Multiple Positive Solutions for a System of Caputo Fractional -Laplacian Boundary Value Problems

Abstract

In this work, we are pleased to investigate multiple positive solutions for a system of Caputo fractional -Laplacian boundary value problems, and we also provide an example for illustrating our main results.

1. Introduction

In this work, we are pleased to discuss the positive solutions for the following system of Caputo fractional -Laplacian boundary value problems:where are the fractional derivatives of Caputo sense with ; is the -Laplacian, i.e., with ; and the constants , and the functions satisfy the following conditions: (H0) , , with  (H1) are nonnegative continuous functions on , where

As a generalization of integer-order equations, fractional-order equations can effectively describe various materials and physical processes with memory and genetic properties. It has a large number of applications in our society, such as biology, chemical kinetics, electromagnetics, transmission and diffusion, and automatic control. Recently, there many researchers pay their attentions to studying the existence of solutions for various types of fractional-order equations by use of fixed point theorems, upper and lower solution methods, and monotone iterative techniques. For instance, we refer the readers to [1–25] and references therein.

In [1], by using the method of upper and lower solutions and the Schauder fixed point theorem, Vong investigated the positive solutions for the following nonlocal fractional boundary value problem:where , and is a function of bounded variation. may be singular at .

In [2], Wang and Yang studied the integral boundary value problem of Caputo sense:

By the Leggett–Williams fixed point theorem, they obtained multiple positive solutions when the nonlinearity is bounded from below. Moreover, when is asymptotically linear at infinity, they also obtained an existence theorem.

In [3], Wang et al. adopted the theory of mixed monotone operators to obtain a unique positive solution for the mixed fractional boundary value problems involving the -Laplacian:where is the Caputo fractional derivative and are the Riemann–Liouville fractional derivatives.

Recently, coupled systems of fractional differential equations have also been investigated by many authors. Some results on the direction can be found in a series of papers [11–25] and the references cited therein.

In [11], Wang utilized the Guo–Krasnosel’skii fixed point theorem to investigate the multiple positive solutions for the mixed fractional -Laplacian differential system:where are the Riemann–Liouville fractional derivatives, the Caputo fractional derivatives, and and are the Riemann–Stieltjes integrals.

In [12], Rao studied the system of fractional -Laplacian differential equations:

When the nonlinearities satisfy some appropriate conditions, the author made use of the Avery–Henderson fixed point theorem and the six functionals’ fixed point theorem to obtain some existence theorems of multiple positive solutions.

Inspired by the aforementioned results, in this work, we study the solvability for (1) and establish the existence results of multiple positive solutions via the six functional fixed point theorem under some bounded conditions for . Finally, we also provide an example to illustrate our main results.

2. Preliminaries

In this section, we only recall the definition of Caputo fractional derivative, for more details, see the book [26].

Definition 1. The fractional derivative of in the Caputo sense is defined aswhere , denotes the integer part of the number .
Now, we calculate Green’s functions associated with (1). Letand then by the boundary conditions in (1), we haveConsequently, substituting (8) and (9) into (1), we obtainWe next translate (10) into an equivalent system of integral equations. By the similar arguments as in [2], we have the following result.

Lemma 1. Problem (10) is equivalent to the following system of Hammerstein-type integral equations:where

Proof. We only need to consider the case (by the similar method, the case can be easily proved). Using Lemma 2.5 of [2], we havewhere . implies that , and by , we haveTherefore,This completes the proof.
Note from (8) and Lemma 1, we haveandwith the boundary conditionsBy Lemma 2.5 of [2], we have the result:

Lemma 2. Problems (17) and (18) is equivalent to the following Hammerstein-type integral equation:where

Lemma 3 (see [2], Lemma 2.8). The functions have the following properties:(i)(ii) where

Let . Then, is a real Banach space and is a cone on . Moreover, is a Banach space with the norm , and a cone on . From Lemmas 1 and 2, we define operators and as follows:

Then, we obtain that , are completely continuous operators, and if there exists such that , i.e., , , then is a positive solution for (1).

Lemma 4 (see [27]). Let be a cone in a real Banach space . Suppose that , and are nonnegative continuous concave functionals on , are nonnegative continuous convex functionals on , and there are nonnegative constants , and such that is a completely continuous operator and (B1) is a bounded set (B2) and are disjoint subset of  (B3)  (B4)  (B5)

If the following claims hold: (C1) with , and  (C2) with , and  (C3) with , and  (C4) with , and  (C5) with and  (C6) with and

Then, has at least three fixed points , and in such that with , and with .

3. Main Results

We first define some notations:where and . Then, , and are the concave functionals on , and , and are the convex functionals on .

Theorem 1. Suppose that there exist positive real numbers , and such that satisfy the following conditions: (H2) for all and . (H3) for all and . (H4) for all and , where :

Then, (1) has at least three positive solutions , and in .

Proof. Note, from a standard calculus argument, we obtain the set is bounded. Since if , and then Let , and with . Then, if , we have . This means thatand . Also, it can easily be shown thatHence, these sets are nonempty. As a result, (B1)–(B5) of Lemma 4 are satisfied.
Now, we verify the functional conditions.

Claim 1. , for all with and . Note, from Lemma 3 (ii), we have

Claim 2. , for all with . At this case, we have , for , and from (H4), we obtain

Claim 3. , for all , and . Then, we have

Claim 4. , for all with . Therefore, by Lemma 3 (ii) and (H3), we have

Claim 5. , for all with , and . Then, we obtain

Claim 6. , for all with . Therefore, by Lemma 3 (ii) and (H2), we have

Remark 1. Our Green’s functions satisfy some special inequalities (see Lemma 3 (ii)). So, in the Claims 1 and 2 of Theorem 1, cannot reach zero. Therefore, we use the Caputo derivative for the considered scheme. This also explains the reason that we do not use other forms of fractional derivatives (including the Riemann–Liouville type and the Hadamard type).

Example 1. Consider the example:where .
In order to obtain the constants in Theorem 1, we need to calculateand we haveMoreover, we also obtain that , , , , , and . If we put , , , andThen, satisfies the following conditions:(i) for all and (ii) for all and (iii) for all and Therefore, all assumptions in Theorem 1 are satisfied, and the system of (32) has at least triple positive solutions.

4. Conclusion

In this paper, we use the six functional fixed point theorem to study the system of (1) under some bounded conditions for . We first transform the original fractional differential system into the equivalent system of Hammerstein-type integral equations, and then with the help of properties of Green’s functions, we provide some sufficient conditions to guarantee the existence of multiple positive solutions for our system. Finally, an example is presented to illustrate the effectiveness of the main result.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11771017), China Postdoctoral Science Foundation (2019M652348), Key Research of Henan Higher Education Institutions (17A11001, 19A110018, and 20B110006), Technology Research Foundation of Chongqing Educational Committee (KJQN201900539), Fundamental Research Funds for the Universities of Henan Province (NSFRF180320), and Henan Polytechnic University Doctor Fund (B2016-58).