Research Article | Open Access
The Existence and Uniqueness of a Class of Fractional Differential Equations
By using inequalities, fixed point theorems, and lower and upper solution method, the existence and uniqueness of a class of fractional initial value problems, are discussed, where is the standard Riemann-Liouville fractional derivative, . Some mistakes in the literature are pointed out and some new inequalities and existence and uniqueness results are obtained.
Once the models of fractional differential equation for the actual problem have been established, people immediately faced the problem of how to solve these models. In many cases, it is very difficult to obtain the exact solution of the fractional differential equation. So it requires researchers to find as many characteristics of the solution of the problem as possible. For example, does the equation have a solution? If there is one solution, is the solution unique? How can we compare the size of the solution? We noted that although there were many works with respect to fractional differential equations, which were shown in [1–10] and the references therein, the basic theory of the problem is still not perfect.
Al-Bassam  (1965) first considered the following Cauchy-type initial value problem (IVP): in the space of continuous functions provided that is real-valued, continuous, and Lipschitzian in a domain such that . Applying the operator he reduced problem (1) to the Volterra nonlinear integral equation: By the use of the method of successive approximation he established the existence of the continuous solution of (2). He probably first indicated that the method of contracting mapping can be applied to prove the uniqueness of the solution of (2) and gave such a formal proof. However, from (2) one has , so in space , the Cauchy-type problem (1) cannot be reduced to the integral equation (2) except that .
Delbosco and Rodino  (1996) considered the nonlinear fractional differential equation Using Schauder’s fixed point theorem to the integral operator in (2) with , they proved that the equation considered has at least one continuous solution for a suitable provided that is continuous on for some . Applying the contractive mapping method, they showed that if additionally then (3) has a unique solution . Clearly, the solution satisfies . They also proved that if is such that and the Lipschitz condition holds, then the weighted Cauchy-type IVP has a unique solution such that for any .
In  (2008), Lakshmikantham and Vatsala considered the IVP for fractional differential equations given by The basic theory for the IVP of fractional differential equations was discussed by employing the classical approach. The theory of inequalities, local existence, extremal solutions, comparison result, and global existence of solutions was considered. The idea of this paper is very interesting.
In  (2009), Zhang considered the existence and uniqueness of the solution of the following IVP for fractional differential equation: using the method of upper and lower solutions and its associated monotone iterative technique. However, the paper did not explain why the pointwise convergence can be used instead of the convergence with norm in the space .
We refer the readers to monographs [8, 10] for other arguments about the fractional IVP. We noted that on one hand there are some confusions about the initial value of the solution in some of the above works. On the other hand there is no contribution about the basic theory for the following fractional differential equation IVP: where , is the standard Riemann-Liouville fractional derivative, . This problem is very important in many models of physics phenomena [7, 9, 10, 14–16], so it is worth studying the parallel theory to the known theory of ordinary differential equations.
The rest of the paper is organized as follows. In Section 2, some related basic lemmas and definitions are given. Section 3 contains the uniqueness result by means of contracting mapping. The existence of the minimal and maximal solutions is given in Section 4 using lower and upper solution method.
Lemma 2 (see ). The following relation holds in the case of(1);(2);(3).
Lemma 3 (see ). Supposing that , then(1)(2)
Lemma 4 (see ). Suppose that is an ordered Banach space, is an increasing completely continuous operator, and . Then the operator has a minimal fixed point and a maximal fixed point . If one lets then
By the use of the continuity of and Lemma 1, the IVP (8) is equivalent to the following Volterra integral equation : Define the space For , define an operator by Then the fixed point of solves IVP (8) and vice versa.
Definition 5. A function is called a lower solution of problem (8), if it satisfies
Definition 6. A function is called an upper solution of problem (8), if it satisfies
If one of the above inequalities is strict, then we call it as a strict lower (upper) solution.
Remark 7. Clearly, if functions are lower and upper solutions (or strict) of IVP (8), then there are (or the inequality is strict).
3. The Uniqueness of the Solution
Many methods can be applied to study the existence of solution. However, generally speaking, it is nothing more than two ways. One is based on the method of the approximate solution of exact solution to prove the existence of the solution, namely, classical successive approximation method. A. Cauchy, R. Lipschitz, G. Peano, and so forth used this method to solve the existence of some special types of differential equations. In 1893, C. Picard applied this method to study the general nonlinear differential equation and established the existence and uniqueness of solutions, named the Cauchy-Picard Theorem. This method itself also contains a structural method to obtain the exact solution and thus provides a way for the approximate solution. Another method is transforming the solution into the fixed point of some maps. Although the method cannot give the approximate solution, it is the abstraction and generalization of the former method and is simple to use. In this section, we will establish the uniqueness of the solution for fractional IVP (8) by the use of the second method.
Theorem 8. Assume that is continuous and Lipschitzian with respect to the second and the third variables; that is to say, there exist constants such that for all Then the fractional IVP (8) has a unique solution .
Proof. For , the norm is defined as
where is a positive constant such that
Then is a Banach space.
Clearly, the operator defined by (18) maps to .
Now we prove that operator is a compressed map on . Let ; then, for ,
Taking into account that the function is Lipschitzian, by the use of the Cauchy-Schwartz inequality, we have According to the definition of , we know that is a compressed map. Banach fixed point theorem shows that there exists a unique such that ; equivalently, IVP (8) has a unique solution .
Remark 9. Similar to paper , we can permit function to have some singularity on .
Remark 10. The study about the following problem is meaningful: where is the standard Riemann-Liouville fractional derivative, .
4. Some Inequalities and the Existence of the Solution
Firstly, let us discuss the result about the strict inequalities for fractional IVP.
Theorem 11. Assume that the functions are lower and upper solutions of problem (8) and at least one of them is strict. For every is nondecreasing about . Then Furthermore, the fractional IVP (8) has a minimal solution and a maximal solution such that
Proof. Without loss of generality, suppose that . Let . By the use of Lemma 3 and the definition of lower solution , one has
Integration from to yields
Similarly, let ; we get
Suppose for contradiction that conclusion (27) is not true. Combining the fact that , is continuous on and , there exists such that
Taking into account that for and , by the use of the monotonicity of integral operator , one has With inequalities (33) and (34), is nondecreasing and above arguments give which is a contradiction to (33). Thus, conclusion (27) holds. Furthermore, combining and the monotonicity of integral yields that (28) also holds.
A standard proof can show that is an increasing completely continuous operator. Setting , by the use of Lemma 4, the existence of is obtained. The proof is complete.
The following conclusion is about the nonstrict inequalities.
Theorem 12. Assume that the functions are lower and upper solutions of problem (8). If there exist two real numbers such that, for , , there holds then implies Furthermore, the fractional IVP (8) has a minimal solution and a maximal solution such that
Proof. Given , let . Then, for ,
From (39) and condition (36), combining the fact that with the condition , one has This inequality combined with shows that is a strict upper solution of problem (8).
For , by the use of Theorem 11, we get . As is arbitrary, (37) holds.
The rest of the proof is just similar to Theorem 11.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by NNSF (61174078, 61201431), SDNSF (ZR2010AM035), a project of Shandong Higher Educational Science and Technology program (J11LA07), the Taishan Scholar project, Research Award Fund for Outstanding Young Scientists of Shandong (BS2012SF022), and SDUST Research Fund (2011KYTD105). The authors thank the referee for his/her valuable comments and constructive suggestions.
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