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# Nonlocal Boundary Value Problem for Impulsive Differential Equations of Fractional Order

*Advances in Difference Equations*
**volumeÂ 2011**, ArticleÂ number:Â 404917 (2011)

## Abstract

We study a nonlocal boundary value problem of impulsive fractional differential equations. By means of a fixed point theorem due to O'Regan, we establish sufficient conditions for the existence of at least one solution of the problem. For the illustration of the main result, an example is given.

## 1. Introduction

Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in various fields, such as physics, mechanics, aerodynamics, chemistry, and engineering and biological sciences, involves derivatives of fractional order. Fractional differential equations also provide an excellent tool for the description of memory and hereditary properties of many materials and processes. In consequence, fractional differential equations have emerged as a significant development in recent years, see [1â€“3].

As one of the important topics in the research differential equations, the boundary value problem has attained a great deal of attention from many researchers, see [4â€“11] and the references therein. As pointed out in [12], the nonlocal boundary condition can be more useful than the standard condition to describe some physical phenomena. There are three noteworthy papers dealing with the nonlocal boundary value problem of fractional differential equations. Benchohra et al. [12] investigated the following nonlocal boundary value problem

where denotes the Caputo's fractional derivative.

Zhong and Lin [13] studied the following nonlocal and multiple-point boundary value problem

Ahmad and Sivasundaram [14] studied a class of four-point nonlocal boundary value problem of nonlinear integrodifferential equations of fractional order by applying some fixed point theorems.

On the other hand, impulsive differential equations of fractional order play an important role in theory and applications, see the references [15â€“21] and references therein. However, as pointed out in [15, 16], the theory of boundary value problems for nonlinear impulsive fractional differential equations is still in the initial stages. Ahmad and Sivasundaram [15, 16] studied the following impulsive hybrid boundary value problems for fractional differential equations, respectively,

Motivated by the facts mentioned above, in this paper, we consider the following problem:

where , is a continuous function, and are continuous functions, with ,,,,,, and are two continuous functions. We will define in Section 2.

To the best of our knowledge, this is the first time in the literatures that a nonlocal boundary value problem of impulsive differential equations of fractional order is considered. In addition, the nonlinear term involves . Evidently, problem (1.5) not only includes boundary value problems mentioned above but also extends them to a much wider case. Our main tools are the fixed point theorem of O'Regan. Some recent results in the literatures are generalized and significantly improved (see Remark 3.6)

The organization of this paper is as follows. In Section 2, we will give some lemmas which are essential to prove our main results. In Section 3, main results are given, and an example is presented to illustrate our main results.

## 2. Preliminaries

At first, we present here the necessary definitions for fractional calculus theory. These definitions and properties can be found in recent literature.

The Riemann-Liouville fractional integral of order of a function is given by

where the right side is pointwise defined on .

The Caputo fractional derivative of order of a function is given by

where , denotes the integer part of the number , and the right side is pointwise defined on .

Let , then the fractional differential equation has solutions

where ,,.

Let , then one has

where ,,.

Second, we define

, and exist with .

. Let ; it is a Banach space with the norm , where .

Like the definition 2.1 in [16], we give the following definition.

Definition 2.5.

A function with its Caputo derivative of order existing on is a solution of (1.5) if it satisfies (1.5).

To deal with problem (1.5), we first consider the associated linear problem and give its solution.

Lemma 2.6.

Assume that

For any , the solution of the problem

is given by

Proof.

By Lemmas 2.3 and 2.4, the solution of (2.6) can be written as

where . Taking into account that , for , we obtain

Using , we get

If , then we have

where . In view of the impulse conditions , , we have

Repeating the process in this way, the solution for can be written as

Applying the boundary condition , we find that

Substituting the value of into (2.10) and (2.13), we obtain (2.7).

Now, we introduce the fixed point theorem which was established by O'Regan in [22]. This theorem will be applied to prove our main results in the next section.

Denote by an open set in a closed, convex set of a Banach space . Assume that . Also assume that is bounded and that is given by , in which is continuous and completely continuous and is a nonlinear contraction (i.e., there exists a nonnegative nondecreasing function satisfying for , such that for all ), then either

has a fixed point , or

there exists a point and with , where represent the closure and boundary of , respectively.

## 3. Main Results

In order to apply Lemma 2.7 to prove our main results, we first give ,, as follows. Let ,

Clearly, ,

Now, we make the following hypotheses.

is continuous. There exists a nonnegative function with on a subinterval of . Also there exists a nondecreasing function such that for any .

There exist two positive constants such that . Moreover, , and

are continuous. There exists a positive constant such that

Let

where .

Now, we state our main results.

Theorem 3.1.

Assume that ,, and are satisfied; moreover, , where , then the problem (1.5) has at least one solution.

Proof.

The proof will be given in several steps.

Step 1.

The operator is completely continuous.

Let . In fact, by , can be replaced by . For any , we have

These imply that , where is a positive constant, that is, is uniformly bounded. In addition, for any , for all , , we can obtain

Taking into account the uniform continuity of the function on , we get that is equicontinuity on . By the Lemma 5.4.1 in [23], we have as relatively compact. Due to the continuity of , , , it is clear that is continuous. Hence, we complete the proof of Step 1.

Step 2.

is bounded.

From , it follows that there exists a positive constant , such that

Now, we verify the validity of all the conditions in Lemma 2.6 with respect to the operator ,, and . Let . From , we have

Combining with the property that is bounded (Step 1), we have bounded on . Hence, we can assume that , is a constant.

Step 3.

is a nonlinear contraction.

Let ,,. By , we obtain , and ,. Since , we have , that is, is a nonlinear contraction .

Step 4.

in Lemma 2.7 does not occur.

To this end, we perform the argument by contradiction. Suppose that holds, then there exist ,, such that . Hence, we can obtain and

Therefore, . However, it contradicts with (3.8).

Hence, by using Steps 1â€“4, Lemmas 2.6 and 2.7, has at least one fixed point , which is the solution of problem (1.5).

Next, we will give some corollaries.

Corollary 3.2.

Assume that , , and are satisfied; moreover, , where ; then the problem (1.5) has at least one solution.

Assume that,

()(sublinear growth), is continuous. There exists a nonnegative function with on a subinterval of . Also there exists a constant , such that for any .

Corollary 3.3.

Assume that , , and are satisfied, then the problem (1.5) has at least one solution.

Assume that

is continuous. There exists a nonnegative function with on a subinterval of . Also there exists a constant such that for any ,

there exist two positive constants such that . Moreover, , and

are continuous. There exists a positive constant , such that

Corollary 3.4.

Assume that , , and are satisfied, then the problem (1.4) has at least one solution.

Assume that

is continuous. There exists a nonnegative function with on a subinterval of . for any .

Corollary 3.5.

Assume that ,, and are satisfied, then the problem (1.4) has at least one solution.

Remark 3.6.

Compared with Theorem 3.2 in [16], Corollary 3.5 does not need conditions , and . Moreover, we only need .

Example 3.7.

Consider the following problem:

where . Here, ,,. Let and , then we can see that holds. Choosing ,, we can easily obtain that holds. Let , then we have that also holds. Moreover, ,. Hence, we get for any given . Therefore, By Theorem 3.1, the above problem (3.13) has at least one solution for .

## References

Kilbas AA, Srivastava HM, Trujillo JJ:

*Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies*.*Volume 204*. Elsevier, Amsterdam, The Netherlands; 2006:xvi+523.Ross B (Ed):

**The fractional calculus and its applications**In*Lecture Notes in Mathematics*.*Volume 475*. Springer, Berlin, Germany; 1975.Podlubny I:

*Fractional Differential Equations, Mathematics in Science and Engineering*.*Volume 198*. Academic Press, New York, NY, USA; 1999:xxiv+340.Bai Z, LÃ¼ H:

**Positive solutions for boundary value problem of nonlinear fractional differential equation.***Journal of Mathematical Analysis and Applications*2005,**311**(2):495-505. 10.1016/j.jmaa.2005.02.052Jiang D, Yuan C:

**The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application.***Nonlinear Analysis. Theory, Methods & Applications*2010,**72**(2):710-719. 10.1016/j.na.2009.07.012Benchohra M, Graef JR, Hamani S:

**Existence results for boundary value problems with non-linear fractional differential equations.***Applicable Analysis*2008,**87**(7):851-863. 10.1080/00036810802307579Gejji VD:

**Positive solutions of a system of non-autonomous fractional differential equations.***Journal of Mathematical Analysis and Applications*2005,**302**(1):56-64. 10.1016/j.jmaa.2004.08.007Kaufmann ER, Mboumi E:

**Positive solutions of a boundary value problem for a nonlinear fractional differential equation.***Electronic Journal of Qualitative Theory of Differential Equations*2008, (3):11.Zhang S:

**Positive solutions for boundary-value problems of nonlinear fractional differential equations.***Electronic Journal of Differential Equations*2006, (36):12.Bai Z:

**On positive solutions of a nonlocal fractional boundary value problem.***Nonlinear Analysis. Theory, Methods & Applications*2010,**72**(2):916-924. 10.1016/j.na.2009.07.033Li CF, Luo XN, Zhou Y:

**Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations.***Computers & Mathematics with Applications*2010,**59**(3):1363-1375.Benchohra M, Hamani S, Ntouyas SK:

**Boundary value problems for differential equations with fractional order and nonlocal conditions.***Nonlinear Analysis. Theory, Methods & Applications*2009,**71**(7-8):2391-2396. 10.1016/j.na.2009.01.073Zhong W, Lin W:

**Nonlocal and multiple-point boundary value problem for fractional differential equations.***Computers & Mathematics with Applications*2010,**59**(3):1345-1351.Ahmad B, Sivasundaram S:

**On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order.***Applied Mathematics and Computation*2010,**217**(2):480-487. 10.1016/j.amc.2010.05.080Ahmad B, Sivasundaram S:

**Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations.***Nonlinear Analysis. Hybrid Systems*2009,**3**(3):251-258. 10.1016/j.nahs.2009.01.008Ahmad B, Sivasundaram S:

**Existence of solutions for impulsive integral boundary value problems of fractional order.***Nonlinear Analysis. Hybrid Systems*2010,**4**(1):134-141. 10.1016/j.nahs.2009.09.002Zhang X, Huang X, Liu Z:

**The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay.***Nonlinear Analysis. Hybrid Systems*2010,**4**(4):775-781. 10.1016/j.nahs.2010.05.007Tian Y, Bai Z:

**Existence results for the three-point impulsive boundary value problem involving fractional differential equations.***Computers & Mathematics with Applications*2010,**59**(8):2601-2609. 10.1016/j.camwa.2010.01.028Agarwal RP, Ahmad B:

**Existence ofsolutions for impulsive anti-periodic boundary value problems of fractional semilinear evolution equations.**to appear in*Dynamics of Continuous Discrete and Impulsive Systems. Series A*Abbas S, Benchohra M:

**Upper and lower solutions method for impulsive partial hyperbolic differential equations with fractional order.***Nonlinear Analysis. Hybrid Systems*2010,**4**(3):406-413. 10.1016/j.nahs.2009.10.004Wang G, Ahmad B, Zhang L:

**Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order.***Nonlinear Analysis. Theory, Methods and Applications*2011,**74**(3):792-804. 10.1016/j.na.2010.09.030O'Regan D:

**Fixed-point theory for the sum of two operators.***Applied Mathematics Letters*1996,**9**(1):1-8. 10.1016/0893-9659(95)00093-3Guo D, Sun J, Liu Z:

*Functional Methods of Nonlinear Ordinary Differential Equation*. Shandong Science and Technology Press; 2005.

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Yang, L., Chen, H. Nonlocal Boundary Value Problem for Impulsive Differential Equations of Fractional Order.
*Adv Differ Equ* **2011**, 404917 (2011). https://doi.org/10.1155/2011/404917

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DOI: https://doi.org/10.1155/2011/404917