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.
Keywords:fractional differential equations; existence results; fixed point theorem; positive solution
In this paper, we are concerned with the existence of positive solutions for the fractional equation
with the boundary value conditions
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 .
In , Xu et al. discussed the problem
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.
In , Liang and Zhang studied the following nonlinear fractional boundary value problem:
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.
In , Agarwal and Ahmad studied the solvability of the following anti-periodic boundary value problem for a nonlinear fractional differential equation:
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.
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.
Thus, by the boundary value condition (2.2), we can obtain
From (2.11), we have
Substituting (2.12) into (2.10), we get
So, by (2.13), (2.12), and (2.9), we have
Thus, from (2.8), we have
Hence, from (2.7) together with (2.12)-(2.15), it follows that
by Definition 2.1, we have
For the forthcoming analysis, we need to introduce some new notations.
We also need the following lemma, which will play an important role in obtaining our main results in Section 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.
Similarly, we can deduce that
(3) The proof is divided into four steps.
Summing up the above analysis (i)-(ii), we obtain
So, by (2.23)-(2.24), we have
Step 3. Now, we show that
Summing up the above analysis (i)-(ii), and noting Step 2 of the proof as before, it follows that
Step 4. It remains to show that
Similarly, we can obtain that
The proof is complete. □
Now, we introduce a cone as follows:
We define an operator T on P as follows:
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.
3 Main results
We first prove the following lemma to obtain our main results.
Similarly, we can obtain
Now, we show that the operator T is compact on P.
In terms of Lemma 2.4, it follows from (2.26)-(2.27) that
According to (2.4)-(2.5) and by applying the mean value theorem, we have
So, (3.5) together with (3.6)-(3.8) implies that there exists a constant N such that the inequality
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.
Thus, in view of Lemma 2.4, from (2.26)-(2.27) and (3.9), it follows that
Consider the following ancillary BVP:
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.
Step 1. We first show that
Thus, from (2.27) and Lemma 2.4, it follows that
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.
Thus, from (3.12), (3.14), it follows that
Hence, (2.26) together with (3.15) implies that
Similarly, we can obtain
So, the relation (3.13) holds.
Therefore, from (2.26) and in view of Lemma 2.4, we have
Similarly, we can obtain
So, the relation (3.19) holds.
Example 3.1 Consider the following BVP:
So, by Theorem 3.1, BVP (3.23) has at least one positive solution.
Example 3.2 Consider the following BVP:
The author declares that he has no competing interests.
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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