Abstract
This paper considers a system of fractional differential equations involving -Laplacian operators and two parameters where , , and are the standard Riemann-Liouville derivatives, , , , , and and and are two positive parameters. We obtain the existence and uniqueness of positive solutions depending on parameters for the system by utilizing a recent fixed point theorem. Furthermore, an example is present to illustrate our main result.
1. Introduction
During the past several decades, many fractional problems with differential equations have been paid much attention, see [1–10] for example. Also, much attention has been focused on the existence of positive solutions for such equations, see [3–29] and the references therein. As we know, the -Laplacian operator has very a important position in theoretical research and engineering applications. In 1945, to discuss turbulent flow in a porous medium, a basic mechanical problem, Leibenson [30] introduced a differential equation with a -Laplacian operator:
Since then, there are many papers investigating differential equations with -Laplacian operators. Recently, the study of fractional equations with a -Laplacian operator has also gained plenty of attention, see [19, 20, 31–40] for instance. In [35], the authors studied a fractional equation with a -Laplacian operator: where , , and denote the Riemann-Liouville derivatives, , and . Based on Schauder’s fixed point theorem and by using the upper-lower solution method, they obtained the existence and uniqueness of solutions.
Recently, fractional differential systems have been also studied by many people because of their great application value, see [5, 20, 23–30]. So, the results on fractional systems with -Laplacian operator are many, see [11, 41–45]. For example, Rodica [41] discussed a fractional differential system: where , , , , , , , , , for all , , for all , , , , and . The existence of solutions was obtained via Guo-Krasnosel’skii’s fixed point theorem.
From literature, we see that most results are the existence of solutions, but the uniqueness is scarce. Inspired by [34], we discuss the following system of fractional differential equations with -Laplacian operators: where , , and denote the standard Riemann-Liouville derivatives, , , , and , and and are two positive parameters. It should be pointed out, in [45], that Hao et al. investigated the existence of solutions for system (4) without considering the uniqueness. They used Guo-Krasnosel’skii’s fixed point theorem to get some existence results for positive solutions under different values of and . In this paper, based upon a recent fixed point theorem, we aim to present the existence and uniqueness of positive solutions for system (4) depending on fixed positive constants and . Our results can tell us that the unique positive solution exists in a product set and can be approximated by giving an iterative sequence for any initial point in the product set. Therefore, our result is an extension and improvement of the previous works. At the end, an example is given to illustrate the result.
2. Preliminaries
Lemma 1 (see [45]). Assume ,. If , then the unique solution of the following problem: is where For convenience, we can easily give the following Lemma by using Lemma 1.
Lemma 2. Let , , ,. If , then has a unique solution where
By Lemmas 4 and 5 in [45], the following conclusion is clear.
Lemma 3. The functions defined by (7) and (12) have several properties: (i) is continuous on and for (ii)(iii) where
Suppose that is a real Banach space with a partial order induced by a cone . For any , the notation denotes that there exist and such that . For (i.e., and ), define a set . Evidently, . For with . Suppose , then . If is normal, then is normal.
Lemma 5 (see [47]). Let be a normal cone in a Banach space and with . Operators are increasing and satisfy the following:
There exist such that
where , , ;
There is such that , .
Then,
(a), , and exist , , such that and(b)for any given , the equation has a unique solution in . Moreover, take any fixed point , letthen , , as .
3. Positive Solutions Depending on Parameters
Let , a Banach space with the norm . We study (4) in the product space . For , let . Then, is a Banach space. Let , , then is a cone and is normal, and the space has a partial order:
Lemma 6. Let , be continuous. By using Lemmas 1 and 2 and some results in [45], is a positive solution of (4) if and only if is a solution of the following equations:
Theorem 7. Let , , , . Assume that
and ,
are increasing with respect to the second, third variables, i.e., , for ,
for , there is , such that and
for ,
Then
(a)there are , , such that andwhere , are the Green functions in Lemmas 1 and 2(b)System (4) has a unique positive solution depending on in , where , (c)Take any initial point , letthen , as
Proof. We consider three operators and defined by where , , , and are defined by (7) and (12). From Lemma 3 and , it is clear that and . From our above discussion, we can easily claim that is a solution of system (4) if and only if is a fixed point of operator . Next, we only need to prove that all assumptions of Lemma 5 are satisfied for operators .
We first show that are increasing. To do this, for , with , , one has , , and by and Lemma 3,
That is, and .
Second, we indicate that satisfy condition of Lemma 5. Let , . Then, for , by , we have
Similarly, . For and , by , we obtain
That is, , for , .
Set , where , , . Then, . Now, we prove that , . In view of and Lemma 3, for , we have
Noting that and guarantee that and , . Because , , then
So, we have
By the definition of , it is clear that ; then, we have , so , , that is, . Similarly, from Lemma 3 and , we get .
Now, by Lemma 5, we obtain the following conclusions: (1)We can find , , such that and that is, (2)The operator equation has a unique solution depending on in , where , . That is, . So, system (4) has a unique positive solution in (3)Taking any initial point , let then , as
Taking we have the following conclusion.
Corollary 8. Let , , and , . Assume that , , and hold. Then, the following system: has a unique positive solution in . In addition, take any given point , construct then , as .
4. An Example
Considering the following system:
where , , , , , , , , and
Obviously, and
Note that and are increasing in , it implies that , are increasing with respect to the second and third variables for . Moreover, set , , . Then, , , , , and for , . Hence, all conditions of Theorem 7 are satisfied. Then, Theorem 7 shows that system (34) has a unique positive solution in , where , , and taking any given point , let then , as , where
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgments
This paper was supported financially by the Shanxi Province Science Foundation (201901D111020) and Graduate Science and Technology Innovation Project of Shanxi (2019BY014).