Abstract
This article deals with the differential equations of fractional order on the halfline. By the recent LeggettWilliams normtype theorem due to O’Regan and Zima, we present some new results on the existence of positive solutions for the fractional boundary value problems at resonance on unbounded domains.
MSC: 26A33, 34A08, 34A34.
Keywords:
fractional order; halfline; coincidence degree; at resonance1 Introduction
In this article, we are concerned with the fractional differential equation
where is the RiemannLiouville fractional derivative, , and satisfies the following condition: (H) = is continuous and for each , there exists satisfying and , such that
.
The problem (1.1) happens to be at resonance in the sense that the kernel of the linear operator is not less than onedimensional under the boundary value conditions.
Fractional calculus is a generalization of the ordinary differentiation and integration. It has played a significant role in science, engineering, economy, and other fields. Some books on fractional calculus and fractional differential equations have appeared recently (see [13]); furthermore, today there is a large number of articles dealing with the fractional differential equations (see [415]) due to their various applications.
In [8], the researchers dealt with the existence of solutions for boundary value problems of fractional order of the form
where and is continuous. The results are based on the fixed point theorem of Schauder combined with the diagonalization method.
In [9], Su and Zhang studied the following fractional differential equations on the halfline using Schauder’s fixed point theorem
Employing the LeraySchauder alternative theorem, in [12], Zhao and Ge considered the fractional boundary value problem
However, the articles on the existence of solutions of fractional differential equations on the halfline are still few, and most of them deal with the problems under nonresonance conditions. And as far as we know, recent articles, such as [4,6,7], investigating resonant problems are on the finite interval.
Motivated by the articles [1620], in this article we study the differential equations (1.1) under resonance conditions on the unbounded domains. Moreover, we have successfully established the existence theorem by the recent LeggettWilliams normtype theorem due to O’Regan and Zima. To our best knowledge, there is no article dealing with the resonant problems of fractional order on unbounded domains by the theorem.
The rest of the article is organized as follows. In Section 2, we give the definitions of the fractional integral and fractional derivative, some results about fractional differential equations, and the abstract existence theorem. In Section 3, we obtain the existence result of the solution for the problem (1.1) by the recent LeggettWilliams normtype theorem. Then, an example is given in Section 4 to demonstrate the application of our result.
2 Preliminaries
First of all, we present some fundamental facts on the fractional calculus theory which we will use in the next section.
The RiemannLiouville fractional integral of order of a function is given by
provided that the righthand side is pointwise defined on .
The RiemannLiouville fractional derivative of order of a continuous function is given by
where , provided that the righthand side is pointwise defined on .
Lemma 2.2 ([9])
for some, , whereNis the smallest integer greater than or equal toν.
Now, let us recall some standard facts and the fixed point theorem due to O’Regan and Zima, and these can be found in [16,17,2123].
Let X, Z be real Banach spaces. Consider an operation equation
where is a linear operator, is a nonlinear operator. If and ImL is closed in Z, then L is called a Fredholm mapping of index zero. And if L is a Fredholm mapping of index zero, there exist linear continuous projectors and such that , and , . Then it follows that is invertible. We denote the inverse of this map by . For ImQ is isomorphic to KerL, there exists an isomorphism .
It is known that the coincidence equation is equivalent to
A nonempty convex closed set is called a cone if
Note that C induces a partial order ⪯ in X by
The following lemma is valid for every cone in a Banach space.
LetCbe a cone in the Banach spaceX. Then for every, there exists a positive numbersuch that
Let be a retraction, i.e., a continuous mapping such that for all . Denote
and
LetCbe a cone inXand let, be open bounded subsets ofXwithand. Assume that: 1^{∘} = Lis a Fredholm operator of index zero;; 2^{∘} = is continuous and bounded andis compact on every bounded subset ofX;; 3^{∘} = for alland;; 4^{∘} = γmaps subsets ofinto bounded subsets ofC;; 5^{∘} = , wherestands for the Brouwer degree;; 6^{∘} = there existssuch thatfor, whereandis such thatfor every;; 7^{∘} = ;; 8^{∘} = ..Then the equationhas a solution in the set.
Let
with the norm
and
equipped with the norm
Remark 2.1 It is easy for us to prove that and are Banach spaces.
Set
Define
and
Then the multipoint boundary value problem (1.1) can be written by
Definition 2.3 is called a solution of the problem (1.1) if and u satisfied Equation (1.1).
Next, similar to the compactness criterion in [12,24], we establish the following criterion, and it can be proved in a similar way.
Lemma 2.4is a relatively compact set inXif and only if the following conditions are satisfied:
(a) is uniformly bounded, that is, there exists a constantsuch that for each, .
(b) The functions fromare equicontinuous on any compact subinterval of, that is, let J be a compact subinterval of, then, there existssuch that for, ,
(c) The functions fromare equiconvergent, that is, given, there existssuch that
3 Main results
In this section, we will present the existence theorem for the fractional differential equation on the halfline. In order to prove our main result, we need the following lemmas.
Lemma 3.1Let. Thenis the solution of the following fractional differential equation:
if and only if
and
Proof In view of Lemmas 2.1 and 2.2, we can certify the conclusion easily, so we omit the details here. □
Lemma 3.2The operatorLis a Fredholm mapping of index zero. Moreover,
and
Proof It is obvious that Lemma 3.1 implies (3.1) and (3.2). Now, let us focus our minds on proving that L is a Fredholm mapping of index zero.
where . Evidently, , , and is a continuous linear projector. In fact, for an arbitrary , we have
that is to say, is idempotent.
Let , where is an arbitrary element. Since and , we obtain that . Take , then can be written as , , for . Since , by (3.2), we get that , which implies that , and then . Therefore, , thus, .
Now, , and observing that ImL is closed in Z, so L is a Fredholm mapping of index zero. □
It is clear that is a linear continuous projector and
Also, proceeding with the proof of Lemma 3.2, we can show that .
Note that
and
Define the linear isomorphism as
where
Then, it is easy to verify that
Now, we state the main result on the existence of the positive solutions to the problem (1.1) in the following.
Theorem 3.1Letsatisfy the condition (H). Assume that there exist six nonnegative functions (), () andsuch that
and
where, , andis defined by (3.12), is bounded on, , , ,
and
Then the problem (1.1) has at least one positive solution in domL.
Proof For the simplicity of notation, we denote
and
Consider the cone
Set
where , . Clearly, and are an open bounded set of X.
Step 1: In view of Lemma 3.2, the condition 1^{∘} of Theorem 2.1 is fulfilled.
Step 2: By virtue of Lemma 2.4, we can get that is continuous and bounded and is compact on every bounded subset of X, which ensures that the assumption 2^{∘} of Theorem 2.1 holds.
Step 3: Suppose that there exist and such that .
Since
we have
From (3.7) and (3.8), we get that
and
On account of the fact that
and considering (3.14) and (3.15), we have
and
Thus,
and
By (3.9), (3.10) and (3.13), we obtain that
which is a contradiction to . Therefore, 3^{∘} is satisfied.
Step 4: Let , then we can verify that is a retraction and 4^{∘} holds.
Step 5: Let , then . Inspired by Aijun and Wang [5], we set
Define homeomorphism by , then
It is obvious that implies that by (3.8) and (3.11).
Take , then . Suppose that , , then we have that . Also, in view of (3.8),
It is a contradiction. Besides, if , then , which is impossible. Hence, for , , .
Therefore,
which shows that 5^{∘} is true.
Therefore, combining (3.6), (3.8) and (3.11), we get that
Thus, for all . So, 6^{∘} holds.
Step 7: For , from (3.8) and (3.11), we have
which implies that . Hence, 7^{∘} holds.
Step 8: For , by (3.6), (3.8) and (3.11), we obtain that
Thus, , that is, 8^{∘} is satisfied.
Hence, applying Theorem 2.1, the problem (1.1) has a positive solution in the set . □
4 Examples
To illustrate our main result, we will present an example.
Example 4.1
and
It is easy for us to certify that f satisfies the condition (H).
Noting that
and
Meanwhile, by simple computation we can get that
Thus, to sum up the points which we have just indicated, by Theorem 3.1, we can conclude that the problem (4.1) has at least one positive solution.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All the authors typed, read, and approved the final manuscript.
Acknowledgement
This project is supported by the Hunan Provincial Innovation Foundation For Postgraduate (NO. CX2011B079) and the National Natural Science Foundation of China (NO. 11171351).
References

Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006)

Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York (1993)

Lakshmikantham, V, Leela, S, Vasundhara Devi, J: Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge (2009)

Bai, Z, Zhang, Y: The existence of solutions for a fractional multipoint boundary value problem. Comput. Math. Appl.. 60, 2364–2372 (2010). Publisher Full Text

Yang, A, Wang, H: Positive solutions of twopoint boundary value problems of nonlinear fractional differential equation at resonance. Electron. J. Qual. Theory Differ. Equ.. 71, 1–15 (2011)

Jiang, W: The existence of solutions to boundary value problems of fractional differential equations at resonance. Nonlinear Anal.. 74, 1987–1994 (2011). Publisher Full Text

Bai, Z: Solvability for a class of fractionalpoint boundary value problem at resonance. Comput. Math. Appl.. 62, 1292–1302 (2011). Publisher Full Text

Arara, A, Benchohra, M, Hamidi, N, Nieto, JJ: Fractional order differential equations on an unbounded domain. Nonlinear Anal.. 72, 580–586 (2010). Publisher Full Text

Su, X, Zhang, S: Unbounded solutions to a boundary value problem of fractional order on the halfline. Comput. Math. Appl.. 61, 1079–1087 (2011). Publisher Full Text

Kou, C, Zhou, H, Yan, Y: Existence of solutions of initial value problems for nonlinear fractional differential equations on the halfaxis. Nonlinear Anal.. 74, 5975–5986 (2011). Publisher Full Text

Liang, S, Zhang, J: Existence of three positive solutions of mpoint boundary value problems for some nonlinear fractional differential equations on an infinite interval. Comput. Math. Appl.. 61, 3343–3354 (2011). Publisher Full Text

Zhao, X, Ge, W: Unbounded solutions for a fractional boundary value problems on the infinite interval. Acta Appl. Math.. 109, 495–505 (2010). Publisher Full Text

Yang, L, Chen, H: Unique positive solution for boundary value problem of fractional differential equations. Appl. Math. Lett.. 23, 1095–1098 (2010). Publisher Full Text

Chai, G: Positive solutions for boundary value problem of fractional differential equation with pLaplacian operator. Bound. Value Probl.. 2012, (2012)

Ahmad, B, Nieto, JJ: RiemannLiouville fractional integrodifferential equations with fractional nonlocal integral boundary conditions. Bound. Value Probl.. 2011, (2011)

O’Regan, D, Zima, M: LeggettWilliams normtype theorems for coincidences. Arch. Math.. 87, 233–244 (2006). Publisher Full Text

Franco, D, Infante, G, Zima, M: Second order nonlocal boundary value problems at resonance. Math. Nachr.. 284, 875–884 (2011). Publisher Full Text

Infante, G, Zima, M: Positive solutions of multipoint boundary value problems at resonance. Nonlinear Anal.. 69, 2458–2465 (2008). Publisher Full Text

Yang, L, Shen, C: On the existence of positive solution for a kind of multipoint boundary value problem at resonance. Nonlinear Anal.. 72, 4211–4220 (2010). Publisher Full Text

Yang, L, Shen, C: Positive solutions for second order fourpoint boundary value problems at resonance. Topol. Methods Nonlinear Anal.. 38, 1–16 (2011)

Mawhin, J: Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces. J. Differ. Equ.. 12, 610–636 (1972). Publisher Full Text

Santanilla, J: Some coincidence theorems in wedges, cones, and convex sets. J. Math. Anal. Appl.. 105, 357–371 (1985). Publisher Full Text

Petryshyn, WV: On the solvability of in quasinormal cones with T and Fkset contractive. Nonlinear Anal.. 5, 589–591 (1981)

Agarwal, RP, O’Regan, D: Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic, Dordrecht (2001)