The Existence and Uniqueness of a Class of Fractional Differential Equations

and Applied Analysis 3 Define the space C q [0, T] := {x | x ∈ C [0, T] , D q−1 0+ x ∈ C [0, T]} . (17) For x ∈ C[0, T], define an operator A : C[0, T] → C[0, T] by (Ax) (t) = x0 Γ (q) t q−1 + 1 Γ (q)


Introduction
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][2][3][4][5][6][7][8][9][10] and the references therein, the basic theory of the problem is still not perfect.
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 lim  → + () = ∞, so in space [, ], the Cauchy-type problem (1) cannot be reduced to the integral equation (2) except that  1 = 0. Delbosco and Rodino [6] (1996) considered the nonlinear fractional differential equation Using Schauder's fixed point theorem to the integral operator in (2) with  =  1 = 0, they proved that the equation considered has at least one continuous solution  ∈ [0, ] for a suitable 0 ≤  ≤ 1 provided that   (, ) is continuous on [0, 1] × R for some (0 ≤  <  < 1).Applying the contractive mapping method, they showed that if additionally then (3) has a unique solution () ∈ [0, 1].Clearly, the solution satisfies (0) = 0.They also proved that if (, ) = () is such that (0) = 0 and the Lipschitz condition holds, then the weighted Cauchy-type IVP has a unique solution () such that  1− () ∈ [0, ℎ] for any ℎ > 0.
In [12] (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 [13] (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  1− [0, ].
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 This problem is very important in many models of physics phenomena [7,9,10,[14][15][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.
If one of the above inequalities is strict, then we call it as a strict lower (upper) solution.
Remark 7. Clearly, if functions V,  are lower and upper solutions (or strict) of IVP (8), then there are V ≤ V,  ≥  (or the inequality is strict).

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) where  is a positive constant such that Then Clearly, the operator  defined by (18 Now we prove that operator  is a compressed map on 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  : where  ∈ ([0,] × R 2 , R),   0+ () is the standard Riemann-Liouville fractional derivative, 1 <  <  < 2.

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 V,  ∈   [0, ] are lower and upper solutions of problem (8) and at least one of them is strict.For every  ∈ [0, ], (, , ) is nondecreasing about , .Then V () <  () , 0 <  ≤ . (28) Furthermore, the fractional IVP (8) has a minimal solution  * and a maximal solution  * such that Proof.Without loss of generality, suppose that Suppose for contradiction that conclusion (27) is not true.Combining the fact that Taking into account that which is a contradiction to (33).Thus, conclusion (27) holds.Furthermore, combining V(0) = (0) = 0 and the monotonicity of integral  −1 0+ yields that (28) also holds.A standard proof can show that  :   [0, ] →   [0, ] is an increasing completely continuous operator.Setting  := [V, ], by the use of Lemma 4, the existence of  * ,  * is obtained.The proof is complete.
The following conclusion is about the nonstrict inequalities.