We consider the nonlinear fractional differential equation
with the nonlocal boundary conditions
where , , , , for all , , , denotes the Riemann-Liouville derivative of order k (for ), the integrals from the boundary condition (BC) are Riemann-Stieltjes integrals with
non-decreasing functions, , and for all .
We study the existence of nonnegative solutions for problem (E)-(BC) by using the Banach contraction mapping principle and the Krasnosel’skii fixed point theorem for the sum of two operators. Equation (E) supplemented with the multi-point boundary conditions
where for all , ( ), , , , was investigated in the paper . The last condition in
(BC1) can be written as , where is the step function defined by
So (BC) is a generalization of (BC1), because in (BC) we have a sum of Riemann-Stieltjes integrals and various orders for the fractional derivatives. In the paper , the authors investigated the existence of nonnegative solutions for the Caputo fractional differential equation
with the boundary conditions
where , , , ( ), is the Caputo fractional derivative, and the operators A and B are defined as the operators from our problem, given above. In the paper , the authors studied the existence and multiplicity of positive solutions for the Riemann-Liouville fractional differential equation , subject to the boundary conditions (BC), where f is a sign-changing function that can be singular in the points and/or in the variable x. In addition, the methods used in the proofs of the main results in  are different than those used in the present paper, namely, in  the authors used various conditions which contain height functions of the nonlinearity defined on special bounded sets, and two theorems from the fixed point index theory. For some recent results on the existence, nonexistence and multiplicity of solutions for fractional differential equations and systems of fractional differential equations subject to various boundary conditions we refer the reader to the monographs   and the papers  - . We also mention the books  - , and the papers  -  for applications of the fractional differential equations in various disciplines.
2. Preliminary Results
We present in this section some auxiliary results from  that we will use in the proof of the main results. We consider the fractional differential equation
with the boundary conditions (BC), where . We denote by
Lemma 1 If , then the unique solution of problem (1)-(BC) is given by
Lemma 2 If , then the solution x of problem (1)-(BC) given by (2) can be written as
for all , .
By using some properties of the functions given by (5) from , we obtain the following lemma.
Lemma 3 We suppose that . Then the function G given by (4) is a continuous function on and satisfies the inequalities:
a) for all , where , , and , ;
b) for all ;
c) , for all , where .
Lemma 4 We suppose that , and for all . Then the solution x of problem (1)-(BC) given by (3) satisfies the inequality for all , where , and so for all .
In the proof of our main theorems, we use the Banach contraction mapping principle and the Krasnosel’skii fixed point theorem for the sum of two operators presented below.
Theorem 1 (see  ) If is a nonempty complete metric space with the metric d, and is a contraction mapping, then T has a unique fixed point ( ).
Theorem 2 (  ) Let M be a closed, convex, bounded and nonempty subset of a Banach space X. Let and be two operators such that
a) for all ;
b) is a completely continuous operator (continuous, and compact, that is, it maps bounded sets into relatively compact sets);
c) is a contraction mapping.
Then there exists such that .
3. Main Results
In this section we study the existence of nonnegative solutions for our problem (E)-(BC). We present now the assumptions that we will use in the sequel.
(I1) , , , , for all , , , , are nondecreasing functions, an
(I2) is measurable with respect to t on , ( ).
(I3) There exist the functions such that
a.e. and for all .
(I4) There exists the function such that
(I5) , , and .
We denote by and .
Theorem 3 We suppose that assumptions (I1)-(I5) hold. If , where , then problem (E)-(BC) has at least one nonnegative solution on .
Proof. By (I4) we obtain that the function is Lebesgue integrable on for all and , the function is Lebesgue integrable on for all , and the function is Lebesgue integrable on for all , and .
We consider now the integral equation
By Lemma 2 we easily deduce that if x is a solution of Equation (6) (or equivalently (7)), then x is a solution of problem (E)-(BC).
Let . We define the operator on by
If x is a fixed point of operator , then x is a solution of Equation (6) (or (7)), and hence x is a solution of problem (E)-(BC). Therefore we will study the existence (and uniqueness) of the fixed points of operator by using the Banach contraction mapping principle.
We firstly show that if , then . Indeed, we have
Hence is a continuous function. By (I1), (I2) and Lemma 4, we obtain for all , and then .
In addition, for any and all , we deduce
and then for all , so .
We show now that is a contraction mapping on . For , and any , by using (I3), we find
Therefore we obtain the inequality
Because , we deduce that is a contraction mapping. By Theorem 1, we conclude that has a unique fixed point, which is a nonnegative solution of problem (E)-(BC).
In what follows, we denote by
Theorem 4 We suppose that assumptions (I1),
(I2)' is a continuous function and (I3), (I4), (I5) hold. If , then problem (E)-(BC) has at least one nonnegative solution on .
Proof. We define , where
and is given by (8). We consider the set . Then is a closed, convex and nonempty set of . We define the operators and on by
By (I1), (I2)' and Lemma 4, we have for all . For any , by using (I3), we find
Then for any and all , we obtain by using the above inequality
because and for all .
Therefore for any and all , we deduce by using (I4) that
Hence for and , we find
Therefore for and , by using (I3) we obtain
So, we deduce
where is given by (8). Because , we conclude that is a contraction mapping.
By using assumptions (I2)' and (I5), we deduce that is a continuous mapping. In addition, is uniformly bounded on , because for any , we find
and then for all .
The operator is also equicontinuous on . Indeed, let , , with . We have
where . Then we obtain that as .
By using the Arzela-Ascoli theorem, we deduce that is relatively compact. By Theorem 2, we conclude that operator has at least one fixed point, and so problem (E)-(BC) has at least one nonnegative solution.
4. An Example
Let ( ), , , , , for all , .
We consider the fractional differential equation
with the boundary conditions
where and for all
, with for all with , and for all . Then we obtain and . So assumptions (I1) and (I5) are satisfied.
We define the function
for all and with . We deduce that and . Besides we obtain the inequalities
for all , , and
We define , , , and , for all . We have and . So assumptions (I2), (I3), (I4) are also satisfied.
In addition, we find , and so . Therefore, by Theorem 3, we conclude that problem (E0)-(BC0) has at least one nonnegative and nontrivial solution.
In this paper, we investigated the existence of nonnegative solutions for the Riemann-Liouville fractional differential equation with integral terms (E) supplemented with the boundary conditions (BC) which contain Riemann-Liouville fractional derivatives of different orders and Riemann-Stieltjes integrals, by using the Banach contraction mapping principle and the Krasnosel’skii fixed point theorem for the sum of two operators. For some future research directions, we have in mind the study of the existence, nonexistence and multiplicity of solutions or positive solutions for fractional differential equations subject to other boundary conditions.
The authors thank the referee for his/her valuable comments and suggestions.
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