Periodic BVPs for fractional order impulsive evolution equations
© Yu and Wang; licensee Springer. 2014
Received: 26 November 2013
Accepted: 16 January 2014
Published: 7 February 2014
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.
Keywordsfractional order impulsive evolution equations periodic BVPs
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.  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 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 .
where p is a given function and is a small parameter perturbation that may be caused by some adaptive control algorithms or parameter drift.
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
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 ()
provided the right side is point-wise defined on , where is the gamma function.
Definition 2.2 ()
Definition 2.3 ()
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.
Lemma 2.5 (see Lemma 2.9 )
For any fixed , and are linear and bounded operators, i.e., for any , and .
Both and are strongly continuous.
For every , and are compact operators if is compact.
We present sufficient conditions for the existence and boundedness of the operator B.
Lemma 2.6 (see Theorem 3.3 and Remark 3.4 )
- (i)is compact for each and the homogeneous linear nonlocal problem
has no non-trivial PC-mild solutions.
If , then the operator is invertible and .
If for , then as and the operator is 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).
where and .
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).
where and . The desired results are obtained. □
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.
and χ is a nonnegative function.
We introduce the assumption (HP):
(HP1): is measurable in t.
The following result shows that given a periodic motion we can design periodic motion controllers that are robust with respect to parameter drift.
and uniformly on where is the mild solution of (1).
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
(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 thatwhere
(HI1): is continuous and there exists a constant such that for all , .
(HI2): is continuous and there exists a constant such that , for all , .
Clearly, Q is well defined on due to our assumptions.
Then, we only need to show that Q is a contraction on .
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 4.2 Suppose that (HA), (HB), (HI2), and (HF2) and (HF3) are satisfied. Then for every , (3) has at least a PC-mild solution on J.
Thus, we see that .
Just like the proof in our previous work , 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. □
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 .
Thus, one can choose such that (11) holds. Therefore, (17) has a unique PC-mild solution on .
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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