1. Introduction
We consider the system of fractional differential equations
with the nonlocal boundary conditions
where
,
,
,
,
,
,
for all
,
,
,
for all
,
,
,
,
,
,
, the functions
f and
g are nonnegative and they may be singular at
and/or
, the integrals from the boundary conditions (
2) are Riemann–Stieltjes integrals with
,
and
,
functions of bounded variation, and
denotes the Riemann–Liouville fractional derivative of order
of function
u (for
for
,
for
). The fractional derivative
is defined by
,
, where
,
stands for the largest integer not greater than
, and
, is the gamma function (the Euler function of second type). This work is motivated by the application of
p-Laplacian operator in several fields such as nonlinear elasticity, fluid flow through porous media, glaciology, nonlinear electrorheological fluids, etc., for details, see [
1] and the references cited therein.
Under some assumptions on the functions
f and
g, we present existence and multiplicity results for the positive solutions of problem (
1) and (
2). By a positive solution of problem (
1) and (
2) we mean a pair of functions
, satisfying the system (
1) and the boundary conditions (
2), with
for all
, or
for all
, (
). In the proof of our main theorems we use the Guo–Krasnosel’skii fixed point theorem (see [
2]). The existence and nonexistence of positive solutions for the system (
1) with two positive parameters
and
, and nonsingular and nonnegative nonlinearities, supplemented with the multi-point boundary conditions
where
,
,
,
,
,
for all
(
),
,
for all
(
),
, was investigated in [
3], by applying the Guo–Krasnosel’skii theorem. In the paper [
4], the authors studied the system (
1) with positive parameters, and nonsingular and nonnegative nonlinearities, subject to the nonlocal coupled boundary conditions
where
,
for all
,
,
,
for all
,
,
.
In [
5], by applying the fixed point theorem for mixed monotone operators, the authors proved the existence of positive solutions for the multi-point boundary value problem for nonlinear Riemann–Liouville fractional differential equations
where
,
,
,
, and
f is a nonnegative function which may be singular at
. In [
6], the authors investigated the existence and uniqueness of positive solutions for the fractional boundary value problem
where
,
,
,
denotes the Caputo fractional derivative of order
of function
u defined by
, for
, and
, for
, and the nonlinear terms
f and
h may be singular on the time variable and space variables. The authors used in [
6] the theory of mixed monotone operators, and they also discussed there the dependence of solutions upon a parameter.
Systems with fractional differential equations without
p-Laplacian operators, with parameters or without parameters, subject to various multi-point or Riemann–Stieltjes integral boundary conditions were studied in the last years in [
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
25,
26,
27]. For various applications of the fractional differential equations in many scientific and engineering domains we refer the reader to the books [
28,
29,
30,
31,
32,
33,
34], and their references.
The paper is organized as follows. In
Section 2, we study two nonlocal boundary value problems for fractional differential equations with
p-Laplacian operators, and we present some properties of the associated Green functions.
Section 3 contains the main existence theorems for the positive solutions for our problem (
1) and (
2), and in
Section 4, we give two examples which illustrate our results.
2. Auxiliary Results
We consider firstly the nonlinear fractional differential equation
with the boundary conditions
where
,
,
,
,
,
for all
,
,
,
are bounded variation functions, and
. We denote by
Lemma 1. If , then the unique solution of problem (3) and (4) is given bywhere the Green function is given bywith Proof. We denote by
. Then problem (
3) and (
4) is equivalent to the following two boundary value problems
and
For the first problem (
8), the function
is the unique solution
of (
8). For the second problem (
9), if
then by [
7] (Lemma 2.2), we deduce that the function
where
is given by (
6), is the unique solution
of problem (
9). Now, by using relations (
10) and (
11), we find formula (
5) for the unique solution
of problem (
3) and (
4). □
Next we consider the nonlinear fractional differential equation
with the boundary conditions
where
,
,
,
,
,
for all
,
,
,
are bounded variation functions, and
. We denote by
In a similar manner as above we obtain the following result.
Lemma 2. If , then the unique solution of problem (12) and (13) is given bywhere the Green function is given bywith By using the properties of the functions
,
,
,
,
,
given by (
7) and (
16) (see [
7,
17]), we obtain the following properties of the Green functions
and
that we will use in the next section.
Lemma 3. Assume that , , and , are nondecreasing functions and . Then the Green functions and given by (6) and (15) have the properties: - (a)
are continuous functions;
- (b)
for all, where
with
;
- (c)
for all;
- (d)
for all, where
with
;
- (e)
for all.
By similar arguments used in the proof of [
17] (Lemma 2.5), we deduce the next lemma.
Lemma 4. Assume that , and , are nondecreasing functions, , , , , for all , for all . Then the solutions u and v of problems (3), (4), (12) and (13), respectively, satisfy the inequalities , for all . In addition, we have the inequalities , for all . 3. Existence of Positive Solutions
In this section, we investigate the existence of positive solutions for problem (
1) and (
2) under various assumptions on the functions
f and
g which may be singular at
and/or
. We present the basic assumptions that we will use in the main theorems.
- (I1)
, , , , , , for all , , , for all , , , , are nondecreasing functions, , , , , , , .
- (I2)
The functions
and there exist the functions
and
,
, with
such that
where
,
.
Remark 1. We present below two cases in which ; for other cases see the examples from Section 4. a) If , that is for all , , , then the inequalities (17) are satisfied with equality. In addition, the conditions are also satisfied, because in this nonsingular case, we obtainwhere is the beta function (the Euler function of first type), with . In a similar manner we have . b) If , , , and , then by using the Cauchy inequality we findwhere is the norm of in the space . In a similar manner we obtain . By using Lemmas 1 and 2 (the relations (
5) and (
14)),
is a solution of problem (
1) and (
2) if and only if
is a solution of the nonlinear system of integral equations
We consider the Banach space
with supremum norm
, and the Banach space
with the norm
. We define the cone
by
We also define the operators
and
by
and
. Then
is a solution of problem (
1) and (
2) if and only if
is a fixed point of operator
.
Lemma 5. Assume that and hold. Then is a completely continuous operator (continuous, and it maps bounded sets into relatively compact sets).
Proof. We denote by
,
. Using
and Lemma 3, we deduce that
,
. In addition, we find
By Lemma 3 we conclude that maps into .
We will show that
maps bounded sets into relatively compact sets. Suppose
is an arbitrary bounded set. Then there exists
such that
for all
. By the continuity of
and
we deduce that there exists
such that
. By using Lemma 3, for any
and
, we obtain
Then , for all , and so , and are bounded.
We will prove next that
is equicontinuous. By using Lemma 1, for
and
we deduce
Hence for any
we find
Then for any
we obtain
Therefore for any
we deduce
We compute the integral of function
, by exchanging the order of integration, and we have
For the integral of the function
, we obtain
We conclude that
. Hence for any
with
and
, by (
18) and (
19), we find
By (
19), (
20) and the absolute continuity of the integral function, we deduce that
is equicontinuous. By a similar approach, we obtain that
is also equicontinuous, and so
is equicontinuous. Using the Ascoli–Arzela theorem, we conclude that
and
are relatively compact sets, and so
is also relatively compact. Besides, we can prove that
and
are continuous on
(see [
16] (Lemma 1.4.1)). Then
is a completely continuous operator on
. □
Under the assumptions and , by using Lemma 4, we obtain , and so (denoted again by ) is also a completely continuous operator. For we denote by the open ball centered at zero of radius r, and by and its closure and its boundary, respectively.
Theorem 1. Assume that and hold. In addition, the functions and g satisfy the conditions
- (I3)
There exist and
such that - (I4)
There exists,
such that
Then problem (1) and (2) has at least one positive solution .
Proof. We consider the above cone
. By
we deduce that for
and
, there exists
such that
where
are defined in the proof of Lemma 5. Then by (
21) and Lemma 3, for any
and
, we obtain
By
, we suppose that
(in a similar manner we can study the case
). Then for
, where
, there exists
such that
Then by (
23), for any
and
, we find
We can choose
and then we conclude
By using Lemma 5, the relations (
22), (
24), and the Guo–Krasnosel’skii fixed point theorem, we deduce that
has a fixed point
, that is
, and
and
for all
. Then
or
, that is
for all
or
for all
. Hence
is a positive solution of problem (
1) and (
2). □
Remark 2. Theorem 1 remains valid if the functions and f satisfy the inequalities (21) and (23), instead of and . Theorem 2. Assume thatandhold. In addition the functionsand g satisfy the conditions
- (I5)
- (I6)
There exist
,
,
and
such that
Then problem (1) and (2) has at least one positive solution.
Proof. We consider again the cone
. By
we deduce that for
,
, there exist
,
such that
By using
and (
25), for any
, we obtain
Then we find
and so
for all
. We choose
and then we deduce
The choosing of
above is based on the inequalities
for
and
, and
for
and
. Here
or
. We explain the above inequality (
27) in one case, namely
and
. In this situation, by using (
26), and the relations
,
(from the definition of
and
), we have the inequalities
In a similar manner we treat the cases: and ; and ; and .
By
, we suppose that
(in a similar manner we can study the case
). We deduce that for
, where
, there exists
such that
Then by using (
28), for any
and
, we find
Therefore
for all
, and then
By Lemma 5, the relations (
27), (
29), and the Guo–Krasnosl’skii fixed point theorem, we conclude that
has at least one fixed point
, that is
, which is a positive solution of problem (
1) and (
2). □
Remark 3. Theorem 2 remains valid if the functions and g satisfy the inequalities (25) and (28), instead of and .
Theorem 3. Assume that, , andhold. In addition, the functionsandsatisfy the condition
- (I7)
where
Then problem (1) and (2) has at least two positive solutions.
Proof. We consider the operators
, and the cone
defined in this section. If
,
and
hold, then by the proof of Theorem 1, we deduce that there exists
(we can consider
) such that
If
,
and
hold, then by the proof of Theorem 2 we find that there exists
(we can consider
) such that
We consider now the set
. By
, for any
and
, we obtain
So
, for all
,
. Then
Therefore, by (
30), (
32) and the Guo–Krasnosel’skii fixed point theorem, we conclude that problem (
1), (
2) has one positive solution
with
. By (
31), (
32) and the Guo–Krasnosel’skii fixed point theorem, we deduce that problem (
1), (
2) has another positive solution
with
. Then problem (
1) and (
2) has at least two positive solutions
. □
Remark 4. Theorem 3 remains valid if the functions f and g satisfy the inequalities (23) and (28), instead of and . 4. Examples
Let , , , (), , (), , , , , , , , , , , , for all , , .
We consider the system of fractional differential equations
with the nonlocal boundary conditions
We obtain here
and
. We also find
Example 1. We consider the functionswhere , , , . Here , , , for all , , for all . By using the Hölder inequality, we obtain Hence assumptions and are satisfied.
In addition, in
, for
, we obtain
, and in
for
we have
(and
). Then by Theorem 1, we conclude that problem (
33) and (
34) with the nonlinearities (
35) has at least one positive solution
.
Example 2. We consider the functionswhere , , , , , . Here we have , , , . By using a computer program, we obtainwhere is the regularized hypergeometric function. So , and then assumptions and are satisfied. For
, we find
, and if we consider
we obtain
, and then assumptions
and
are satisfied. After some computations we deduce
In addition, we find
. If
and
, then the inequalities
and
are satisfied (that is, assumption
is satisfied). For example, if
and
, and
and
, then the above inequalities are satisfied. By Theorem 3, we conclude that problem (
33) and (
34) with the nonlinearities (
36) has at least two positive solutions
.