Existence results of positive solutions for boundary value problems of fractional differential equations
© Chai; licensee Springer. 2013
Received: 15 October 2012
Accepted: 15 April 2013
Published: 29 April 2013
In this paper, we are concerned with the following fractional equation:
with the boundary value conditions
where is the standard Caputo derivative with and δ, γ are constants with , . By applying a new fixed point theorem on cone and Krasnoselskii’s fixed point theorem, some existence results of positive solution are obtained.
MSC: 34A08, 34B15, 34B18.
Keywordsfractional differential equations existence results fixed point theorem positive solution
where is the standard Caputo derivative with and δ, γ are constants with , .
Differential equations of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetism, etc. (see [1–5]). There has been a significant development in the study of fractional differential equations and inclusions in recent years, see the monographs of Podlubny , Kilbas et al. , Lakshmikantham et al. , Samko et al. , Diethelm , and the survey by Agarwal et al. . For some recent contributions on fractional differential equations, see [9–25] and the references therein.
On the other hand, it is well known that the fourth-order boundary value problem describes the deformations of an elastic beam in equilibrium state. Owing to its importance in physics, the existence of solutions to this problem has been studied by many authors; see, for example, [26–30] and references therein. Recently, there have been a few papers dealing with the existence of solutions for fractional equations of order .
where and is nonnegative, is the Riemann-Liouville fractional derivative of order α. The existence results of positive solutions are obtained by applying the Leray-Schauder nonlinear alternative theorem.
where , is nondecreasing relative to u, is the Riemann-Liouville fractional derivative of order α. By means of the lower and upper solution method and fixed point theorems, some results on the existence of positive solutions were obtained.
where . The existence results were obtained by the nonlinear alternative theorem.
Inspired by above work, the author will be concerned with the boundary value problem (BVP for short in the sequel) (1.1)-(1.2). To the best of our knowledge, no contribution exists concerning the existence of solutions for BVP (1.1)-(1.2). In the present paper, by applying a new fixed point theorem on cone and Krasnoselskii’s fixed point theorem, some existence results of positive solution for BVP (1.1)-(1.2) are obtained. It is worth to point out that the results in this paper are also new even for relative to the corresponding literature with regard to the fourth-order boundary value problem. In addition, the conditions imposed in this paper are easily verified.
The organization of this paper is 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 give two examples to demonstrate our main results.
In this section, we introduce some preliminary facts which are useful throughout this paper.
Let ℕ be the set of positive integers, ℝ be the set of real numbers, , and . Let . Denote by the Banach space endowed with the norm , where for .
Definition 2.1 
Definition 2.2 
where , denotes the integer part of α.
Definition 2.3 
Lemma 2.1 
holds on .
Lemma 2.2 
holds on .
For convenience, we first list some hypotheses which will be used throughout this paper.
() , , .
We have the following lemma, which is important in this paper.
hold for , and u satisfies the boundary condition (2.2).
Again, from (2.16) and Lemma 2.1, we have that , . In addition, noting that , it is easy to see that from (2.19). □
For the forthcoming analysis, we need to introduce some new notations.
It is easy to verify that , and noting that , , .
We also need the following lemma, which will play an important role in obtaining our main results in Section 3.
Lemma 2.4 Under the assumption (), Green’s function has the following properties:
(1) G is continuous on ;
(2) , ; , , ;
(3) , ; , , ;
, , ; , , , .
Proof (1) Observing the expression of Green’s function given by (2.4)-(2.6), the conclusion (1) of Lemma 2.4 is obvious.
(2) We first show that , , .
Owing to the fact that and , we have that . Thus, we immediately obtain that for from (2.20) together with the condition , .
for , and so for .
To summarize, for all with .
Now, since for , and for with , it follows that for all .
(3) The proof is divided into four steps.
Step 1. We show that for .
Step 2. We show that for and .
where , because .
So, by (2.23)-(2.24), we have
for , and .
The proof is complete. □
It is easy to check that the above set P is a cone in the space , which will be used in the sequel.
where , , , , and , are given by (2.20)-(2.21), respectively.
A function is a positive solution of BVP (1.1)-(1.2) if , , , and u satisfies BVP (1.1)-(1.2).
By Lemma 2.3, it is easy to know that a function is a positive solution of BVP (1.1)-(1.2) iff is a nonzero fixed point of T. So, we can focus on seeking the existence of a nonzero fixed point of T in P.
Finally, for the remainder of this section, we give the following two theorems, which are fundamental in the proof of our main results.
Let X be a Banach space, and let be a cone. Suppose that the functions α, β satisfy the following condition:
(D) are continuous convex functionals satisfying , for , ; for , and for with , where is a constant.
Lemma 2.5 
Set . Suppose that is a completely continuous operator satisfying
() , ; , ;
() , ;
() there is a such that and for all and .
Then T has at least one fixed point in .
Lemma 2.6 
Assume that , are two open subsets of X with , and let be a completely continuous operator such that either
(i) , ; , , or
(ii) , ; , .
Then T has a fixed point in .
3 Main results
We first prove the following lemma to obtain our main results.
Lemma 3.1 Suppose that ()-() hold. Then the operator T defined by (2.26) maps P into P, and T is completely continuous.
Proof It is well known that the norms and are equivalent on . So, we can consider that the Banach space is equipped with the norm in the following proof.
In addition, . Thus, .
Now, we show that the operator T is compact on P.
In fact, let U be an arbitrary bounded set in P. Then there exists a positive number L such that for all , and so such that , for all .
Because the functions and are integrable on I, the formulae (3.3)-(3.4) yield that , , where . So, . That is, TU is uniformly bounded.
holds for any and with . That is, the set TU is equicontinuous.
Similarly, we can deduce that the set is also equicontinuous in terms of (2.27).
So, as a consequence of the Arzelà-Ascoli theorem, we have that TU is a compact set.
Now, we come to prove the operator T is continuous on P.
for all , whenever .
whenever . That is, T is continuous on P. □
We are now in a position to state and prove the first theorem in the article. Let constants , satisfy , .
Theorem 3.1 Suppose that ()-() hold. In addition, there are two constants , with such that f satisfies the following condition:
() , for ;
, for .
Then BVP (1.1)-(1.2) has at least one positive solution u satisfying and .
Proof We already know that is completely continuous by Lemma 3.1.
Let , for . It is easy to verify that the functions α, β satisfy the condition (D).
Choose a constant L large enough so that , where , and , . Set , , . Define the function on as , , where , .
Obviously, the function is continuous on according to the continuity of f. Thus, by an argument similar to that in Lemma 3.1, the operator given by is also completely continuous on P and maps P into P.
We will prove that T has at least one nonzero fixed point in P by applying Lemma 2.5. The approach is divided into four steps.
In fact, for any , owing to the condition , we have that , , and so .
keeping in mind that , for .
Step 2. Now, we come to verify that the conditions corresponding to () in Lemma 2.5 hold.
Thus, , noting that the assumption . That is, .
Thus, , noting that . That is, .
Step 3. We verify that the conditions corresponding to () in Lemma 2.5 hold.
Hence, , and so from the choice of L. Thus, .
for any , .
So, the conditions corresponding to () in Lemma 2.5 hold.
Summing up the above steps 1-4 and applying Lemma 2.5, we obtain that BVP (3.10) has at least one positive solution . That is, . , and so from the fact that and by (3.13). Thus, , , , , and . Hence, , , and so u is a positive solution of BVP (1.1)-(1.2). The proof is complete. □
Now, we state another theorem in this paper. Let us begin with introducing some notations.
Let . Denote with , and with . Put , , where , , , and () are given in Lemma 2.4.
Theorem 3.2 Assume that ()-() hold. If , , then BVP (1.1)-(1.2) has at least one positive solution.
holds when with , .
So, the relation (3.13) holds.
whenever with , .
So, the relation (3.19) holds.
Summing up (3.13) and (3.19), applying Lemma 2.6, the operator T has at least one fixed point . Thus u is a positive solution of BVP (1.1)-(1.2). The proof is complete. □
where constants , are two positive numbers. Then BVP (3.23) has at least one positive solution.
holds for .
So, by Theorem 3.1, BVP (3.23) has at least one positive solution.
where constants , are two positive numbers and a constant . Then BVP (3.24) has at least one positive solution.
In fact, observing that , , the conclusion follows from Theorem 3.2.
The author sincerely thanks the anonymous referees for their valuable suggestions and comments which have greatly helped improve this article. Article is supported by the Natural Science Foundation of Hubei Provincial Education Department (D20102502).
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