Existence results for nonlinear fractional differential equations involving different Riemann-Liouville fractional derivatives
© Wang et al.; licensee Springer 2013
Received: 9 May 2013
Accepted: 15 August 2013
Published: 4 October 2013
By applying an iterative technique, a necessary and sufficient condition is obtained for the existence of the unique solution of nonlinear fractional differential equations involving two Riemann-Liouville derivatives of different fractional orders. Finally, an example is also given to illustrate the availability of our main results.
Keywordsdifferent fractional-order nonlinear fractional differential equations Riemann-Liouville derivative monotone iterative technique
where (), , D is the standard Riemann-Liouville fractional derivative, , and . It is worthwhile to indicate that the nonlinear term f involves the unknown function’s Riemann-Liouville fractional derivatives with different orders.
The method of upper and lower solutions coupled with the monotone iterative technique is an interesting and powerful mechanism. The importance and advantage of the method needs no special emphasis [6, 7]. There have appeared some papers dealing with the existence of the solution of nonlinear Riemann-Liouville-type fractional differential equations [8–18] or nonlinear Caputo-type fractional differential equations [19–22] by using the method. For example, by employing the method of lower and upper solutions combined with the monotone iterative technique, Lakshmikanthan and Vatsala , McRae  and Zhang  successfully investigated the initial value problems of Riemann-Liouville fractional differential equation , where .
However, in the existing literature [8–18], only one case when is considered. The research, involving Riemann-Liouville fractional derivative of order , proceeds slowly and there appear some new difficulties in employing the monotone iterative method. To overcome these difficulties, we apply a substitution . Note that the technique has been discussed for fractional problems in papers [10, 11]. To the best of our knowledge, it is the first paper, in which the monotone iterative method is applied to nonlinear Riemann-Liouville-type fractional differential equations, involving two different fractional derivatives and .
We organize the rest of this paper as follows. In Section 2, by using the monotone iterative technique and the method of upper and lower solutions, the minimal and maximal solutions of an equivalent problem of (1.1) are investigated and two explicit monotone iterative sequences, converging to the corresponding minimal and maximal solution, are given. In addition, the uniqueness of the solution for fractional differential equations (1.1) is discussed. In Section 3, an example is given to illustrate our results.
2 Existence results
has a unique solution , where I is the fractional integral and , , and .
for some .
By the condition , it follows that . Therefore, we have .
Conversely, by a direct computation, we can get and . It is easy to verify satisfies (2.1).
This completes the proof. □
where , and , are the standard fractional integrals.
Now, we list for convenience the following condition:
where , .
where , .
Thus, we can obtain , . That is, . Similarly, we can prove that . Then, (a) holds.
Hence, we have , . That is, . Then, (b) holds.
Employing the same arguments used in Ref. , we see that , converge to their limit functions , , respectively. That is, and . Moreover, , are solutions of (2.5) in . (2.7) is true.
Thus, taking limit in (2.12) as , we have . That is, , are the minimal and maximal solution of (2.5) in the ordered interval , respectively.
This completes the proof. □
then problem (2.5) has a unique solution .
Now, we are going to show that problem (2.5) has a unique solution x, i.e., .
which implies that . Since , then it holds . That is, . Therefore, problem (2.5) has a unique solution . □
Let be the unique solution of (2.5). Noting that and , we can easily obtain the following theorem.
Theorem 2.3 Let all conditions of Theorem 2.2 hold. Then problem (1.1) has a unique solution , .
Hence, condition (H1) holds.
Then, all conditions of Theorem 2.3 are satisfied. In consequence, the problem (3.1) has a unique solution .
The authors would like to thank the referees for their useful comments and remarks. This work is supported by the NNSF of China (No.61373174) and the Natural Science Foundation for Young Scientists of Shanxi Province, China (No. 2012021002-3).
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