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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 615230, 14 pages
http://dx.doi.org/10.1155/2012/615230
Research Article

Existence and Uniqueness of Solutions for Initial Value Problem of Nonlinear Fractional Differential Equations

1School of Mathematical Sciences, University of Jinan, Shandong, Jinan 250022, China
2Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, USA
3School of Control Science and Engineering, University of Jinan, Shandong, Jinan 250022, China

Received 8 May 2012; Revised 12 June 2012; Accepted 13 June 2012

Academic Editor: Dumitru Baleanu

Copyright © 2012 Qiuping Li 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.

Abstract

We discuss the initial value problem for the nonlinear fractional differential equation , where , , and , , is the standard Riemann-Liouville fractional derivative and is a given continuous function. We extend the basic theory of differential equation, the method of upper and lower solutions, and monotone iterative technique to the initial value problem. Some existence and uniqueness results are established.

1. Introduction

Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary noninteger order, so fractional differential equations have wider application. Fractional differential equations have gained considerable importance; it can describe many phenomena in various fields of science and engineering such as control, porous media, electrochemistry, viscoelasticity, and electromagnetic.

In the recent years, there has been a significant development in fractional calculus and fractional differential equations; see Kilbas et al. [1], Miller and Ross [2], Podlubny [3], Baleanu et al. [4], and so forth. Research on the solutions of fractional differential equations is very extensive, such as numerical solutions, see El-Mesiry et al. [5] and Hashim et al. [6], mild solutions, see Chang et al. [7] and Chen et al. [8], the existence and uniqueness of solutions for initial and boundary value problem, see [930], and so on.

With the deep study, many papers that studied the fractional equations contained more than one fractional differential operator; see [1620].

Babakhani and Daftardar-Gejji in [16] considered the initial value problem of nonlinear fractional differential equation By using Banach fixed point theorem and fixed point theorem on a cone some results of existence and uniqueness of solutions are established.

Zhang in [17] studied the singular initial value problem for fractional differential equation by nonlinear alternative of Leray-Schauder theorem:

In above two equations, is defined , where , and , , is the standard Riemann-Liouville fractional derivative.

McRae in [14] studied the initial value problem by the method of upper and lower solutions and monotone iterative technique:

In this paper, we use similar method as in [16] to consider the initial value problem: where , , and , , is the standard Riemann-Liouville fractional derivative and is a given continuous function.

Since is assumed continuous, the IVP (1.4) is equivalent to the following Volterra fractional integral equation:

In Section 2, we give some definitions and lemmas that will be useful to our main results. In Section 3, we will use the basic theory of differential equation, the method of upper and lower solutions, and monotone iterative technique to investigate the initial value problem (1.4), and some existence and uniqueness results are established. In Section 4, an example is presented to illustrate the main results.

2. Preliminaries

In this section, we need the following definitions and lemmas that will be useful to our main results. These materials can be found in the recent literatures; see [1, 11, 16].

Definition 2.1 (see [1]). Let be a finite interval on the real axis . The Riemann-Liouville fractional integrals and of order are defined by respectively. Here is the Gamma function. These integrals are called the left-sided and the right-sided fractional integrals. We denote by in the following paper.

Definition 2.2 (see [1]). Let be a finite interval on the real axis . The Riemann-Liouville fractional derivatives and of order are defined by respectively, where , means the integral part of . These derivatives are called the left-sided and the right-sided fractional derivatives. We denote by in the following paper.

Definition 2.3. Letting be locally Hölder continuous with exponent , we say that is an upper solution of (1.4) if and is a lower solution of (1.4) if

Next, we will list the following lemma from [11] that is useful for our main results.

Lemma 2.4 (see [11, Lemma 2.1]). Let be locally Hölder continuous with exponent such that for any , we have Then it follows that .

Corollary 2.5. Let be locally Hölder continuous with exponent such that for any , we have Then it follows that provided .

Lemma 2.6. Let be a family of continuous functions on , for each , where , and for . Then the family is equicontinuous on .

Proof. Since is a family of continuous functions on , there exists such that for .
Let . For , , we get Thus, is equicontinuous on .

Lemma 2.7 (see [16, Theorem 4.2]). Let be continuous and Lipschitz with respect to second variable with Lipschitz constant . Let satisfy Then IVP (1.4) has a unique solution.

Lemma 2.8. Let be locally Hölder continuous with exponent , and one of the nonstrict inequalities being strict. Then implies , .

Proof. Suppose that , is not true. We suppose the inequality . Letting , there exists such that , , and . Then by Corollary 2.5, we can obtain . From the conditions and the definition of , we have This is a contradiction to . The proof is complete.

Lemma 2.9. Assume that the conditions of Lemma 2.8 hold with nonstrict inequalities (2.3) and (2.4). Furthermore, suppose that Then implies , provided .

Proof. Let . For small , we have Then, from (2.11) and (2.12) we get Applying Lemma 2.8, we obtain , . By the arbitrariness of , we can conclude that . The proof is complete.

Corollary 2.10. The function , where , is admissible in Lemma 2.9 to yield on .

3. Main Results

In this section, we establish the existence and uniqueness criteria of solutions for initial value problem (1.4).

Theorem 3.1. Assume that , where and . Then IVP (1.4) possesses at least one solution on , where .

Proof. Let be a continuous function on , , such that , and , where , are the continuous fractional derivatives. For , we define the function on and on , where . We observe that , exist for and If , we can employ (3.1) to extend as a continuously fractional differentiable function on , such that holds. Continuing this process, we can define over so that ; it has a continuous fractional derivative and satisfies (3.1) on the same interval . Furthermore, , since on . Therefore, from Lemma 2.6, the family is an equicontinuous and uniformly bounded function. An application of Ascoli-Arzela Theorem shows the existence of a sequence such that as , and exists uniformly on . Due to being uniformly continuous, we can obtain which uniformly tends to , and uniformly tends to as . Therefore, term by term, integration of (3.1) with , yields This proves that is a solution of IVP (1.4) and the proof is complete.

Theorem 3.2. Let be lower and upper solutions of the IVP (1.4) which are locally Hölder continuous with exponent such that , and , where . Furthermore, suppose that Then there exists a solution of IVP (1.4) satisfying on .

Proof. For the need of proof, we define function as Therefore, defines a continuous extension of to which is also bounded because is bounded on . Then by Theorems 3.1 and 3.2, we can obtain that the initial value problem has a solution on .
Clearly, from the definition of function , we know that if IVP (3.6) exits a solution satisfying on , then is also a solution of IVP (1.4). In the following, we will prove that the solution of IVP (3.6) satisfies on .
For any , we consider Then, we get Therefore, it follows that . Next, we will show that , . Suppose that it is not true. Then there exists such that Therefore, , and . Letting , we have and , . Then from Corollary 2.5, we can obtain and which is a contradiction. The other case can be proved similarly.
Hence, we get on . Letting , we obtain on . The proof is complete.

Now, we will give the existence of maximal and minimal solutions of initial value problem (1.4).

Theorem 3.3. Let , , be lower and upper solutions of (1.4) such that on . Furthermore, suppose that and satisfy Then there exist monotone sequences and such that , as uniformly on , where and are minimal and maximal solutions of IVP (1.4), respectively.

Proof. For any satisfying , we consider the following linear fractional differential equation: Obviously, the right hand side of (3.13) satisfies the Lipschitz condition. From (3.11) and Lemma 2.7, it is clear that for every , there exists a unique solution of (3.13) on . Define the operator by and use it to construct the sequences , . We need to prove the following propositions hold: (i), ; (ii) is a monotone operator on the segment
To prove (i), let , where is the unique solution of (3.13) with . Letting , we have By Corollary 2.10, we can obtain that on , that is, .
Similarly, we can get .
To prove (ii), let , such that . Assume that and . Setting , then using the condition (3.11), we have From Corollary 2.10, we can obtain that on , which implies . And (ii) is proved.
Therefore, we can define the sequences , . From the previous discussion, we can get
Clearly, the sequences , are uniformly bounded on . From (3.13), we have , which are also uniformly bounded. By Lemma 2.6, we know that , are equicontinuous on . Then applying Ascoli-Arzela Theorem, there exist subsequences , that converge uniformly on . From (3.17), we can see that the entire sequences , converge uniformly and monotonically to , , respectively, as . It is now easy to show that , are solutions of IVP (1.4) by the corresponding Volterra fractional integral equation for (3.13).
In the following, we will prove that and are the minimal and maximal solutions of IVP (1.4), respectively. We need to show that if is any solution of IVP (1.4) satisfying on , then we have on .
We assume that for some , on and letting , we have which implies . Similarly, we have on . Since on , this proves for all by induction. Letting , we conclude that on and the proof is complete.

Theorem 3.4. Suppose that the conditions of Theorem 3.3 hold. In addition, we assume Then is the unique solution of IVP (1.4) provided .

Proof. We have proved in Theorem 3.3, so we just need to prove . Letting , we get From Corollary 2.10, we obtain on , which implies . Hence, is the unique solution of IVP (1.4).

4. Examples

In this paper, we will present an example to illustrate the main results.

Example 4.1. Consider the initial value problem of fractional differential equation
Choose , ; then we can obtain That is, and are the lower and upper solutions of initial value problem (4.1). Furthermore, and are locally continuous with exponent .
Since then by Theorem 3.2, there exists a solution of initial value problem (4.1) satisfying .
Next, we will prove the existence of maximal and minimal solutions for initial value problem (4.1) by using Theorem 3.3.
Let and be lower and upper solutions of (4.1). Furthermore, for any , we have Then let . We get Thus, from Theorem 3.3, there exist monotone sequences and such that , as uniformly on , where and are minimal and maximal solutions of initial value problem (4.1), respectively.
In addition, Hence, by Theorem 3.4, initial value problem (4.1) has a unique solution.

5. Conclusion

In this paper, we considered the initial value problem of nonlinear fractional differential equation The basic theory of differential equation, the method of upper and lower solutions, and monotone iterative technique have been applied for the existence and uniqueness of solutions of the initial value problem. And several results were obtained. Besides, we studied the existence of minimal and maximal solutions. In Section 4, we also give an example to illustrate our results.

Acknowledgments

This research is supported by the Natural Science Foundation of China (11071143, 60904024, 61174217), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119), supported by Shandong Provincial Natural Science Foundation (ZR2010AL002, ZR2009AL003), and supported by Natural Science Foundation of Educational Department of Shandong Province (J11LA01).

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