This article deals with the differential equations of fractional order on the half-line. By the recent Leggett-Williams norm-type 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; half-line; coincidence degree; at resonance
In this article, we are concerned with the fractional differential equation
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 [1-3]); furthermore, today there is a large number of articles dealing with the fractional differential equations (see [4-15]) due to their various applications.
In , the researchers dealt with the existence of solutions for boundary value problems of fractional order of the form
In , Su and Zhang studied the following fractional differential equations on the half-line using Schauder’s fixed point theorem
Employing the Leray-Schauder alternative theorem, in , Zhao and Ge considered the fractional boundary value problem
However, the articles on the existence of solutions of fractional differential equations on the half-line 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 [16-20], 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 Leggett-Williams norm-type 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 Leggett-Williams norm-type theorem. Then, an example is given in Section 4 to demonstrate the application of our result.
First of all, we present some fundamental facts on the fractional calculus theory which we will use in the next section.
Lemma 2.2 ()
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 .
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 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.
with the norm
equipped with the norm
Remark 2.1 It is easy for us to prove that and are Banach spaces.
Then the multi-point boundary value problem (1.1) can be written by
3 Main results
In this section, we will present the existence theorem for the fractional differential equation on the half-line. In order to prove our main result, we need the following lemmas.
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,
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.
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.
Then the problem (1.1) has at least one positive solution in domL.
Proof For the simplicity of notation, we denote
Consider the cone
Step 1: In view of Lemma 3.2, the condition 1∘ of Theorem 2.1 is fulfilled.
From (3.7) and (3.8), we get that
On account of the fact that
and considering (3.14) and (3.15), we have
By (3.9), (3.10) and (3.13), we obtain that
Step 5: Let , then . Inspired by Aijun and Wang , we set
which shows that 5∘ is true.
Therefore, combining (3.6), (3.8) and (3.11), we get that
To illustrate our main result, we will present an example.
It is easy for us to certify that f satisfies the condition (H).
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.
The authors declare that they have no competing interests.
All the authors typed, read, and approved the final manuscript.
This project is supported by the Hunan Provincial Innovation Foundation For Postgraduate (NO. CX2011B079) and the National Natural Science Foundation of China (NO. 11171351).
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