Abstract
In this paper, the existence of positive solutions in terms of different values of two parameters for a system of conformable-type fractional differential equations with the p-Laplacian operator is obtained via Guo-Krasnosel’skii fixed point theorem.
1. Introduction
In this paper, we study the existence of positive solutions for the following system of fractional differential equations: subject to the following boundary condition:where are real numbers; and are the conformable fractional derivative; , ; are continuous; and are positive parameters.
Fractional differential equations have many applications in various fields such as biological science, chemistry, physics, and engineering. Many authors have made large achievements about the study of fractional differential equations boundary value problems. Most results have adopted the Riemann-Liouville and Caputo-type fractional derivatives; we can see [1–28] and the references therein; for example, in [28], by using Guo-Krasnosel’skii fixed point theorem, the authors obtained the various existence results for positive solutions about a system of Riemann-Liouville type fractional boundary value problems with two parameters and the p-Laplacian operator. As we know, there is another kind of fractional derivative which is conformable fractional derivative. Recently, in [29], the authors Khalil R. et al. first introduced a new simple well-behaved definition of the fractional derivative called conformable fractional derivative. They first presented the definition of conformable fractional derivative of order and generalized the definition to include order . In [30], Abdeljawad proceeded on to develop the definitions and set the basic concepts in this new simple interesting fractional calculus. Since then, there are a few authors to study the boundary value problems for conformable-type fractional differential equations; for example, we can see [31–33] and the reference therein. In [33], the authors applied approximation method and fixed point theorems on cone to consider the existence and multiplicity of positive solution about the following fractional differential equation with the p-Laplacian operator: where is a real number, is the conformable fractional derivative, , and is continuous.
There are few papers about the system of fractional differential equations concerning conformable fractional derivative. System (1), (2) is a new type of conformable fractional differential equations. Motivated by the recent papers [28, 33, 34], we consider the existence of positive solutions of the system for conformable fractional differential equations (1), (2). By using Guo-Krasnosel’skii fixed point theorem, we establish some sufficient conditions on for the existence of at least one positive solutions of system (1), (2) for appropriately chosen parameters.
The organization of this paper is as follows. In Section 2, we recall some concepts about the conformable fractional derivative and give some lemmas with respect to the corresponding Green’s function. In Section 3, we give some results about the existence of positive solutions of system (1), (2). In Section 4, we summarize the main results of the third section.
2. Preliminaries
For the convenience of the reader, we give the following concepts and lemmas of conformable fractional calculus, and some auxiliary results that will be used to prove our main theorems (see [29–33]).
Definition 1. Let and be a n-differential function at , then the fractional conformable derivative of order at is given by provided the limit of the right hand side exists. If is -differentiable in some , where , and exists, then define
Definition 2. Let . The fractional integral of order at of a function is given by where denotes the integration operator of order .
Lemma 3. Let and be -differentiable at a point . Then(i), .(ii), where .(iii), for all constant functions .(iv).(v).(vi)If, in addition, is differentiable, then (vii)If, in addition, is differentiable at , then
Lemma 4. Given and a continuous function defined in the domain of , one has that for .
Lemma 5 (mean value theorem). Let and be a given function that satisfies(i) is continuous on ,(ii) is -differentiable for some . Then, there exists such that
Lemma 6. Given and an -differentiable function, one has that if and only if , where
Lemma 7. Given and an -differentiable function that belongs to , one has that , for some ,
By Lemma 2.7 in [33], we can obtain the following lemmas.
Lemma 8 (see [33]). Let and . Then the conformable fractional differential equation has a unique solution where and
Lemma 9 (see [33]). The function defined by (9) has the following properties:(i), for all ;(ii), for .
Lemma 10 (see [35]). Let be a Banach space, be a cone, and be bounded open subsets of , , and . Assume that is a completely continuous operator such that (i) or(ii). Then the operator has at least one fixed point in .
3. Main Results
Let with supremum norm . Let with the norm . Then is a Banach space. We define the cone
In the following, we define the operators and : where is defined by (9).
Obviously, the nontrivial fixed points of the operator in P are positive solutions of system (1), (2).
Lemma 11. The operator is a completely continuous operator.
Proof. It is obvious that for . By Lemma 8, we have and Then On the other hand, by Lemma 9, we have and So we have i.e.,
By the paper [33], we know that and are completely continuous operator. It is obvious that is completely continuous. The proof is completed.
Denote
Theorem 12. Assume that , and , then for each and , system (1), (2) has a positive solution , , where
Proof. Let and . Then there exists a number such that , and For the above , we know that there exists such that Let . For any , we have So Similarly, we have So Hence On the other hand, for the above , there exists such that Let . Let For any , we have So by Lemma 9, we have So Similarly, we have So Hence By Lemma 10 and (26) (32), the operator has one fixed point . That is, is a positive solution of system (1), (2).
Similar to the proof of Theorem 12, we can easily obtain the following results.
Theorem 13. Assume that and , then for each and , system (1), (2) has a positive solution
Theorem 14. Assume that and , then for each and , system (1), (2) has a positive solution
Theorem 15. Assume that , then for each and , system (1), (2) has a positive solution
Theorem 16. Assume that or , then for each and , system (1), (2) has a positive solution
Theorem 17. Assume that or , then for each and , system (1), (2) has a positive solution
Theorem 18. Assume that or , then for each and , system (1), (2) has a positive solution
Denote
Theorem 19. Assume that , and , then for each and , system (1), (2) has a positive solution , , where
Proof. Let and . Then there exists a number such that , and For the above , we know that there exists such that Let . For any , we have So by Lemma 9, we obtain So Similarly, we get So Hence We define the functions , , and So it is obvious that and are nondecreasing on for every ; and ; and they satisfy the conditions For the above , there exists such that So , , and
By the definition of , we get , . Let . Let For any , we have So Similarly, we have So Hence By Lemma 10 and (41) (48), the operator has one fixed point . So is a positive solution of system (1), (2).
Similar to the proof of Theorem 19, we can easily obtain the following results.
Theorem 20. Assume that and , then for each and , system (1), (2) has a positive solution
Theorem 21. Assume that and , then for each and , system (1), (2) has a positive solution
Theorem 22. Assume that , then for each and , system (1), (2) has a positive solution
Theorem 23. Assume that or , then for each and , system (1), (2) has a positive solution
Theorem 24. Assume that or , then for each and , system (1), (2) has a positive solution
Theorem 25. Assume that or , then for each and , system (1), (2) has a positive solution
Theorem 26. Assume that or , then for each and , system (1), (2) has a positive solution
4. Conclusion
In this paper, we investigate the existence of positive solutions for a system of conformable-type fractional differential equations with two parameters and the p-Laplacian operator. By employing Guo-Krasnosel’skii fixed point theorem, under different combinations of sublinearity and superlinearity of the nonlinearities , various sufficient conditions for the existence of at least one positive solutions of system (1), (2) are derived in terms of appropriately chosen parameters .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.
Acknowledgments
The project is supported by the Youth Science Foundation of China (11801322), the National Natural Science Foundation of China (11571207), and Shandong Natural Science Foundation (ZR2018MA011).