Existence and Uniqueness of Solution for a Class of Nonlinear Fractional Order Differential Equations

and Applied Analysis 3 Lemma 2.1 see 33, 34 . For α > 0, the general solution of the fractional differential equation Dx t 0 is given by


Introduction
Fractional calculus deals with generalization of differentiation and integration to the fractional order 1, 2 .In the last few decades the fractional calculus and fractional differential equations have found applications in various disciplines 2-6 .Owing to the increasing applications, a considerable attention has been given to exact and numerical solutions of fractional differential equations 2, 6-11 .Many papers were dedicated to the existence and the uniqueness of the fractional differential equations, to the analytic methods for solving fractional differential equations, e.g., Greens function method, the Mellin transform method, and the power series see for example references 2, 6-26 and the references therein .On this line of taught in this manuscript we proved the existence and uniqueness of a specific nonlinear fractional order ordinary differential equations within Caputo derivatives.Very recently in 27-31 , the authors and other researchers studied the existence and uniqueness of solutions of some classes of fractional differential equations with delay.The paper is

Fractional Integral and Derivatives
In this section, we present some notations, definitions, and preliminary facts that will be used further in this work.The Caputo fractional derivative allows the utilization of initial and boundary conditions involving integer order derivatives, which have clear physical interpretations.Therefore, in this work we will use the Caputo fractional derivative D proposed by Caputo in his work on the theory of viscoelasticity 32 .
Let α ∈ R, n − 1 < α ≤ n ∈ N and x ∈ C 0, ∞ , R ; then the Caputo fractional derivative of order α defined by where is the Riemann-Liouville fractional integral operator of order α and Γ is the gamma function.
The fractional integral of x t t − a β , a ≥ 0, β > −1 is given as For α, β ≥ 0, we have the following properties of fractional integrals and derivative 33 .
The fractional order integral satisfies the semigroup property The integer order derivative operator D m commutes with fractional order D α , that is: The fractional operator and fractional derivative operator do not commute in general.Then the following result can be found in 33, 34 .
Lemma 2.1 see 33, 34 .For α > 0, the general solution of the fractional differential equation D α x t 0 is given by where α denotes the integer part of the real number α.
In view of Lemma 2.1 it follows that But in the opposite way we have The proof of the above proposition can be found in 9, page 53 .As a pursuit of this in this paper, we discuss the existence and uniqueness of solution for nonlinear fractional order differential equations satisfying the boundary conditions or satisfying the initial conditions where 1 < α ≤ 2 and 0 < β γ ≤ α.
In the following, we present the existence and the uniqueness results for fractional differential equation 2.9 with boundary conditions 2.10 .

Existence and Uniqueness of Solutions
Lemma 3.1.Assume that f : 0, 1 × R 2 → R is continuous.Then x ∈ C 0, 1 is a solution of the boundary value problem 2.9 and 2.10 if and only if x t is the solution of the integral equation for some constants c 0 , c 1 where G t, s given by

3.3
Proof.Assume that x ∈ C 0, 1 is a solution of the fractional differential equation 2.9 satisfying boundary conditions 2.10 .Then in view of Lemma 2.1 and Proposition 2.2, we have for some constants c 0 and c 1 .Hence using the boundary conditions 2.10 we obtain c 0 −x 0 and

3.5
Substituting c 0 −x 0 and c 1 into 3.4 we get

3.9
We consider the space The space B is a Banach space 35 .
Proof.Define an operator F : B → B by

3.15
In order to show that the boundary value problem 2.9 , 2.10 has a solution, it is sufficient to prove that the operator F has a fixed point.For s ≤ t, from 3.2 , we have

3.17
On the other hand, for s > t, we arrive at same conclusion.Therefore, Define the set Ω {x ∈ B : x ≤ 8R}.For x ∈ Ω, using 3.15 and 3.18 , we obtain From the Caputo derivative and with using 3.12 -3.14 , we have

3.25
Hence, it follows that Fx t 2 − Fx t 1 → 0, as t 2 → t 1 .By the Arzela-Ascoli theorem, F : Ω → Ω is completely continuous.Thus by using the Schauder fixed-point theorem, it was proved that the boundary value problem 2.9 , 2.10 has a solution.Theorem 3.3.Let f : 0, 1 × R 2 → R be continuous.If there exists a constant μ such that |f t, x, y − f t, x, y | ≤ μ |x − x| |y − y| for each t ∈ 0, 1 and all x, x, y, y ∈ R and 4M 3μ ≤ 1, where Then the boundary value problem 2.9 with boundary conditions 2.10 has a unique solution.
Proof.Under condition on f, we have

3.27
Using 3.22 we conclude Abstract and Applied Analysis 11

3.28
Thus, we have Therefore, by the contraction mapping theorem, the boundary value problem 2.9 , 2.10 has a unique solution.
Then the initial value problem 2.9 , 2.10 has a solution.
Proof.In view of Lemma 2.1 and Proposition 2.2, we have x t ρtI α−β x t − ραI α−β 1 x t I α f t, x t , D γ x t − c 0 − c 1 t.

3.31
By initial conditions we have c 0 −x 0 and c 1 −1.Define an operator T : Ω → Ω by Tx t x 0 t ρtI α−β x t − ραI α−β 1 x t I α f t, x t , D γ x t .

3.32
Can be easily to prove that T : Ω → Ω is completely continuous as operator F.
then the initial value value problem 2.9 , 2.11 has a unique solution.
The proof of the Theorem 3.5 is similar to the proof of Theorem 3.3.Note that Then using Proposition 2.2 we have, Example 3.6.Consider the following boundary value problem for nonlinear fractional order differential equation:

3.36
Then, 3.36 with assumed boundary conditions has a solution in Ω.
In Example 3.6 f t, x t , D γ x t

Conclusion
We considered a class of nonlinear fractional order differential equations involving Caputo fractional derivative with lower terminal at 0 in order to study the existence solution satisfying the boundary conditions or satisfying the initial conditions.The unique solution under Lipschitz condition is also derived.In order to illustrate our results several examples are presented.The presented research work can be generalized to multiterm nonlinear fractional order differential equations with polynomial coefficients.