# Periodic BVPs for fractional order impulsive evolution equations

Xiulan Yu1 and JinRong Wang23*

Author Affiliations

1 College of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, Shanxi, 030031, P.R. China

2 Department of Mathematics, Guizhou University, Guiyang, Guizhou, 550025, P.R. China

3 School of Mathematics and Computer Science, Guizhou Normal College, Guiyang, Guizhou, 550018, P.R. China

For all author emails, please log on.

Boundary Value Problems 2014, 2014:35  doi:10.1186/1687-2770-2014-35

 Received: 26 November 2013 Accepted: 16 January 2014 Published: 7 February 2014

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, we study periodic BVPs for fractional order impulsive evolution equations. The existence and boundedness of piecewise continuous mild solutions and design parameter drift for periodic motion of linear problems are presented. Furthermore, existence results of piecewise continuous mild solutions for semilinear impulsive periodic problems are showed. Finally, an example is given to illustrate the results.

MSC: 34B05, 34G10, 47D06.

##### Keywords:
fractional order; impulsive evolution equations; periodic BVPs

### 1 Introduction

In order to describe dynamics of populations subject to abrupt changes as well as other evolution processes such as harvesting, diseases, and so forth, many researchers have used impulsive differential systems to describe the model since the last century. For a wide-ranging bibliography and exposition on this important object see for instance the monographs of [1-4] and the papers [5-12].

Fractional differential equations appear naturally in fields such as viscoelasticity, electrical circuits, nonlinear oscillation of earthquakes etc. In particular, impulsive fractional evolution equations are used to describe many practical dynamical systems in many evolutionary processes models. Recently, Wang et al.[13] discussed Cauchy problems and nonlocal problems for impulsive fractional evolution equations involving the Caputo fractional derivative. However, periodic boundary value problems (BVPs for short) for impulsive fractional evolution equations have not been studied extensively.

In this paper we study the periodic BVPs for impulsive fractional evolution equations. Firstly, we discuss periodic BVPs for impulsive fractional evolution equations:

(1)

in Banach space X, where is the Caputo fractional derivative of order q with the lower limit zero, is the generator of a -semigroup on a Banach space X, is continuous, and are the elements of X, , and and represent respectively the right and left limits of at .

Secondly, we design parameter drift for the above periodic motion. We study the following impulsive periodic BVPs with parameter perturbations:

(2)

where p is a given function and is a small parameter perturbation that may be caused by some adaptive control algorithms or parameter drift.

Finally, we consider semilinear impulsive periodic problems:

(3)

where is continuous and is continuous.

The rest of this paper is organized as follows. In Section 2, the existence and boundedness of the operator are given. In Section 3, the existence and boundedness of PC-mild solutions and the design parameter drift for such a periodic motion are presented. In Section 4, existence results of PC-mild solutions for impulsive periodic problems are showed. Finally, an example is presented to illustrate the theory.

### 2 Existence and boundedness of operator

Suppose , let . Let . We denote by the Banach space of all continuous functions from J into X with the norm . We also introduce the set of functions = { is continuous at , and x is continuous from the left and has right-hand limits at }. Endowed with the norm

it is easy to see is a Banach space.

For measurable functions , define the norm , . We denote by the Banach space of all Lebesgue measurable functions l with .

Definition 2.1 ([14])

The fractional integral of order γ with the lower limit zero for a function f is defined as

provided the right side is point-wise defined on , where is the gamma function.

Definition 2.2 ([14])

The Riemann-Liouville derivative of order γ with the lower limit zero for a function can be written as

Definition 2.3 ([14])

The Caputo derivative of order γ for a function can be written as

Remark 2.4 If f is an abstract function with values in X, then the integrals which appear in Definitions 2.1 and 2.2 are taken in Bochner’s sense.

As in our previous work [13], by a PC-mild solution of (1) we mean the function satisfying

and , where

and

here is a probability density function defined on , that is,

Lemma 2.5 (see Lemma 2.9 [15])

The operatorhas the following properties:

(i) For any fixed, andare linear and bounded operators, i.e., for any, and.

(ii) Bothandare strongly continuous.

(iii) For every, andare compact operators ifis compact.

Suppose that here the bounded operator exists given by

(4)

We present sufficient conditions for the existence and boundedness of the operator B.

Lemma 2.6 (see Theorem 3.3 and Remark 3.4 [16])

The operatorBdefined in (4) exists and is bounded, if one of the following three conditions holds:

(i) is compact for eachand the homogeneous linear nonlocal problem

has no non-trivialPC-mild solutions.

(ii) If, then the operatoris invertible and.

(iii) Iffor, thenasand the operatoris invertible and.

### 3 Linear impulsive periodic problems and robustness

In this section, we consider the existence of PC-mild solutions of (1) and of design parameter drift for (2).

We list the following assumptions.

(HA): is the infinitesimal generator of a -semigroup .

(HF): is strongly measurable for and there exist a constant and a real-valued function such that , for each .

(HB): The operator B defined in (4) exists and is bounded.

We first give an existence theorem of PC-mild solutions of (1).

Theorem 3.1Assume that (HA), (HF), and (HB) are satisfied. Then (1) has aPC-mild solution given by

(5)

where

(6)

Further, we have

(7)

whereand.

Proof We consider an impulsive Cauchy problem,

(8)

It follows from the expression of the initial value that the mild solution of (8) corresponding to the initial value must be the PC-mild solution of (1).

For the estimation of , by Lemma 2.5,

where and . The desired results are obtained. □

Remark 3.2 In Theorem 3.1, we replace (HB) by ; it is a compact semigroup and

has no non-trivial mild solutions. Then one can use the Fredholm alternative theorem to derive that the operator equation has a unique solution . Thus, the PC-mild solution of (1) is unique.

Define with for . It can be seen that endowed with the norm , is a Banach space. Denote

where

and χ is a nonnegative function.

We introduce the assumption (HP):

(HP1): is measurable in t.

(HP2): There exists a nonnegative function ϖ such that and for any , , and , and we have

(HP3): There exists a nonnegative function χ such that , and for any , , and , we have

By a PC-mild solution of (2), we mean the function satisfying

and .

The following result shows that given a periodic motion we can design periodic motion controllers that are robust with respect to parameter drift.

Theorem 3.3Let (HA), (HF), (HB), and (HP) hold. Then there is asuch that, for, (2) has aPC-mild solutionsatisfying

anduniformly onwhereis the mild solution of (1).

Proof By (HB), one can choose

to be fixed. Consider the map on given by

Obviously, .

By the assumption (HP), we can choose a such that

and

For and , one can verify that

(9)

and

This implies that is a contraction mapping on . Then, has a unique fixed point given by

(10)

which is just the PC-mild solution of (2).

From the expressions (9) and (10), one can get . It is easy to see that uniformly on . □

### 4 Semilinear impulsive periodic problems

We impose the following assumptions.

(HF1): is continuous and there exist a constant and a real-valued function such that for all .

(HF2): is continuous and maps a bounded set into a bounded set.

(HF3): For each , there exists a constant such that

where

(HI1): is continuous and there exists a constant such that for all , .

(HI2): is continuous and there exists a constant such that , for all , .

Theorem 4.1Let (HA), (HB), (HI1), and (HF1) be satisfied. Then (3) has a uniquePC-mild solution onJprovided that

(11)

Proof Consider

(12)

where

(13)

and we define an operator Q on :

(14)

Clearly, Q is well defined on due to our assumptions.

Then, we only need to show that Q is a contraction on .

In general, for each , we have

Hence, the condition (11) allows us to conclude, in view of the Banach contraction mapping principle, that Q has a unique fixed point , which is just the unique PC-mild solution of (3). □

Theorem 4.2Suppose that (HA), (HB), (HI2), and (HF2) and (HF3) are satisfied. Then for every, (3) has at least aPC-mild solution onJ.

Proof Consider the mapping

by

where

(15)

is defined in (13) and

(16)

For each , ,

Thus, we see that .

Just like the proof in our previous work [13], one can prove that Q is a continuous mapping from to and it is a compact operator. Now, Schauder’s fixed point theorem implies that Q has a fixed point, which gives rise to a PC-mild solution. □

### 5 Example

We consider impulsive fractional differential equations with periodic boundary conditions,

(17)

in where will be chosen later.

Define for where . Then A is the infinitesimal generator of a -semigroup in . Moreover, is also compact and , . By the Fredholm alternative theorem, exists and is bounded where is defined in Section 2.

Define , , , and is a continuous function, , with . Define , , . for all with .

Moreover,

Thus, one can choose such that (11) holds. Therefore, (17) has a unique PC-mild solution on .

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

This work was carried out in collaboration between all authors. JRW raised these interesting problems in this research. JRW and XLY proved the theorems, interpreted the results and wrote the article. All authors defined the research theme, read and approved the manuscript.

### Acknowledgements

This work is partially supported by Key Support Subject (Applied Mathematics), Key Project on the Reforms of Teaching Contents, Course System of Guizhou Normal College and Doctor Project of Guizhou Normal College (13BS010) and Guizhou Province Education Planning Project (2013A062).

### References

1. Bainov, DD, Simeonov, PS: Impulsive Differential Equations: Periodic Solutions and Applications, Longman, New York (1993)

2. Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations, World Scientific, Singapore (1995)

3. Benchohra, M, Henderson, J, Ntouyas, SK: Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, New York (2006)

4. Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations, World Scientific, Singapore (1989)

5. Abada, N, Benchohra, M, Hammouche, H: Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions. J. Differ. Equ.. 246, 3834–3863 (2009). Publisher Full Text

6. Ahmed, NU: Existence of optimal controls for a general class of impulsive systems on Banach space. SIAM J. Control Optim.. 42, 669–685 (2003). Publisher Full Text

7. Akhmet, MU: On the smoothness of solutions of impulsive autonomous systems. Nonlinear Anal. TMA. 60, 311–324 (2005). Publisher Full Text

8. Fan, Z, Li, G: Existence results for semilinear differential equations with nonlocal and impulsive conditions. J. Funct. Anal.. 258, 1709–1727 (2010). Publisher Full Text

9. Liang, J, Liu, JH, Xiao, T-J: Nonlocal impulsive problems for nonlinear differential equations in Banach spaces. Math. Comput. Model.. 49, 798–804 (2009). Publisher Full Text

10. Liu, J: Nonlinear impulsive evolution equations. Dyn. Contin. Discrete Impuls. Syst.. 6, 77–85 (1999)

11. Battelli, F, Fec̆kan, M: Chaos in singular impulsive O.D.E.. Nonlinear Anal. TMA. 28, 655–671 (1997). Publisher Full Text

12. Wang, J, Fec̆kan, M, Zhou, Y: Ulam’s type stability of impulsive ordinary differential equations. J. Math. Anal. Appl.. 395, 258–264 (2012). Publisher Full Text

13. Wang, J, Fec̆kan, M, Zhou, Y: On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn. Partial Differ. Equ.. 8, 345–361 (2011). Publisher Full Text

14. Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006)

15. Wang, J, Zhou, Y: A class of fractional evolution equations and optimal controls. Nonlinear Anal., Real World Appl.. 12, 262–272 (2011). Publisher Full Text

16. Wang, J, Zhou, Y, Fec̆kan, M: Alternative results and robustness for fractional evolution equations with periodic boundary conditions. Electron. J. Qual. Theory Differ. Equ.. 2011, (2011) Article ID 97