Existence and Uniqueness of Solutions for the System of Nonlinear Fractional Differential Equations with Nonlocal and Integral Boundary Conditions

and Applied Analysis 3 for a ≤ t ≤ b, is called the Riemann-Liouville fractional integral operator of order α. Here Γ · is the Gamma function defined for any complex number z as


Introduction
Differential equations of fractional order have been proved to be valuable tools in the modeling of many phenomena of various fields of science and engineering.Indeed, we can obtain numerous applications in viscoelasticity 1-3 , dynamical processes in self-similar structures 4 , biosciences 5 , signal processing 6 , system control theory 7 , electrochemistry 8 , diffusion processes 9 , and linear time-invariant systems of any order with internal point delays 10 .Furthermore, fractional calculus has been found many applications in classical mechanics 11 , and the calculus of variations 12 , and is a very useful means for obtaining solutions of nonhomogenous linear ordinary and partial differential equations.For more details, we refer the reader to 13 .There are several approaches to fractional derivatives such as Riemann-Liouville, Caputo, Weyl, Hadamar and Grunwald-Letnikov, and so forth.Applied problems require those definitions of a fractional derivative that allow the utilization of physically interpretable with the nonlocal and integral boundary condition where E ∈ R n×n is an identity matrix, B ∈ R n×n is the given matrix, and Here, f t, x t and g t, x t ∈ R n are smooth vector functions, c D α 0 is the Caputo fractional derivative of order α, 0 < α ≤ 1.
The organization of this paper is as follows.In Section 2, we provide necessary background.In Section 3, theorems on existence and uniqueness of a solution are established under some sufficient conditions on nonlinear terms.Finally, in Section 4, a simple example of application of the main result of this paper is presented.

Preliminaries
In this section, we present some basic definitions and preliminary facts which are used throughout the paper.By C 0, T , R n , we denote the Banach space of all vector continuous functions x t from 0, T into R n with the norm a, b and α > 0, then the operator I α a , defined by for a ≤ t ≤ b, is called the Riemann-Liouville fractional integral operator of order α.Here Γ • is the Gamma function defined for any complex number z as Definition 2.2.The Caputo fractional derivative of order α > 0 of a continuous function g : where n α 1, the notation α stands for the largest integer not greater than α .
Remark 2.3.Under natural conditions on g t , the Caputo fractional derivative becomes the conventional integer order derivative of the function g t as α → n.

Main Results
Lemma 3.1.Let 0 < α ≤ 1, y t and g t ∈ C 0, T , R n .Then, nonlocal boundary value problem Proof.Assume that x t is a solution of nonlocal boundary value problem 3.1 and 3.2 , then using Lemma 2.6, we get Applying condition 3.2 and identity 3.6 , we get From condition 1.3 it follows that the inverse of the matrix E B exists.Therefore, we can write Using formulas 3.6 and 3.8 , we obtain which can be written as 3.3 .Lemma 3.1 is proved.

3.12
The application of the fractional differential operator c D α 0 to both sides of 3.12 yields

3.13
Hence, x t solves fractional differential equation 1.1 .Also, it is easy to see that x t satisfies nonlocal boundary condition 1.2 .
The first main statement of this paper is an existence and uniqueness of boundary value problem 1.1 and 1.2 result that it is based on a Banach fixed point theorem.

Theorem 3.3. Assume that:
H1 There exists a constant L > 0 such that for each t ∈ 0, T and all x, y ∈ R n .
H2 There exists a constant M > 0 such that for each t ∈ 0, T and all x, y ∈ R n . If then boundary value problem 1.1 and 1.2 has unique solution on 0, T .
Proof .Transform problem 1.1 and 1.2 into a fixed point problem.Consider the operator

3.18
Clearly, the fixed points of the operator P are solution of problem 1.1 and 1.2 .We will use the Banach contraction principle to prove that under assumption 3.16 operator P has a fixed point.It is clear that the operator P maps into itself and that

3.19
for any x, y ∈ C 0, T , R n and t ∈ 0, T .From condition 1.3 it follows that Then, using estimates 3.19 and 3.20 , we get

3.21
Consequently, by assumption 3.16 operator P is a contraction.As a consequence of Banach's fixed point theorem, we deduce that operator P has a fixed point which is a solution of problem 1.1 and 1.2 .Theorem 3.3 is proved.
The second main statement of this paper is an existence of boundary value problem 1.1 and 1.2 result that it is based on Schaefer's fixed point theorem.Theorem 3.4.Assume that: H4 There exists a constant N 1 > 0 such that |f t, x | ≤ N 1 for each t ∈ 0, T and all x ∈ R n .

H5 The function g
H6 There exists a constant N 2 > 0 such that |g t, x | ≤ N 2 for each t ∈ 0, T and all x ∈ R n .
Then, boundary value problem 1.1 and 1.2 has at least one solution on 0, T .
Proof.We will divide the proof into four main steps in which we will show that under the assumptions of theorem operator P has a fixed point.
Step 1. Operator P under the assumptions of theorem is continuous.Let {x n } be a sequence such that x n → x in C 0, T , R n .Then, for each t ∈ 0, T Since f and g are continuous functions, we have as n → ∞.
Step 2. Operator P maps bounded sets in bounded sets in C 0, T , R n .Indeed, it is enough to show that for any η > 0, there exists a positive constant l such that for each x ∈ B η {x ∈ C 0, T , R n : x ≤ η}, we have P x • ≤ l.By assumptions H4 and H6 , we have for each t ∈ 0, T , Hence, Thus,

3.26
Step 3. Operator P maps bounded sets into equicontinuous sets of 0, T , R n .Let t 1 , t 2 ∈ 0, T , t 1 < t 2 , B η be a bounded set of C 0, T , R n as in Step 2, and let x ∈ B η .Then,

3.27
As t 1 → t 2 , the right-hand side of the above inequality tends to zero.As a consequence of Steps 1 to 3 together with the Arzela-Ascoli theorem, we can conclude that the operator Step 4. A priori bounds.Now, it remains to show that the set Δ {x ∈ C 0, T , R n : x λP x for some 0 < λ < 1} 3.28 is bounded.Let x λ Px for some 0 < λ < 1.Then, for each t ∈ 0, T we have

3.29
This implies by assumptions H4 and H6 as in Step 2 that for each t ∈ 0, T we have Thus, for every t ∈ 0, T , we have Therefore, x ≤ R.

3.32
This shows that the set Δ is bounded.As a consequence of Schaefer's fixed point theorem, we deduce that P has a fixed point which is a solution of problem 1.1 and 1.2 .Theorem 3.4 is proved.Moreover, we will give an existence result for problem 1.1 and 1.2 by means of an application of a Leray-Schauder type nonlinear alternative, where conditions H4 and H6 are weakened.Theorem 3.5.Assume that (H3), (H5) and the following conditions hold.
H7 There exist θ f ∈ L 1 0, T , R and continuous and nondecreasing H8 There exist θ g ∈ L 1 0, T , R and continuous nondecreasing such that |g t, x | ≤ θ g t ψ g |x| for each t ∈ 0, T and all x ∈ R n .
H9 There exists a number K > 0 such that

3.35
Then, boundary value problem 1.1 and 1.2 has at least one solution on 0, T .
Proof.Consider the operator P defined in Theorems 3.3 and 3.4.It can be easily shown that operator P is continuous and completely continuous.Let x t Px t for each t ∈ 0, T .Then, from assumptions H7 and H8 if follows that for each t ∈ 0, T

3.36
Abstract and Applied Analysis 11 Thus, Then, by condition H9 , there exists K such that x / K. Let U {x ∈ C 0, T , R : x < K}.

3.38
The operator P : U → C 0, T , R is continuous and completely continuous.By the choice of U, there exists x ∈ ∂U such that x λP x for some λ ∈ 0, 1 .As a consequence of the nonlinear alternative of Leray-Schauder type 43 , we deduce that P has a fixed point x in U, which is a solution of problem 1.1 and 1.2 .This completes of proof of Theorem 3.5.

An Example
In this section, we give an example to illustrate the usefulness of our main results.Let us consider the following nonlocal boundary value problem for the system of fractional differential equation

4.1
Evidently, Hence, conditions H1 and H2 hold with L M 0, 1.We will check that condition 3.

Conclusion
In this work, some existence and uniqueness of a solution results have been established for the system of nonlinear fractional differential equations under the some sufficient conditions on nonlinear terms.Of course, such type existence and uniqueness results hold under the same sufficient conditions on nonlinear terms for the system of nonlinear fractional differential equations 1.1 , subject to multipoint nonlocal and integral boundary conditions Ex 0 where B j ∈ R n×n are given matrices and J j 1 B j < 1.Here, 0 < t 1 < • • • < t J ≤ T .Moreover, applying the result of the paper 44 the first order of accuracy difference scheme for the numerical solution of nonlocal boundary value problem 1.1 and 5.1 can be presented.Of course, such type existence and uniqueness results hold under the some sufficient conditions on nonlinear terms for the solution system of this difference scheme.
a solution of the boundary value problem 1.1 and 1.2 if and only if it is a solution of the integral equation Proof.If x t solves boundary value problem 1.1 and 1.2 .Then, by the same manner as in Lemma 3.1, we can prove that x t is solution of integral equation 3.10 .Conversely, let x t is solution of integral equation 3.10 .We denote thatv t T 0 G t, s f s, x s ds E B −1 T 0 g s, x s ds.3.11Then, by Lemmas 2.7 and 2.8, we obtain 16is satisfied for appropriate values of 0 < α ≤ 1 with T 1. Indeed, Then, by Theorem 3.3 boundary value problem 4.1 has a unique solution on 0, 1 for values of α satisfying condition 4.3 .For example, if α 0.2 then