Some Existence Results for Impulsive Nonlinear Fractional Differential Equations with Closed Boundary Conditions

and Applied Analysis 3 Definition 2.1. The fractional arbitrary order integral of the function h ∈ L1 J, R of order α ∈ R is defined by I 0h t 1 Γ α ∫ t 0 t − s α−1h s ds, 2.1 where Γ · is the Euler gamma function. Definition 2.2. For a function h given on the interval J , Caputo fractional derivative of order α > 0 is defined by D α 0 h t 1 Γ n − α ∫ t 0 t − s n−α−1h n s ds, n α 1, 2.2 where the function h t has absolutely continuous derivatives up to order n − 1 . Lemma 2.3. Let α > 0, then the differential equation Dh t 0 2.3 has solutions h t c0 c1t c2t · · · cn−1tn−1, ci ∈ R, i 0, 1, 2, . . . , n − 1, n α 1. 2.4 The following lemma was given in 4, 10 . Lemma 2.4. Let α > 0, then ID α h t h t c0 c1t c2t · · · cn−1tn−1, 2.5 for some ci ∈ R, i 0, 1, 2, . . . , n − 1, n α 1. The following theorem is known as Burton-Kirk fixed point theorem and proved in 21 . Theorem 2.5. Let X be a Banach space and A, D : X → X two operators satisfying: a A is a contraction, and b D is completely continuous. Then either i the operator equation x A x D x has a solution, or ii the set ε {x ∈ X : x λA x/λ λD x } is unbounded for λ ∈ 0, 1 . Theorem 2.6 see 22 , Banach’s fixed point theorem . Let S be a nonempty closed subset of a Banach space X, then any contraction mapping T of S into itself has a unique fixed point. 4 Abstract and Applied Analysis Next we prove the following lemma. Lemma 2.7. Let 1 < α ≤ 2 and let h : J → R be continuous. A function x t is a solution of the fractional integral equation: x t ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪⎩ ∫ t 0 t − s α−1 Γ α h s ds Ω1 t Λ1 Ω2 t Λ2, t ∈ J0 ∫ t tk t − s α−1 Γ α h s ds 1 Ω1 t k ∑ i 1 ∫ ti ti−1 ti − s α−1 Γ α h s ds T − tk Ω1 t Ω2 t t − tk k ∑ i 1 ∫ ti ti−1 ti − s α−2 Γ α − 1 h s ds


Introduction
This paper considers the existence and uniqueness of the solutions to the closed boundary value problem BVP , for the following impulsive fractional differential equation: x T ax 0 bT x 0 , Tx T cx 0 dT x 0 , where and Δx t k has a similar meaning for x t , where a, b, c, and d are real constants with Δ : c 1 − b 1 − a 1 − d / 0. The boundary value problems for nonlinear fractional differential equations have been addressed by several researchers during last decades.That is why, the fractional derivatives serve an excellent tool for the description of hereditary properties of various materials and processes.Actually, fractional differential equations arise in many engineering and scientific disciplines such as, physics, chemistry, biology, electrochemistry, electromagnetic, control theory, economics, signal and image processing, aerodynamics, and porous media see 1-7 .For some recent development, see, for example, 8-14 .
On the other hand, theory of impulsive differential equations for integer order has become important and found its extensive applications in mathematical modeling of phenomena and practical situations in both physical and social sciences in recent years.One can see a noticeable development in impulsive theory.For instance, for the general theory and applications of impulsive differential equations we refer the readers to 15-17 .Moreover, boundary value problems for impulsive fractional differential equations have been studied by some authors see 18-20 and references therein .However, to the best of our knowledge, there is no study considering closed boundary value problems for impulsive fractional differential equations.
Here, we notice that the closed boundary conditions in Motivated by the mentioned recent work above, in this study, we investigate the existence and uniqueness of solutions to the closed boundary value problem for impulsive fractional differential equation 1.1 .Throughout this paper, in Section 2, we present some notations and preliminary results about fractional calculus and differential equations to be used in the following sections.In Section 3, we discuss some existence and uniqueness results for solutions of BVP 1.1 , that is, the first one is based on Banach's fixed point theorem, the second one is based on the Burton-Kirk fixed point theorem.At the end, we give an illustrative example for our results.

Preliminaries
Let us set J 0 0, t 1 , The following definitions and lemmas were given in 4 .
Definition 2.1.The fractional arbitrary order integral of the function h ∈ L 1 J, R of order α ∈ R is defined by where Γ • is the Euler gamma function.
Definition 2.2.For a function h given on the interval J, Caputo fractional derivative of order α > 0 is defined by where the function h t has absolutely continuous derivatives up to order n − 1 .
Lemma 2.4.Let α > 0, then The following theorem is known as Burton-Kirk fixed point theorem and proved in 21 .
Theorem 2.5.Let X be a Banach space and A, D : X → X two operators satisfying: a A is a contraction, and b D is completely continuous.

Then either i the operator equation x A x D x has a solution, or
ii the set ε {x ∈ X : x λA x/λ λD x } is unbounded for λ ∈ 0, 1 .Theorem 2.6 see 22 , Banach's fixed point theorem .Let S be a nonempty closed subset of a Banach space X, then any contraction mapping T of S into itself has a unique fixed point.
Next we prove the following lemma.Lemma 2.7.Let 1 < α ≤ 2 and let h : J → R be continuous.A function x t is a solution of the fractional integral equation:

if and only if x t is a solution of the fractional BVP
where Abstract and Applied Analysis 5

2.8
Proof.Let x be the solution of 2.7 .If t ∈ J 0 , then Lemma 2.4 implies that

2.10
for some d 0 , d 1 ∈ R. Thus we have

2.11
Observing that 12 then we have

2.14
If t ∈ J 2 , then Lemma 2.4 implies that

2.15
for some e 0 , e 1 ∈ R. Thus we have

2.16
Similarly we observe that

2.17
Abstract and Applied Analysis 7 thus we have

2.19
By a similar process, if t ∈ J k , then again from Lemma 2.4 we get

2.20
Now if we apply the conditions: x T ax 0 bT x 0 , Tx T cx 0 dT x 0 , 2.21 we have

2.22
In view of the relations 2.8 , when the values of −c 0 and −c 1 are replaced in 2.9 and 2.20 , the integral equation 2.7 is obtained.Conversely, assume that x satisfies the impulsive fractional integral equation 2.6 , then by direct computation, it can be seen that the solution given by 2.6 satisfies 2.7 .The proof is complete.

Main Results
Definition 3.1.A function x ∈ PC 1 J, R with its α-derivative existing on J is said to be a solution of 1.1 , if x satisfies the equation C D α x t f t, x t on J and satisfies the conditions: x T ax 0 bT x 0 , Tx T cx 0 dT x 0 .

3.1
For the sake of convenience, we define The followings are main results of this paper.

Theorem 3.2. Assume that
A1 the function f : J × R → R is continuous and there exists a constant L 1 > 0 such that f t, u − f t, v ≤ L 1 u − v , for all t ∈ J, and u, v ∈ R, A2 I k , I * k : R → R are continuous, and there exist constants L 2 > 0 and Moreover, consider the following:

3.3
Then, BVP 1.1 has a unique solution on J.
Proof.Define an operator F :

3.4
Now, for x, y ∈ PC J, R and for each t ∈ J, we obtain Abstract and Applied Analysis T pL 3 x s − y s .

3.5
Therefore, by 3.3 , the operator F is a contraction mapping.In a consequence of Banach's fixed theorem, the BVP 1.1 has a unique solution.Now, our second result relies on the Burton-Kirk fixed point theorem.
Proof.We define the operators A, D :

3.6
It is obvious that A is contraction mapping for Now, in order to check that D is completely continuous, let us follow the sequence of the following steps.
Step 1 D is continuous .Let {x n } be a sequence such that x n → x in PC J, R .Then for t ∈ J, we have

3.8
Since f is continuous function, we get Dx n t − Dx t −→ 0 as n −→ ∞.

3.9
Step 2 D maps bounded sets into bounded sets in PC J, R .Indeed, it is enough to show that for any r > 0, there exists a positive constant l such that for each x ∈ B r {x ∈ PC J, R : x ≤ r}, we have D x ≤ l.By A3 , we have for each t ∈ J,

3.10
Step 3 D maps bounded sets into equicontinuous sets in PC 1 J, R .Let τ 1 , τ 2 ∈ J k , 0 ≤ k ≤ p with τ 1 < τ 2 and let B r be a bounded set of PC 1 J, R as in Step 2, and let x ∈ B r .Then where This implies that A is equicontinuous on all the subintervals J k , k 0, 1, 2, . . ., p. Therefore, by the Arzela-Ascoli Theorem, the operator D : PC 1 J, R → PC 1 J, R is completely continuous.
To conclude the existence of a fixed point of the operator A D, it remains to show that the set ε x ∈ X : x λA x λ λD x for some λ ∈ 0, 1 T pM 3 .

3.15
Consequently, we conclude the result of our theorem based on the Burton-Kirk fixed point theorem.

An Example
Consider the following impulsive fractional boundary value problem:

4.2
Since the assumptions of Theorem 3.2 are satisfied, the closed boundary value problem 4.1 has a unique solution on 0, 1 .Moreover, it is easy to check the conclusion of Theorem 3.3.

1 . 1
include quasi-periodic boundary conditions b c 0 and interpolate between periodic a d 1, b c 0 and antiperiodic a d −1, b c 0 boundary conditions.