Research Article | Open Access
KumSong Jong, HuiChol Choi, KyongJun Jang, SunAe Pak, "Existence and Uniqueness Results for a Class of Singular Fractional Boundary Value Problems with the -Laplacian Operator via the Upper and Lower Solutions Approach", Journal of Function Spaces, vol. 2020, Article ID 2930892, 15 pages, 2020. https://doi.org/10.1155/2020/2930892
Existence and Uniqueness Results for a Class of Singular Fractional Boundary Value Problems with the -Laplacian Operator via the Upper and Lower Solutions Approach
In this paper, we study the existence and uniqueness of positive solutions to a class of multipoint boundary value problems for singular fractional differential equations with the -Laplacian operator. Here, the nonlinear source term permits singularity with respect to its time variable . Some fixed-point theorems such as the Leray-Schauder nonlinear alternative, the Schauder fixed-point theorem, and the Banach contraction mapping principle and the properties of the Gauss hypergeometric function are used to prove our main results. And by employing the upper and lower solutions technique, we derive a new approach to obtain the maximal and minimal solutions to the given problem. Finally, we present some examples to demonstrate our existence and uniqueness results.
In this paper, we consider the existence of positive solutions of the following -point boundary value problems for singular nonlinear fractional differential equations where , and are the standard Riemann–Liouville derivatives. Here, , is singular at (i.e., ) and is defined as .
Fractional calculus has a history of several hundred years, and many valuable results, which have contributed to the development of mathematical theories and their application to practice, have been created during its historical process (see ). Also, fractional differential equations are one of the powerful tools to model and solve scientific and technological problems arising in physics, chemistry, biology, and mechanics, and it has developed more and more in depth (see ). In particular, after Leibenson’s work dealt with the application of the integer-order differential equation with the -Laplacian operator to the analysis of turbulent flow in porous media (see ), many valuable existence results for this equation have been achieved, and recently, the achievements obtained in this integer-order differential equation are more generalized to the fractional differential equation (see [4–6] and the references therein). However, due to the nonlinearity of the -Laplacian operator, not much has been studied on the solutions to singular fractional differential equations with this operator and many researchers have been paid their attention to those equations. For instance, by using the fixed-point theorem of mixed monotone operators, Jong et al.  proved the existence of positive solutions to the boundary value problem (1), in which the nonlinear source term was singular with respect to its space variable , and proposed a new approach by which the approximate solution of the given problem could be obtained. Unlikely in , this paper deals with the boundary value problem (1), in which the function permits singularity with respect to its time variable .
Many researchers have derived some important results for solutions to boundary value problems of fractional differential equations with singularity with respect to the time variable (see [8–29]). In , Henderson et al. established the existence and multiplicity of positive solutions for a system of nonlinear Riemann–Liouville fractional differential equations with the coupled multipoint boundary conditions where and the functions can be singular at the points and/or . They employed the Guo–Krasnosel’skii fixed-point theorem to prove that their problem has at least one positive solution. And Wu and Zhou  used the upper and lower solutions method to study the existence of positive solutions for the fractional-order eigenvalue problem with the -Laplacian operator where and can be singular at and (for more detailed information about the upper and lower solutions method to solve integral and differential equations, see ).
Taking the previous results together, to our best knowledge, very little is known about the existence and uniqueness of positive solutions of -Laplacian fractional boundary value problems with singularities with respect to their time variable.
Motivated by the above works, in this paper, we first apply the Leray-Schauder nonlinear alternative to establish the existence of solutions to our problem (1) and then use the Schauder fixed-point theorem and upper and lower control functions to derive the upper and lower solutions method to obtain the maximal and minimal solutions. And we prove the uniqueness of solutions to the given problem by using some useful properties of the Gauss hypergeometric function and the Banach contraction mapping principle.
Throughout this paper, we suppose that
For the convenience of the readers, we will give some necessary definitions and lemmas here.
The Riemann-Liouville fractional integral and the Riemann-Liouville fractional derivative of order of a function are given by where , provided that the right-hand sides are pointwise defined on (see ).
Lemma 1 (see ). Assume that with a fractional derivative of order that belongs to . Then, for some , where is the smallest integer greater than or equal to .
Lemma 2 (see ). (Schauder fixed-point theorem). Let be a nonempty, closed, bounded, and convex subset of a Banach space , and suppose is a compact operator. Then, has a fixed point.
Lemma 3 (see ). Let be a Banach space with a closed and convex subset of . Assume is a relatively open subset of , with , and is a compact map. Then, either, (i) has a fixed point in , or(ii)there is a point and , with
Lemma 4. Let be a continuous function such that and define the function as Then, is continuous on .
Proof. From the assumption (5), we can see easily that
Put as follows
Then, we divide the proof of this lemma into the following three cases:
If , the definition of an improper integral yields Take the limit to obtain This implies .
If , we can get In a similar way above, the first term of the right side in Equation (17) can be evaluated as And for the second term of the right side in Equation (17), it can be easily seen that Combining these two inequalities above, we can find For the case , if , then we have Similarly to that given above, we obtain By simple calculation, we can get So for any , it holds that If , the definition of an improper integral also implies . Taking the limit on both sides of the inequality (11), we can obtain Therefore, it follows directly from the inequalities (7) and (10), (11), (12), (13) that is well-defined on . Since , it is obvious that is continuous on . Combining this with the continuity of at , we can prove that is a continuous function on .
Lemma 5. Let be a continuous function such that satisfies (6). Then, the boundary value problem has a unique solution which is given by where in which where
Remark 6. In Lemma 5, a function with a fractional derivative of order that belongs to (i.e., ) is said to be a solution of the boundary value problem (14) if it satisfies the fractional differential equation and the boundary conditions of (12).
Proof. As we can see in the proof of Lemma 4, we have
So, we can get
This implies .
Also, Lemma 4 asserts that Since and , it follows from Lemma 1 that a solution of the boundary value problem (14), , satisfies For the rest of the proof, it is easy to see that doing as in the proof of Lemma 4 in  leads to a conclusion of this lemma.
Lemma 8 (see ). Let . Then, the boundary value problem has a unique solution which is given by where in which where
Lemma 9 (see ). If , then the function in Lemma 8 satisfies the following conditions: (i), for (ii), for where The following useful properties of which will be used later can be found in : (i)If , and , then(ii)If , then
3. Main Results
In this section, we will prove the existence and uniqueness of positive solutions for the boundary value problem (1) and derive the upper and lower solutions method by using some fixed-point theorems.
3.1. The Existence Results for Problem (1)
Definition 10. A function is called a solution of problem (1) if it satisfies the fractional differential equation and the boundary conditions of (1).
The following hypothesis concerned with the function , which permits singularity with respect to time variable, will be used in this article.
(H1). There exist such that for any , Let be the Banach space equipped with the norm and put .
Proof. Suppose that is a solution of the problem (1). Putting , by using Lemma 8, we have
Also putting , by Lemma 5, we obtain
Since , Equations (47) and(48) yields
It is obvious that and is a solution of the integral Equation (46).
Conversely, suppose that is a solution of the integral Equation (46). Put as follows: Then, Equation (36) implies that . From the definition of , we also have So, we can get and it follows that . Combining Equation (36) with we can see This yields .
Since the boundary value problems (14) and (16) have unique solutions by Lemma 5 and Lemma 8, we can find that the function satisfies the fractional differential equation and the boundary conditions of the problem (1). This completes the proof.
Define the operator on as follows: Then, the function is a solution of the integral Equation (46) if and only if the operator has a fixed point in .
Lemma 12. If the hypothesis (H1) holds, then
Proof. The conclusion of this lemma easily follows from Lemma 4, so we omit the details.
For convenience we define the function on as follows: Obviously, we can see that . In fact, using the hypothesis (H1), for any , it holds that and .
Lemma 13. If the hypothesis (H1) holds, then the operator is completely continuous.
Proof. We first prove that the operator is continuous on . For this, choose any and any sequence convergent to . Then, there exists such that . From the continuity of the function , we can put
From the definition of the function , we can see that for any ,
Since , we have
A simple calculation provides that
and therefore, by using the Lebesgue dominated convergence theorem, we know
Combining this with , we can obtain that for any ,
Since , we can get that for any ,
Next, we show that for any bounded set , is relatively compact. To do this, by the Arzela–Ascoli theorem, it is sufficient to prove that is uniformly bounded and equicontinuous. Denote .
From Lemma 7 and Lemma 9, we see that for any and any , Put . Since , we have Therefore, we know This means that is uniformly bounded.
By using the inequality (20), we can get that for any , Combining this inequality with the uniform continuity of on , it holds that for any , there exists such that This implies that is equicontinuous. The proof is completed.
Put and list more hypotheses to be used in this paper.
(H2). There exists a nondecreasing function such that for any .
(H3). There exists such that .
Proof. Putting , it is obvious that . By using Lemma 13, we can know that the operator is completely continuous.
Assume that there exist a point and a number , with . By the hypothesis (H2), we see that for any , Using Lemma 7 and Lemma 9, we have Combining these two inequalities, we obtain Now taking th power on both sides of the above inequality, since , , and is a nondecreasing function, we get This is a contradiction to the hypothesis (H3). Therefore, by Lemma 3, the operator has at least one fixed point in . The proof is completed.
3.2. Derivation of the Upper and Lower Solutions Method
We define the upper control function and the lower control function on as follows:
Then, we know that the functions and are nondecreasing in and for any ,
Definition 15. The functions and are said to be a upper solution and a lower solution of the integral Equation (46) in , respectively, if
Lemma 16. Assume that the hypotheses (H1)-(H3) hold and there exist a upper solution and a lower solution of the integral Equation (46) in . Then, the problem (1) has at least one solution in , where .
Proof. It is obvious that is a nonempty, closed, bounded, and convex subset of the Banach space and the operator is completely continuous. In a similar way to the proof of the Theorem 14, for any , we can get
By the hypothesis (H3), we have
That is, .
Now we prove . In fact, since the functions