Abstract
In this paper, we investigate the existence and uniqueness of solutions for a class of integral boundary value problems of nonlinear fractional differential equations with -Laplacian operator. We obtain some existence and uniqueness results concerned with our problem by using Schaefer’s fixed-point theorem and Banach contraction mapping principle. Finally, we present some examples to illustrate our main results.
1. Introduction
This paper deals with the existence and uniqueness of solutions for the following fractional integral boundary value problem with p-Laplacian operator: where , and are the Caputo fractional derivatives, , and the -Laplacian operator is defined as .
In the past few decades, fractional differential equations have been widely applied to many fields in natural and social sciences, because they are important tools in mathematically describing many phenomena of science and engineering such as aerodynamics, control theory, signal and image processing, plasma dynamics, blood flow phenomena, and viscoelastic and non-Newtonian fluid mechanics (see [1–5] and their references). In 1983, Leibenson [2] proposed the following integer-order differential equation model with -Laplacian operator to study the turbulent flow in a porous medium in his work: where is a -Laplacian operator.
Based on Leibenson’s results, many researchers have generalized his model into a variety of models with fractional order and obtained many valuable existence and uniqueness results for -Laplacian fractional differential equations with two-point boundary conditions (see [6–14]), multipoint boundary conditions (see [15–18]), and nonlocal boundary conditions (see [19–23]). Since the 1st derivatives of unknown functions existed in Leibenson’s model, it became a special study to discuss the existence results for -Laplacian fractional differential equations where the fractional orders were in the neighborhood of 1. So, Chai [7] used the fixed-point theorem on cones to investigate the existence and multiplicity of positive solutions for fractional differential equations with -Laplacian operator: where , and and are the Riemann-Liouville fractional derivatives. Jong [15] established the existence and uniqueness of positive solutions for multipoint boundary value problems of nonlinear fractional differential equations with -Laplacian operator by using the Banach contraction mapping principle where , and .
Integral boundary value problems for differential equations have arisen in the study of various fields such as underground water flow, blood flow problems, and thermoelasticity. So, the study on the existence of solutions for the -Laplacian integral boundary value problems has attracted the attention of many researchers recently (see [20–23]). Zhang et al. [23] considered the existence of symmetric positive solutions of the problem for the following nonlinear fourth-order -Laplacian differential equations with integral boundary conditions: where are the nonnegative, symmetric functions and . Zhang and Cui [22] investigated the existence of positive solutions for nonlinear fourth-order singular -Laplacian differential equations with the integral boundary conditions where , may be singular at , and are nonnegative. The existence results on solutions of problem (2) are established by employing upper and lower solution methods together with maximal principle. Jiang [20] used the generalized continuous theorem to investigate the existence of solutions to the integral boundary value problem of -Laplacian multiterm fractional differential equations at resonance where and .
Summarizing previous results, very few papers dealt with the existence of solutions for integral boundary value problems of -Laplacian fractional differential equations, especially, Zhang and Cui [22] who established the existence of positive solutions for fourth-order singular -Laplacian differential equations under and Jiang [20] who considered the existence of solutions for nonlinear multiterm fractional differential equations with . Moreover, due to the nonlinearity of -Laplacian operator , it is more difficult to study for the case rather than for the case . Motivated by the above facts, this paper deals with the existence and uniqueness of solutions of problem (1) in which the fractional derivatives are Caputo fractional derivatives with .The structure of this paper is organized as follows.
In Section 2, we recall some definitions and lemmas. In Section 3, we prove the existence and uniqueness of solutions for nonlinear integral boundary value problem with -Laplacian operator by using Schaefer’s fixed-point theorem and Banach contraction mapping principle. Finally, we give two examples to illustrate our main results in Section 4.
2. Preliminaries
The Riemann-Liouville fractional integral and the Caputo fractional derivative of order of a function is given by where , provided that the right-hand side is pointwise defined on (see [4, 5]).
Lemma 1 (see [1]). If and , then .
Lemma 2 (see [9]). Assume that . Then, .
For the sake of convenience, put and and assume that and .
Lemma 3. Let . Then, the fractional boundary value problem has a unique solution which is given by where
Proof. In view of Lemma 2, we have that
By means of the property of the fractional integral of a continuous function, we obtain that .
Since , from (12), we obtain
Hence, (12) can be written as
Thus, we can easily get
In the right side of (15), the term can be rewritten as
so we get
Therefore, the unique solution of (9) is given by
Conversely, let be the function which is expressed by (10). Putting
we get
Then, we have that
Since , applying to both sides of (21) and using Lemma 1, we can obtain
On the other hand, multiplying (10) by and integrating on , we have
Hence, (10) can be written as
Since , we can also have that
Therefore, we can know that is a solution of problem (9) and (9) has a unique solution which is given by (10). The proof is completed.
Lemma 4. Let , then -Laplacian integral boundary value problem (1) has a unique solution which is given by where is the number that satisfies .
Proof. The proof of this lemma is divided into two steps. (Step 1)Let and consider the boundary value problemBy using Lemma 2, we have that From (28) and boundary condition of (27) we can obtain Then, (28) can be written as In a similar way to the proof of Lemma 3, it can be easily seen that Hence, we have that the unique solution of (27) is given by Conversely, let be the function which is expressed by (32). Then, from the definition of , (32) can be written as Since and , applying to both sides of (33), we can obtain On the other hand, multiplying (32) by and integrating on , we have that So, we can rewrite (32) as Since , we can get Therefore, we can know that is the solution of (27). (Step 2)Now let be the solution of (1). Putting , then we have thatAlso denoting by , then by Lemma 3, we can see that that Since it is well-known that , combining (38) and (39) yields The proof is completed.
Remark 5. From the definition of and , it is easy to know that those functions are continuous in .
Lemma 6 (see [24]). Schaefer’s fixed-point theorem. Let be the Banach space and be completely continuous operator. If the set is bounded, then has at least one fixed point in .
The basic properties of the -Laplacian operator which will be used in the following studies are listed below (see [10]). (i)If and , then(ii)If and , then
3. Main Results
In this section, we establish the existence and uniqueness of solutions of problem (1) by using Schaefer’s fixed-point theorem and the Banach contraction mapping principle.
Let us consider the Banach space endowed with the norm .
Define an operator by Then, Equation (26) is equivalent to the operator equation
From Lemma 4, the existence of solutions for the problem (1) refers to the existence of fixed points of Equation (44). Therefore, it is sufficient to prove the existence of fixed points of (44).
Lemma 7. The operator is completely continuous.
Proof. Since and is continuous, we can know that is continuous.
Let be a bounded subset, then for any , there exists such that .
We will show that is relatively compact in . Since is a continuous function, there exists such that . Then, we have
And since , evaluating the upper bound of gives
So, we obtain
Then, we can get easily that
In view of the definition of the -Laplacian operator, we have that
This shows that is uniformly bounded in . For any , we have that
And since
the integral term can be divided into the following three parts:
In a similar way to this, we can obtain
These inequalities yield
Therefore, we get
This shows that is equicontinuous in . By using the Arzela-Ascoli theorem, we can see that is relatively compact in . As a consequence of the above discussion, the operator is completely continuous. The proof is completed.
Denote as follows:
In this article, the following hypotheses will be used.
(H1). There exist nonnegative functions such that (i)(ii)
Theorem 8. Assume that the hypothesis (H1) holds, then problem (1) has at least one solution.
Proof. Consider the following set: For any , it can be easily seen that On the other hand, from the condition (i) of the hypothesis (H1) and (49), we have that From (58), we get Since implies , we obtain Therefore, we have that and by using the condition (ii) of the hypothesis (H1), we can see that So, we can know that is bounded. In view of Schaefer’s fixed-point theorem (Lemma 6), the operator has at least one fixed point which is the solution of the problem (1). The proof is completed.
Here, put and list more hypotheses to obtain the uniqueness results for our problem.
(H2). There exists a real number such that For the readers’ convenience, denote as follows:
Theorem 9. Let . Assume that the hypotheses (H1) and (H2) are satisfied and , then problem (1) has a unique solution.
Proof. Put .
Firstly, we will show that . In fact, from the hypothesis (H1) and (49), for any , we have that
Next, we will prove the uniqueness of solutions for problem (1).
For any and any , by the hypothesis (H1), we get
Since , we can see . Hence, from (H2) and one basic property of -Laplacian operator (42), for any and any , we have that
This means that
Since , we can see that is a contraction mapping. By means of the Banach contraction mapping principle, we can prove that has a unique fixed point in . That is, problem (1) has a unique solution. The proof is completed.
4. Examples
The following examples are concerned with the illustration of Theorem 8 and Theorem 9.
Example 1. Consider the following integral boundary value problem:
The problem (70) can be rated as the boundary value problem where and By simple calculation, we have and .
So, testing whether the hypothesis (H1) holds or not, we get
Therefore, by Theorem 8, the problem (70) has at least one solution.
Example 2. Consider the boundary value problem
Comparing with the problem (1), it can be easily seen that , and .
Since and , we obtain
So, we can see that the hypothesis (H1) is satisfied. Calculating the radius , we have
Since , for any and any , we get
From the above discussion, we can see that the hypothesis (H2) holds. Therefore, it follows by Theorem 9 that the problem (72) has a unique solution.
Data Availability
No datasets are generated or analyzed during this study.
Conflicts of Interest
The authors declare that they have no conflict of interest.
Authors’ Contributions
All authors carried out the proof and conceived of the study. All authors read and approved the final manuscript.