Positive solutions of fractional differential equations at resonance on the half-line
© Chen and Tang; licensee Springer 2012
Received: 19 January 2012
Accepted: 30 April 2012
Published: 22 June 2012
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.
The problem (1.1) happens to be at resonance in the sense that the kernel of the linear operator is not less than one-dimensional 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 [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.
where and is continuous. The results are based on the fixed point theorem of Schauder combined with the diagonalization method.
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.
provided that the right-hand side is pointwise defined on .
where , provided that the right-hand side is pointwise defined on .
Lemma 2.2 ()
for some, , where N is the smallest integer greater than or equal to ν.
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 .
for all and ;
The following lemma is valid for every cone in a Banach space.
Let C be a cone in X and let, be open bounded subsets of X withand. Assume that: 1∘ = L is a Fredholm operator of index zero;; 2∘ = is continuous and bounded andis compact on every bounded subset of X;; 3∘ = for alland;; 4∘ = γ maps subsets ofinto bounded subsets of C;; 5∘ = , wherestands for the Brouwer degree;; 6∘ = there existssuch thatfor, whereandis such thatfor every;; 7∘ = ;; 8∘ = ..Then the equationhas a solution in the set.
Definition 2.3 is called a solution of the problem (1.1) if and u satisfied Equation (1.1).
- (b)The functions from are equicontinuous on any compact subinterval of , that is, let J be a compact subinterval of , then , there exists such that for , ,
- (c)The functions from are equiconvergent, that is, given , there exists such that
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.
Lemma 3.1 Let. Thenis the solution of the following fractional differential equation:
Proof In view of Lemmas 2.1 and 2.2, we can certify the conclusion easily, so we omit the details here. □
Lemma 3.2 The operator L is 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.
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. □
Also, proceeding with the proof of Lemma 3.2, we can show that .
Thus, , where .
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.
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 .
which is a contradiction to . Therefore, 3∘ is satisfied.
Step 4: Let , then we can verify that is a retraction and 4∘ holds.
where and .
It is obvious that implies that by (3.8) and (3.11).
It is a contradiction. Besides, if , then , which is impossible. Hence, for , , .
which shows that 5∘ is true.
And we can take .
which implies that . Hence, 7∘ holds.
Thus, , that is, 8∘ is satisfied.
Hence, applying Theorem 2.1, the problem (1.1) has a positive solution in the set . □
To illustrate our main result, we will present an example.
It is easy for us to certify that f satisfies the condition (H).
Evidently, satisfies (3.11).
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.
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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