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 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:
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:
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 PCmild solutions and the design parameter drift for such a periodic motion are presented. In Section 4, existence results of PCmild 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 righthand 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 pointwise defined on , where is the gamma function.
Definition 2.2 ([14])
The RiemannLiouville 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 PCmild solution of (1) we mean the function satisfying
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
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 nontrivialPCmild solutions.
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): is the infinitesimal generator of a semigroup .
(HF): is strongly measurable for and there exist a constant and a realvalued function such that , for each .
(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
Proof We consider an impulsive Cauchy problem,
It follows from the expression of the initial value that the mild solution of (8) corresponding to the initial value must be the PCmild 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 nontrivial mild solutions. Then one can use the Fredholm alternative theorem to derive that the operator equation has a unique solution . Thus, the PCmild 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):
(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 PCmild solution of (2), we mean the function satisfying
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 aPCmild solutionsatisfying
anduniformly onwhereis the mild solution of (1).
Proof By (HB), one can choose
to be fixed. Consider the map on given by
By the assumption (HP), we can choose a such that
and
and
This implies that is a contraction mapping on . Then, has a unique fixed point given by
which is just the PCmild 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 realvalued 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 uniquePCmild solution onJprovided that
Proof Consider
where
and we define an operator Q on :
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 PCmild solution of (3). □
Theorem 4.2Suppose that (HA), (HB), (HI2), and (HF2) and (HF3) are satisfied. Then for every, (3) has at least aPCmild solution onJ.
Proof Consider the mapping
by
where
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 PCmild solution. □
5 Example
We consider impulsive fractional differential equations with periodic boundary conditions,
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 PCmild 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).
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