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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 359452, 13 pages
A Note on Impulsive Fractional Evolution Equations with Nondense Domain
1School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
2School of Mathematics and Statistics, Suzhou University, Suzhou 234000, China
Received 22 March 2012; Revised 28 July 2012; Accepted 29 July 2012
Academic Editor: Juan J. Trujillo
Copyright © 2012 Zufeng Zhang and Bin Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper is concerned with the existence of integral solutions for nondensely defined fractional functional differential equations with impulse effects. Some errors in the existing paper concerned with nondensely defined fractional differential equations are pointed out, and correct formula of integral solutions is established by using integrated semigroup and some probability densities. Sufficient conditions for the existence are obtained by applying the Banach contraction mapping principle. An example is also given to illustrate our results.
The aim in this paper is to study the existence of the integral solutions for the fractional semilinear differential equations of the form where , is the Caputo fractional derivative. is a given function, is continuous everywhere except for a finite number of points at which exist and , and is a real Banach space with the norm . Denoting the domain of by , is nondensely closed linear operator on , . , , and represent the right and left limits at of as usual; we assume . represents the jump in the state at time . Moreover, for any , the histories belong to defined by .
In the past decades, the theory of fractional differential equations has become an important area of investigation because of its wide applicability in many branches of physics, economics, and technical sciences [1–10]. In recent years, many authors were devoted to mild solutions to fractional evolution equations, and there have been a lot of interesting works. For instance, in , El-Borai discussed the following equation in Banach space : where generates an analytic semigroup, and the solution was given in terms of some probability densities. In , Zhou and Jiao concerned the existence and uniqueness of mild solutions for fractional evolution equations by some fixed point theorems. Cao et al.  studied the -mild solutions for a class of fractional evolution equations and optimal controls in fractional powder space. For more information on this subject, the readers may refer to [14–16] and the references therein.
Research on integer order differential evolution equations including a nondensely defined operator was initialed by Da Prato and Sinestrari  and has been extensively investigated by many authors [18–25]. The main methods used in their work are based on integrated semigroup theory. Recently, existence results for integral solutions of nondensely defined fractional evolution equations were established in some papers [9, 26]. But there are some errors in transforming integral solution into an available form. For example, definition of integral solution  is given by Here and . will be introduced in next section. If we let take values in , then (1.3) becomes According to , integral solution should be mild solution in this case. But as pointed in , (1.4) is not the mild solution.
Motivated by these papers and the fact that impulse effects exist widely in the realistic situations, we give the definition of integral solution and prove the existence results for impulsive semilinear fractional differential equations with nondensely defined operators. The rest of the paper will be organized as follows. In Section 2, we will recall some basic definitions and preliminary facts from integrated semigroups and fractional derivation and integration which would be used later. Section 3 is devoted to the existence of integral solutions of problem (1.1). We present an example to illustrate our results in Section 4. At last, we end the paper with a conclusion.
In this section, we introduce notations, definitions, and preliminary results which would be used in the rest of the paper.
We denote by the Banach space of all continuous functions from into with the norm For the norm of is defined by denotes the Banach space of bounded linear operators from into , with the norm where and . Let be the space of -valued Bochner function on with the norm In order to define an integral solution of problem (1.1), we will introduce the set of functions Endowed with the norm , is a Banach space.
Seting then is a Banach space with the norm
Definition 2.1 (see ). Letting be a Banach space, an integrated semigroup is a family of operators of bounded linear operators on with the following properties:(i);(ii) is strongly continuous;(iii) for all .
Definition 2.2 (see ). An operator is called a generator of an integrated semigroup, if there exists such that , and there exists a strongly continuous exponentially bounded family of linear bounded operators such that and for all .
Proposition 2.3 (see ). Let be the generator of an integrated semigroup . Then for all and ,
Definition 2.4 (see ). We say that a linear operator satisfies the Hille-Yosida condition if there exists and such that and
Here and hereafter, we assume that satisfies the Hille-Yosida condition. Let us introduce the part of in on. Let be the integrated semigroup generated by . We note that is a -semigroup on generated by and , , where and are the constants considered in the Hille-Yosida condition [28, 30].
Let ; then for all , as . Also from the Hille-Yosida condition it is easy to see that .
For more properties on integral semigroup theory the interested reader may refer to [18, 30].
Definition 2.5 (see ). The Riemann-Liouville fractional integral of order of a function is defined by provided the right-hand side is pointwise defined on and where is the gamma function.
Remark 2.6. According to , holds for all , .
Definition 2.7 (see ). The Caputo fractional derivative of order of a function is defined by
3. Main Results
In this section we will establish the existence and uniqueness of integral solution for problem (1.1).
Definition 3.1. A function is said to be an integral solutions of (1.1) if(i) for ,(ii),(iii)
Lemma 3.2. If is an integral solution of (1.1), then for all , . In particular, , ,, belong to .
Proof. Using Remark 2.6, for each , since . Consequently, for such that , because . Hence, we deduce that . The proof is completed.
Lemma 3.3 (see ). Let , ; then is a one-sided stable probability density function, and its Laplace transform is given by
Lemma 3.4. For , the integral solution in Definition 3.1 is given by where where is the probability density function defined on .
Proof. From the definition, for we have
Consider the Laplace transform
Note that for each , , then we have . Applying (3.6) to (3.5) yields
where is the identity operator defined on .
From (3.2), we get According to (3.7) and (3.8), we have Inverting the last Laplace transform, we obtain In view of for and Lemma 3.2, we have
For , , we can prove the results by the similar methods used previously. The proof is completed.
Let us list the following hypotheses.(H1) satisfies the Hille-Yosida condition, and assume that .(H2) For is strongly measurable.(H3) There exists a constant and such that (H4) There exists such that (H5) There exists a constant such that
Theorem 3.6. Assuming that hypotheses – hold, then problem (1.1) has a unique integral solution provided that .
Proof. Define by
and for ,
Firstly we check that is well defined on .
For each , take . It is obvious that is well defined. Direct calculation shows that , for and . Let Then for , we have From (H1), (3.12), (3.19), and the fact that , we get It means that is Lebesgue integrable with respect to for all . Therefore is Bochner integrable with respect to for all .
From , we know exists; then exists. Therefore we get which is well defined on .
For , similar discussion could obtain is well defined. Hence, is well defined on .
Secondly, we will prove operator is a contraction.
For and , by the hypotheses and , we get Now take , and : In view of , we have that the operator is a contraction. By the Banach contraction principle we have that has a unique fixed point , which gives rise to a unique integral solution to the problem (1.1). The proof is finished.
Remark 3.7. For impulsive Caputo fractional differential equations, its integral solutions (or mild solutions; see ) can be expressed only by using piecewise functions. Thus Definition 2.3 given in  is unsuitable.
4. An Example
As an application of our results we consider the following fractional differential equations of the form
Consider endowed with the supnorm and the operator defined by Now, we have . As we know from  that satisfies the Hille-Yosida condition with and , . Hence, operator satisfies (H1) and .
Then the system (4.1) can be reformulated as where , , , .
An essence error of the formula of solutions which appeared in the recent work on the nondensely defined fractional evolution differential equations is reported in this work. A correct formula of integral solutions for nondensely defined fractional evolution equations could be obtained from the results in this paper.
In view of the complicated definitions for integral or mild solutions for impulsive fractional evolution equations, many fixed point theorems related to completely continuous operators are hard to be used to establish the existence results. As far as we know, only  applied Leray Schauder Alternative theorem to the existence of mild solutions of impulsive fractional differential equations. But there is a mistake in proving that is equicontinuous (page 2009, Step 3). How to overcome this difficulty is our next work.
This work was partially supported by National Natural Science Foundation of China (11171122) and University Science Foundation of Anhui Province (KJ2012B187).
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