#### Abstract

In this paper, we consider the differential equations with right-sided Caputo and left-sided Riemann-Liouville fractional derivatives. Furthermore, the expression of Green’s function is derived, and its properties are investigated. By the fixed-point theorem for both -concave operators and mixed monotone operators, we get the existence and uniqueness of the solution, respectively. As applications, some examples are provided to illustrate our main results.

#### 1. Introduction

Fractional differential equations are generalization of the ordinary differential equations to a nonintegral order, and they have been widely used in other fields of mathematics (as discussed in [1–10]). In recent decades, many authors devoted themselves to fractional equations. More fractional boundary value problems have been applied to physics, biology, medicine, software engineering, neural network, and other related sciences (see [11–16]). With the publication of works discussed in [17–22], the theory of fractional boundary value problems are gradually enriched and systematized.

In [23], Song and Cui concerned the existence of solutions of nonlinear mixed fractional differential equation with the integral boundary value problem under resonance: where is the left Caputo fractional derivative of order , and is the right Riemann-Liouville fractional derivate of order . The coincidence degree theory is the main theoretical basis to prove the existence of solutions of such problems. In recent years, there have been some studies on the existence of solutions for mixed fractional differential equations (see [24, 25]).

By using the fixed-point theorem for mixed monotone operators, Jong et al. [26] dealt with the existence of positive solutions of the following multipoint boundary value problems for nonlinear fractional differential equations where , , and are the standard Riemann-Liouville derivatives with . The fixed-point theorem for mixed monotone operators was also used to prove the existence of solutions of boundary value problems [27, 28].

Moreover, many researchers focused on the -concave operators in [29–31] and its applications in [32].

Based on above works, this paper investigates the existence of solutions for the fractional differential equations where is the right-sided Caputo fractional derivative with is the left-sided Riemann-Liouville fractional derivative with and . Here, is continuous, and is a constant real number.

In Section 2, there are some definitions, properties, and lemmas related to this article. Then, we obtain the Green’s function and prove some lemmas of Green’s function. In Section 3, two important theorems are obtained. In Theorem 13, set is defined. According to the fixed-point theorem of increasing concave operator, the existence of the unique solution of boundary value problem (3) is obtained. In Theorem 14, set is defined, the existence of solutions is also obtained by the fixed-point theorem of mixed monotone operators. In the last section, some examples are given to illustrate the validity of the theorems.

#### 2. Preliminaries

In this part, we present some basic definitions, properties, and lemmas.

*Definition 1 (see [20]). *The left-sided and right-sided Riemann-Liouville fractional integrals of order of a function are given by
where is the Gamma function.

*Definition 2 (see [20]). *The left-sided Riemann-Liouville fractional derivative and right-sided Caputo fractional derivative of order of a function are given by
where .

*Property 3 (see [20]). *Let and . If , then
where is arbitrary constant.

Lemma 4. *Let α ∈ (0,1], β ∈ (1,2]. For y ∈ C[0,1], then the unique solution of the fractional differential equation
is where
*

*Proof. *Applying the right-sided fractional integral to both sides of the Equation (7) and by Property 3, we can obtain that
where is arbitrary constant. Applying the left-sided fractional integral to both sides Equation (9) above and by Property 3, we can obtain that
where . By and , we get , then
Finding the derivative of (11), we have
Since , it follows
Exchanging the order of the above double integral, we have
Substitute into (11), we know that
Exchange the order of the first double integral in the above formula, we get that
Calculated that the Green’s function of the fractional differential Equation (7) is
The proof is completed.

Consequently, the boundary value problem (3) has a unique solution if and only if satisfies the integral equation:

Lemma 5. *The Green’s functions defined by Lemma 4 satisfy the following properties:
*(1)*, for all *(2)*, for all , where .*

*Proof. *First, we prove that the function is nonnegative.

For any ,
For any ,
In a word, for any , .

Then, we prove .

For any ,
Finally, we prove that .

For any ,
The proof is completed.

Next, we summarize two fixed-point lemmas and some basic concepts in ordered Banach space.

Let be a real Banach space which is partially ordered by a cone , i.e., if and only if . If and , then we denote or . denotes the zero element of . is called normal if there exists such that, for all , implies ; in this case, is called the normality constant of . We say that an operator is increasing if implies [29].

Given and , we define the set

*Remark 6. *If , let , we have .

Let and , we define the set

*Definition 7 (see [29, 31]). *Let be a given operator. For any and , there exists such that . Then, is called a generalized concave operator.

*Definition 8 (see [29]). *Let be a given operator. For any and , there exists such that
Then, is called a -concave operator.

Lemma 9 (see [29]). *Let be normal and be an increasing -concave operator with . Moreover, for any , making the sequence , then we obtain as .*

*Definition 10 (see [28]). *Let . Operator is said to be mixed monotone if is nondecreasing in and nonincreasing in , i.e., implies .

*Definition 11 (see [28]). *Let a mixed monotone operator. Assume that for all , there exists such that
holds for all ; then, is called a mixed monotone model operator.

Lemma 12 (see [28]). *Let . is a mixed monotone operator. Then, has exactly one fixed-point in . Moreover, constructing successively the sequences
for any initial point , we have and as .*

#### 3. Main Results

We consider the space with the usual maximum norm . Clearly, is a Banach space. Set . Obviously, is a normal cone and , the normality constant is 1.

Theorem 13. *Let and are defined by , where and , is the Green’s functions of (3) defined by Lemma 4. Suppose the following conditions hold:**( H1). is continuous and increasing with respect to the second variable, that is, for any , we get , where *

*(*

*H*2). For any there is such that , where*(*

*H*3). with for .*Then, the fractional differential equations (3) have a unique nontrivial solution in . Moreover, for any given initial value , making the sequence then we obtain as .*

*Proof. *For any , it is easy to see , that is, .

Further,
Hence, .

Let
The boundary (3) has an integral formulation given by
So, is the solution of the problem (3) if and only if is the fixed point of the operator of .

Firstly, it is apparent from the definition of that is .

From (H2), we know that
By Definition 8, is -concave operator.

Secondly, for and , we get
So, is increasing.

Thirdly, we would prove that .

Let It is not difficult to verify that
That is to say, . Then, we have . So, .

Consequently, by using Lemma 9, the operator has a unique fixed-point in , i.e., . And for any , there exists the sequence satisfies as ,
The proof is completed.

Theorem 14. *Let , where and is defined by Theorem 13. Suppose the following conditions hold:**( H4). , where are continuous functions and for any fixed , is nondecreasing in , and is nonincreasing in *

*(*

*H*5). For , there exists such that . For , from above inequality, we can get that , where and ;*(*

*H*6)*Then, the boundary value problem (3) has a unique positive solution in .*

*Proof. *According to definition of in Theorem 13, is normal cone. Let
From Remark 6, for any , there exist two positive constants and such that . Let , it is easy to know that and
By (*H*5), we have
Hence, for any , we have the followings:
It follows from the above (*H*6) that .
Suppose ,
then
So, .

By (*H*4), we obtain that for any and ,
For any and ,
Consequently, is a mixed monotone operator.

For any and in (*H*5),
So, we get
From Definition 11, is a mixed monotone model operator. By Lemma 12, has exactly one fixed-point in . Constructing successively the sequences
For any initial point , we have and as .

The proof is completed.

#### 4. Application

Now, we give two concrete examples to illustrate our main theorems.

*Example 1. *Consider the mixed fractional differential equation
Here, ; therefore, , .
(1)It is obvious that is continuous and increasing with respect to (2)Taking for any and , we have
(3)Obviously, , and , for all .Therefore, it follows from Theorem 13 that the problem (48) has a unique nontrivial solution , where and .

*Example 2. *Consider the mixed fractional differential equation
Here, , therefore, .
(1)It is obvious that is continuous. Letting , it is easy to know that for any , is nondecreasing in , and is nonincreasing in (2)For , there exists such that , , where and (3)Obviously, for , ,