Abstract
In this paper, we consider a fractional differential system with multistrip and multipoint mixed boundary conditions involving -Laplacian operator and fractional derivatives. The existence result of positive solutions is established by the Leggett-Williams fixed point theorem. Also, an example is presented to illustrate our main result.
1. Introduction
Fractional calculus is a significant branch of mathematical analysis which concerns derivatives and integrals. Fractional differential equations are caused by the intensive development of the theory of fractional calculus and have gained great interest currently, because of their crucial applications in various fields of physics, biology, and engineering such as control, porous media, electromagnetic, and other fields [1–4].
Moreover, many mathematical formulations of physical and biological phenomena lead to fractional differential equations with operator, such as memory, fluid dynamics, predator harvesting, and hereditary properties of various materials, which are difficult to achieve in classical integer-order models. Hence, in recent years, more and more researchers have considered the fractional differential equations with -Laplacian operator, by applying various mathematical theory and tools; refer to [5–9].
For example, authors in [10] discussed the following differential system with -Laplacian operator where , , , , , is a operator: , and . Authors established the conclusion of the existence and uniqueness of solution to the above problem by using topological degree theory.
By reading the existing literatures [11–24], we note that the fractional differential system involving -Laplacian operator and multistrip and multipoint boundary conditions has not been studied yet. Thus, in this paper, we first pay close attention to the following fractional differential system, involving -Laplacian operator and lower fractional derivatives:with multistrip and multipoint boundary conditions:where and are the Riemann-Liouville fractional derivatives of order and , is a continuous function. For , , , , , , , , are nonnegative constants. . And , are nondecreasing functions of bounded variation in .
We emphasize that, despite the complication of the discrete and integral boundary conditions (3) and (4), operator T can be presented in a concise form based on the piecewise form of Green’s function. And we can accurately estimate the upper and lower bounds of its value, which is fully prepared for the establishment of the main theorem.
Accordingly, the conclusion we reached is an extensive result and a meaningful supplement to the theory of the fractional differential systems with -Laplacian operator.
Theorem 1 (the Leggett-Williams fixed point theorem). Let be a cone in a real Banach space , , be a nonnegative continuous concave function on P such that for all , and . Suppose that is completely continuous and there exist constants such that ; ; Then has at least three fixed points , , and which satisfy
2. Preliminaries
In this section, we will put forward some crucial definitions and theorems, preparing for establishing the existence of solutions.
Definition 2. The Riemann-Liouville fractional integral of order of a function is given by provided the right-hand side is pointwise defined on , where is the Euler gamma function defined by , for
Definition 3. The Riemann-Liouville fractional derivative of order for a function is given by , where , stands for the largest integer not greater than .
According to the definitions of Riemann-Liouville’s derivative, the following lemmas can be achieved.
Lemma 4. For , if we assume that , then we have for some , while is the smallest integer greater than or equal to .
Remark 5. The following properties are useful for our discussion:(i)As a basic example, we quote for , (ii).
For convenience, we denote Besides, it should be noted that .
Lemma 6. For , the equationswith boundary conditions (3) and (4) have a unique solution as follows: whereand is a piecewise function; the expression of the th piece is as follows Meanwhile, is also a piecewise function; the expression of th piece is as follows
Proof. We shall reduce BVP (11), (3), and (4) to two fractional differential problems. To this goal, first, by means of transformation, we convert problem (11) intoIn view of Lemma 4, the solution of problem (17) can be written aswhere , are arbitrary real constants.
By , we find . Setting in (19), together with the boundary condition , we have Accordingly, we can easily find outCombining (21) with (19), we can obtain Next, considering the multipoint boundary conditions, we divide the domain of integration into pieces. For , the unique solution of (17) can be expressed as When , we observe that the th integrand in (23) can be taken as follows: where is also noted by 0.
As a result, the Green function is always valid and can exactly express each integrand in the above formula, for . Thus, we can take (23) as a unified form is a piecewise function such that the form of the th piece is given by (15).
We take (25) into (18); then From Lemma 4, we havewhere , are arbitrary real constants.
By , we easily find . Setting t=1 in (27), together with the boundary condition , we have Set and into (27); we find outFurthermore, by integrating and summing on both sides of (29), it is easy to see that Then, we get Take (31) into (29); we conclude that the unique solution of BVP (11), (3), and (4) is Analogically, the unique solution of BVP (11), (3), and (4) is where is also a piecewise function, the th piece is defined by (16), and the calculation process is similar.
This completes the proof.
Denote
Lemma 7. For and , it holds that
Proof. For and , we have , while is a monotone increasing function. Therefore, we can infer that . For , we finally find out , which implies and , for , .
This completes the proof.
Lemma 8. For , the functions defined by (14) have the following properties:(1), , for ;(2);
Proof. (1) The positive of and is easily checked.
(2) For , we have For , we have The other inequality is achieved similarly. All these complete the proof of the lemma.
For , denote and the constant is given by the maximum value of , , , and .
Lemma 9. For , the functions , , and defined by Lemma 6 satisfy the following results:(1), , are continuous, , , ;(2), ;(3), .
Proof. The nonnegativeness of , , and is easily checked.
(2) According to the property (2) in Lemma 8, we have For , we have Because of the randomicity of , we have According to the above, from the property (2) of Lemma 8, we easily get This completes the proof of the lemma.
Define the space and is endowed with the norm , where ,
Then we introduce the product space endowed with the norm and define a partial order where , , for .
Lemma 10. is a Banach space.
Proof. Let be a Cauchy sequence in the space . Then clearly and are Cauchy sequences in the space . Therefore, and converge to some and on uniformly and . Then, we need to prove .
Note that By the convergence of , we getuniformly for . On the other hand, by Remark 5, one has , for and some . Furthermore, we can getFrom (44) and (45),we obtain Taking the -order derivative on both sides of yields, we have In view of Remark 5 and Lemma 4, we find that Above all, we can achieve that is a Banach space. This completes the proof.
Define the cone byLet the nonnegative continuous concave functional on the cone be defined by where is a positive constant, satisfying .
For convenience, for , we denote where .
For any , we introduce the integral operator by whereConsequently, by calculation where
Lemma 11. is completely continuous.
Proof. Firstly, we prove that . In view of the nonnegativeness and continuity of , and , is continuous and for . Hence, .
Next, we show is uniformly bounded. Let be bounded; that is, there exists a positive constant such that , , , for all . Let . Then for , from the Lemma 9, we have Hence, is uniformly bounded. Then, we show is equicontinuous. Indeed, for any , , , we have Moreover, Therefore, the above process implies that is equicontinuous for all .
By means of the Arzela-Ascoli theorem, is completely continuous. Likewise, the proof of follows the same method as . Thus, is completely continuous.
3. Existence Result
For convenience, we denote
Theorem 12. There exist nonnegative numbers , and satisfies the following conditions:, for ;, for ;, for Then BVP (2)-(4) has at least three positive solutions , , and such that
Proof. We will verify that conditions of Theorem 1 are satisfied.
Let We first prove that . For any , , we have , , , . Consequently, combining with properties (2) and (3) of Lemma 9, Thus, and further to get . Therefore is completely continuous. The proof of is the same as . Thus, is completely continuous.
By the above, condition of Theorem 1 can be satisfied, using assumption .
Finally, in order to verify , we set , for . It is easy to find that Hence, If , we have , , , , for all . Then That is, for all . This shows that condition of Theorem 1 is also satisfied.
From the above, BVP (2)-(4) has at least three positive solutions , , and such that The proof is completed.
Example 13. To verify the feasibility of our conclusion, here we consider the boundary value problem of nonlinear fractional differential system as follows:with boundary conditions In this model, we set , and Let , for , It is obvious that () is continuous. Take ; by calculation, we obtain For , set , , and , which satisfy , such that , for ;, for ;, for ,Thus, all the conditions in Theorem 12 are achieved. Consequently, BVP (66) has at least three positive solutions , , and such that , , and with .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All authors contributed equally to the manuscript, read and approved the final draft.
Acknowledgments
The work is supported by National Training Program of Innovation (No. 201910019172). The funding body plays an important role in the design of the study and analysis, calculation, and writing the manuscript.