Abstract and Applied Analysis

Volume 2014 (2014), Article ID 131548, 14 pages

http://dx.doi.org/10.1155/2014/131548

Research Article

The Existence of Positive Solutions for Fractional Differential Equations with Integral and Disturbance Parameter in Boundary Conditions

Mei Jia and Xiping Liu

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

Received 13 November 2013; Accepted 17 January 2014; Published 24 March 2014

Academic Editor: Paul Eloe

Copyright © 2014 Mei Jia and Xiping Liu. 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.

We study the existence and nonexistence of the positive solutions for the integral boundary value problem of the fractional differential equations with the disturbance parameter in the boundary conditions and the impact of the disturbance parameter on the existence of positive solutions. By using the upper and lower solutions method, fixed point index theory and the Schauder fixed point theorem, we obtain sufficient conditions for that the problem has at least one positive solution, two positive solutions and no solutions. Under certain conditions, we also obtain the demarcation point which divides the disturbance parameters into two subintervals such that the boundary value problem has positive solutions for the disturbance parameter in one subinterval while no positive solutions in the other.

1. Introduction

In this paper, we are concerned with the existence and nonexistence of positive solutions for the boundary value problem of the fractional differential equations

where ,  ,  ,  ,  ,  ,  ,  , disturbance parameter , and is the Caputo fractional derivative of order .

In the recent decades, since fractional differential equations have been applied widely and successfully in the description of complex dynamics, they have been regarded as a valuable tool being used in the fields to handle viscoelastic, physics, chemistry, electrical engineering, biology aspects, and so forth; see [1–12] and references therein. And besides, the boundary value problems for the differential equations appear in many applications; see [13–16]. As a result, the boundary value problems for the fractional differential equations are one of the most active fields in the researches of nonlinear differential equations theories and plenty of meaningful achievements have been gained in the related fields; see [1–8, 17–29] and the references therein. Due to that the boundary value problems with the integral boundary conditions include two-, three-, and multipoint boundary value problems as special cases and they can better describe the actual phenomenon; more and more emphases have been put on the researches of them; see [19, 20, 30–34] and the references therein.

At the same time, while using the methods of the differential equations to solve the actual problems, it is inevitable that there always exists disturbance which will have great influence on the existence of the solutions. In paper [35, 36], the authors studied nonlinear nonlocal boundary value problem with nonhomogeneous boundary conditions

where they discussed the impact of disturbance parameters on the existence of the solution and obtained some meaningful conclusions. And then, the authors further studied the 2nth order nonlinear nonlocal boundary value problem with nonhomogeneous boundary conditions; see [37].

The purpose of this paper is to study the impact of the disturbance parameter on the existence of positive solutions and obtain sufficient conditions for the boundary value problem (1) to have at least one positive solution, at least two solutions, and no solutions. Under certain conditions, we obtain that there exists a constant , which separates into two disjoint subintervals and such that the boundary value problem (1) has at least two positive solutions for each , at least one positive solution for and , and no positive solutions for . The main tools we applied are the upper and lower solutions method, fixed point index theory, and the Schauder fixed point theorem.

This paper is organized as follows. In Section 2, we introduce the basic definitions and the basic properties of the integral kernel. In Section 3, we study the comparison principles and the basic lemmas. In Section 4, we consider the existence and nonexistence of the positive solutions of the boundary value problem (1), and we study the impact of the disturbance parameter on the existence of positive solutions.

2. Preliminaries

In this section, we give some basic definitions and lemmas which play an important role in our research.

Definition 1 (See [2, 3]). Let , for a function . The Riemann-Liouville fractional integral operator of order of is defined by

provided the integral exists.

The Caputo derivative of order for a function is given by

provided the right side is pointwise defined on , where and .

If , then .

Definition 2. Let be the space of functions which are absolutely continuous on . We denote by the set of functions which have continuous derivatives up to order on such that . In particular, .

Lemma 3 (See [2]). If , then the Caputo fractional derivative exists almost everywhere on , where is the smallest integer greater than or equal to .

Lemma 4 (See [2]). Suppose and . Then

where is the smallest integer greater than or equal to .

Definition 5. One says is a solution of the boundary value problem (1) if ,   and satisfies (1). One says is a positive solution of the boundary value problem (1) if is a solution of the boundary value problem (1) and and for .

Throughout this paper, we assume the following conditions hold.

(H1),   and , where

(H2)There exists a function and such that ,   and .(H3) is monotonically increasing with respect to and .

For , firstly we consider the boundary value problem

Lemma 6. Suppose (H1) holds. Then the boundary value problem (7) has the unique solution

where the function is given by

where

Proof. According to Lemma 4, is equivalent to the following equation:

where .

By the boundary condition and , we can show

Hence, we can obtain

It follows the definition Riemann-Liouville fractional integral that

that is,

We multiply by the function on both sides of (16), integrate from 0 to 1,

and we can easily get

So,

where .

Hence,

It follows from (12) that

We can obtain and .

Similarly, we can obtain the following lemma.

Lemma 7. Suppose (H1) holds and . Then the boundary value problem

has the unique solution

Lemma 8. Suppose (H1) holds. Then the solution and of the boundary value problem (1) is equivalent to the solution of the integral equation

Proof. If and is the solution of the boundary value problem (1), by Lemma 4, is equivalent to the equation

where .

For convenience, we denote .

Similar to the proof of Lemma 6, we can obtain

Substituting and into (25), we can obtain

We multiply by the function on both sides of (27), integrate from 0 to 1, and get

Since , we have

That is,

On the other hand, if is the solution of the integral equation (24), we have

It is easy to see that .

Hence, and .

We can easily verify that satisfies the boundary value problem (1).

Therefore, and is the solution of the boundary value problem (1).

Lemma 9. Suppose (H1) holds. Then the functions ,  , and have the following properties;

(1), for and there exists a constant such that for

(2) for and there exists a constant such that for

(3) for and for

where .

Proof. By the expression of , it is easy to see .

Since (H1) holds, we can show that for

For , it follows

For , by (10), we have

It is easy to see that

(H1) implies that . By (38), we have . Hence,

For , we can show that

from (36) and (39).

We denote . It is obvious that .

By (35) and (39), for , we have

Hence, .

By the expression of , see (11), it is easy to see for and

where .

By and , we have for and

For the sake of the reader, we state the fixed point index theorem and Schauder’s fixed point theorem which will be used later.

Lemma 10 (See [38]). Let be a cone of a real Banach space ,   be a bounded open set in , and . Suppose is a completely continuous operator. If for any and , then

Lemma 11 (Schauder’s fixed point theorem, see [39]). Let be a real Banach space, and let be nonempty closed bounded and convex; compact. Then has a fixed point.

3. Comparison Principle and the Existence of Solutions

Definition 12. Let and . One says that is a lower solution of the boundary value problem (1), if

Where

Let and . We say that is an upper solution of the boundary value problem (1), if

Where

where is defined in (H2).

The following comparison principle will play a very important role in our main results analysis.

Lemma 13. Let (H1) hold. Suppose that ,  , and satisfies

Then for .

Proof. Denote

then for . Let such that

By Lemma 8, we can get that the boundary value problem

has unique solution

where and .

It follows that for from Lemma 9.

Lemma 14. Let (H1) and (H2) hold. Suppose that , , and satisfies

Then for .

Proof. Let

Since (H2) holds, we have for , , , and .

Hence, by (54), we have

In view of Lemma 13, for , which implies that .

We can easily obtain the following lemma from the definition of , where is defined by Lemma 7.

Lemma 15. Suppose (H1) holds. is a positive solution of the boundary value problem (1) if and only if is a positive solution of the boundary value problem

Let be the Banach space with the norm ,

and let

be cones in , and , where is defined by Lemma 9.

Lemma 16. Suppose (H1) holds. If is a positive solution of the boundary value problem (57), then

Proof. By Lemma 6, we can show that the solution of the boundary value problem (57) satisfies

In view of Lemma 9,

We define by

By Lemma 16, we have . Since and are nonnegative, then is a positive solution of (1) if and only if is a fixed point of the operator .

Theorem 17. Suppose (H1), (H2), and (H3) hold and there exist a nonnegative lower solution and an upper solution of the boundary value problem (1) such that . Then the boundary value problem (1) has at least one solution such that .

Proof. Let

We consider the boundary value problem

By Lemma 15, is a positive solution of the boundary value problem (65) if and only if is a positive solution of the following boundary value problem:

We define by

Then is a solution of (66) if and only if is a fixed point of the operator . It is easy to see .

Next we can prove that is completely continuous.

Let be a bounded set, and there exists a constant such that for . Because is continuous, there exists a constant such that . We have

So is uniformly bounded.

We denote

Hence,

where is defined by (11).

Since is continuous on , we have which is uniformly continuous on . It implies that for any , there exists , when ; whenever and , we can obtain

By (H1), it is easy to see . We take .

Therefore, as , whenever and , we can show that

Thus, we have proved is equicontinuous.

By Arzela-Ascoli theorem, we know that is relatively compact.

We can easily show that is continuous since is continuous. Hence, is completely continuous.

Since is bounded and is completely continuous, we can get has at least one fixed point by Schauder fixed point theorem, that is, there exists a solution of the boundary value problem (66).

Then is a solution of the boundary value problem (65).

Finally, we prove , for .

We can prove that if each solution of the boundary value problem (65) satisfies for , then is a solution of the boundary value problem (1).

Let , for .

If , since is monotonically increasing with respect to and is a lower solution of (1), we have

It follows that on from Lemma 13.

If , we have

By (H3), we can get , that is,

So,

It follows that on from Lemma 14.

Hence, we show on .

Similarly, we can get on .

Therefore, each solution of the boundary value problem (65) satisfies for . That is, , and is a solution of the boundary value problem (1).

Theorem 18. Suppose (H1), (H2), and (H3) hold:

if there exists a constant such that the boundary value problem

has a positive solution , then for each with , the boundary value problem (1) has a positive solution and , where for ;

if there exists a constant such that the boundary value problem (77) does not have positive solutions, then for each , the boundary value problem (1) does not have positive solutions.

Proof. Let for . By Lemma 15, is a positive solution of the boundary value problem

which implies .

Since , we take and .

We can easily verify that and are a lower solution and an upper solution of the boundary value problem (1) and .

By Theorem 17, we have that the boundary value problem (1) has a positive solution and .

If there exists a constant such that the boundary value problem (77) has a positive solution, by (1), we can show that for each with , the boundary value problem (1) has a positive solution. So, the boundary value problem (1) has a positive solution, for , which is a contradiction.

Therefore, if there exists a constant such that the boundary value problem (77) does not have positive solutions, then for each , the boundary value problem (1) does not have positive solutions.

4. Impact of Disturbance Parameter on the Existence of Solutions

For convenience, we give the following notations:

We can see that from (H1) and we denote .

Lemma 19. Suppose that (H1) holds, , and is a bounded set. Then for each , there exists a constant such that the solution of the boundary value problem (1) satisfies .

Proof. Since is a bounded set, there exists a constant such that each , and we have .

Since , there exists a constant such that

for any , and .

By Lemma 15, is a positive solution of the boundary value problem (1) if and only if is a positive solution of the boundary value problem (57).

In view of Lemma 16, . We can get . Otherwise, if , we have

for . Hence,

which is a contradiction.

We take , then

Theorem 20. Suppose (H1) holds. If one of the following conditions is satisfied, then the boundary value problem (1) does not have positive solutions:

(1),   and is small enough;(2),   and is large enough;(3)there exist constants and such that

for , and .

Proof. If there exists and is small enough such that the boundary value problem (1) has a positive solution , we have that is a positive solution of the boundary value problem (57) by Lemma 15, and by Lemma 16.

Because , there exists a constant such that

for any , and .

We take and is small enough such that

for . Hence,

which is a contradiction.

If there exists a constant and is large enough such that the boundary value problem (1) has a positive solution , we have that is a positive solution by Lemma 15.

By , there exists a constant such that

for any , and . We take , then

for . Hence,

which is a contradiction.

Since there exist constants and such that

for , we have

which is a contradiction.

Theorem 21. Suppose (H1) holds

(1)If , then the boundary value problem (1) has at least one positive solution when and is small enough.(2)If , then the boundary value problem (1) has at least one positive solution when .

Proof. Because , there exists a constant such that

for any , and .

Let , where for . So for .

Let , then .

For any , we have

That is . By the Schauder fixed point theorem, we can get that has at least one fixed point on . In view of Lemma 15, is a positive solution of the boundary value problem (1).

Since , there exist constants and such that

for any and , where satisfies .

Let . We have

We choose and , where for . Then, for any ,

That is . By the Schauder fixed point theorem, we can get that has at least one fixed point on . In view of Lemma 15, is a positive solution of the boundary value problem (1).

Theorem 22. Suppose that (H1), (H2), and (H3) hold. If and , then there exists a constant with the following properties.

   separates into two disjoint subintervals and .

The boundary value problem (1) has at least two positive solutions for each , has at least one positive solution for and , and does not have positive solutions for each .

Proof. We have the following four steps to prove the conclusions of Theorem 22.

Step  1. Let

By Theorem 21, we have .

For each , denote

Then, in view of Theorem 18,    if and only if .

It follows that is a bounded set from and Theorem 20.

Let , then is a bounded set. Hence, the set has the supremum; we denote and we have from Theorem 21.

Step  2. We prove that the boundary value problem (1) has at least one positive solution when .

Since , there exist and such that as .

Let be a solution of the boundary value problem (1) with replaced by . By Lemma 8, the boundary value problem (1) with is equivalent to

In view of Lemma 9, is uniformly bounded and equicontinuous. By Arzela-Ascoli, has a convergent subsequence; we also denote , and converge to .

By Lebesgue dominated convergence theorem, we can get

Hence, is a positive solution of the boundary value problem (1) with .

Step  3. By Theorem 18, when , the boundary value problem (1) has at least one positive solution. That is, .

Let . Then separates into two disjoint subintervals and ; we have that the boundary value problem (1) has at least one positive solution for each and does not have positive solutions for each .

Step  4. We prove the boundary value problem (1) has at least two positive solutions when .

For each , there exist , , such that .

Let be a solution of the boundary value problem (1) with replaced by . By Theorem 18, the boundary value problem (1) has a positive solution with .

Similarly, let be a solution of the boundary value problem (1) with replaced by and . Hence, .

In fact, we can easily show that from and (H1).

Let and . We can easily verify and are a lower solution and an upper solution of the boundary value problem (1), respectively, and .

We choose , then .

Define by

Let

and define by

Similar to the proof of Theorem 17, we can prove and are completely continuous for each . By Lemma 8, is a positive solution of the boundary value problem (1) if and only if .

By Lemma 19, there exists a constant such that the fixed point of satisfies for each .

Let

Clearly, is a nonempty open-bounded subset of .

Since is bounded, there exists a constant such that for any . Let ; it is obvious that and for , and . Hence, by Lemma 10, for each , we have

Because does not have a fixed point on , then for each ,

We notice that , by the excision property of the fixed point index, (106) and (107), for each , we have

Since , we have for any . So,

We define by

It is obvious that is completely continuous.

We can prove for .

Otherwise, if there exists such that , that is

Hence, is a solution of the boundary value problem (1) with replaced by . We can get , which is a contradiction.

According to the homotopy invariance of the fixed point index and (109), it follows that

By the additivity property of the fixed point index, (108) and (112), we obtain

Therefore, the boundary value problem (1) has a solution in and from . That is, the boundary value problem (1) has at least two positive solutions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to the referees for their valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (no. 11171220) and the Foundation of the Education Department of Shanghai (no. 10ZZ93).

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