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 BVPs1 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 wideranging bibliography and exposition on this important object see for instance the monographs of [14] and the papers [512].
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:
in Banach space X, where
Secondly, we design parameter drift for the above periodic motion. We study the following impulsive periodic BVPs with parameter perturbations:
where p is a given function and
Finally, we consider semilinear impulsive periodic problems:
where
The rest of this paper is organized as follows. In Section 2, the existence and boundedness
of the operator
2 Existence and boundedness of operator
B
=
[
I
−
T
(
T
)
]
−
1
Suppose
it is easy to see
For measurable functions
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 pointwise defined on
Definition 2.2 ([14])
The RiemannLiouville derivative of order γ with the lower limit zero for a function
Definition 2.3 ([14])
The Caputo derivative of order γ for a function
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 PCmild solution of (1) we mean the function
and
and
here
Lemma 2.5 (see Lemma 2.9 [15])
The operatorhas the following properties:
(i) For any fixed
(ii) Both
(iii) For every
Suppose that here the bounded operator
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)
has no nontrivialPCmild solutions.
(ii) If
(iii) If
3 Linear impulsive periodic problems and robustness
In this section, we consider the existence of PCmild solutions of (1) and of design parameter drift for (2).
We list the following assumptions.
(HA):
(HF):
(HB): The operator B defined in (4) exists and is bounded.
We first give an existence theorem of PCmild solutions of (1).
Theorem 3.1Assume that (HA), (HF), and (HB) are satisfied. Then (1) has aPCmild solution given by
where
Further, we have
where
Proof We consider an impulsive Cauchy problem,
It follows from the expression of the initial value
For the estimation of
where
Remark 3.2 In Theorem 3.1, we replace (HB) by
has no nontrivial mild solutions. Then one can use the Fredholm alternative theorem
to derive that the operator equation
Define
where
and χ is a nonnegative function.
We introduce the assumption (HP):
(HP1):
(HP2): There exists a nonnegative function ϖ such that
(HP3): There exists a nonnegative function χ such that
By a PCmild solution of (2), we mean the function
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 a
and
Proof By (HB), one can choose
to be fixed. Consider the map on
Obviously,
By the assumption (HP), we can choose a
and
For
and
This implies that is a contraction mapping on
which is just the PCmild solution of (2).
From the expressions (9) and (10), one can get
4 Semilinear impulsive periodic problems
We impose the following assumptions.
(HF1):
(HF2):
(HF3): For each
where
(HI1):
(HI2):
Theorem 4.1Let (HA), (HB), (HI1), and (HF1) be satisfied. Then (3) has a uniquePCmild solution onJprovided that
Proof Consider
where
and we define an operator Q on
Clearly, Q is well defined on
Then, we only need to show that Q is a contraction on
In general, for each
Hence, the condition (11) allows us to conclude, in view of the Banach contraction
mapping principle, that Q has a unique fixed point
Theorem 4.2Suppose that (HA), (HB), (HI2), and (HF2) and (HF3) are satisfied. Then for every
Proof Consider the mapping
by
where
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
5 Example
We consider impulsive fractional differential equations with periodic boundary conditions,
in
Define
Define
Moreover,
Thus, one can choose
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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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, Article ID 97 (2011)