equicontinuous in
.
Let , then we have
For every
, we have
(37)
(37) and
(38)
(38) Then we obtain by Lesbegue's theorem that
(39)
(39) Now, let
, we have
Using the fact that for
,
(40)
(40) and
(41)
(41) we get again by Lesbegue's theorem that
(42)
(42) Furthermore, for
, we have
(43)
(43) By similar arguments as above, we deduce that
(44)
(44) Finally, we conclude that the familly
is equicontinuous in
. Hence, by Ascoli's theorem, we deduce that
is relatively compact in
.
Proposition 3.3
Assume and suppose that there exist a non-negative function θ in
such that
(45)
(45) and
(46)
(46) where
for
.
Then problem (Equation1(1)
(1) ) has a unique solution
satisfying for each
(47)
(47)
Proof.
Let and θ be a non-negative function satisfying for each
(48)
(48) Put
. We consider the closed convex set given by
(49)
(49) Using Lemma 3.2 and Proposition 2 (ii), we easily see that the function
belongs to
and satisfies (Equation48
(48)
(48) ). So Y is not empty. In order to use the Schauder's fixed point theorem, we denote
and we define the operator T on Y by
(50)
(50) We need to check that the operator T has a fixed point v in Y. For this choice of
, we will prove that T map Y into itself. Indeed, let
, by using (Equation48
(48)
(48) ), we have
and
Furthermore, using (Equation48
(48)
(48) ), we have for each
(51)
(51) This implies by Proposition 2 (ii) that
(52)
(52) Hence, it follows from Lemma 3.2 that the family TY is relatively compact in
. So Y is invariant under T.
Next, we shall prove the continuity of T. Let be a sequence in Y which converges uniformly to v in Y.
For , we have
For
, we have
(53)
(53) and for every
,
(54)
(54) Using Proposition 2 (ii) and (Equation45
(45)
(45) ), we obtain by Lesbegue's theorem that
(55)
(55) Since
is continuous, we deduce that
(56)
(56) We have
(57)
(57) Using (Equation48
(48)
(48) ), we obtain by Lesbegue's theorem that for
(58)
(58) Since TY is a relatively compact in
, we have the uniform convergence, namely
(59)
(59) Thus, we have proved that T is a compact mapping from Y into itself. It follows by the Schauder fixed point theorem that there exists
such that Tv=v. Put
(60)
(60) then
and u satisfies the equation
(61)
(61) Then due to Lemma 3.1, u is a positive continuous solution of problem (Equation1
(1)
(1) ).
Finally, let us prove that u is the unique positive continuous solution satisfying (Equation47(47)
(47) ). To this aim, we assume that
has two positive solutions u and v satisfying (Equation47
(47)
(47) ). Then, there exists a constant m>1 such that
(62)
(62) This implies that the set
(63)
(63) is not empty. Now, put
then we aim to show that
Suppose that c>1. Then by simple calculus, we obtain that
(64)
(64) We conclude by Proposition 2 (iii) that
(65)
(65) Then we have
(66)
(66) Which implies with the fact that
is increasing monotone that
(67)
(67) Using again Proposition 2 (iii), we conclude that
(68)
(68) By symmetry, we obtain that
. So
. Since
and c>1, we have
. This yields to a contradiction with the fact that
. Hence c=1 and consequently u=v.
3.1. Proof of Theorem 2
Suppose that and
satisfy
(69)
(69) Let θ be the function defined on
by
(70)
(70) Then, for
We conclude by Lemma 2.9 that for
(71)
(71) This implies by Proposition 2 (ii) that
and for
, we have
(72)
(72) Using again
and by Lemma 2.9, we deduce that for
(73)
(73) Hence it follows from Proposition 3 that problem (Equation1
(1)
(1) ) has a unique positive solution u in
satisfying for
,
(74)
(74) As an application of our main result, we give the following example.
3.2. Example 1
Let , p>1 and
. Let a and ψ be two positive continuous functions on
such that
(75)
(75) and
(76)
(76) where
and
. Therefore by Theorem 2, problem (Equation1
(1)
(1) ) has a unique positive solution
satisfying for each
(77)
(77)
Acknowledgments
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the manuscript.
Disclosure statement
No potential conflict of interest was reported by the authors.
ORCID
Adnane Hamiaz http://orcid.org/0000-0001-5605-8728
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Funding
References
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