Existence of Positive Solutions to a Boundary Value Problem for a Delayed Nonlinear Fractional Differential System
© Zigen Ouyang et al. 2011
Received: 14 November 2010
Accepted: 24 February 2011
Published: 14 March 2011
Though boundary value problems for fractional differential equations have been extensively studied, most of the studies focus on scalar equations and the fractional order between 1 and 2. On the other hand, delay is natural in practical systems. However, not much has been done for fractional differential equations with delays. Therefore, in this paper, we consider a boundary value problem of a general delayed nonlinear fractional system. With the help of some fixed point theorems and the properties of the Green function, we establish several sets of sufficient conditions on the existence of positive solutions. The obtained results extend and include some existing ones and are illustrated with some examples for their feasibility.
In the past decades, fractional differential equations have been intensively studied. This is due to the rapid development of the theory of fractional differential equations itself and the applications of such construction in various sciences such as physics, mechanics, chemistry, and engineering [1, 2]. For the basic theory of fractional differential equations, we refer the readers to [3–7].
Recently, many researchers have devoted their attention to studying the existence of (positive) solutions of boundary value problems for differential equations with fractional order [8–23]. We mention that the fractional order involved is generally in with the exception that in [12, 23] and in [8, 17]. Though there have been extensive study on systems of fractional differential equations, not much has been done for boundary value problems for systems of fractional differential equations [18–20].
On the other hand, we know that delay arises naturally in practical systems due to the transmission of signal or the mechanical transmission. Though theory of ordinary differential equations with delays is mature, not much has been done for fractional differential equations with delays [24–31].
where is the standard Riemann-Liouville fractional derivative of order for some integer , for , for , and is a nonlinear function from to . The purpose is to establish sufficient conditions on the existence of positive solutions to (1.1) by using some fixed point theorems and some properties of the Green function. By a positive solution to (1.1) we mean a mapping with positive components on such that (1.1) is satisfied. Obviously, (1.1) includes the usual system of fractional differential equations when for all and . Therefore, the obtained results generalize and include some existing ones.
The remaining part of this paper is organized as follows. In Section 2, we introduce some basics of fractional derivative and the fixed point theorems which will be used in Section 3 to establish the existence of positive solutions. To conclude the paper, the feasibility of some of the results is illustrated with concrete examples in Section 4.
We first introduce some basic definitions of fractional derivative for the readers' convenience.
provided that the integral exists on , where is the Gamma function.
provided that the right-hand side is pointwise defined on , where .
It is well known that if then . Furthermore, if and then for .
The following results on fractional integral and fractional derivative will be needed in establishing our main results.
Lemma 2.3 (see ).
where , .
Lemma 2.4 (see ).
for some , .
Now, we cite the fixed point theorems to be used in Section 3.
Lemma 2.5 (the Banach contraction mapping theorem ).
Let be a complete metric space and let be a contraction mapping. Then has a unique fixed point.
Let be a closed and convex subset of a Banach space . Assume that is a relatively open subset of with and is completely continuous. Then at least one of the following two properties holds:
(i) has a fixed point in ;
(ii)there exists and with .
Let be a cone in a Banach space . Assume that and are open subsets of with and . Suppose that is a completely continuous operator such that either
(i) for and for
(ii) for and for .
Then has a fixed point in .
3. Existence of Positive Solutions
In this section, we always assume that .
This completes the proof.
The following two results give some properties of the Green functions .
For is continuous on and for .
Note that and for . It follows that and hence for .
Therefore, for and the proof is complete.
If , then for .
If , then for .
- (i)Obviously, for . Now, for , we have(3.14)
- (ii)Again, one can easily see that for . When , we have in this case that(3.15)
which implies that for . To summarize, we have proved (ii) and this completes the proof.
Now, we are ready to present the main results.
then (1.1) has a unique positive solution.
This, combined with Lemma 3.3 and (3.17) and (3.18), immediately implies that is a contraction. Therefore, the proof is complete with the help of Lemmas 3.1 and 2.5.
The following result can be proved in the same spirit as that for Theorem 3.4.
then (1.1) has a unique positive solution.
then (1.1) has at least one positive solution.
Let and be defined by (3.19) and (3.20), respectively. We first show that is completely continuous through the following three steps.
which implies that is continuous.
Immediately, we can easily see that is a bounded subset of .
Now the equicontituity of on follows easily from the fact that is continuous and hence uniformly continuous on .
Similarly, we can have if . To summarize, , a contradiction to . This proves the claim. Applying Lemma 2.6, we know that has a fixed point in , which is a positive solution to (1.1) by Lemma 3.1. Therefore, the proof is complete.
As a consequence of Theorem 3.6, we have the following.
If all , , are bounded, then (1.1) has at least one positive solution.
Suppose that there exist and positive constants with such that
(ii) , for ,
where . Then (1.1) has at least a positive solution.
Therefore, we have verified condition (ii) of Lemma 2.7. It follows that has a fixed point in , which is a positive solution to (1.1). This completes the proof.
In this section, we demonstrate the feasibility of some of the results obtained in Section 3.
It follows from Theorem 3.4 that (4.1) has a unique positive solution on .
By now we have verified all the assumptions of Theorem 3.8. Therefore, (4.6) has at least one positive solution satisfying .
Supported partially by the Doctor Foundation of University of South China under Grant no. 5-XQD-2006-9, the Foundation of Science and Technology Department of Hunan Province under Grant no. 2009RS3019 and the Subject Lead Foundation of University of South China no. 2007XQD13. Research was partially supported by the Natural Science and Engineering Re-search Council of Canada (NSERC) and the Early Researcher Award (ERA) Pro-gram of Ontario.
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