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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 493164, 8 pages
Monotone Iterative Solutions for Nonlinear Boundary Value Problems of Fractional Differential Equation
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China
Received 27 April 2013; Accepted 7 August 2013
Academic Editor: Yong Zhou
Copyright © 2013 Chaozhu Hu et al. 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.
By means of the method of quasi-lower and quasi-upper solutions and monotone iterative technique, we consider the nonlinear boundary value problems with Caputo fractional derivative and introduce two well-defined monotone sequences of quasi-lower and quasi-upper solutions which converge uniformly to the actual solution of the problem, and then the existence results of the solution for the problems are established. A numerical iterative scheme is introduced to obtain an accurate approximate solution and to give one example to demonstrate the accuracy and efficiency of the new approach.
We are interested in the existence and uniqueness of solution of the following nonlinear boundary value problem of fractional order where , , and are a continuous function, , and is the Caputo fractional derivative of order .
It is well known that the differential equations with fractional order are generalization of ordinary differential equations to noninteger order, they occur more frequently in different research areas and engineering, such as physics, control of dynamical systems, chemistry and so forth. We also remark that several kinds of fractional derivatives were introduced to investigate the fractional differential equation, see, for example, [1, 2], and references therein.
Roughly speaking, it is a difficult task to give exact solutions for fractional differential equations. Recently, there are a number of numerical and analytical techniques to concerned with such problems, for instance, the homotopy analysis method, the Adomian decomposition method and the critical point theory have been discussed the fractional differential equations, such as [3–9]. On the other hand, very recently, the monotone iterative technique, combined with the method of lower and upper solutions were introduced to study the problems [10–13]. In [10, 11], the authors used the method of lower and upper solutions to investigate the existence of solutions for a class of fractional initial value problems. In  the authors considered fractional boundary value problem and proved the existence of solution. In , the author discussed the Monotone iterative technique for boundary value problems of a nonlinear fractional differential equation with deviating arguments. Specially, here it is worth mentioning, that Al-Refai and Ali Hajji  introduce two well-defined monotone sequences of lower and upper solutions which converge uniformly to the actual solution of the problem. However, the existence results in  mainly depend upon a restrictive condition; that is, It is a critical condition in order to discuss the monotone iterative sequences. It is, therefore, natural to ask how to discuss the problem (1), if is not necessary decreasing with respect to . So, being directly inspired by , the purpose of this paper is to study the nonlinear boundary value problems of fractional order with Dirichlet boundary conditions. We introduce a method based on quasi-lower and quasi-upper solutions to prove the existence of a unique solution with no claim on the strictwe monotone. And also give an algorithm to construct two monotone sequences of quasi-lower and quasi-upper solutions. Moreover, the constructed sequences are proved to converge uniformly to the unique solution of the problem.
The paper is organized as follows. Preliminaries are in Section 2. Then in Section 3, we construct the monotone sequences of quasi-lower and quasi-upper solutions and prove their uniform convergence to the unique solution of the problem. Finally, in Section 4, we establish the numerical approach employed to obtain accurate numerical solution, and give one example to demonstrate the accuracy and efficiency of the new approach.
In this sections, we present the definition of a pair of quasi-lower and quasi-upper solutions and some lemmas which will be needed in the next section.
Definition 1. The functions are called a pair of quasi-lower and quasi-upper solutions of the problem (1), if it satisfies If the above inequalities are equalities, we call and a pair of quasi-solutions of the problem (1).
Lemma 2 (see ). Let and is a constant. If satisfies the relations then for .
Lemma 3 (see ). A function is a solution of the linear fractional BVP problems if and only if it is a solution of integral equation where
Lemma 4. For all , we have where
Proof. The proof of this lemma is easy, so we omit it.
In the rest of the paper, for convenience sake, we define the set let and list some conditions. , is a pair of quasi-lower and quasi-upper solutions of the problem (1) and , . The function is increasing with respect to , and is decreasing with respect to . And there exists a constant such that
Lemma 5. Let and hold; then for any fixed , the linear problems have a unique solution , .
Proof. Firstly, we prove that if is a solution of (17), then . From , and if is a quasi-lower solution of the problem (1), then we have
Thus from (18) and if is a solution of (17), set ; then
By Lemma 2, we have that
That is, . Similarly, we show . Therefore, .
Next, we show that (17) has a unique solution.
From Lemma 3, the problems (17) are equivalent to the following integral equation where .
Let For any , by Lemma 4 and condition , we have So By the Banach fixed point theorem, the operator has a unique point. This shows that (17) has a unique solution , .
3. Monotone Sequences of Quasi-Lower and Quasi-Upper Solutions
In this sections we construct the monotone sequences of quasi-lower and quasi-upper solutions and prove their uniform convergence to the unique solution of the problem.
Theorem 6. Let and hold, and assume that is a pair of solutions of Then one has the following. (i) The sequence , is a pair of quasi-lower and quasi-upper solutions of (1). Moreover, (ii) The sequence , converges uniformly to and , respectively, with . Moreover, and are a pair of minimal-maximal quasi-solutions of (1) in . (iii) Suppose further that there exist constants such that , and Then is the actual solution of (1) in .
Proof. (i) For any fixed , consider the linear problems
By Lemma 5, the linear problems (28) have a unique solution . Define operator as follows:
Again, it is easy that to prove that , are a pair of quasi-solutions of the problem (1) if and only if
Furthermore, from (28) and condition , we can prove is increasing with respect to , and is decreasing with respect to ; that is,
In fact, setting , , by (28) and condition , we have
Thus, by setting , then
By Lemma 2, we have that
That is, , or the first inequality holds in (31). Similarly, the second inequality holds in (31) too.
Again, similar to the previous argument, we can show that
Now, let Thus, from (31), (35), and (36), we have Moreover, it is clear that , satisfy (36) if and only if , are a pair of solutions of (25).
To prove , we only need to show that , are a pair of quasi-lower and quasi-upper solutions of (1). In fact, from (25) and condition , we can obtain Hence, , is a pair of quasi-lower and quasi-upper solutions of (1). This proves (i).
(ii) Since the sequence is monotone nondecreasing and is bounded from above mention, the sequence is monotone nonincreasing and is bounded from below, therefore the pointwise limits exist and these limits are denoted by and . The sequence , are sequences of continuous functions defined on the compact set , hence By Dini’s theorem , the convergence is uniform. This is uniformly on .
Moveover, from (25), we have This show that , are a pair of quasi-solutions of the problem (1).
Now, we prove that and are a pair of minimal-maximal quasi-solutions of (1).
Let be a pair of quasi-solutions of the problem (1), in the following, we show this using induction arguments. In fact, we have Assume that is true. Thus from (31) and (36), we can obtain Therefore, Thus, taking limit in (44) as , we have That is, and are a pair of minimal-maximal quasi-solutions of (1) in .
(iii) Since , , it is sufficient to prove that , . In fact, from (40) and supposition (iii), we have (notice that ) Thus, by setting , from (46), we obtain By Lemma 2 (notice that ), we have that Therefore, , . Hence is the actual solution of (1) in . Thus, we complete this proof.
4. Numerical Implementation
In this section, the numerical procedures are introduced to obtain an approximate solution of (1).
For the problem (25), from Lemma 3, we have Thus by Theorem 6, the unique solution to problem (1) is given by Setting Discretize the interval with the nodes , , . Using Simpson’s quadrature rule, we can obtain the function , and , to approximate the integral and on the right hand sid of (49), where where are the coefficients in composite Simpson’s quadrature rule: , , , .
Remark 7. Applying the above numerical scheme, we can solve iteratively for and up to some large enough . For a given accuracy , we take and as -accurate approximations of and , respectively. And we use to describe the error.
Example 8. Consider the fractional BVP Let It can be easily verified that are a pair of ordered quasi-lower and quasi-upper solutions of (58). Moreover, Thus Hence for this example, we found that, for , it has iterations for . the graph of and , for some values of , is in Figure 1. Table 1 displays versus for selected values of .
This work was partially supported by NNSF of China (Grant no. 11171122).
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