International Journal of Differential Equations

Volume 2012 (2012), Article ID 842813, 17 pages

doi:10.1155/2012/842813

Research Article

Generalized Monotone Iterative Technique for Caputo Fractional Differential Equation with Periodic Boundary Condition via Initial Value Problem

J. D. Ramírez¹ and A. S. Vatsala²

¹Department of Mathematics, Lamar University, P.O. Box 10047, Beaumont, Texas 77710, USA

²Department of Mathematics, University of Louisiana Lafayette, P.O. Box 41010, Lafayette, LA 70504, USA

Received 23 May 2012; Revised 27 July 2012; Accepted 30 July 2012

Academic Editor: Shaher Momani

Copyright © 2012 J. D. Ramírez and A. S. Vatsala. 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.

We develop a generalized monotone method using coupled lower and upper solutions for Caputo fractional differential equations with periodic boundary conditions of order , where . We develop results which provide natural monotone sequences or intertwined monotone sequences which converge uniformly and monotonically to coupled minimal and maximal periodic solutions. However, these monotone iterates are solutions of linear initial value problems which are easier to compute.

1. Introduction

The study of fractional differential equations has acquired popularity in the last few decades due to its multiple applications, see [1–5] for more information. However, it was not until recently that a study on the existence of solutions by using upper and lower solutions, which is well established for ordinary differential equations in [6], has been done for fractional differential equations. See [3, 7–16] for recent work.

In this paper we recall a comparison theorem from [3] for a Caputo fractional differential equation of order , , with initial condition. We will use coupled lower and upper solutions combined with a generalized monotone method of initial value problems to prove the existence of coupled minimal and maximal periodic solutions. The results developed provide natural sequences and intertwined sequences which converge uniformly and monotonically to coupled minimal and maximal periodic solutions. Instead of the usual approach as in [6, 17] where the iterates are solutions of linear periodic boundary value problems we have used a generalized monotone method of initial value problems. This idea was presented in [18] for integrodifferential equations. The advantage of this method, compared to what was developed in [3, 15], is that it avoids computing the solution of the linear periodic boundary value problem using the Mittag-Leffler function at every step of the iterates. We also modify the comparison theorem which does not require the Hölder continuity condition as in [3].

2. Preliminary Definitions and Comparison Results

In this section we state some definitions and recall some results for a Caputo initial value problem which we need in our main results. Consider the initial value problem of the form:

Here, is the Caputo derivative of order for , which is defined in [1–3, 5] as

In this paper we will denote .

The relation between the Caputo fractional derivative and the Riemann-Liouville fractional derivative, , is given by

Throughout this paper we consider the Caputo derivative of order , for and . We start by showing some comparison results relative to initial value problems with the Caputo fractional derivative of order .

Lemma 2.1. Let . If there exists such that and on , then it follows that

Proof. Let , the using the relation (2.3) we have that

Since this lemma was proven in [8] for the Riemann Liouville derivative, we have that implies , and the proof is complete.

Remark 2.2. In [3] they proved the above result by assuming that is Hölder continuous of order . Although the proof is correct, it is not useful in the monotone method or any iterative method because we will not be able to prove that each of those iterates are Hölder continuous of order .

Now we are ready to establish the following comparison theorem.

Theorem 2.3. Let , and for the following inequalities hold true,

Suppose further that satisfies the following Lipschitz condition,

then implies that

Proof. Assume first without loss of generality that one of the inequalities in (2.6) is strict, say , and , where and . We will show that for .

Suppose, to the contrary, that there exists such that for which

Setting it follows that and for . Then by hypothesis and Lemma 2.1 we have that . Thus

which is a contradiction to the assumption . Therefore .

Now assume that the inequalities (2.6) are nonstrict. We will show that .

Set , where and , where is the one parameter Mittag-Leffler function. This implies that and for .

Using (2.6) and the Lipschitz condition (2.7), we find that

Here we have utilized the fact that is the solution of the initial value problem

Clearly there is no assumption on the growth of . Applying now the result for strict inequalities to , we get that for , for every and consequently making , we get that for .

The following corollary will be useful in our main results.

Corollary 2.4. Let be such that

Then we have from the previous theorem the estimate

The result of the above corollary is still true even if , which we state separately and prove it.

Corollary 2.5. Let on . Then , if .

Proof. By definition of and by hypothesis,

which implies that on . Therefore, on .

Note that the above result may not be true for the Riemann Liouville derivative.

We recall a comparison result from [3] for periodic boundary conditions. As in Theorem 2.3, the proof does not require Hölder continuity.

Theorem 2.6. Let , , , and for ,

Suppose further that is strictly decreasing in for each , then

Proof. If the conclusion is not true; that is, if there exists such that , then there exists an such that

Setting we find that if , then

By Lemma 2.1 we have that . Thus, . We now obtain by hypothesis and by the strictly decreasing nature of in ,

which is a contradiction.

If , then

so , and by the above argument we also get a contradiction.

Hence for and the proof is complete.

Two important cases of this theorem are the following which are useful to prove the uniqueness of the solution of a Caputo fractional differential equation with periodic boundary conditions.

Corollary 2.7. Let be such that

for and . Then for .

Similarly, if

for and . Then for .

Remark 2.8. It is to be noted that in the proof of these equivalent results from [3] we use Lemma 2.1 which does not require the Hölder continuity assumption.

A generelized monotone method for periodic boundary value problems was recently developed in [3, 15]. However it uses the approach established in [6] for ordinary differential equations where the iterates are solutions of the linear periodic boundary value problem,

where and .

The explicit solution of this equation is given by

where and are Mittag-Leffler functions with one and two parameters, respectively.

This poses a problem to compute the linear iterates since it involves Mittag-Leffler functions. The advantage of our method is that it does not require the Mittag-Leffler function in our computations.

Pandit et al. used the initial value problem to obtain a generalized monotone method in [18] for nonlinear integrodifferential equations with periodic boundary conditions. In the next section we develop a monotone method using this idea.

3. Generalized Monotone Method for the Nonlinear Periodic Boundary Value Problem via Initial Value Problem

In this section, we will develop a generalized monotone method for the nonlinear periodic boundary value problem (3.2), given below, by using coupled upper and lower solutions and the corresponding initial value problem (2.1), where does not depend on ,

For that purpose consider the nonlinear periodic boundary value problem of the form:

where and .

If satisfies the fractional differential equation

and is such that for , then is a periodic solution of (3.2).

Furthermore, throughout this paper, we will assume that is increasing in and is decreasing in for .

Here below we provide the definition of coupled lower and upper solutions of (3.2).

Definition 3.1. Let . Then and are said to be,

(i)natural lower and upper solutions of (3.2) if

(ii)coupled lower and upper solutions of Type I of (3.2) if

(iii)coupled lower and upper solutions of Type II of (3.2) if

(iv)coupled lower and upper solutions of Type III of (3.2) if,

We will state the following four theorems related to coupled lower and upper solutions of Type I and Type II, respectively. We develop the generalized monotone method for the periodic boundary value problem via the initial value problem approach. We obtain natural sequences and intertwined sequences which converge uniformly and monotonically to coupled minimal and maximal periodic solutions of (3.2).

In the next theorem, we use coupled lower and upper solutions of Type I and obtain natural sequences which converge uniformly and monotonically to coupled minimal and maximal periodic solutions of (3.2).

Theorem 3.2. Assume that

(A1) are coupled lower and upper solutions of Type I for (3.2) with on ; (A2), is increasing in and is decreasing in .

Then the sequences defined by

are such that and in uniformly and monotonically, such that and are coupled minimal and maximal solutions of (3.2), respectively, provided that , where is any periodic solution of (3.2). That is, and satisfy the coupled system

such that .

Proof. By hypothesis, . We will show that .

It follows from (3.5) that

and by (3.8), we get that

Therefore, . If we let , then and,

Since and , by Corollary 2.5 we have that and, consequently, on . By a similar argument we can show that , , and . Thus, .

Now we will show that for .

Assume that

for .

Let . Then

so . By the increasing nature of and the decreasing nature of it follows that

Similarly, by Corollary 2.5 we have that and consequently .

By a similar argument we can show that . Using the hypothesis that on , the above argument and induction we can show that . Therefore for ,

Now we have to show that the sequences converge uniformly. We will use Arzela-Ascoli theorem by showing that the sequences are uniformly bounded and equicontinuous.

First we show uniform boundedness. By hypothesis both and are bounded on , then there exists such that for any , , and . Since for each , it follows that

and consequently is uniformly bounded. By a similar argument is also uniformly bounded.

To prove that is equicontinuous, let . Then for ,

Since and are uniformly bounded and and are continuous on , there exists independent of such that

Thus, for any there exists independent of such that for each ,

provided that .

Similarly we can prove that is equicontinuous and uniformly bounded.

This proves that and are uniformly bounded and equicontinuous on . Hence by Arzela-Ascoli’s theorem there exist subsequences and which converge uniformly to and , respectively. Since the sequences are monotone, the entire sequences converge uniformly.

We have shown that the sequences converge in . In order to show that they converge in , observe that since each is constructed as follows

we get that

Taking limits when , we obtain by the Lebesgue dominated convergence theorem that

Hence in . Furthermore, the above expression is equivalent to

By a similar argument in and it can be shown that

Since on for all , we get that on which shows that and are minimal and maximal periodic solutions of (3.2), respectively. This completes the proof.

Remark 3.3. In [3] the uniqueness is shown by making additional assumptions to Theorem 3.2 and using Corollary 2.7. However the iterates are solutions of the form (2.25). In our result, we have proved the existence of coupled minimal and maximal periodic solutions of (3.2), in particular if we get minimal and maximal periodic solutions of (3.2).

The next result also uses coupled upper and lower solutions of Type I. However, we obtain intertwined sequences which converge to coupled minimal and maximal periodic solutions of (3.2). The proof is similar to Theorem 3.2, hence we do not provide the proof.

Theorem 3.4. Assume that conditions (A1) and (A2) of Theorem 3.2 are true. Then the iterative scheme given by

give alternating monotone sequences and satisfying

for each on , provided that .

Furthermore and in , where and are coupled minimal and maximal periodic solutions of (3.2), respectively; that is, if then , and and satisfy the coupled system

In the previous two theorems we assumed the existence of coupled lower and upper solutions of type I. We can state two more results involving coupled lower and upper solutions of Type II, however they require an additional assumption in order to obtain natural or intertwined sequences converging to coupled minimal and maximal periodic solutions of problem (3.2).

Theorem 3.5. Assume that

(B1) and are coupled lower and upper solutions of Type II for (3.2) with on ,(B2), is increasing in , and is decreasing in .

If is a solution of (3.2) such that . Then the sequences defined by (3.27) give alternating monotone sequences and satisfying

for each on , provided that .

Furthermore and in , where and are coupled minimal and maximal solutions of (3.2), respectively; that is, if then , and and satisfy the coupled system

Theorem 3.6. Assume that conditions (B1) and (B2) of Theorem 3.5 are true. Then the sequences defined by (3.8) and (3.9) are such that and in uniformly and monotonically, provided that , where and are coupled minimal and maximal solutions of (3.2), respectively; that is, and and satisfy the coupled system

Remark 3.7. The proof of Theorems 3.5 and 3.6 follow on the same lines as Theorem 3.2. However, it is easy to compute coupled lower and upper solutions of Type II.

4. Numerical Examples

In this section we present some numerical examples which are application of Theorem 3.5.

Example 4.1. Consider the problem

Clearly and are solutions of the equation.

Since is increasing in for and decreasing for , we let

and and be lower and upper solutions of type II, respectively, because

We construct our sequences according to Theorem 3.5 and show graphically in Figure 1 that .

Figure 1: Dashed: . Solid: .

Example 4.2. Consider the problem

Since is increasing in for and decreasing for , we let

and and be lower and upper solutions of type II, respectively, because

where .

Using Theorem 3.5 we show in Figure 2 four steps of and four steps of . Observe that converges more quickly to the periodic solution.

Figure 2: and .

Example 4.3. Consider

Since is increasing in for and decreasing for ,

and and are lower and upper solutions of type II, respectively, because

where , which implies that

Once again using Theorem 3.5 we computed in Figure 3 four steps of and 4 steps of . As in the previous examples converges more quickly to the periodic solution.

Figure 3: and .

5. Concluding Remarks

In our main results as well as in our numerical results we have not used the assumption needed to obtain unique solution of (3.2). If we make appropriate uniqueness assumption on the nonlinear terms and then our linear iterates will require the computation of the Mittag-Leffler function. We plan to take up this work in the near future.

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