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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 793263, 5 pages
Generalized Quasilinearization for the System of Fractional Differential Equations
1College of Electronic and Information Engineering, Hebei University, Baoding 071002, China
2College of Mathematics and Computer Science, Hebei University, Baoding 071002, China
Received 9 December 2012; Accepted 8 February 2013
Academic Editor: Yongsheng S. Han
Copyright © 2013 Peiguang Wang and Ying Hou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper considers the initial value problems of the system of fractional differential equations and constructs two monotone sequences of upper and lower solutions. By using quasilinearization technique, monotone sequences of approximate solutions that converge quadratically to a solution are obtained.
In this paper, we consider the system of Caputo fractional differential equations: where , , , , and .
The theories and properties of fractional differential equations have received attention from some researchers because many mathematical modeling appeared in the fields of physics, chemistry, engineering and biological sciences and so on. For examples and details, we can refer to the, monographs of Miller and Ross , Podlubny , Kilbas et al. , and West et al  and the papers of Debnath , Rossikhin and Shitikova , and Ferreira et al . There are many results on the basic theory of initial value and boundary value problems for fractional differential equations, which can be found in [8–10]. Meanwhile, there are some qualitative and numerical solutions for various fractional equations with delay and impulsive effects. For details, see some recent papers [11–18] and the references therein.
It is well known that the monotone iterative technique is an ingenious method providing a constructive approach to find solutions for the nonlinear problem via linear iterates. Lakshmikantham and Vatsala , and McRae  investigated the existence of minimal and maximal solutions of fractional differential equations by establishing a comparison result and using the monotone method, respectively; Benchohra and Hamani  used a monotone iterative technique in the presence of lower and upper solutions to discuss the existence of solutions to impulsive fractional differential inclusions.
Quasilinearization  provides an elegant and easier approach to obtain a sequence of approximate solutions with quadratic convergence, and the method has been extended to fractional differential equations in [23, 24]. However, to the best of our knowledge, there are few results for the system of fractional differential equations, especially results on the convergence of the system. In the present paper, we will discuss the approximate solutions of the system of fractional differential equations through the application of quasilinearization. The significance of this work lies in the fact that the system of fractional differential equations can also obtain a monotone sequence of approximate solutions converging uniformly to the solution of the problem and possessing quadratic convergence.
The nonhomogeneous linear system of Caputo fractional differential equations is given by where is an nth order matrix over complex field, and is an n-dimensional locally integrable column vector function on .
Using the method of successive approximations, we get the solution of (2) as where are Mittag-Leffler functions of one parameter and two parameters, respectively.
Now, we present the following definition and lemma which help to prove our main result.
Definition 1. Let be lower and upper solutions of (1) if they satisfy the inequalities respectively, for .
Lemma 2. Suppose that are lower and upper solutions of (1), and (H1) are quasimonotone nondecreasing in for each and for each ,
where , is a constant.
Then, implies that .
Proof. Firstly, suppose that
and . We will prove that . Suppose the conclusion is not true, then the set
Let . Certainly, . Since the set is closed, and consequently there exists a such that . Moreover, , for , and Hence, it easily follows that This together with the quasimonotonicity of yields which leads to a contradiction.
In order to prove the case of nonstrict inequalities, consider the functions where is sufficiently small constant. Then using (6), we have Also . Now using the result corresponding to strict inequalities, we get Letting , we obtain the required result and the proof is complete.
Corollary 3. The function where is admissible in Lemma 2 to yield .
3. Main Result
Theorem 4. Assume that are lower and upper solutions of (1) such that , and (H1) are quasimonotone nondecreasing in for , , , , exist and are continuous on , satisfying , ; (H2), where is an matrix given by
Then, there exist monotone sequences , which converge uniformly to the solution of (1) and the convergence is quadratic.
Proof. (H1) in Theorem 4 implies for any , ,
And for any
since by assumption .
It is also clear that for ,
Let be the solutions of IVPs: where . We will prove that . To do this, let , so that . Then using (20), we obtain
Since is quasimonotone nondecreasing by assumption to and it follows from Corollary 3 that , proving that .
Now we let and note that . Also, since , using (16), we get
In view of , we have which yields Hence, we obtain this implies that , using Corollary 3. As a result, we have In a similar way, we can prove that To show , we use (16), (19), and : Using a similar argument, it is easy to show that And, therefore, by Lemma 2 and (19), it follows that by (19), we have is Lipschitzian in on . This proves that Now assume that for some , We now aim to show that where and are the solutions of linear IVPs: Now, set so that and . It follows from Corollary 3 and using (16) that
On the other hand, letting yields Since , (16) and give, as before, which shows that This proves that using Corollary 3 and (16), since . Hence, we get Similarly, we can prove that Also, by (16), (35), and the fact that , we obtain
Using a similar argument, we have, as before, , and hence Lemma 2 shows that By (19), we have that is in on and is quasimonotone nondecreasing in . This proves (34). Therefore, by induction, we have for all :
Employing the standard procedure, it is now easy to prove that the sequences and converge uniformly and monotonically to the unique solution of (1) on .
We will now show that the convergence of and to is quadratic. First set so that . Then using integral mean value theorem and the fact that and , we get So, we get where , , , and in , and , , and are positive matrices. Using (2) to (4), we can get where . Thus, we have Similarly, we have The proof is complete.
The authors would like to thank the reviewers for their valuable suggestions and comments. This work is supported by the National Natural Science Foundation of China (11271106) and the Natural Science Foundation of Hebei Province of China (A2013201232).
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