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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 316850, 19 pages
Existence Results of Nondensely Defined Fractional Evolution Differential Inclusions
1School of Mathematics and Statistics, Huazhong University of Science and Technology, Hubei, Wuhan 430074, China
2School of Mathematics and Statistics, Suzhou University, Anhui, Suzhou 234000, China
Received 26 March 2012; Accepted 20 April 2012
Academic Editor: Yonghong Yao
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 deals with the existence results of integral solutions for nondensely defined fractional evolution differential inclusions. Our approach is based on integrated semigroup theory and a fixed point theorem for condensing map due to Martelli. An example is also given to illustrate our results.
In the past decades, the theory of fractional differential equations and inclusions has become an important area of investigation because of its wide applicability in many branches of physics, economics, and technical sciences [1–10].
Our aim in this paper is to study the existence of the integral solutions for the fractional semilinear differential inclusions, of the form where is the Caputo fractional derivative of order , . is a nondensely closed linear operator on , is a real Banach space with the norm . is a nonempty, bounded, closed, and convex multivalued map, and denotes the family of all nonempty subsets of .
It is well known that one important way to introduce the concepts of mild solutions for fractional evolution equations is based on some probability densities and Laplace transform. This method was initialed by El-Borai  and developed by Zhou and Jiao . Since then, many interesting existence results of mild solutions for fractional evolution equations appeared [13–16]. We will point out that the unbounded operators in their papers were assumed to be densely defined and generate a strongly continuous semigroup.
However, as indicated in , we sometimes need to deal with nondensely defined operators and there are extensive work on this subject when equations involve the integral-order derivative, see monograph [18–23] and references therein. Very recently, Wang and Zhou  considered problem (1.1) in the case when is densely defined and generates a strongly continuous semigroup. As far as we know, there are few papers dealing with semilinear fractional differential systems with nondense domain. Motivated by this, we discuss the integral solution to problem (1.1) by using probability densities and integral semigroup. We turn the integral solutions of problem (1.1) to a new formula something like the mild solutions. This new formula of integral solutions is firstly introduced even in fractional evolution equations. Thus, our work can be seen as a supplement to work  and a contribution to this emerging field of fractional differential equations with nondense domain.
This paper will be organized as follows. In Section 2, we recall some basic definitions and preliminary facts for integrated semigroup, fractional calculus, and multivalued map which will be used later. Section 3 is devoted to the existence results of integral solutions for problem (1.1). We will present in Section 4 an example which illustrates our main theorem.
In this section, we introduce notations, definitions, and preliminary results which are used in the rest of the paper.
We denote by the Banach space of all continuous functions from into with the norm denotes the Banach space of bounded linear operators from into , with the norm where and .
Assume that and . For a measurable function , define the norm where is the Lebesgue measure on . Let be the Banach space of all Lebesgue measurable functions with .
Lemma 2.1 (HöLder inequality). Assume that and . If , , then for , and
Lemma 2.2 (Bochner theorem). A measurable function is Bochner’s integrable if is Lebesgue integrable.
Definition 2.3 (see ). Let be a Banach space; an integrated semigroup is a family of operators of bounded linear operators on with the following properties:; is strongly continuous; for all .
Definition 2.4 (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 2.5 (see ). Let be the generator of an integrated semigroup . Then for all and ,
Definition 2.6 (see ). We say that linear operator satisfies the Hille-Yosida condition if there exist 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 ([19, 27]).
Let ; then for all , as . Also from the Hille-Yosida condition it is easy to see that .
Definition 2.7 (see ). The Riemann-Liouville fractional integral of order of a function is defined by provided the right-hand side is pointwise defined on , where is the gamma function.
Remark 2.8. According to , holds for all , .
Lemma 2.10 (see ). Suppose , is a nonnegative, function locally integrable on and is a nonnegative, nondecreasing continuous function defined on , (constant), and suppose is nonnegative and locally integrable on with on this interval. Then
We also introduce some basic definitions and results of multivalued maps. See  for more details.
Let be a metric space; denotes the family for all nonempty subsets of . We use the following notations:
A multivalued map is convex (closed) valued if is convex (closed) for all and is bounded on bounded sets if is bounded in for all , that is, . is called upper semicontinuous (u.s.c. for short) on if for each the set is nonempty, closed subset of , and for each open set of containing , there exists an open neighborhood of such that . is said to be completely continuous if is relatively compact for every .
If the multivalued map is completely continuous with nonempty compact valued, the is u.s.c. if and only if has closed graph, that is, , , imply .
Definition 2.12 (see ). An upper semicontinuous map is said to be condensing if for any bounded subset with , one has , where denotes the Kuratowski measure of noncompactness.
We remark that a completely continuous multivalued map is the easiest example of a condensing map.
Theorem 2.13 (see ). Let be a compact interval and a Banach space. Let , be measurable with respect to for each , upper semicontinuous with respect to for each . Moreover, for each fixed the set is nonempty. Also let be a linear continuous mapping from to ; then the operator is a closed graph operator in .
Theorem 2.14 . (Martelli, ). Let be a Banach space and a condensing map. If the set is bounded, then has a fixed point.
3. Existence of Integral Solutions
Definition 3.1. One says that a continuous function is an integral solution of problem (3.1) if(i) for ,(ii), .
Lemma 3.2. If is an integral solution of (1.1), then for all , . In particular, .
Proof. By Remark 2.8 and , for each , we get that . From we have for , . 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. The integral solution in Definition 3.1 is given by where is the probability density function defined on .
Proof. From the definition, 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 (3.3), we get According to (3.7), (3.8), and (3.9), we have Inverting the last Laplace transform, we obtain In view of for and Lemma 3.2, we have The proof is completed.
Definition 3.6. One says that a continuous function is an integral solution of problem (1.1) if(i) for ,(ii) and there exists such that for a.e. and
We are now in a position to state and prove our main results of the existence of solutions for problem (1.1).
Let us list the following hypotheses:
satisfies the Hille-Yosida condition; the operator is compact in whenever and satisfies , where is a constant;, for each , is measurable and for each , is upper semicontinuous; for each fixed , the set = is not empty; for each , there exist and such that where .
Theorem 3.7. Assume that hypotheses (H1)–(H4) hold; then problem (1.1) has an integral solution .
Proof. Denote , which is a closed subset of . Obviously, with the same norm in is also a Banach space. Transform the problem (1.1) into a fixed point problem. Consider the multivalued operator defined by
where . Obviously, the fixed points of the operator are integral solutions of problem (1.1). Now we will show that satisfies all conditions of Theorem 2.14. The proof would be divided into the following steps.Step 1 ( is convex for each ). Indeed, if and belong to , then there exist , such that for each , we have
Let ; then for each , we have Since is convex, we have .Step 2 ( maps bounded sets into bounded sets in ). Indeed, it is enough to show that there exists a positive constant such that for each , one has .
Let ; then there exists such that for , we have From (H2) and the fact that , for we have From Lemma 2.1 and (H4), for we have where , , .
Then from (3.20) and (3.21), we get that Step 3 ( maps bounded sets into equicontinuous sets of ). Let , , , and be a bounded set of . For each and , there exists such that Then, where By using analogous argument performed in (3.20) and (3.21), we can conclude that Hence and .
On the other hand, (H2) implies that for is continuous in the uniform operator topology; then from the Lebesgue dominated convergence theorem, we get and Consequently, independently of as , which means that is equicontinuous.Step 4 (For each , is relatively compact in ). Obviously, is relatively compact in . Let be fixed. For and , there exists such that
For arbitrary and , define an operator on by Then from the compactness of , , we get that the set is relatively compact in for each and . Moreover, for every , we have In view of (3.21), we have From and , we get that there are relatively compact sets arbitrarily close to the set , . Hence the set , , is also relatively compact in .Step 5 ( has a closed graph). Let , , and as ; we will prove that . means that there exists such that
We must prove that there exists such that Consider the linear continuous operator defined by We can easily see that is continuous. On the other hand, From Theorem 2.13, it follows that is a closed graph operator. Moreover, we have that Since , it follows from Theorem 2.13 that there exists such that Thus, This implies that .
Therefore is a completely continuous multivalued map, u.s.c. with convex closed values. In order to prove that has a fixed point, we need one more step.Step 6 (The set is bounded). Let ; then for some . Thus there exists such that for , From (H4), for each we have where , .
Then from Corollary 2.11, we have Therefore, we obtain that This shows that is bounded.
As a consequence of Theorem 2.14, we conclude that has a fixed point which is the integral solution of problem (1.1). This completes the proof.
4. An Example
As an application of our results we consider the following fractional differential inclusions of the form where , satisfies semi-continuous assumptions (H3) and (H4).
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), (H2), and .
Then the system (4.1) can be reformulated as where , .
This work was partially supported by National Natural Science Foundation of China (11171122) and Anhui Province College Natural Science Foundation (KJ2012A265, KJ2012B187, KJ2011B176).
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