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Existence of positive solutions for a fractional high-order three-point boundary value problem
Advances in Difference Equations volume 2014, Article number: 90 (2014)
Abstract
In this paper, the authors consider the following fractional high-order three-point boundary value problem: , , , , where , , , , is the standard Riemann-Liouville derivative of order α, and is continuous. By using some fixed point index theorems on a cone for differentiable operators, the authors obtain the existence of positive solutions to the above boundary value problem.
MSC:34A08, 34B15.
1 Introduction
In this paper, we investigate the existence of solutions for the following fractional high-order equation:
with the three-point boundary value conditions
where , , , , is the standard Riemann-Liouville derivative of order α and is continuous.
Differential equations with fractional order are a generalization of the ordinary differential equations to non-integer order. This generalization is not a mere mathematical curiosity but rather has interesting applications in many areas of science and engineering such as electrochemistry, control, porous media, electromagnetism, etc. (see [1–5]). There has been a significant development in the study of fractional differential equations in recent years; see for example [6–27]. Furthermore, several kinds of the high-order boundary value problems of fractional equations have been studied; see [6–10, 28–31] for example. In [28], using the Guo-Krasnosel’skii fixed point theorem, Goodrich discussed the existence of positive solutions for the following fractional boundary value problem:
where , .
Moreover, Goodrich [29] investigated the existence of a positive solution to system of fractional boundary value problems and extended his previous study in [28].
Recently, motivated by the above work of Goodrich, Xu et al. [6] investigated the existence and uniqueness of positive solution for the following fractional boundary value problem:
where , and .
More recently, by the method of upper and lower solution together with Schauder fixed position theorem, Vong [7] studied the existence of positive solutions of the nonlocal boundary value problem for fractional equation:
where , , and is a function of bounded variation.
Moreover, Waug et al. [30], El-Shahed and Shammakh [31], Yang et al. [8], Zhang and Han [9], Wu et al. [10] also studied similar problems.
It is worth pointing out that the fixed point index theorems on cone for differentiable operators are the effective tools to investigate positive solutions of fractional equation. However, to the author’s knowledge, such theorems are rarely used in the literature. Different from the literature mentioned above, in the present paper, the authors apply some fixed point theorems for differentiable operators to establish the existence results on positive solutions to the fractional nonlocal boundary value problem (1.1)-(1.2). That is one of the features of this paper. Another feature of this paper is that some spectral properties of a correlative linear integral operator are introduced to obtain some positive eigenvector.
The rest of this paper is organized as follows. In Section 2, we present some necessary definitions and preliminary results that will be used to prove our main results. In Section 3, we put forward and prove our main results. Finally, we will give two examples to demonstrate our main results.
2 Preliminaries
In this section, we introduce some preliminary facts which are used throughout this paper.
Let ℕ be the set of positive integers, ℝ be the set of real numbers. be the set of real positive numbers. Denote by the Banach space endowed with the norm . Let .
Definition 2.1 ([3])
The Riemann-Liouville fractional integral of order of a function is given by
Definition 2.2 ([3])
The Riemann-Liouville fractional derivative of order of function is given by
where , denotes the integer part of α.
Lemma 2.1 ([32])
Let . If with a fractional derivative of order α that belongs to , then
for some , , .
To study the existence of solutions to the boundary value problems (BVPs, for short), we first consider the following auxiliary BVP:
where , and α, η, k, n are given in (1.2).
We have the following lemma.
Lemma 2.2 For , the BVP (2.1) has a unique solution given by
where
Proof By Lemma 2.1, it follows from that there exist some constants () such that
The boundary value conditions in (2.1) imply that , and so
Hence
Thus,
The condition together with (2.4) and (2.5) yields
and so
from (2.3).
-
(1)
If , then
(2.6) -
(2)
If , then
(2.7)
So, we always have
where Green’s function is given by (2.2). The proof is complete. □
Now, we give some properties of .
Lemma 2.3 The Green function G has the following properties:
-
(1)
for all ;
-
(2)
for all , where and .
Proof It is easy to see that the conclusion (1) of Lemma 2.3 is true from the expression of in (2.2). So, it remains to show that the conclusion (2) of Lemma 2.3 is true. Our proof is divided into two steps.
Step 1. In this step, we show that
Let , , . Then we have the following cases to consider.
-
(i)
If , then
-
(ii)
If , then
because
-
(iii)
If , then
So, from the above analysis, we know that the inequality (2.8) holds.
Step 2. Now, we show that
Let , . Similar to the proof in Step 1, we deduce the relation (2.9).
-
(i)
If , then . Owing to that , it is easy to know that
and so
-
(ii)
If , then
noting that for and .
-
(iii)
If , then
noting that for .
-
(iv)
If , then
From
we have
noting that .
Summing up the above discussion, we know that the inequality (2.9) holds.
Now, from (2.8) and (2.9), the conclusion (2) of Lemma 2.3 follows. The proof is complete. □
We introduce a cone as follows:
Define an operator by
We establish the following lemma, which will be used in the next section.
Lemma 2.4 is completely continuous.
Proof Let B be an arbitrary bounded set in E. Then there exists such that for any . First, we show that the set is equicontinuous on .
In fact, for an arbitrary and any as well as with , there are three cases to consider.
Case 1. If , then from (2.6) and (2.10), it follows that
Therefore, there is a such that
Case 2. If , then by a similar argument to (2.11), from (2.7) and (2.10), we have
Thus, there exists a such that
Case 3. If with , then from (2.12)-(2.13), it follows that
noting that .
Summing up the above analysis on Cases 1-3, we conclude that when , for , that is, G is equicontinuous on .
Now, we show that G is bounded in E.
In fact, from the fact that , , we immediately have
So, by the Arzela-Ascoli theorem, we know that is a compact operator. Again, because T is a bounded operator on E owing to (2.14), is continuous, and therefore T is completely continuous on E.
Finally, we apply the Lemma 2.3 to obtain
and
for any . So, , that is, . The proof is complete. □
For the remainder of this section, we introduce the following lemmas, which will be used to obtain our main result in the next section.
Lemma 2.5 ([33])
Let P be a cone in a Banach space E, be completely continuous, and . Suppose that A is differentiable at θ along P 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
where .
Lemma 2.6 ([33])
Let P be a cone in a Banach space E, be completely continuous. Suppose that A is differentiable at ∞ along P 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
where .
Lemma 2.7 ([34])
Let be nonnegative on , and let the operator be completely continuous, where K is defined as . If the spectral radius , then K has a positive eigenfunction corresponding to its first eigenvalue , i.e. there exists with , , , .
3 Main results
Let , . Define an operator on as
for , where is the Green function (2.2), whose domain is restricted on .
Let , where is given in Lemma 2.3. Obviously, is a cone in . We have the following lemma.
Lemma 3.1 is completely continuous. Moreover, the spectral radius .
Proof Since the proof of the complete continuity of is similar to that in Lemma 2.4, we omit it. Here, we only show that .
Let , . Because
and
we have
For any , from
we have
and so
Hence, , and so , which implies that . So, we obtain
The proof is complete. □
Let us list the following assumptions, which will be used later.
(H1) .
(H2) , , , , , where the partial derivative . Moreover, such that is continuous on .
(H3) , where , .
(H4) There exists such that holds uniformly on with respect to t. Moreover, , .
(H5) , where .
Define an operator A on P as
Obviously, the following lemma is true in view of Lemma 2.4.
Lemma 3.2 Let (H1) hold. Then is completely continuous.
We need the following two lemmas, which will play an important role to obtain the existence results.
Lemma 3.3 Let (H1)-(H2) hold. Then the operator A is differentiable at θ along P, and , , , where
Proof For any , by the mean value theorem, there exists such that
that is,
Again, due to the fact that is uniformly continuous on , for arbitrary , there exists such that
when , for all .
So, from (3.1)-(3.2), it follows that
when , for all .
Consequently, for any with , from (3.3) and (2.9), it follows that
So, , that is, , . The proof is complete. □
Lemma 3.4 Let (H1)-(H3) hold. Then has no positive eigenvectors corresponding to an eigenvalue greater than or equal to one.
Proof If not, then there exist a and with , and so
Thus,
and so
Because , , , , and is continuous on , the following inequality:
holds. Immediately, from (3.4) it follows that , which contradicts (H3). So, the conclusion of Lemma 3.4 is true. The proof is complete. □
Lemma 3.5 Let (H1), (H4), and (H5) hold. Then A is differentiable at ∞ along P and , , where
Proof From (H4), it follows that, for arbitrary , there exists such that
when , and so
when .
Let . Then
holds for any .
Now, for any , by the above inequality, we have
where . Thus . So, when , that is, . The proof is complete. □
Lemma 3.6 Let (H1), (H4), and (H5) hold. Then has no positive eigenvectors corresponding to an eigenvalue greater than or equal to one.
Proof The proof is similar to the proof of Lemma 3.4. In fact, if not, then there exist a and such that , and so
Thus,
and so
Because and , , , the relation
holds, and therefore , which contradicts (H5). The proof is complete. □
Let . We are in a position to state our main result in the present paper.
Theorem 3.1 Let (H1)-(H3) hold. If , then BVP (1.1)-(1.2) has a positive solution.
Proof In view of Lemmata 3.2-3.4 and by applying Lemma 2.5, we conclude that there exist a such that
where , and P is defined as before.
By Lemma 2.7 and Lemma 3.1, we know that there exists a with , , satisfying
So, from Lemma 2.3, it follows that
Thus, , , that is, .
Let
It is easy to see that .
On the other hand, by , there exists a such that
when . Take . Set . Then for any , the inequality , implies
We show that the following relation holds:
In fact, if not, then there exist a and a such that
Obviously, we can assume that . From (3.8), it follows that
because .
Let . Then and
Again, from (3.8), (3.6), and (3.9), for , we have
which contradicts the definition of . Hence, the relation (3.7) holds. So, in terms of the fixed point index theorem on a cone, we have
Thus, (3.5) and (3.10) imply that
So, A has a fixed point , that is, is a positive solution of BVP (1.1)-(1.2). The proof is complete. □
Let , where is given as in Lemma 2.3. We state another result in this paper.
Theorem 3.2 Let (H1), (H4), and (H5) hold. Assume that there exists such that
Then BVP (1.1)-(1.2) has a positive solution.
Proof We show that
where .
In fact, for any , from (3.11) we have
owing to
Thus, by Lemma 2.3,
So,
because
Therefore, the relation (3.12) holds. Consequently, applying the fixed point index theorem, we get
On the other hand, by Lemma 3.2, Lemma 3.5, Lemma 3.6, and Lemma 2.6, we know that there exists such that
So, by (3.13) and (3.14), we have
Therefore, A has a fixed point , that is, is a positive solution of BVP (1.1)-(1.2). The proof is complete. □
Example 3.1 Consider the following boundary value problem:
To obtain the existence result, we will apply Theorem 3.1 with , , , and function , , . Clearly, the function satisfies , , and , . Further, and . So, all the assumptions of Theorem 3.1 are satisfied and therefore BVP (3.15) has at least one positive solution.
Example 3.2 Consider the following boundary value problem:
To obtain the existence result, we will apply Theorem 3.2 with , , , and function , , . Clearly, the function satisfies , holds uniformly on with respect to t. Further, . On the other hand, it is easy to see that exists a such that , , noting that , , and , , , . So, all the assumptions of Theorem 3.2 are satisfied and therefore BVP (3.16) has at least one positive solution.
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Acknowledgements
The authors sincerely thank the anonymous referees for their valuable suggestions and comments, which have greatly helped improve this article. This work is supported by the Natural Science Foundation of Hubei Provincial Education Department (Q20132505).
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Chai, G., Hu, S. Existence of positive solutions for a fractional high-order three-point boundary value problem. Adv Differ Equ 2014, 90 (2014). https://doi.org/10.1186/1687-1847-2014-90
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DOI: https://doi.org/10.1186/1687-1847-2014-90