Journal of Function Spaces
Volume 2019 (2019), Article ID 2787569, 11 pages
https://doi.org/10.1155/2019/2787569

Research Article

Multiple Positive Solutions for Nonlinear Fractional Differential Equations with Integral Boundary Value Conditions and a Parameter

Juan Mao¹ and Daliang Zhao²

¹Department of Basic Courses, Shandong Polytechnic, Jinan, 250104, China
²School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, China

Correspondence should be addressed to Daliang Zhao; dlzhao928@sdnu.edu.cn

Received 20 January 2019; Accepted 28 March 2019; Published 2 May 2019

Academic Editor: Pasquale Vetro

Copyright © 2019 Juan Mao and Daliang Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

By using the theory of fixed-point index on cone for differentiable operators, spectral radii of some related linear integral operators, and properties of Green’s function, the existence of multiple positive solutions to a nonlinear fractional differential system with integral boundary value conditions and a parameter is established. At last, some examples are also provided to illustrate the validity of our main results.

1. Introduction

In recent years, the interest in the study of fractional differential equations has been growing rapidly since it has many applications in biology, mechanics, electrochemistry, and dynamical processes in self-similar structures, etc.; see [1–3], for instance. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes. Fractional-order models have proved to be more accurate than integer order models and have more applications. In consequence, many meaningful results in these fields have been obtained. For details, see [4–14] and references therein.

In addition, several new types of fractional derivatives are also investigated in some excellent papers. In [15], Yang studied a class of fractional derivatives of constant and variable orders for the first time. In [16], the authors introduced a new fractional derivative without singular kernel, and this was an extension of the Riemann-Liouville fractional derivative with singular kernel and had some important applications in the modeling of the fractional-order heat flow.

Meanwhile, there are a lot of papers (see [3, 17–28], for instance) dealing with the existence of positive solutions of nonlinear fractional differential equations by use of fixed-point theorems, monotone iterative technique, and upper and lower solution method. For example, in [25], by using the method of upper and lower solutions and Schauder fixed theorem, Vong investigated the positive solutions for the following nonlocal fractional BVP:

where , and is a function of bounded variation. may be singular at

In [26], The authors studied a class of higher order fractional BVP. In view of monotone iterative method, they got an existence and uniqueness result of positive solution.

Goodrich [8] dealt with a problem similar to the system in [26] but with local conditions. By deriving properties of the Green’s function and by means of Guo-Krasnoselskii’s fixed-point theorem, the author established some existence results of at least one positive solution provided that satisfies some growth conditions.

However, compared with the methods of fixed-point theorems, Leray-Schauder theory, and monotone iterative technique, there is little work investigating positive solution of fractional integral boundary value problem by use of the differentiable operator method. On the other hand, the results about such system with a parameter are relatively scarce.

Motivated by all the above-mentioned works, we aim to establish some existence criteria of multiple positive solutions for the following nonlinear fractional BVP with integral boundary conditions and a parameter:

where , , , , is the Caputo fractional derivative, and is continuous. It is worth mentioning that there is no paper studying the multiple positive solutions for the fractional differential systems by using the differentiable operator method. Besides, to the best of our knowledge, no contribution exists investigating the existence of positive solutions for BVP (2). It should be noted that is more general than those in [23, 25, 26], in which equals one.

Contributions of this article are as follows. Firstly, we derive the corresponding Green’s function for fractional BVP (2) and argue its important properties which will play an important role in our proof. Also, this is the key to establish an appropriate cone. Secondly, by using the differential operator method, spectrum theory, and the properties of Green’s function, we firstly investigate some existence results of multiple positive solutions for the considered fractional system. The methods used in the present paper are different from all the above-mentioned works.

This paper is organized in the following way. In Section 2, we give some necessary preliminaries and lemmas. Section 3 establishes sufficient conditions for the existence of multiple positive solutions to system (2). In Section 4, some examples are given to highlight the value of the new contributions.

2. Preliminaries and Some Lemmas

We now recall some results about fractional calculus in common use. These materials can be found in the recent literature; see [2, 3, 29, 30].

Definition 1. The Riemann-Liouville fractional integral of order of a continuous function is given by

provided that the right side integral is pointwise defined on .

Definition 2. The Caputo fractional derivative of order of a continuous function is given by

provided that the right side integral is pointwise defined on

Lemma 3. Let Then

for some

Let be endowed with the norm for each Clearly, is a Banach space.

Lemma 4. Suppose function . Then the unique solution of the following system

where , , , , is given by

where is the Green’s function of system (6) given by

Proof. From Lemma 3, it follows that

where .
So, one has

From , we get

So we have,

By direct integration to (13), it is easy to get

Through the conditions , (12), and (14), one has

Then, we obtain

Therefore, we can get the following solution of (6) by (13),

In the following, we divide the proof into two cases.
Case 1

Case 2

The conclusion of this lemma follows

Next, we give some important properties about Green’s function belonging to system (6).

Lemma 5. The Green’s function of system (6) satisfies the following inequalities:
;
;
;
.

Proof. We denote and take .
For , one has

and

This means that is concave on . So we get
For , we have

and

These three facts can also indicate
For , we have , and

So, we can get that
For , we obtain , and

Hence, we also have
Consequently, one gets the validity of and . The establishment of and is obvious.

Lemma 6 (see [29]). Let be a Banach space, be a cone in , and be a bounded open subset in . Suppose that is a completely continuous operator. Then the following results hold:
(i) if there exists such that , for any , then ;
(ii) if , , for any , then

Lemma 7 (see [29]). Let be a cone in a Banach space , be completely continuous, and . Suppose that is differentiable at along and 1 is not an eigenvalue of corresponding to a positive eigenvector. Moreover, if has no positive eigenvectors corresponding to an eigenvalue greater than one, then there exists such that , for , where

Lemma 8 (see [29]). Let be a cone in a Banach space and be completely continuous. Suppose that is differentiable at along and 1 is not an eigenvalue of corresponding to a positive eigenvector. Moreover, if has no positive eigenvectors corresponding to an eigenvalue greater than one, then there exists such that , for , where

Define a cone by

where Let

Define integral operators and by

Lemma 9 (see [14]). is completely continuous and the spectral radius

By a similar process in the proof of Lemma 9, we have the following lemma.

Lemma 10. Let , and then operator maps cone into and is completely continuous.

3. Main Results

For convenience in the following discussion, we present the following notations and assumptions:

(H1), and , (H2)(H3)There exists , such that uniformly holds with respect to on , and .

Adopting regular approaches similar to that in [14], we give the following two lemmas.

Lemma 11. Suppose that (H1) and (H2) hold. Thus is differentiable at along , and

In addition, has no positive eigenvectors corresponding to an eigenvalue greater than or equal to one.

Proof. Obviously, by . From mean value theorem, it follows that

for some . Notice , and we can obtain that for any , there exists a such that and

So for any , noticing (31) and Lemma 5, we get

which means that

Now, we are ready to show that has no positive eigenvectors corresponding to an eigenvalue greater than or equal to one. Otherwise, there exist and such that Then

Notice , and it follows

By direct integration to (35) on , we obtain

Notice , from which we infer that

which contradicts (H2). The proof is thus completed.

Lemma 12. Suppose that (H3) holds. Thus is differentiable at along and

In addition, has no positive eigenvectors corresponding to an eigenvalue greater than or equal to one.

Proof. From (H3), it follows that

where , and is a constant corresponding to
Thus, for any , it has

which implies that

when Therefore,
Suppose that has positive eigenvectors corresponding to an eigenvalue greater than or equal to one. Then we can choose a function and a constant satisfying , and thus

By means of , one has

Integrating (43) with respect to on gives

From the condition , it follows that

Notice (H3); this is a contradict. Hence, Lemma 12 holds.

The following theorems are main results in this paper.

Theorem 13. Suppose that (H1), (H2), and hold. Then BVP (2) has at least one positive solution.

Proof. From Lemmas 9, 11, and 7, it follows that

where is a suitable constant.
Let

Standard arguments can show that is completely continuous and Let , and The proof of follows by arguing as in [14], so we omit the proof. Hence, there exist and such that ; that is

According to Krein-Rutman theorem, we can obtain some with such that

Then, for all , one has

Thus, , which means
Moreover, using , we can choose a constant such that

Choose Thus for any , it follows

If we can choose a function and a constant such that

then we know
Denote

Obviously, and Noticing (48), (49), (51), and (53), we obtain

This is in contradiction with the definition of Then we have

Using Lemma 6 yields that This together with (46) gives

Therefore, system (2) has at least one positive solution.

Theorem 14. Suppose that (H3) and hold. Then BVP (2) has at least one positive solution.

Proof. Since , we know that there exists a positive constant such that

By a similar way in Theorem 13, there exists a function such that

Next, we are ready to show

In fact, if not, then there exist and such that

which indicates
Denote

Obviously, and , for any This together with (58), (59), and (61) gives

By means of the definition of , we get a contradiction. Thus, from Lemma 6, one gets

One the other hand, by Lemmas 9, 12, and 8, we can choose a constant such that From this and (64), we get

Consequently, BVP (2) has at least one positive solution.

Theorem 15. Suppose that (H1), (H2), and hold. In addition, suppose there exists such that

where . Then BVP (2) has at least two positive solutions.

Proof. By Lemmas 9, 11, and 7, we can choose a constant such that

Let be the eigenfunction of corresponding to . For , one has , for any Thus, it follows

which yields . Then, for , we can obtain This means , for By Lemma 6, we get

On the other hand, choose such that Then by , there exists such that

Let . Thus, we get

Choose Now, we are ready to show

Otherwise, suppose and such that Then,

and, noticing that , we get that the inverse operator of exists and equals

from which one has This together with (73) gives

In addition, by and Lemma 5, we have

which is a contradiction. By Lemma 6, we get This together with (67) and (69) guarantees that

Therefore, BVP (2) has at least two positive solutions.

Theorem 16. Suppose that (H3) and hold. In addition, suppose there exists such that

where Then BVP (2) has at least two positive solutions.

Proof. Let be the eigenfunction of corresponding to . For , it yields

which means . Thus, we can easily get , for This together with Lemma 6 gives

By , there exists a constant such that

where satisfies
If there exist and such that , we have

Since , we know that is reversible and

By (82), one has

Noticing , we get a contradiction. That is to say,

By Lemma 6, we obtain

On the other hand, in view of Lemmas 9, 12, and 8, we can choose a constant such that Combining this with (86) and (80), we easily get

Hence, BVP (2) has at least two positive solutions.

Theorem 17. Suppose that (H1) and (H2) hold. In addition, suppose that there exist with such that

hold, where are the same as in Theorems 15 and 16. Then BVP (2) has at least two positive solutions.

Proof. Similar to the proof of Theorem 15, we can choose such that

Moreover, for , we get

which gives This implies that for all By Lemma 6, we have This together with (89) guarantees

Therefore, BVP (2) has at least two positive solutions.

4. Examples

Example 1. Consider the following system:

Then BVP (92) has at least one positive solution.

Proof. BVP (92) can be regarded as a BVP of the form (2), where

and Obviously, , and , By direct calculation, we have

and Thus, (H1) and (H2) hold. By Theorem 13, BVP (92) has at least one positive solution.

Example 2. Consider the following system of fractional differential equations:

Then BVP (95) has at least one positive solution.

Proof. (95) is a BVP of type (2), where

and By direct calculation, we get that uniformly holds with respect to on , , and
Clearly, satisfies conditions (H3). Hence, our conclusion follows from Theorem 14.

Example 3. Consider the following system of fractional differential equations:

Here is a positive constant which satisfies Then BVP (97) has at least two positive solutions.

Proof. We will apply Theorem 15. Corresponding to the problem (2), we take

and
Obviously, , and , Thus,

and This means (H1) and (H2) hold.
In addition, by Lemma 5, we get By direct calculation, we have Taking , we obtain , for any .
By Theorem 15, system (97) has at least two positive solutions.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the project of Shandong Province Higher Educational Science and Technology Program of China under grant J18KA233 and the National Natural Science Foundation of China under grant 61873150.

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