Journal of Function Spaces and Applications

Volume 2013, Article ID 793263, 5 pages

http://dx.doi.org/10.1155/2013/793263

## Generalized Quasilinearization for the System of Fractional Differential Equations

^{1}College of Electronic and Information Engineering, Hebei University, Baoding 071002, China^{2}College 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.

#### Abstract

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.

#### 1. Introduction

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 [1], Podlubny [2], Kilbas et al. [3], and West et al [4] and the papers of Debnath [5], Rossikhin and Shitikova [6], and Ferreira et al [7]. 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 [19], and McRae [20] 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 [21] 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 [22] 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 *n*th 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.

#### 2. Preliminaries

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 *(H_{1})* 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
is nonempty.

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 *(H_{1})* are quasimonotone nondecreasing in for , , , , exist and are continuous on , satisfying , ; *(H_{2})*, where is an matrix given by
**where .**Then, there exist monotone sequences , which converge uniformly to the solution of (1) and the convergence is quadratic.*

* Proof. * (H_{1}) in Theorem 4 implies for any ,??,
And for any
we have
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.

#### Acknowledgments

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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